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MANOVA (Multivariate Analysis of Variance) – Method & Examples

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MANOVA (Multivariate Analysis of Variance)

Multivariate Analysis of Variance, often abbreviated as MANOVA, is a statistical test that extends the capabilities of the Analysis of Variance ( ANOVA ) by allowing for the simultaneous analysis of multiple continuous dependent variables.

When we use ANOVA, we analyze the difference between different group means for a single dependent variable . However, there might be scenarios where we need to analyze multiple dependent variables. In such cases, MANOVA can be a valuable tool.

Here’s a brief rundown of how MANOVA works:

• Formulation of Hypotheses: In a MANOVA, you have two types of hypotheses – null and alternative. The null hypothesis posits that the population means of the dependent variables are equal across different groups, while the alternative hypothesis suggests that at least one dependent variable’s mean is different.
• Analysis and Computation: With your hypotheses in place, you would run your data through a MANOVA test. This involves complex computations, usually done via statistical software like R, SPSS, or SAS.
• Interpretation of Results: The results of your MANOVA will include a number of values, including F values, p-values, and degrees of freedom for each dependent variable and for the model as a whole. These will help you determine whether or not to reject your null hypothesis.

One important thing to note about MANOVA is that it makes several assumptions, such as normality, linearity, homogeneity of variance-covariance matrices, and absence of multicollinearity, among others. Violations of these assumptions can lead to inaccurate results, so it’s important to ensure these conditions are met before conducting the analysis.

MANOVA Methodology

Here is a simplified step-by-step methodology of a MANOVA:

• Formulation of Hypotheses: First, establish your hypotheses. The null hypothesis usually states that the mean differences between the groups on the set of dependent variables are zero. The alternative hypothesis states that at least one mean difference between the groups on the dependent variables is not zero.
• Selection of Level of Significance: The level of significance is the maximum chance you are willing to take of rejecting the null hypothesis when it is true. Typically, this is set to 0.05, but it can be lower if you want to be more certain.
• Selection and Measurement of Variables: Choose the variables you wish to analyze. You will need one or more categorical independent variable(s) and two or more continuous dependent variables.
• Check the Assumptions: Verify that your data meets the assumptions for a MANOVA test. These include normality (each combination of levels of independent variables comes from a multivariate normal distribution), homogeneity of variance-covariance matrices, absence of multicollinearity, and more.
• Perform the MANOVA: Run the MANOVA test using statistical software. This will provide a Pillai’s Trace, Wilks’ Lambda, Hotelling’s Trace, and Roy’s Greatest Root value for the model. Each of these values is a test statistic used to determine whether the group means are different. The statistic you use will depend on the specifics of your data and your hypothesis.
• Interpret the Results: You will look at the p-value associated with your chosen test statistic. If it’s less than your chosen level of significance, you will reject the null hypothesis, concluding that there are significant differences among the group means.
• Post Hoc Analyses: If the MANOVA is significant, this is followed by determining which pairs of means of dependent variables are significantly different. This can be done using a variety of tests such as pairwise t-tests or Tukey’s test.
• Report the Results: Finally, report your findings in the appropriate format. Make sure to include details of the analysis, such as the test statistic used, the resulting p-value, and the means and standard deviations of the groups. If post hoc tests were performed, these results should be included as well.

MANOVA Formulas

The formulation of MANOVA involves matrix algebra and is generally computed using statistical software, but here’s a brief overview of the formulas involved:

• Total Sum of Squares and Cross Products (SSCP) Matrix: This matrix, denoted as T, is an extension of the Total Sum of Squares (SS) in univariate ANOVA to multivariate analysis. It’s calculated by taking the sum of the cross product of the deviation scores for each observation from the grand mean (the mean of all observations).
• Between-groups SSCP Matrix (H matrix): This matrix represents the between-group variation. It is calculated by multiplying the cross product of the deviation of each group mean from the grand mean by the number of observations in each group, and then summing these across all groups.
• Within-groups SSCP Matrix (E matrix): This matrix represents the within-group variation. It is calculated by finding the sum of the cross products of the deviation scores within each group, then summing these across all groups.
• Wilks’ Lambda: One of the most common test statistics used in MANOVA is Wilks’ Lambda (Λ), which is the ratio of the determinant of the E matrix to the determinant of the T matrix. Smaller values of Wilks’ Lambda indicate greater differences between the groups.
• Pillai’s Trace, Hotelling’s Trace, and Roy’s Greatest Root: These are other test statistics that can be used in MANOVA, each calculated using different formulas based on the eigenvalues of the H and E matrices.

Because of the complexity of these calculations, they’re generally not done by hand except in the simplest cases. Instead, they’re typically carried out using statistical software such as R, SPSS, or SAS.

Examples of MANOVA

Examples of MANOVA are as follows:

Suppose a psychologist wants to study the effects of different treatments on two different outcomes: anxiety levels and self-esteem scores among adults with social phobia. The psychologist decided to compare three treatments: cognitive behavioral therapy (CBT), medication, and a control group (no treatment). In this case, the treatment type is the independent variable (with three levels: CBT, medication, and control), and the two dependent variables are anxiety levels and self-esteem scores.

Here’s how a MANOVA would come into play:

• Hypotheses Formulation: The psychologist would start by forming a null hypothesis stating that there is no difference in the multivariate means of the anxiety and self-esteem scores between the three treatment groups. The alternative hypothesis would state that there is a difference in the multivariate means of at least one of the dependent variables (anxiety levels or self-esteem scores) between the three groups.
• Data Collection: The psychologist would then carry out the treatments and collect data on the patients’ anxiety levels and self-esteem scores after a defined period.
• Assumptions Checking: Before running the MANOVA, the psychologist needs to check that the data meet the assumptions of the test, including multivariate normality, homogeneity of covariance matrices, and the absence of multicollinearity among the dependent variables.
• Perform the MANOVA: Using statistical software, the psychologist would perform the MANOVA test, specifying the independent and dependent variables.
• Interpretation of Results: After running the MANOVA, the psychologist would look at the results. If the p-value associated with the chosen test statistic (e.g., Wilks’ Lambda) is less than the chosen significance level (typically 0.05), they would reject the null hypothesis and conclude that there is a significant difference in the multivariate means of the dependent variables (anxiety levels and/or self-esteem scores) between the three treatment groups.
• Post Hoc Testing: If a significant difference is found, the psychologist may proceed with post hoc tests to determine which specific groups differ from each other on the dependent variables.
• Report the Findings: Finally, the psychologist would write up the results, noting the methodology, the test statistic and its associated p-value, and the implications of the findings for treatment of adults with social phobia.

Suppose an education researcher wants to investigate the impact of different teaching methods on students’ learning outcomes. They are interested in comparing traditional classroom instruction, online learning, and blended learning (a mix of classroom and online instruction).

They measure learning outcomes using two different metrics: students’ final exam scores and their self-reported understanding of the course material. In this case, the teaching method is the independent variable (with three levels: traditional, online, and blended learning), and the two dependent variables are final exam scores and self-reported understanding.

• Hypotheses Formulation: The researcher would start by forming a null hypothesis that there is no difference in the multivariate means of the final exam scores and self-reported understanding between the three teaching methods. The alternative hypothesis would state that there is a difference in the multivariate means of at least one of the dependent variables (final exam scores or self-reported understanding) between the three teaching methods.
• Data Collection: The researcher would then collect data on the students’ final exam scores and self-reported understanding after a semester of instruction using one of the three teaching methods.
• Assumptions Checking: Before running the MANOVA, the researcher would need to check that the data meet the assumptions of the test, including multivariate normality, homogeneity of covariance matrices, and the absence of multicollinearity among the dependent variables.
• Perform the MANOVA: Using statistical software, the researcher would then perform the MANOVA, specifying the independent and dependent variables.
• Interpretation of Results: After running the MANOVA, the researcher would examine the results. If the p-value associated with the chosen test statistic (e.g., Pillai’s Trace) is less than the chosen significance level (typically 0.05), they would reject the null hypothesis and conclude that there is a significant difference in the multivariate means of the dependent variables (final exam scores and/or self-reported understanding) between the three teaching methods.
• Post Hoc Testing: If a significant difference is found, the researcher may conduct post hoc tests to figure out which specific groups differ from each other on the dependent variables.
• Report the Findings: Finally, the researcher would write up the results, noting the methodology, the test statistic and its associated p-value, and what the findings mean for educational practice.

When To Use MANOVA

Here are some scenarios when you would use MANOVA:

• Multiple Measurements: If your research involves multiple measurements that are related, MANOVA could be appropriate. For instance, if you are studying the impact of a diet regimen on body mass index (BMI), cholesterol levels, and blood pressure, MANOVA allows you to assess these three dependent variables (BMI, cholesterol levels, blood pressure) simultaneously.
• Protecting Against Type I Errors: When you conduct multiple ANOVA tests separately for each dependent variable, the probability of making a Type I error (i.e., falsely rejecting the null hypothesis) increases. MANOVA allows you to conduct one analysis, thus controlling for this inflation of error.
• Understanding Inter-relationships Among Dependent Variables: MANOVA can provide insights into the inter-relationships among dependent variables and how these relationships are influenced by the independent variables. For instance, it can help understand how the diet regimen differentially impacts BMI, cholesterol levels, and blood pressure together.
• Exploring Multivariate Effects: Sometimes, an independent variable may not have significant effects on individual dependent variables when considered separately, but it may have a significant combined effect. MANOVA can detect these multivariate effects that univariate tests (like ANOVA) cannot.

Applications of MANOVA

Multivariate Analysis of Variance (MANOVA) is a powerful statistical technique used in a variety of fields. Here are a few examples of how it’s applied:

• Psychology: In psychology, MANOVA can be used to investigate the effect of various factors (e.g., treatment methods, environmental conditions) on multiple psychological measures (e.g., anxiety levels, self-esteem scores, depression inventory scores).
• Biology and Medicine: MANOVA can be used to compare the impact of different treatments or conditions on multiple related biological or medical outcomes. For example, a researcher might use MANOVA to compare the effects of different drugs on several measures of cardiovascular health.
• Education: In education research, MANOVA could be used to compare the effects of different teaching methods on multiple learning outcomes, such as students’ test scores, self-reported understanding, and engagement in class.
• Business: In marketing and business research, MANOVA might be used to compare customer perceptions (e.g., product satisfaction, likelihood of recommending a product, perceived value) across different product groups or marketing strategies.
• Social Science: In sociology or political science, a researcher might use MANOVA to compare public opinions across different demographics on several related measures, such as trust in government, political engagement, and political efficacy.
• Environmental Science: MANOVA can be used to compare different ecological or environmental conditions (e.g., different pollution levels or climate zones) on multiple ecological measures (e.g., species diversity, plant growth, soil health).

Advantages of MANOVA

Multivariate Analysis of Variance (MANOVA) has several advantages over univariate methods:

• Multiple Outcome Variables: Unlike univariate methods like ANOVA, MANOVA allows you to examine the effect of one or more independent variables on two or more dependent variables simultaneously. This allows for a more comprehensive view of the data.
• Protection against Type I Errors: When multiple ANOVAs are performed separately for each dependent variable, the likelihood of making a Type I error (falsely rejecting the null hypothesis) increases. By considering multiple dependent variables simultaneously, MANOVA helps control this error rate.
• Interactions among Dependent Variables: MANOVA allows for the examination of potential interactions between dependent variables, providing insights into how they may collectively be influenced by the independent variables.
• Detection of Multivariate Effects: There may be instances where the effect of an independent variable isn’t significant on any individual dependent variable but is significant when all dependent variables are considered together. MANOVA can help detect such multivariate effects, which could be missed by separate ANOVAs.
• Economical: MANOVA is more economical in terms of sample size requirements as it incorporates multiple dependent variables into a single analysis, unlike separate ANOVAs which require larger sample sizes to maintain power.

Disadvantages of MANOVA

While Multivariate Analysis of Variance (MANOVA) offers many advantages, there are also several potential challenges and limitations to consider:

• Assumptions: MANOVA makes several assumptions, such as multivariate normality, homogeneity of covariance matrices, and absence of multicollinearity among dependent variables. If these assumptions are violated, the results of the MANOVA may be invalid. Checking and meeting these assumptions can be more complex and challenging than for univariate tests like ANOVA.
• Interpretation Difficulty: MANOVA results can be difficult to interpret, especially if there are interactions between factors or if there is more than one dependent variable. The relationships between multiple dependent variables can be complex, and understanding the multivariate test statistics (such as Wilks’ Lambda or Pillai’s Trace) requires a solid understanding of multivariate statistics.
• High Computational Requirements: Because it involves multiple dependent variables, the computations for MANOVA are more complex and demanding than for ANOVA. This may be a limitation in situations where computational resources are limited.
• Requires Large Sample Sizes: MANOVA typically requires larger sample sizes than univariate ANOVA to achieve the same statistical power. This is because the more dependent variables you have, the more challenging it becomes to detect an effect.
• Risk of Overfitting: With multiple dependent variables, there’s a greater risk of overfitting the model, especially if there’s a high degree of correlation between dependent variables.
• Less Robust to Missing Data: Like most statistical methods, MANOVA is less robust when dealing with missing data. Depending on the amount and nature of the missing data, it may impact the validity of the results.

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MANOVA: A Practical Guide for Data Scientists

You will learn how MANOVA empowers data scientists with multi-dimensional analysis for deeper insights.

Introduction

Multivariate Analysis of Variance (MANOVA)  is an extension of the Analysis of Variance (ANOVA) technique, used extensively in data science and statistics. Unlike ANOVA, which examines the impact of one or more independent variables on a single dependent variable, MANOVA allows for the examination of multiple dependent variables simultaneously. This comprehensive approach is beneficial in scenarios where variables are interrelated and offers a more nuanced understanding of data sets where various outcomes are of interest.

In the ever-evolving field of  data science , understanding and applying MANOVA is crucial. Its capability to analyze multiple dimensions of data in a single model makes it an invaluable tool for data scientists often tasked with extracting meaningful insights from complex and multi-dimensional datasets. MANOVA not only enhances the accuracy of the analysis but also provides a more efficient means of understanding the relationships and interactions between variables.

Throughout this article, readers will gain a thorough understanding of MANOVA, beginning with its basic concepts and the situations that warrant its use. We will guide you through a  step-by-step process  of performing MANOVA, including data preparation, execution, and interpretation of results. Additionally, the article will cover advanced applications of MANOVA in various fields of data science, providing insights into innovative uses and future trends in multivariate analysis. By the end of this guide, you will be well-equipped with the knowledge and skills to effectively implement MANOVA in your data science projects, enhancing your analytical capabilities and contributing to the pursuit of truth and knowledge in the field.

• Enhanced Discriminative Power : MANOVA identifies subtle differences across multiple dependent variables, surpassing the capabilities of multiple ANOVAs in multivariate contexts.
• Critical Role of Data Normalization : Emphasize the importance of data normalization in MANOVA to ensure result accuracy, highlighting its role in maintaining data integrity.
• Comprehensive Variable Analysis : Highlight MANOVA’s unique ability to simultaneously analyze interdependent variables, offering a complete picture of the data landscape.
• Nuanced Interpretation of Results : Stress the importance of a nuanced approach in interpreting MANOVA results, balancing statistical significance with real-world applicability.

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Understanding the Basics of MANOVA

MANOVA (Multivariate Analysis of Variance) , an advanced statistical method, extends the principles of ANOVA (Analysis of Variance) to multiple dependent variables. This technique is pivotal in data science for its robust approach to dissecting complex datasets.

Definition and Key Concepts

MANOVA is designed to analyze the impact of one or more independent variables on two or more dependent variables simultaneously. Critical concepts in MANOVA include:

• Dependent and independent variables.
• Multivariate normality.
• Homogeneity of variance-covariance matrices.
• The importance of interaction effects.

The method is particularly effective in situations where dependent variables are not only related to the independent variables but also to each other.

When and Why to Use MANOVA

The primary scenarios where MANOVA becomes crucial include:

• Situations where multiple dependent variables are interrelated and need a joint analysis.
• Studies aiming to control Type I error rates when multiple ANOVAs could lead to false positives.
• Research that necessitates the examination of the interaction effects among variables.

Understanding when to apply MANOVA helps make informed decisions and obtain more reliable and comprehensive results from multivariate data.

Comparison with Other Statistical Methods

While ANOVA examines the impact on a single dependent variable, MANOVA extends this to multiple outputs, thus providing a more detailed analysis. Compared to separate ANOVAs for each dependent variable, MANOVA reduces the risk of Type I errors. Additionally, unlike regression analysis, focusing on predicting one dependent variable based on independent variables, MANOVA explores how independent variables distinctly affect multiple outcomes.

In summary, MANOVA stands out for its ability to provide a holistic view of complex datasets where the same factors influence multiple outcomes. This thorough understanding of MANOVA’s basics, including its purpose and comparison with other methods, lays a solid foundation for exploring its more advanced applications in data science.

Step-by-Step Guide to Performing MANOVA

Performing a Multivariate Analysis of Variance (MANOVA) requires a systematic approach, ensuring accurate analysis and meaningful interpretation of complex data. This section provides a detailed guide on executing MANOVA, from data preparation to effective execution.

Data Prerequisites and Preparation

Before conducting MANOVA, specific prerequisites must be met:

• Data Collection : Ensure the data collected is relevant to the research question and includes multiple dependent variables to be analyzed simultaneously.
• Data Cleaning : Address missing values, outliers, and inconsistencies in the dataset.
• Variable Selection : Identify the independent and dependent variables. The dependent variables should be metric (interval or ratio scale), and the independent variables should be categorical.
• Data Normalization:  Normalize the data to ensure uniformity and comparability across different scales and units of measurement. This step is vital when dependent variables vary significantly in scale or units, as it prevents skewed results due to scale differences. Standard normalization techniques include min-max scaling, z-score normalization, or log transformation. The choice of method depends on the data’s characteristics, ensuring that each variable contributes equally to the analysis and maintaining result accuracy.
• Checking Assumptions : Verify that the data meets MANOVA’s assumptions, including multivariate normality, homogeneity of variance-covariance matrices, and the independence of observations.

Detailed Walkthrough of the MANOVA Process

• Define the Hypothesis : Clearly state the null and alternative hypotheses regarding the relationships between the independent and dependent variables.
• Choose the MANOVA Test : Select the appropriate MANOVA test based on the study design and hypothesis. Standard tests include Wilks’ Lambda, Pillai’s Trace, Hotelling’s Trace, and Roy’s Largest Root.  *See the upcoming section ‘When to Use Specific MANOVA Tests’ for detailed guidance.
• Data Analysis Setup : Using statistical software (such as R, Python, SPSS, or SAS), input the data correctly and specify the MANOVA model.
• Run the Test : Execute the MANOVA procedure and record the output.
• Post-hoc Analysis : If significant results are found, conduct post-hoc tests to understand where the differences lie.

When to Use Specific MANOVA Tests

• Wilks’ Lambda : Best suited for small sample sizes or when the assumption of equal covariance matrices is met. It’s the most commonly used test due to its robustness and reliability across various conditions. Use Wilks’ Lambda when your data is well-behaved and follows the assumptions of MANOVA closely.
• Pillai’s Trace : Preferred when dealing with unequal sample sizes and violation of assumptions regarding the homogeneity of variances and covariances. Pillai’s Trace is considered the most robust test against violations of these assumptions, making it a safer choice for less ideal datasets.
• Hotelling’s Trace : Effective in scenarios where you have a larger sample size and relatively equal group sizes. This test is more sensitive than Wilks’ Lambda to differences between groups. It is beneficial when you expect substantial group differences and have sufficient data to support this analysis.
• Roy’s Largest Root (Greatest Characteristic Root) : Ideal for situations where the focus is on the largest eigenvalue, and you are interested in the most significant multivariate effect. However, it’s less commonly used due to its sensitivity to violations of assumptions. It is generally recommended when you have a strong rationale for focusing on the principal eigenvalue.

In summary, the choice of test in MANOVA depends on your sample size, group sizes, and the robustness of your data to the assumptions of MANOVA. Wilks’ Lambda is a good general-purpose choice, while Pillai’s Trace offers more robustness against assumption violations. Hotelling’s Trace is suitable for larger, well-balanced datasets, and Roy’s Largest Root is specific for focusing on the most significant multivariate effect.

Tips for Effective Execution

• Data Understanding : A thorough understanding of your data and its structure is crucial. Consider conducting exploratory data analysis (EDA) before MANOVA.
• Software Proficiency : Familiarize yourself with the statistical software you are using. Each software has specific ways of implementing MANOVA.
• Interpretation Skills : Learn to interpret the MANOVA output effectively, focusing on understanding what the results mean in the context of your research question.
• Documentation : Keep a detailed record of all steps and decisions made during the analysis to ensure reproducibility and transparency.

By following this guide, you can perform MANOVA with a clear understanding of each step. Remember, the key to practical MANOVA analysis lies in meticulous data preparation, a firm grasp of statistical principles, and the ability to interpret results within the broader context of your research.

Interpreting MANOVA Results

Correctly interpreting the results of a Multivariate Analysis of Variance (MANOVA) is crucial for extracting meaningful insights from your data analysis. This section delves into understanding the output, addressing common pitfalls, and illustrating with a case study.

Understanding Output and Results

After running MANOVA, the output typically includes several vital statistical measures:

• Wilks’ Lambda : A measure of how well each function separates cases into groups. Lower values indicate more group separation.
• Pillai’s Trace : This is another measure of group separation, with higher values indicating more differentiation.
• Hotelling’s Trace  and  Roy’s Largest Root  provide additional insights into group differences.

Each of these measures has an associated F-value and p-value, which indicate the statistical significance of the results. A significant p-value (usually <0.05) suggests significant differences between group means on the combined dependent variables.

Common Pitfalls and How to Avoid Them

• Overlooking Assumptions : Not checking for multivariate normality and homogeneity of variance-covariance can lead to incorrect conclusions. Always test these assumptions before running MANOVA.
• Misinterpretation of Results : Avoid jumping to conclusions based on significant p-values alone. Understand the context and practical significance of your findings.
• Inadequate Post-hoc Analysis : If you find significant results, conduct post-hoc tests to explore where these differences lie. This helps in understanding the specific relationships between variables.

Case Study Example

Consider a study assessing the effectiveness of a new teaching method on student performance. The dependent variables are scores in mathematics, science, and language. The independent variable is the teaching method (traditional vs. new method).

After conducting MANOVA, suppose we find a significant p-value for Wilks’ Lambda. This suggests that there are overall differences in performance scores between the two teaching methods. Post-hoc analysis reveals that the new teaching method significantly improves mathematics and science scores but not language scores. This nuanced understanding assists in evaluating the teaching method’s effectiveness across different subjects.

Interpreting MANOVA results requires statistical insight and a deep understanding of the research context. By carefully examining the output and considering both statistical and practical significance, one can draw comprehensive and accurate conclusions from the MANOVA analysis. This approach ensures that the insights gained are statistically valid but also meaningful and actionable in real-world scenarios.

As we conclude this comprehensive guide on Multivariate Analysis of Variance (MANOVA), let’s recap the key takeaways and encourage the application of MANOVA in data science projects.

Key Takeaways:

• Versatility and Depth : MANOVA stands out for its ability to analyze multiple dependent variables simultaneously, providing a deeper and more nuanced understanding of data sets.
• Accuracy and Efficiency : By addressing the interrelations between variables, MANOVA enhances the accuracy of statistical analysis and offers efficient insight into complex data sets.
• Critical Thinking in Data Preparation and Analysis : The success of MANOVA hinges on proper data collection, preparation, and testing of assumptions, underscoring the importance of thorough and systematic approaches in data science.
• Statistical Significance and Practical Relevance : Understanding MANOVA results requires a grasp of statistical significance and an appreciation of their practical implications in real-world scenarios.
• Continuous Learning and Adaptation : The field of data science is ever-evolving. MANOVA represents both a classic and adaptable tool for researchers and practitioners alike.

Encouragement to Apply MANOVA:

Data science is a field where theory meets practice, and MANOVA perfectly embodies this intersection. Whether you are exploring new patterns in biomedical research, assessing marketing strategies, or delving into social science inquiries, MANOVA can provide insightful analyses that transcend the capabilities of more straightforward methods like ANOVA.

As data scientists and researchers, we encourage you to integrate MANOVA into your analytical toolkit. Embrace its complexity as an opportunity for growth and discovery. Let the insights you gain from MANOVA advance your projects and contribute to your field’s broader pursuit of knowledge and truth.

As we continue to explore and innovate in data science and statistics, let us remember that tools like MANOVA are not just methodologies — they are windows into understanding the complex tapestry of our world.

Next Steps:

As you embark on your journey with MANOVA, consider exploring further resources and case studies. Delve into advanced applications and continuously update your skills with the latest software and analytical techniques. Remember, the path of a data scientist is one of lifelong learning and curiosity. Happy analyzing!

Recommended Articles

Are you interested in more in-depth analysis techniques? Discover other related articles on our blog and expand your data science toolkit!

• ANOVA and T-test: Understanding the Differences
• ANOVA versus ANCOVA: Breaking Down the Differences
• Common Mistakes to Avoid in One-Way ANOVA Analysis

Frequently Asked Questions (FAQs)

Q1: What is MANOVA?  It’s a statistical method to analyze the differences in multiple dependent variables across different groups.

Q2: How does MANOVA differ from ANOVA?  Unlike ANOVA, which examines one dependent variable, MANOVA assesses multiple dependent variables simultaneously.

Q3: When should you use MANOVA?  Use MANOVA to understand the impact of independent variables on two or more dependent variables.

Q4: What are the assumptions for MANOVA?  Assumptions include multivariate normality, homogeneity of variance-covariance matrices, and independence of observations.

Q5: How do you interpret MANOVA results?  Interpretation involves examining Pillai’s Trace, Wilks’ Lambda, Hotelling’s Trace, and Roy’s largest root to understand group differences.

Q6: What are common pitfalls in using MANOVA?  Common pitfalls include ignoring assumptions, misinterpreting results, and improper data scaling or transformation.

Q7: Can MANOVA be used with categorical data?  Yes, but the categorical variables must be appropriately coded as dummy variables in the analysis.

Q8: How does MANOVA handle correlated dependent variables?  MANOVA is specifically designed to manage and analyze correlated dependent variables, providing more accurate results than separate ANOVAs.

Q9: What software can be used for MANOVA?  Several statistical software packages like R, Python (with libraries like Pandas and StatsModels), SPSS, and SAS offer MANOVA capabilities.

Q10: Are there any prerequisites for learning MANOVA?  A basic understanding of statistics, ANOVA, and multivariate calculus helps grasp the concepts of MANOVA.

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JABSTB: Statistical Design and Analysis of Experiments with R

Chapter 38 multivariate analysis of variance (manova).

MANOVA is a procedure to analyze experimental data involving simultaneous measurements of two or more dependent variables in response to two or more predictor groups.

The basic MANOVA designs are no different than the various t-test or ANOVA designs. Paired, unpaired, one-way (one-factor), two-way (two-factor) and even three-way (three-factor) (or more way) MANOVA experiments are possible. They can be structured either as independent (completely randomized) or intrinsically-linked (related/repeated measures) or a mixture of the two.

What differs is MANOVA designs collect measurements for more than one dependent variable from each replicate.

The statistical jargon for such experiments is they are “multivariate,” whereas ANOVA is “univariate.” This term is owed to a mathematical feature of underlying MANOVA which involves calculating linear combinations of these dependent variables to uncover latent “variates.” The statistical test is actually performed on these latent variates.

Statistical jargon is very confusing most of the time. When we have experiments that involve more than one treatment variable (each at many levels) and only one outcome variable, such as a two-way ANOVA, we call these “multivariable” experiments. But they are also known as “univariate” tests because there is only one dependent variable. When we have experiments that have one or more treatment variables and multiple outcome variables, such as one- or two-way MANOVA, we call these multivariate tests. I’m sorry.

38.1 What does MANOVA test?

Exactly what is that null hypothesis?

Like ANOVA, MANOVA experiments involve groups of factorial predictor variables (it is possible to run MANOVA on only two groups). The null hypothesis addresses whether there are any differences between groups of means. As in ANOVA, this is accomplished by partitioning variance. Therefore, as for ANOVA, the test is whether the variance in the MANOVA model exceeds the residual variance.

However, in MANOVA this operates on the basis of statistical parameters for a composite dependent variable, which is calculated from the array of dependent variables in the experiment.

You’ll see that the MANOVA test output per se generates an F statistic. In that regard the inferential procedure is very much like ANOVA. If the F value is extreme, the model variance exceeds the residual variance, and it has a low p-value. A p-value falling below a preset type1 error threshold favors a rejection of null hypotheses.

However, this is a funny F statistic.

First it is an approximated F, which is mapped to the F test statistic distribution.

MANOVA data sets have one column for each dependent variable, which creates a matrix when there are multiple dependent variables.

The MANOVA test statistic operates on a matrix parameter called the eigenvalue. Eigenvalues are scaling factors for eigenvectors, which are also matrix parameters. Eigenvectors are ways of reducing the complexity of matrix data. Eigenvectors represent latent variates within these multivariate datasets. Note the plural. There can be more than one eigenvector in a multivariate dataset, the number for which is related to the number of dependent and independent variables.

There are four different MANOVA test statistics: Wilks’ Lambda, Hotelling-Lawley Trace, Pillai’s Trace, and Roy’s Largest Root. Each are calculated differently. Given the same input data, each test statistic is then used to generate an F statistic.

Like a lot of things in statistics, they represent different ways to achieve the same objective, but generate different results in the process. Their eponymous inventors believe their’s offers a better mousetrap.

Choosing which of the MANOVA test statistics to use for inference is not unlike choosing which multiple correction method to use following ANOVA. It can be confusing.

It is not uncommon for the test statistics to yield conflicting outcomes. Therefore, it is imperative for the test statistic to be chosen in advance.

38.2 Types of experiments

We imagine a gene expression study that measures the transcripts for several different genes simultaneously within every replicate. Each transcript represents a unique dependent variable since they come from different genes, each with its own network of transcription factors. But we can also imagine underlying latent relationships between some of the genes. For example, some of the genes may be regulated by a common transcript. We might be interested in how different genes are expressed over time and in the absence or presence of certain stimuli. Time and stimuli are each predictor variables. The sample source informs whether the design is completely randomized or related/repeated measure.

Or we are interested in quantifying several different proteins simultaneously by western blot technique. We might be interested in testing how different groups of stimuli or certain mutations affects each of their levels. Or we are able to measure multiple proteins simultaneously in cells using differentially colored fluorescent tags. Each protein (each color) is a unique dependent variable.

Or we’ve placed a set of animals through some protocol comparing three or more groups of treatments. The animals are run through some behavioral assay, such as a latency test. Afterwards, specimens are collected for bloodwork, and also for quantitative histochemical and gene expression analysis. The latency test, the various measures from the blood, and the markers and genes assayed post -mortum are each a unique dependent variable. There might be a dozen or more of these outcome variables measured in each replicate!

38.3 MANOVA is not many ANOVAs

In one respect, you can think of experiments involving multiple dependent variables as running a bunch of ANOVA experiments.

In fact, that’s the mistake most researchers make. They treat each dependent variable as if it came from a distinct experiment involving separate replicates, when in fact they all come from just one experiment. They subsequently run a series of ANOVAs in parallel, one each for all of the dependent variables.

The most common mistake is holding each ANOVA at the standard 5% error threshold, following each with multiple post hoc comparisons. The family-wise error rate (FWER) for this single experiment explodes. A cacophony of undeserved asterisks are splattered all over the chart. It doesn’t help that the data are plotted in a way to make the dependent variables look like they are predictor variables. Maybe they have been treated as predictors?! What a mess!

Minimally, MANOVA provides a handy way to manage the FWER when probing several different dependent variables simultaneously. As an omnibus test MANOVA provides fairly strong type1 error protection in cases where many dependent variables are assessed simultaneously.

But it is important to recall what was mentioned above. MANOVA is not really testing for the signal-to-noise for the effects of the independent variables on each of the dependent variables. MANOVA tests whether there are treatment effects on a combination of the outcome variables. The advantage is that this is performed in a way that maximizes the treatment group differences.

38.3.1 Why MANOVA?

For experiments that measure multiple dependent variables the main alternative to MANOVA is to run separate ANOVAs, each with a multiple comparison correction to limit the FWER. For example, if there are five dependent variables, use the Bonferroni correction to run the ANOVA for each under at type1 error threshold of 1%, meaning we only reject the null when the F p-value < 0.01.

There are good reasons to run MANOVA instead.

The first is as an omnibus test offering protection against inflated type1 error, as when running a series of ANOVA’s for each dependent variable. Particularly when the latter are uncorrected for multiple comparison.

The second is testing more dependent variables increases the chances of identifying treatment effects. For example, a given stimulus condition may not affect the expression of three genes, but it does affect that for a fourth. Had the fourth gene not been added to the analysis we might have concluded that stimulus condition is ineffective.

Third, because it operates on a composite variable, sometimes MANOVA detects effects that ANOVA misses. Imagine studying the effect of diet on growth. We could measure just the weight (or length) of a critter and be done with it. However, with only a little extra effort measuring both weight and height, we have measures in two dimensions instead of just one. In some cases the effects of diet on weight alone or on height alone may be weak, but stronger when assessed in combination.

Select variable to measure judiciously. MANOVA works best when dependent variables are negatively correlated or modestly correlated, and does not work well when they are uncorrelated or strongly positively correlated.

38.4 Assumptions

Some of these should look very familiar by now:

All replicates are independent of each other (of course, repeated/related measurements of one variable my be collected from a single replicate, but this must be accounted for). Data collection must involve some random process.

Here are some proscriptions unique to MANOVA but with univariate congeners:

The dependent variables are linearly related. The distribution of residuals are multivariate normal. *The residual variance-covariance matrices of all groups are approximately equal or homogeneous.

As a general rule MANOVA is thought to be more sensitive to violations of these latter three assumptions than is ANOVA.

Several adjustments can help prevent this:

Strive for balanced data (have equal or near-equal sample sizes in each group). Transform variables with outliers or that have nonlinearity to lessen their impact (or establish other a priori rules to deal with them if they arise). *Avoid multicolinearity.

38.4.1 Multicolinearity

Multicolinearity occurs in a couple of ways. When one dependent variable is derived from other variables in the set (or if they represent two ways to measure the same response) they may have a very high \(R^2\) value. More rarely, dependent variables may be highly correlated naturally. In these cases one of the offending variables should be removed. For example, multicolinearity is likely to happen if including a body mass index (BMI) as a dependent variable along with variables for height and weight, since the BMI is calculated from the others.

Multicolinearity collapse will also occur when the number of replicates are fewer than the number of dependent variables being assessed.

Don’t skimp on replicates and avoid the urge to be too ambitious in terms of the numbers of dependent variables collected and the number of treatment groups. As a general rule, if the dataset rows are replicates and the columns are dependent variables, we want the dataset to be longer than it is wide.

38.5 Calculating variation

There are many parallels between ANOVA and MANOVA.

Both are based upon the general linear model \[Y=\beta X+\epsilon\]

However, for MANOVA, \(Y\) is an \(n \times m\) matrix of dependent variables, \(X\) is an \(n \times p\) matrix of predictor variables, \(\beta\) is an \(p \times m\) matrix of regression coefficients and \(\epsilon\) is a \(n \times m\) matrix of residuals.

Least squares regression for calculating the \(SS\) for each dependent variable is performed in MANOVA as for ANOVA. In addition, variation is also tabulated from cross products between all possible combinations of dependent variables. As for ANOVA, the conservation of variation law applies for cross products just as it does for \(SS\) ,

\[CP_{total}=CP_{model}\ +CP_{residual}\]

For illustration, consider the simplest MANOVA experiment with only two dependent variables ( \(dv1, dv2\) ). The cross product for total variation is:

\[CP_{total}= \sum_{i=1}^n(y_{i,dv1}-\bar y_{grand, dv1})(y_{i,dv2}-\bar y_{grand, dv2}) \]

The cross product for variation associated with the model (group means) is:

\[CP_{model}= \sum_{j=1}^kn\times(\bar y_{group_j,dv1}-\bar y_{grand, dv1})(\bar y_{group_j,dv2}-\bar y_{grand, dv2}) \]

And the cross product, \(CP\) , for residual variation is:

\[CP_{residual}= \sum_{i=1}^n(y_{i,dv1}-\bar y_{group_j, dv1})(y_{i,dv2}-\bar y_{group_j, dv2}) \]

For partitioning of the overall variation, these cross products, along with their related \(SS\) are assembled into \(T\) , \(H\) and \(E\) matrices. These letters reflect a historical MANOVA jargon representing total, hypothesis and error variation. These correspond to the total, model and residual terms we’ve adopted in this course for discussing ANOVA.

\[T = \begin{pmatrix} SS_{total,dv1} & CP_{total} \\ CP_{total} & SS_{total,dv2} \end{pmatrix}\]

\[H = \begin{pmatrix} SS_{model,dv1} & CP_{model} \\ CP_{model} & SS_{model,dv2} \end{pmatrix}\]

\[E = \begin{pmatrix} SS_{residual,dv1} & CP_{residual} \\ CP_{residual} & SS_{residual,dv2} \end{pmatrix}\]

The most important take away is that MANOVA not only accounts for the variation within each dependent variable via \(SS\) in the usual way, the \(CP\) computes the variation associated with all possible relationships between each of the dependent variables.

Note: When experiments have even more dependent variables, there is more variation to track. For example an experiment with three independent variables has a T matrix of 9 cells with 3 unique cross-product values, each duplicated:

\[T = \begin{pmatrix} SS_{total,dv1} & CP_{total,dv1\times dv2} & CP_{total,dv1\times dv3} \\ CP_{total, dv2\times dv1} & SS_{total,dv2} & CP_{total,dv2\times dv3}\\ CP_{total,dv3\times d1} & CP_{total,dv3\times dv2} & SS_{total, dv3} \end{pmatrix}\]

The conservation of variation rule applies to these matrices just as in univariate ANOVA. The total variation is equal to the sum of the model and the residual variation. The same applies in MANOVA, which is expressed by the simple matrix algebraic relationship: \(T=H+E\) .

38.5.1 Eigenvectors and eigenvalues

To deal with this mathematically all of the computations necessary to sort out whether anything is meaningful involve matrix algebra, where the focus is upon “decomposing” these matrices into their eigenvectors and eigenvalues.

What is an eigenvector? The best way I’ve been able to answer this is an eigenvector represents a base dimension in multivariate data, and the eigenvalues serve as the magnitude of that dimension.

MANOVA datasets have multiple response variables, \(p\) , and also multiple predictor groups, \(k\) . How many dimensions can these datasets possess? They will have either \(p\) or \(k-1\) dimensions, whichever is smaller.

The mathematics of these are beyond where I want to go on this topic. If interested in learning more the Kahn Academy has a nice primer . Here is a good place to start for an introduction to R applications . Here’s a graphical approach that explains this further .

It is not necessary to fully understand these fundamentals of matrix algebra in order to operate MANOVA for experimental data. However, it is worth understanding that the MANOVA test statistics operate on something that represents a dimension of the original data set.

38.5.2 MANOVA test statistics

Recall in ANOVA the F statistic is derived from the ratio of the model variance with \(df1\) degrees of freedom to residual variance with \(df2\) degrees of freedom. In MANOVA these variances are essentially replaced by matrix determinants.

The congeners to ANOVA’s model and residual variances in MANOVA are the hypothesis \(H\) and error \(E\) matrices, which have \(h\) and \(e\) degrees of freedom, respectively. There are \(p\) dependent variables. Let the eigenvalues for the matrices \(HE^{-1}\) , \(H(H+E)^{-1}\) , and \(E(H+E)^{-1}\) be \(\phi\) , \(\theta\) , and \(\lambda\) , respectively.

From these let’s catch a glimpse of the four test statistic options available to the researcher when using MANOVA. When we read a MANOVA table in R the test stat column will have values calculated from the parameters below.

38.5.2.1 Pillai

\[V^{(s)}=\sum_{j=1}^s\theta_j \] where \(s=min(p,h)\) $V^{(s)} is then used in the calculation of an adjusted F statistic, from which p-values are derived. Calculation not shown^.

38.5.2.2 Wilks

\[\Lambda = \prod_{j=1}^p(1-\theta_j) \] \(\Lambda\) is then used to calculate an adjusted F statistic, from which p-values are derived. Calculation not shown.

38.5.2.3 Hotelling-Lawley

\[T_g^s = e\sum_{j=1}^s\theta_j \] where \(g=\frac{ph-2}{2}\)

\(T_g^s\) is then used to calculate an adjusted F statistic, from which p-values are derived. Calculation not shown.

38.5.2.4 Roy

\[F_{(2v_1+2, 2v_2+2)}=\frac{2v_1+2}{2v_2+2}\phi_{max} \] where \(v1=(|p-h|-1)/2\) and \(v2=(|e-p|-1)/2\)

Unlike the other MANOVA test statistics, Roy’s greatest root ( \(\phi_max\) ) is used in a fairly direct calculation of an adjusted F statistic, so that is shown here. Note how the degrees of freedom are calculated.

One purpose of showing these test statistics is to illustrate that each are calculated differently. Why? The statisticians who created these worked from different assumptions and objectives, believing their statistic would perform well under certain conditions.

A second reason to show these is so the researcher avoids freeze when looking at MANOVA output: “OMG! What the heck??!”

Yes, the numbers in the MANOVA table can be intimidating at first, even if when we have a pretty good idea of what we are up to. There are three columns for degrees of freedom and two for test statistics. Then there is the p-value. And then there is a MANOVA table for each independent variable. And if we don’t argue a specific test, we might get all four!! Yikes.

38.5.3 Which MANOVA test statistic is best?

That’s actually very difficult to answer.

In practice, the researcher should go into a MANOVA analysis with one test in mind, declared in a planning note, ideally based upon some simulation runs. This is no different than running a Monte Carlo before a t-test experiment, or an ANOVA experiment. Knowing in advance how to conduct the statistical analysis is always the most unbiased approach.

Otherwise, the temptation will be to produce output with all four tests and choose the one that yields the lowest p-value. In most instances the four tests should lead to the same conclusion, though they will not always generate the same p-values.

The most commonly used test seems to be Wilk’s lambda. The Pillai test is held to be more robust against violations of testing conditions assumptions listed above and is a reasonable choice.

38.6 Drug clearance example

Mimche and colleagues developed a mouse model to test how drug metabolism is influenced by a malaria-like infection (Plasmodium chaubadi chaubadi AS, or PccAS).

On the day of peak parisitaemia an experiment was performed to derive values for clearance (in units of volume/time) of four drugs. Clearance values are derived from a regression procedure on data for repeated measurements of drug levels in plasma at various time points following an injection. Any factor causing lower clearance indicates drug metabolism is reduced.

Thus, the clearance values for each drug are the dependent variables in the study. All four drugs were assessed simultaneously within each replicate. The independent variable is treatment, which is at two levels, naive or infection with PccAS, a murine plasmodium parasite. Ten mice were randomly assigned to either of these two treatments.

38.6.1 Rationale for MANOVA

The overarching scientific hypothesis is that parasite infection alters drug metabolism.

The drug panel tested here surveys a range of drug metabolizing enzymes. Limiting the clearance study to only a single drug would preclude an opportunity to gain insight into the broader group of drug metabolizing systems that might be affected by malaria.

Thus, there is good scientific reason to test multiple dependent variables. Together, they answer one question: Is drug metabolism influenced by the parasite? Clearance values are also collected in one experiment in which all four drugs are injected as a cocktail and measured simultaneously. This cannot be treated as four separate experiments statistically.

The main reason to use MANOVA in this case is as an omnibus test to protect type1 error while considering all of the information for the experiment simultaneously.

38.6.2 Data structure

Read in the data. Note that each dependent variable is it’s own column, just as for the predictor variable. An ID variable is good data hygiene.

38.6.3 MANOVA procedure

The Manova function in the car package can take several types of models as an argument. In this case, we have a two group treatment variable so a linear model is defined using the base R lm .

Recall the linear model, \[Y=\beta_o+\beta_1X+\epsilon\] Here we have one model for each of the four dependent variables.

In this experiment we have four \(Y\) variables, each corresponding to one of the drugs. The same \(X\) variable (at two levels) applies to all four dependent variables.

\(\beta_0\) is the y-intercept, and \(\beta_1\) is the coefficient. Each dependent variable will have an intercept and a coefficient. \(\epsilon\) is the residual error, accounting for the variation in the data unexplained by the model parameters for that dependent variable.

By including each of the dependent variable names as arguments within a cbind function we effectively instruct lm to treat these columns as a matrix and run a linear model on each.

Here is the omnibus test. The Pillai test statistic is chosen because it is more robust than the others to violations of the uniform variance assumptions:

The approximated F statistic of 11.943 is extreme on a null distribution of 4 and 5 degrees of freedom with a p-value of 0.009017. Reject the null hypothesis that the variance associated with the combination of linear models is less than or equal to the combined residual error variance.

Since this is a linear model for independent group means this result indicates that there are difference between group means.

Additional information about the Manova test can be obtained using the summary function. Note the matrices for the error and hypothesis, which are as described above.

Also not the output for the four different test statistics. In this simple case of one independent variable with two groups, they all produce an identical approximate F and the p-values that flows from it. The test values do differ, however, although the Hotelling-Lawley and the Roy generate the same values. More complex experimental designs tend to shatter this uniform agreement between the tests.

38.6.3.1 Now what?

Here are the data visualized. The red lines connect group means. This is presented as a way to illustrate the linear model output.

Figure 38.1: Drug clearance in a mouse model for malaria.

Print the linear model results to view the effect sizes as regression coefficients:

Considering the graph above it should be clear what the intercept values represent. They are the means for the naive treatment groups. The ‘treatmentpccas’ values are the differences between the naive and pccas group means. Their values are equivalent to the slopes of the red lines in the graph. In other words, the ‘treatmentpccas’ values are the means of the pccas groups subtracted from the means of the naive groups.

Intercept and treatmentpccas correspond respectively to the general linear model parameters \(\beta_0\) and \(\beta_1\) .

Another way to think about this is the intercept ( \(\beta_0\) ) is an estimate for the value of the dependent variable when the independent variable is without any effect, or \(X=0\) . In this example, if pccas has no influence on metabolism, then drug clearance is simply \(Y= \beta_0\)

In regression the intercept can cause confusion. In part, this happens because R operates alphanumerically by default. It will always subtract the second group from the first. Scientifically, we want the pccas means to be subtracted from naive….because our hypothesis is that malaria will lower clearance. Fortunately, n comes before p in the alphabet, so the group means for p will be subtracted from group means for n and this happens automagically. For simplicity, it is always a good idea to name your variables with this default alphanumeric behavior in mind.

38.6.3.2 MANOVA post hoc

In this the data passes and omnibus MANOVA. It tells us that somewhere among the four dependent variables there are differences between the naive and pccas group means. The question is which ones?

As a follow on procedure we want to know which of the downward slopes are truly negative, and which are no different than zero.

We answer that in the summary of the linear model. There is a lot of information here, but two things are most important. First, the estimate values for the treatmentpccas term. These are differences in group means between naive and pccas. Second, statistics and p-values associated with the ‘treatmentpccas’ effect. These are a one-sample t-test.

These each ask, “Is the Estimate value, given its standard error, equal to zero?” For example, the estimate for caffeine is -0.07269, which is the slope of the red line above. The t-statistic value is -2.344 and it has a p-value just under 0.05.

We’re usually not interested in the information on the intercept term or the statistical test for it. It is the group mean for the control condition, in this case, demonstrating the background level of clearance. Whether that differs from zero not important to know.

Therefore to make the p-value adjustment we should only be interested in the estimates for the ‘treatmentpccas’ terms. We shouldn’t waste precious alpha on the intercept values.

Here’s how we do that:

Given a family-wise type1 error threshold of 5%, we can conclude that PccAS infection changes tolbutamide and buproprion clearance relative to naive animals. There is no statistical difference for caffeine and midazolam clearance between PccAS-infected and naive animals.

It should be noted that this is a very conservative adjustment. In fact, were the Holm adjustment used instead all four drugs would have shown statistical differences.

38.6.3.3 Write up

The drug clearance experiment evaluates the effect of PccAS infection on the metabolism of caffeine, tolbutamide, buproprion and midazolam simultaneously in two treatment groups, naive and PccAS-infected, with five independent replicates within each group. Clearance data were analyzed by a multiple linear model using MANOVA as an omnibus test (Pillai test statistic = 0.905 with 1 df, F(4,5)=11.943, p=0.009). Posthoc analysis usig t-tests indicates PccAS the reductions in clearance relative to naive for tolbutamide (-0.006 clearance units) and buproprion (-0.654 clearance units) are statistically different than zero (Bonferroni adjusted p-values are 0.0199 and 0.0009, respectively).

38.7 Planning MANOVA experiments

Collecting several types of measurements from every replicate in one experiment is a lot of work. In these cases, front end statistical planning can really pay off.

We need to put some thought into choosing the dependent variables.

MANOVA doesn’t work well when too many dependent variables in the dataset are too highly correlated and all are pointing in the same general direction. For example, when several different mRNAs increase in the same way in response to a treatment. They will be statistically redundant, which can cause computational difficulties and the regression fails to converge to a solution. The only recovery from that is to add the variables back to a model one at a time until the offending variable is found. Which invariably causes the creepy bias feeling.

Besides, it seems wasteful to measure the same thing many different ways.

Therefore, omit some redundant variables. Be sure to offset positively correlated variables with negatively correlated variables in the dataset. Similarly, MANOVA uncorrelated dependent variables should be avoided.

Use pilot studies, other information, or intuition to understand relationships between the variables. How are they correlated? Once we have that we can calculate covariances?

38.7.1 MANOVA Monte Carlo Power Analysis

The Monte Carlo procedure is the same as for any other test: 1)Simulate a set of expected data. 2)Run an MANOVA analysis, defining and counting “hits.” 3)Repeat many times to get a long run performance average. 4)Change conditions (sample size, add/remove variables, remodel, etc) 5)Iterate through step 1 again.

38.7.1.1 Design an MVME

Let’s simulate a minimally viable MANOVA experiment (MVME) involving three dependent variables. For each there are a negative and positive control and a test group, for a total of 3 treatment groups.

For all three dependent variables we have decent pilot data for the negative and positive controls. We have a general idea of what we would consider minimally meaningful scientific responses for the test group.

One of our dependent variables represents a molecular measurement (ie, an mRNA), the second DV represents a biochemical measurement (ie, an enzyme activity), and the third DV represents a measurement at the organism level (ie, a behavior). We’ll make the organism measurement decrease with treatment so that it is negatively correlated with the other two.

We must assume each of these dependent variables in normally distributed, \(N(\mu, \sigma^2)\) . We have a decent idea of the values of these parameters. We have a decent idea of how correlated they may be.

All three of the measurements are taken from each replicate. Replicates are randomized to receiving one and only one of the three treatment groups.

The model is essentially one-way completely randomized MANOVA; in other words, one factorial independent variable at three levels, with three dependent variables.

38.7.1.2 Simulate multivariate normal data

The rmvtnorm function in the mvtnorm package provides a nice way to simulate several vectors of correlated data simultaneously, the kind of data you’d expect to generate and then analyze with MANOVA.

But it takes a willingness to wrestle with effect size predictions, correlations, variances and covariances, and matrices to take full advantage.

rmvtnorm takes as arguments the expected means of each variable, and sigma, which is a covariance matrix.

First, estimate some expected means and then their standard deviations. Then simply square the latter to calculate variances.

To predict standard deviations, it is easiest to think in terms of coefficient of variation: What is the noise to signal for the measurement? What is the typical ratio of standard deviation to mean that you see for that variable? 10%, 20%, more, less?

For this example, we’re assuming homoscedasticity, which means that the variance is about the same no matter the treatment level. But when we know there will be heterscedasticity, we should code it in here.

38.7.1.2.1 Building a covariance matrix

For the next step we estimate the correlation coefficients between the dependent variables. We predict the biochem measurements and the organ will be reduced by the positive control and test treatments. That means they will be negatively correlated with the molec variables, where the positive and test are predicted to increase the response.

Where do these correlation coefficients come from? They are estimates based upon pilot data. But they can be from some other source. Or they can be based upon scientific intuition. Estimating these coefficients is no different than estimating means, standard deviations and effect sizes. We use imagination and scientific judgment to make these predictions.

We pop them into a matrix here. We’ll need this in a matrix form to create our covariance matrix.

Next we can build out a variance product matrix, using the variances estimated in the table above. The value in each cell in this matrix is the square root of the product of the corresponding variance estimates ( \(\sqrt {var(Y_1)var(Y_2)}\) . We do this for every combination of variances, which should yield a symmetrical matrix.

Here’s why we’ve done this. The relationship between any two dependent variables, \(Y_1\) and \(Y_2\) is

\[cor(Y_1, Y_2) = \frac{cov(Y_1, Y_2)}{\sqrt {var(Y_1)var(Y_2)}}]\]

Therefore, the covariance matrix we want for the sigma argument in the rmvnorm function is calculated by multiplying the variance product matrix by the correlation matrix.

Since we assume homescedasticity we can use sigmas for each of the three treatment groups. Now it is just a matter of simulating the dataset with that one sigma and the means for each of the variables and groups.

Parenthetically, if we are expecting heteroscedasticity, we would need to calculate a different sigma for each of the groups.

Pop it into a data frame and summarise by groups to see how well the simulation meets our plans. Keep in mind this is a small random sample. Increase the sample size, \(n\) , to get a more accurate picture.

Now we run a the “sample” through a manova. We’re switching to base R’s manova here because it plays nicer than Manova with Monte Carlo

Assuming this Manova test gives a positive response, we follow up with look post hoc at the key comparison, because we’d want to power up an experiment to ensure this one is detected.

Our scientific strategy is to assure the organ effect of the test group. Because one has nothig without the animal model, right?

So out of all the posthoc comparisons we could make, we’ll focus in on that.

The comparison in the 9th row is between the test and the negative control for the organ variable. The estimate means its value is -11.46 units less than the negative control. The unadjusted p-value for that difference is 0.0041. Assuming we’d also compare the test to negative controls for the other two variables, were’s how we can collect that p-value while adjusting it for multiple comparisons simultaneously.

The code below comes from finding a workable way to create test output so the parameter of interest is easy to grab. Unfortunately, output from the Manova function is more difficult to handle.

38.7.2 An MANOVA Monte Carlo Power function

The purpose of this function is to determine the sample size of suitable power for an MANOVA experiment.

This function pulls all of the above together into a single script. The input of this script is a putative sample size. The output is the power of the MANOVA, and the power of the most important posthoc comparison: that between the test group and negative control for the organ variable.

Here’s how to run manpwr . It’s a check to see the power for a sample size of 5 replicates per group.

The output is interesting. Although the MANOVA test is overpowered, the critical finding of interest is underpowered at this sample size. Repeat the function at higher values of n until the posthoc power is sufficient.

It is crucial to understand that powering up for the posthoc results can be so important.

38.8 Summary

MANOVA is an option for statistical testing of multivariate experiments. The dependent variables are random normal The test is more senstive than other parametrics to violations of normality and homogeneity of variance. MANOVA tests whether independent variables affect an abstract combination of dependent variables. For most, use MANOVA as an omnibus test followed by post hoc comparisons of interest to control FWER. Care should be taken in selecting the dependent variables of interest.

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Statistics By Jim

Making statistics intuitive

Multivariate ANOVA (MANOVA) Benefits and When to Use It

By Jim Frost 181 Comments

Multivariate ANOVA (MANOVA) extends the capabilities of analysis of variance (ANOVA) by assessing multiple dependent variables simultaneously. ANOVA statistically tests the differences between three or more group means. For example, if you have three different teaching methods and you want to evaluate the average scores for these groups, you can use ANOVA. However, ANOVA does have a drawback. It can assess only one dependent variable at a time. This limitation can be an enormous problem in certain circumstances because it can prevent you from detecting effects that actually exist.

MANOVA provides a solution for some studies. This statistical procedure tests multiple dependent variables at the same time. By doing so, MANOVA can offer several advantages over ANOVA.

In this post, I explain how MANOVA works, its benefits compared to ANOVA, and when to use it. I’ll also work through a MANOVA example to show you how to analyze the data and interpret the results.

ANOVA Restrictions

Regular ANOVA tests can assess only one dependent variable at a time in your model. Even when you fit a general linear model with multiple independent variables, the model only considers one dependent variable. The problem is that these models can’t identify patterns in multiple dependent variables.

This restriction can be very problematic in certain cases where a typical ANOVA won’t be able to produce statistically significant results. Let’s compare ANOVA to MANOVA.

To learn more about ANOVA tests, read my ANOVA Overview .

Comparison of MANOVA to ANOVA Using an Example

MANOVA can detect patterns between multiple dependent variables. But, what does that mean exactly? It sounds complex, but graphs make it easy to understand. Let’s work through an example that compares ANOVA to MANOVA.

Suppose we are studying three different teaching methods for a course. This variable is our independent variable. We also have student satisfaction scores and test scores. These variables are our dependent variables. We want to determine whether the mean scores for satisfaction and tests differ between the three teaching methods. Here is the CSV file for the MANOVA_example .

The graphs below display the scores by teaching method. One chart shows the test scores and the other shows the satisfaction scores. These plots represent how one-way ANOVA tests the data—one dependent variable at a time.

Both of these graphs appear to show that there is no association between teaching method and either test scores or satisfaction scores. The groups seem to be approximately equal. Consequently, it’s no surprise that the one-way ANOVA P-values for both test and satisfaction scores are insignificant (0.923 and 0.254).

Case closed! The teaching method isn’t related to either satisfaction or test scores. Hold on. There’s more to this story!

How MANOVA Assesses the Data

Let’s see what patterns we can find between the dependent variables and how they are related to teaching method. I’ll graph the test and satisfaction scores on the scatterplot and use teaching method as the grouping variable. This multivariate approach represents how MANOVA tests the data. These are the same data, but sometimes how you look at them makes all the difference.

The graph displays a positive correlation between Test scores and Satisfaction. As student satisfaction increases, test scores tend to increase as well. Moreover, for any given satisfaction score, teaching method 3 tends to have higher test scores than methods 1 and 2. In other words, students who are equally satisfied with the course tend to have higher scores with method 3. MANOVA can test this pattern statistically to help ensure that it’s not present by chance.

In your preferred statistical software, fit the MANOVA model so that Method is the independent variable and Satisfaction and Test are the dependent variables.

The MANOVA results are below.

Even though the one-way ANOVA results and graphs seem to indicate that there is nothing of interest, MANOVA produces statistically significant results—as signified by the minuscule P-values. We can conclude that there is an association between teaching method and the relationship between the dependent variables.

When MANOVA Provides Benefits

Use multivariate ANOVA when your dependent variables are correlated. The correlation structure between the dependent variables provides additional information to the model which gives MANOVA the following enhanced capabilities:

• Greater statistical power : When the dependent variables are correlated, MANOVA can identify effects that are smaller than those that regular ANOVA can find.
• Assess patterns between multiple dependent variables : The factors in the model can affect the relationship between dependent variables instead of influencing a single dependent variable. As the example in this post shows, ANOVA tests with a single dependent variable can fail completely to detect these patterns.
• Limits the joint error rate : When you perform a series of ANOVA tests because you have multiple dependent variables, the joint probability of rejecting a true null hypothesis increases with each additional test. Instead, if you perform one MANOVA test, the error rate equals the significance level .

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Reader Interactions

July 25, 2024 at 10:28 am

Hi Jim, What follow-up statistical tests are available to confirm that method-group 3 has significantly higher test scores or satisfaction scores than groups 1 or 2?. The post-hoc tests in SPSS MANOVA are univariate and produce non-significant group differences.

August 1, 2024 at 7:59 pm

That’s a great question. Unfortunately, I don’t have a great answer. I don’t know of post-hoc tests for MANOVA offhand. The fact that SPSS includes only univariate post-hoc tests makes me wonder if multivariate versions are uncommon or non-existent? I wish I had a better answer for you.

July 25, 2024 at 10:06 am

Jim, What post hoc mean comparison tests are available to show that method 3 results in superior test scores and satisfaction? Or, besides the scatterplot (with group markers) what statistical tests can confirm specific group differences? – Anthony

July 22, 2024 at 1:56 pm

PS: or would you still recommend the ANOVA because the DV is the same but just measured at different time points (i.e. DV with 4 levels?), is that possible? Thank you!

July 22, 2024 at 12:35 pm

Hi Jim, what an awesome explanation, thank you so much!

I have a new dataset from my mentor who asked me to analyse it. Based on what I have, I started with an ANOVA and found out, that it doesn´t make sense.

In this dataset, I have 2 IV´s with 2 levels each: (m/f) and condition (a/b). All participants went through both conditions (2 trials p.person; two x two) with outcome measures (DV) being taken during each trial at 4 timepoints. The two-way ANOVA doesn´t allow for multiple DV´s, hence your explanation brought be to the conclusion, I might have to use a MANOVA test here. Would you confirm my assumption?

Thank you so much, kind regards, Anne

August 1, 2024 at 8:12 pm

From your description, it sounds like you are in potentially a tricky scenario where you have multiple DVs and repeated measures. You can likely perform repeated measures using MANOVA and that would be the best method if your DVs are correlated. However, I don’t have experience doing that and couldn’t offer much practical advance. I’d actually consult a statistician at your organization who could give it the time your study deserves.

Alternatively, you could perform separate repeated measures ANOVA on one DV at a time. If your DVs aren’t correlated or only slightly correlated, this approach is good. If they are correlated, you should consider the approach above.

I hope that helps!

May 11, 2024 at 10:53 am

Hello Jim! I have 12 dependent variables from a mental skills questionnaire and one interdependent variable (age: group 1, group 2 group 3, group 4).

Is appropriate to perform ANOVA or MANOVA?

The dependent variables are correlated

Thank you in advance!

May 14, 2024 at 3:24 am

Hello Pulen!

Because the DVs are correlated, you might gain extra power and detect additional types of relationships using MANOVA. But it is valid to use either.

April 15, 2024 at 8:06 am

I have three subscales of which participants have responded to using a questionnaire.

I want to compare if there is a difference in the three subscales between two groups of participants.

i just want to see if one group has a higher score in each subscale than the other.

is this a good use of manova?

(have already conducted a indeoendent samples t-test to compare overall scores)

April 12, 2024 at 5:31 am

I have several IVs (e.g., family education background, student academic background) and several dependent variables with Likert type questions (e.g., different types of barriers (visa, financial etc)). Can I use MANOVAs to test the relationship? What test would you suggest?

April 12, 2024 at 1:53 pm

Because your dependent variables are ordinal data (i.e., Likert scale), MANOVA and other typical regression or ANOVA analyses aren’t appropriate. Instead, consider using ordinal logistic regression, which is designed for ordinal DVs. You’ll have to fit separate models for each DV.

April 1, 2024 at 5:27 am

Thank you for the prompt reply. Have a good day

March 31, 2024 at 5:14 pm

Thank you so much for your help Jim! Much appreciated

March 31, 2024 at 6:59 pm

You’re very welcome!

March 31, 2024 at 4:29 pm

Hello Jim! I have 3 dependent variables (stress, depression, locus of control) and one interdependent variable (age: group 1, group 2).

Is appropriate to perform MANOVA?

The 2 variables variables are factors from the same questionnaire (stress, depression) And locus of control from another questionnaire. The dependent variables are correlated

March 31, 2024 at 6:50 pm

Yes, assuming that all the dependent variables are measured on the same set of subjects, it sounds like a great use for MANOVA.

March 19, 2024 at 7:32 pm

The post really helped clear things up for me, thanks a bunch! I’m still a bit unclear on one point though, as i’ve seen different explanations of MANOVA across different sources.

If I have one dependent variable & will be running a public goods experiment across 3 groups (1 control & 2 treatments) & would like to compare the mean contributions across groups, would MANOVA be the way to go, or should I use the Kruskal-Wallis test?

Please do advise if I should use a better test that I haven’t mentioned!

Thanks again!

March 20, 2024 at 12:18 am

Hi Ghanimag,

Because you have only one dependent variable, you can’t use MANOVA. That analysis requires multiple DVs. Instead, consider regular ANOVA. You can use one-way ANOVA for your scenario with 1 DV and one independent variable that has three levels. For more information, read my post on One-Way ANOVA .

After performing a one-way ANOVA, you’d likely need to use a Post Hoc Test to evaluate specific differences between group means. Because you have a control group and two experimental groups, I’d consider using Dunnett’s Method, which I discuss in this article about Post Hoc Tests for ANOVA .

Those two articles should help point you in the right direction for your data!

March 14, 2024 at 4:50 pm

Is it appropriate to use MANOVA when doing a Quasi Experiment design?

Ex- two groups (not random assignment), one control and one treatment however, the desire is to measure at least 3 completely different metrics from each group (i.e.- time, satisfaction of product, frequency of errors)

The question in mind that led me to this post was, “what to do with your experiment design when you have multiple DV to measure on outcome?”

Thanks for any help.

March 14, 2024 at 5:29 pm

Yes, it’s a good analysis to use for that situation because you’ve measured the three outcomes together.

Because you haven’t used random assignment, controlling for confounding variables becomes more challenging, and it might be harder to confidently attribute observed differences directly to the treatment or intervention. But that’s a problem for any analysis you might use. Unfortunately, that’s a fact of life with non-randomized experiments.

February 28, 2024 at 1:38 am

I have measured chemical concentrations of 10 parameters in drinking water used by two communities, one affected by a kidney disease and the other with no evidence of the disease. If I want to investigate the possible effects of these chemical parameters on incidence of the disease, what would be the most suitable multivariate statistical test to use? I have separately compared the concentrations in two groups, separately using Welch’s T test, Mann-Whitney Test, ANOVA etc.. Would something like PCA be appropriate?

Thanking for very useful contributions,

February 28, 2024 at 2:04 pm

I’m not entirely sure what data you have specifically. It sounds like you have measured concentrations of elements in drinking water for two communities. And those two communities have differing levels of kidney disease. And you want to link the concentrations to the difference in prevalence in those two communities.

If that’s case, it sounds like you have two data points, one for each community, which is far too low for a rigorous study. If you have multiple water samples from each community, you might have more than two data points within a community, but it doesn’t sound like you can link that directly to specific people and their water consumption, which leads to my next point.

If you have multiple random water samples, you can use t-tests to determine if the concentration differences between communities are statistically significant, but that won’t link it to disease prevalence.

If you can link an individuals kidney disease outcome to their specific drinking water, you would be able to use binary logistic regression. That would allow you to control for confounding variables. But it doesn’t sound like you have that level of data.

PCA won’t help you with this. It is useful for data reduction and to identify patterns in data and can help in understanding the underlying structure of your chemical parameters. However, PCA by itself does not directly establish a relationship with the disease incidence.

Too me, it sounds like you have data suitable for a case study . You can report the statistically significant differences between the communities (assuming they exist), discuss the known role of the contaminants with kidney disease that’s in the scientific literature, and discuss other possible confounding factors. This could lead up to a more thorough study.

December 6, 2023 at 9:33 am

Thanks for the clear explanation! I do have a question regarding the assumptions. I have a dataset looking at soil parameters. The independent variables are contour (top, slope, depression) and depth (0-10, 10-30, 30-60, 60-90 cm) and the dependent variables are pH, density, and conductivity. What grouping would I use to test for normality and variance? Do I run normality tests on all of the measurements for each dependent variable? Or would I break the measurements down according to each level (3 for contour and 4 for depth), or would I look at the combination of contour and depression together (i.e. 12 combinations)? I’m assuming that the same groupings I use for univariate normality (testing pH, density, and conductivity separately) would be the ones I use for multivariate normality (testing a composite variable consisting of a linear combination of the 3 dependent variables)? Thank you much!

November 8, 2023 at 2:05 pm

Hi, I am doing a very simple research project for my dissertation. I have 4 DVs that I am doing a pre- and post-test on. I’ve run a MANOVA and there is significance. Can you tell me where to find a guide for writing the results? I am having trouble locating anything on that. Thank you!

September 10, 2023 at 12:43 am

Thank you so much for this informative post! I was just hoping to get your thoughts on whether MANOVA might be an appropriate way to approach my data?

I’m hoping to examine the effect of pubertal timing on the relationship between social anxiety and interpretation bias (which are theoretically correlated). Both DVs (social anxiety and interpretation bias) are continuous, and pubertal timing is categorical with 3 levels (early/normal/late). I also plan to control for 4 other continuous covariates by including them as IVs (I think this is the right way to approach this?). The trouble is that I really have 2 IVs of interest, as I’m looking to compare this relationship between a measure of objective pubertal timing, and a measure of perceived pubertal timing. Obviously these two IVs are inherently related, so I am wondering if multicollinearity would be too much of an issue to proceed with MANOVA? If it helps at all, the pubertal timing IVs were both initially continuous, so this could be an option if there is a more appropriate analysis approach that needs continuous IVs. Hoping you might have some insight!

July 28, 2023 at 9:41 am

Hi Jim, I have 5 IVs with three levels each and 3 DVs. Is it possible to use MANOVA in order to determine the most significant factor? How?

Thank you, Samuel

July 31, 2023 at 10:20 pm

It is always difficult determining the most significant factor in a model, whether it’s regression or MANOVA. Please read that post where I write about Identifying the Most Important Variable . It is written for the regression context but it should apply to MANOVA as well and will help you understand the issues involved.

June 24, 2023 at 2:54 am

I have two continuous DVs and three independent variables and I want to check how the independent variables affect the collective DVs? Is MANOVA appropriate since all the IVs are continuous?

June 24, 2023 at 10:00 pm

Yes, you can perform MANOVA with continuous IVs. You need to enter them as covariates. If your DVs are correlated, then you have the potential to learn more about their relationship and potentially more statistical power. If the DVs are not correlated, then there isn’t much reason to use MANOVA rather than individual regression models that might be easier to interpret.

November 18, 2022 at 10:17 am

Hi sir, i have assessed 15 samples of mineral for 10 different parameters by giving 1 to 10 scores to the each parameters by 33 different field experts. Which statistical test i should use to compare the data.

November 18, 2022 at 4:05 pm

Hello Dr., it depends on what your ultimate goal is. To me, it sounds like you might need to perform an inter-rater reliability analysis .

September 18, 2022 at 6:09 pm

I think that you have to use some type of transformation – log, square, etc – so to “normalise” your data before applying these test.

September 19, 2022 at 1:38 am

No, performing a transformation is not usually necessary. These transformations are sometimes required as a last resort when your data don’t satisfy the assumptions. However, there are better methods for resolving those problems that you should try first.

When your data and model satisfy the assumptions, you don’t need a transformation. When not all assumptions are satisfied, there’s often a better way to fix it than a transformation.

August 9, 2022 at 8:58 am

I am looking at water quality data (dissolved oxygen, temperature, conductivity, pH etc) from four treatments. It has been suggested to me that I should use MANOVA rather than ANOVA however my data is not normally distributed. Does MANOVA have a nonparametric equivalent that I could use instead?

Thank you Lia

May 13, 2022 at 11:47 pm

thanks very much. this page is so helpful. please after finding significant effect in MANOVA, what is usually the next step

March 28, 2022 at 7:30 pm

Hi Jim, how are we able to determine the DVs are correlated? I am measuring the effect of a news article (3 levels) on attitudes towards mental illness(dangerousness, blame, unpredictability, and support for recovery and outcomes).

March 28, 2022 at 8:58 pm

Hi Marrissa,

Just graph the DVs on scatterplots . And calculate Pearson’s correlations for them.

March 15, 2022 at 3:53 pm

I have 3 independent variables (types of climbing: Trad, Sport, and Bouldering) and 3 dependant variables (motives: sensation seeking, emotion regulation, and agency. I don’t know if a MANOVA would work, and I would be using SPSS to analyse. If it wouldn’t work, what do you recommend?

Many thanks, Matt

March 16, 2022 at 2:01 am

It looks like MANOVA is a good possibility for your data. Ask yourself if the DVs are correlated. If they are, then MANOVA can provide some real advantages. If they’re not correlated, you’d be served just as well by using regular ANOVA. In other words, MANOVA provides most of its benefits when your DVs are correlated.

Regular ANOVA can be easier to interpret, so I only recommend MANOVA when the DVs are correlated.

March 3, 2022 at 2:24 pm

Thanks for your posts, Jim they are all very helpful. I have 2 interventions with 4 different outcomes measures over time (pre-test, 1 month, & 4 months) to assess for retention of effects overtime. I am a little stuck on which test to do. I have chosen a MANOVA repeated measures, but find in SPSS I can only run one DV at a time. When I try to run more than 1 I get too many variables to define and cannot run the test. Thank you for any help you can provide.

March 3, 2022 at 2:30 pm

Unfortunately, I haven’t used SPSS to perform MANOVA so I can’t be of much help here. It does sound like a major limitation! I suppose you could always perform separate ANOVA analyses on each dependent variable, but that won’t help you find any changes in the multivariate response patterns if you have correlated DVs. If the DVs are not correlated, the single DV models would not be a problem.

November 6, 2021 at 10:58 am

Thanks for writing this post! There are two questions I can’t understand: First, You said “When DVs are correlated, MANOVA is beneficial”, but why? I just saw some mentions on textbook that it is not good if DVs are highly correlated. Then, if MANOVA is significant, what should I do next? Continue to do some ANOVA of each DV to analyze the effect of IV? or to do the discriminant analysis? Which one is better? I hope your help, thanks!

Best, Amber

November 8, 2021 at 12:43 pm

It’s independent variables (IVs) that shouldn’t be correlated. When IVs are correlated, it’s known as multicollinearity and, yes, it can be a problem. Click the link to read my post about that.

Most types of models can only handle one DV. In those cases, correlation amongst DVs isn’t possible because you need at least two for correlation. MANOVA is one of the exceptions for models that can handle multiple DVs simultaneously. It is particularly valuable when the DVs are correlated.

As for significant MANOVA models, you can interpret those directly to see what is happening. I do some of that in this post. You can also perform separate ANOVAs as well to get the more standard single DV type of results. But, in some cases, you’ll be missing significant effects. I show that in this post. So, see what works for your data.

October 18, 2021 at 5:33 pm

I need your help about what kind of statistical tests should I use for my research.

I have income before and after the war as an independent variable, and Communication Satisfaction and job satisfaction measured also before and after the war as dependent variables.

I was thinking of MANOVA but I am not sure it is the right test for this situation.

Could you please help?

August 23, 2021 at 9:53 am

Hi, pls I need your advice I’m trying to find out the difference that exist among 7 groups using an independent variable. I used Anova and then ran a post hoc analysis, pls I want to ask is Anova not suitable? My Lecturer is suggesting I use MANOVA. Pls help.

July 31, 2021 at 9:55 am

What’s wrong with Bonferroni correction: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1112991/

July 13, 2021 at 2:15 pm

Mr. Frost: I took your data and loaded it into SPSS and performed a Hotelling’s T MANOVA on the data and indeed found multivariate significance in the three methods on the dependent variables. However, SPSS provides Post Hoc Default Analysis using T-Test with a Bonferroni correction and neither method comparisons came out to be not significant. I am wondering if you suggest the next step would be to perform a Discriminant Function Analysis to get to the root of the group differences.

June 18, 2021 at 5:18 pm

What a fantastic resource you have here.

I am analyzing survey responses from each US state and have a much higher response rate from two of them. The issue I am running into is – these two states are leaders in this field and I want to see if they are statistically different than the others or if they change the collective of a little over 1,300 responses. I am isolating only the nine Likert-type 5-point questions. To compare these – would this work and would I have to do each question separately or would it work with one state vs. the others?

I appreciate any tips or links to articles – I am using SPSS v27 with very little experience.

With Gratitude – Rachel

June 19, 2021 at 3:47 pm

This is a tricky problem for several reasons. I don’t have specific answer for you but can at least raise several issues and provide some information.

One difficult aspect is the fact that you’re using Likert scale data, which are ordinal data. Many analyst consider averages to be inappropriate for this type of data. However, there is some evidence that the regular parameteric tests are ok to use with them. For more information, read my post about analyzing Likert data . Note that the discussion is in context of using t-tests. I’m don’t whether the same details apply to ANOVA, which you’d need to use because you have more groups. But ANOVA is an extension of the t-test.

However, if you average several Likert items together (or add them), then you have stronger grounds for considering it to be a regular continuous variables. Something to consider.

Another point is that you’ll be comparing many groups. You’ll need to consider using a post hoc test to control the family-wise error rate. For more details on that, read my post about using post hoc tests with ANOVA . That will help you answer your questions about comparing states.

I hope that gives you enough information to get you started!

June 1, 2021 at 12:18 pm

I enjoyed your discussion about the usefulness of MANOVA in that it can assess patterns between multiple dependent variables. my question is this – is there any statistic result in the MANOVA output that directly indicates what is that pattern among the DV’s? To me, the output only tells you that there is a relationship among the DV’s. It doesn’t tell you what the nature of that relationship among the DV’s would be. Is the only way to detect a pattern is to plot out the DV’s and heuristically infer the pattern, as you demonstrated with your Excel plots? Thank you.

June 3, 2021 at 2:13 pm

First, I wouldn’t underestimate the power of graphs to represent the results! Often, the patterns are clearly visible in the graphs and you just need to know whether they’re statistically significant.

You can also use Eigen values and Eigen vectors to help you understand MANOVA results. Use these values to determine how the IV means differ between the levels of the different model terms. Focus on the Eigen vectors that correspond to high Eigen values. They’ll point you towards the factor levels associated with the largest mean differences.

I don’t go into that here, but you can look those up for more information!

May 14, 2021 at 1:28 pm

hi there if i have 3 IV (depression, gender and ethnicity) and many DVs like (loneliness, internet use , social media followers etc) is a MANOVA best for this? thanks Ciara

May 11, 2021 at 6:27 am

Hi Madelyn,

You can use python language or even Microsoft excel.

Please find the below link for python code example: https://www.marsja.se/python-manova-made-easy-using-statsmodels/

And if you need any help you can mail me 🙂 [email protected]

I would be happy to help you and even those who have doubts in Data Science can contact me.

May 7, 2021 at 11:21 pm

I recently discovered this website and damn, this is pure gold. Thank you!

May 9, 2021 at 9:15 pm

You’re very welcome, Moksh! So glad it’s helpful!

May 5, 2021 at 2:20 am

Thank you for your article. This help me a lot explaining MANOVA to my advisees on when and why they need to use MANOVA rather than the other alternatives.

April 23, 2021 at 10:24 am

This is excellent! Thank-you so much for the summarization of the differences between ANOVAs and MANOVAs. This was the most concise article I could find and will be helpful in my thesis defense!

April 23, 2021 at 10:53 pm

You’re very welcome, Lease. I’m thrilled to hear it was so helpful! 🙂

April 21, 2021 at 3:06 am

I would suggest you to use jamovi, it is a free, open source software. It is ‘user friendly’ and you can find support for you analysis if needed, and it is intuitive to use.

Good luck fo you project !

April 21, 2021 at 2:04 pm

Thanks so much for the recommendation! I hadn’t heard of Jamovi before and I’ll look into it! Hopefully, it is helpful for Madelyn too!

April 19, 2021 at 6:11 pm

Hi, I’m trying to run a MANOVA for a research project for my class (I’m a senior in high school). I’m wondering what software would be best for me. Ideally, it would be free and fairly simple to use as I am not the most versed in this kind of analysis. Any recommendations? Thank you, Madelyn

April 18, 2021 at 8:55 am

Hi Jim, Thank you for your really informative post. I encountered with the problematic in SPSS, that post hoc test is not being calculated as the IV has only two groups. (I have two sub-groups under the IV, and four correlated DVs to analyse). I wonder if the results of the MANOVA can be still accepted (significant Pillai`s trace etc.)? How the lack of post hoc tests effect the results and overall is it an issue for MANOVA that its IV has only two sub-groups? Hope my question makes sense. Thank you very much Andrea

April 18, 2021 at 10:46 pm

Hi Andrea, if you’re just comparing two groups, then you there’s no need for post hoc tests. Post hoc tests are for cases where you have multiple comparisons. You need three or more groups. With two groups, you have only one comparison.

April 13, 2021 at 8:17 am

I can’t figure out if my scenario is a MANova or another type of ANova (possibly 2 way).

I have test results for two separate groups of students – one from 2010 and another 2020. The same test was given to both groups, so I know fluid and crystallised intelligence. I’m now looking to test the Flynn effect and whether the effect applies to just one type of intelligence, or both.

I thought the two factors would be 2010 and 2020 and that the dependent variable was intelligence type.

But the more I read the more confused I get – do my two intelligence results mean I have multiple DV’s.

All the 2 way ANova videos I’ve seen are looking at two factors – gender and age I just have the year – 2010 ir 2020.

Thanks in advance,

March 15, 2021 at 8:55 am

Hello, I conducted a MANOVA for 2 independent variable (age (3 levels) / pysical activity level). Due to physical activity level was a continuous variable, I included it like if it’s a covariate. I’m confused about how to report the results. I reported the main effect of age between the groups. But how can I report the interaction effect? Is it enough to say the interaction effect was significant or should we detail it? And if we should, how can we do it?

Thank you so much

March 10, 2021 at 4:50 am

Hi Jim, Thank you so much for your help! I’ve done mixed ANOVA in SPSS, in which you can add the time point DVs into a ‘time’ IV — can’t find that when conducting MANOVA, so I’ll continue to investigate.

In terms of the DV correlations, they’re mixed… But I’m scoring people at 2 timepoints on 3 DVs, and predicting big changes for the intervention group, so doesn’t it make sense that say…. T0 stress won’t correlate with T1 anxiety??

My correlations are: Stress T0 – Stress T1: .548** Stress T0 – Anxiety T0: .577** Stress T0 – Anxiety T1: .127 Stress T0 – Depression T0: .325* Stress T0 – Depression T1: .063 Anxiety T0 – Depression T0: .492** Anxiety T0 – Depression T1: .173 Anxiety T1 – Depression T0: .141 Anxiety T1 – Depression T1: .558**

So the non-sig correlations are between T0 measure for one DV, and the T1 measures for the other DVs…

**Does this mean that MANOVA is no good???? Do they all have to correlate?**

Also, my Shapiro-Wilks is sig. for all but one DV, even with log transformed data and 3 x outliers removed… Though thankfully my Levene’s is non-sig.

**Does the SW failure mean I can’t continue with MANOVA, or can I just ‘proceed with caution’?**

Because I haven’t been able to include a repeated measures factor in my SPSS, my output hits a wall, so I’ve only done an ‘exploratory’ MANOVA just to see what output might happen with my logTr data and outliers removed etc..

So at this stage, I’m still feeling my way around. My supervisor said that 3 separate ANOVAs would inflate my results, but at this stage it’s a change I’m willing to take!

Thank you again, wish you were in my department!!

March 9, 2021 at 4:49 pm

Hello, I’m having a little trouble working out what to do and hoping you can help! I’m very new to stats, so please go easy on me!

My supervisor has told me I need to use a MANOVA rather than ANOVA, because I have 3 DVs, but I’ve been given no information on MANOVA or how to perform it in SPSS…. I’ve tried but I’m struggling. I can’t find any tutorials online that replicate my study, or are similar.

My study is an intervention: Intervention Group versus control group Both groups were tested on stress, anxiety and depression at 2 timepoints; pre- and post-intervention.

I have 3 DVs (stress, anxiety, depression), 1 between-subjects IV (group), 1 within-subjects IV (time).

I have some questions: Is MANOVA the right analysis? Would it be called a mixed MANOVA?

In SPSS, I have 8 columns; P#, Group, stress T0, stress T1, anxiety T0, anxiety T1, depression T0, depression T1. The problem I have here is that I don’t know how to create the time IV — because in all the tutorials I’ve found, there are no repeated measures…

That’s the only specific question I have, but generally do you know of any online tutorials that would walk me through this type of study, because I’m really lost and only have 3 weeks until dissertation submission.

Sadly, my supervisor has left it until now to tell me I shouldn’t be using ANOVA 🙁

Many thanks!!

Best wishes,

March 9, 2021 at 7:56 pm

I haven’t used repeated measures in MANOVA. However, I’d imagine it is similar to how you set up repeated measures models in ANOVA. I’ve written a post about repeated measures ANOVA that might be helpful. Unfortunately, it’s been a looooogn time since I’ve used SPSS and can’t provide guidance there.

If the three DVs are correlated, you do gain benefits by using MANOVA instead of ANOVA. If they’re not correlated, those benefits go away. Even if you do go with MANOVA, you can still fit the ANOVA models to help you understand what is happening. Those can some times be easier to interpret. Although you do lose some information as I show in this post. But, the ANOVA results can help fill in some details even when you use MANOVA.

I know that wasn’t super specific, but I hope it helps some!

March 4, 2021 at 7:32 am

Hi Jim, thank you so much for this introduction. I was also wondering whether to perform 2 ANOVAs or a MANOVA. I test 3 different sales tactics (IVs) in an online store and I observed the Conversion Rates and the Abandoned Shopping Carts. I have two seperate sets of hypotheses for the DVs. (e.g. Sales Tactic 1 has higher Conversion Rate than Sales Tactic 2)

– Would you recommend to perform one ANOVA per DV? Or one MANOVA for all DVs?

What is the deciding criterium – is it the Correlations of DVs? Becuase i dont know whether they are correlated and if yes, it is probably not a positive correlation (does it still work?)

Thanks and best, Marcus

March 5, 2021 at 12:06 am

If you want to determine whether your DV means are equal vs not equal (Conversion rates vs Abandoned shopping carts) given the levels of your DV, then you’ll need to use MANOVA! You’d be seeing that relationship between those two DVs change based on the IVs. Look the scatterplot in this post to see the type of differences you’re assessing.

However, if all you need to know is whether the mean the individual DVs vary by the levels of the IVs and you’re not assessing the pattern of relationships between conversion rates and abandoned carts, then separate ANOVA models are ok. Although, if conversion rates and abandoned carts are related, you still gain some power benefits by performing MANOVA rather than ANOVA.

February 24, 2021 at 5:35 am

Τhank you very much Mr. Frost. So I presume the equality refers to all the DV means between the different data sets under comparison i.e. the combined DV (satisfaction + test scores) means for teaching method 1 are equal to the combined DV (satisfaction + test scores) means for teaching method 2 and method 3?

February 25, 2021 at 4:05 pm

It’s all one dataset but with multiple DVs. In the graph, if the results hadn’t been significant, the data points for the different methods would have all followed the same line instead of separate lines. However, because the results are significant, you can see that each method (IV) produces a different relationship between the DVs.

February 23, 2021 at 5:28 am

Thank you very much for this information. I am confused with setting the null hupothesis in the case of MANOVA i.e Ho; regarding the particular treatments, two or more groups do not differ in the mean response of two or more dependent variables?

February 23, 2021 at 3:43 pm

MANOVA interpretation can be less intuitive! MANOVA determines whether the multiple dependent variable (DV) means are different for the values of the independent variable (IV). Therefore:

H0: DV means are equal for the values of the IV. HA: DV means are not equal for the values of the IV.

In the blog post, you can see an example of significant results (rejecting H0) in the Scatterplot of Test vs. Satisfaction. Each method (the IV) has mean test scores and satisfaction levels (the two DVs) that are significantly different. Read the interpretation below the graph for more details.

February 20, 2021 at 8:07 am

Hey Mr. Frost, Great website. I hope this post is in the right place, but my question may be more broad. I am surveying teams with a questionnaire that test team characteristics (inputs and processes like clear roles, common purpose, trust,etc) and team cohesion. My goal is to see which of the team characteristics correlate with teams that are considered cohesive. What stats tests would I use with the data I get returned from the questionnaire to get results (I already know what I will use to aggregate individual responses to group-level)? Thanks for any help. I will try to find a post on the subject of my another question. I thought some sort of anova but not sure. Do I include normal distribution test? (Someone suggested anderson-darling)?

January 30, 2021 at 4:45 am

Hi! Thank you for this helpful article. I am currently designing a study where I want to explore the effect of colour (IV = 4 colour conditions) on competence perception (1st DV) and voting intention (2nd DV). However, I don’t really understand if I conduct a within-participants MANOVA, does the ” within” condition only apply to IV or do the DV scores have to also come from all participants?

Thank you in advance, Celine

November 24, 2020 at 2:07 pm

How would this differ from running an ANOVA with test score as DV, and the teaching method and satisfaction scores as IV’s with an interaction between the two?

November 24, 2020 at 10:46 pm

With a regular ANOVA you have just one DV and multiple IVs (in your example). You can have an interaction between the IVs. For MANOVA, you’d have multiple DVs. You can also have multiple IVs and you can include the interaction term if needed. The main difference is the ability to include multiple DVs in the same model. If those DVs are correlated, you can gain the benefits that I describe in this post. The presence or absence of an interaction term is not a consideration in this context. Both of these analyses can handle interaction terms.

October 26, 2020 at 11:13 pm

In my study I have 3DV and 3IDV. Is Manova suitable for this?

October 26, 2020 at 11:26 pm

If your three dependent variables are correlated, then using MANOVA is a good idea. If the IVs are not correlated, use regular ANOVA and fit a separate model for each DV. Most of the benefits you get from MANOVA versus ANOVA occur when the DVs are correlated.

October 20, 2020 at 12:50 pm

Firstly, thank you for posting this. I am really needing your help/expertise:

I am working on a research study whereby I have collected quarterly SEC data for 10 companies which I plan to study for 2015-2019. All of these companies were affected by passage of new legislation which became effective 01/01/19, thus I have 16 quarters of data in the ‘before’ group and 4 quarters of data in the ‘after’ group. My research question is centered around “has passage of the new legislation caused a decrease in the 5 DVs in the study?

My plan is to use a one-way MANOVA (1 IV, 5 DVs) to compare the means of the before and after groups and test for stat significance. My stats prof says that because I’m using quarterly data, it represents ‘panel data’ and I may not be able to use a MANOVA. I was thinking that even though I have multiple quarters in the ‘before’ and ‘after’ groups, they’re all only really there to provide a more accurate mean for each DV in the ‘before’ and ‘after’ groups. I have searched high and low and can’t find an answer…..can you help???

Cheers, Phil

October 18, 2020 at 2:44 pm

Hi Mr. Frost,

Thank you for the concise article. If you were to cite a few references in support, what might they be? Bray & Maxwell, 1985? Tabachnick & Fidell, 2006?

Thanks again, Kyle

October 13, 2020 at 1:06 pm

hi, if the situation is, i have 1set of questionnaire on three domain afective, psychomotor and cognitive and im going to test to a male subject can i use manova? 3dv and 1 iv?

October 13, 2020 at 2:46 pm

If the the three DVs are correlated, then MANOVA can be a good method to use. However, if you’re assessing only one subject, be aware that you probably can’t generalize the results beyond that subject. It might be good for a pilot study.

September 25, 2020 at 11:17 pm

Hi Jim, I was thinking of running a two-way MANOVA for my project but when running the assumptions, I found that my DVs are multicolinear(90% correlation). References I have read so far recommend that I either average the two DVs or remove one DV from the analysis? Can you please advise me how to deal with it? Should I go ahead and run a MANOVA or run separate ANOVAs with the DVs? I would really appreciate your expert advice.

September 21, 2020 at 2:49 pm

Thank you so much for the great work you’re doing. Your post has been so much helpful to me.

I need your assistance again to clarify a little confusion. I am trying to know the relationship between multiple IVs and DVs. My IVs (which are basically socioeconomic data) contain all possible measurement levels (interval, nominal, and ordinal data types) while my DVs are mainly categorical data types (nominal and ordinal).

I have done some reading online but not clear on the right statistical test to conduct; some seem to point at canonical correlation and others at factor analysis, but my little knowledge is pointing at MANOVA, even though no article I’ve read has explicitly considered the measurement levels of the variables.

Thank you so much, Jim.

August 1, 2020 at 10:27 am

Hi Jim, I am trying to use G*Power software to calculate an a priori sample size for a Hotelling’s T^2. How do I calculate the a priori effect size? Can I do this in SPSS? I have two independent groups of one categorical independent variable, and 3 continuous scale DVs. Thank you for pointing me to the best resource!

July 31, 2020 at 12:16 am

Respected Sir, Please I want a answer below mention question. Using MANOVA rather than performing several one-way ANOVAs: a. Allows for a greater number of independent variables b. Ignores any correlation between dependent variables c. Increases the Type 1 error rate d. Decreases the Type 1 error rate

July 31, 2020 at 1:33 am

Hi Haradhan,

This is obviously a test question and I don’t like just giving people answers. Particularly when the article you’re posting this comment in provides you with the answer. So, I’ll give you a hint about where in this article you should read more carefully because the answer is there. Read the section titled “When MANOVA Provides Benefits.” Pay particular attention to the last bullet point! 😉

July 21, 2020 at 7:43 pm

I’ve been asked to think in a MANOVA for my data, related to tourists’ surveys. I am a little stuck though. Here it goes:

As independent variables I have tourists’ characteristics (age, country, gender, education levels) and trip characteristics (like experience of guides, itinerary and activities, educational opportunities, measured in an 11 point Likert Scale. As dependent variables I have a series of responses on learning outcomes. Learning outcomes includes 3 variables: 1) knowledge (1-0 scale because is True/false), 2) attitudes, and 3) behavior intentions (measured in an 11 point Likert Scale). Each one of the dependent variables are formed of 7-8 different sub questions. Besides that, in the middle, I have perceived learning (measured in an 11 point Likert Scale) that sometimes act as an independent variable and others as a dependent. My questions are: 1. Data does not meet normality assumptions. It is ok to rescale, transform and log? 2. I have PRE and POST responses, do you recommend me to use the change values in the data set to run MANOVA? –Rescaled, log and transform-? 3. Do you recommend me to group the responses of the sub questions to have 3 dependent variables instead of having a lot of sub questions within each dependent variable? 4. Do I have to use the perceived learning as a dependent variable? And is it ok to compare, let’s say: age X gender X trip characteristics with attitudes and perceived learning? 5. How do I know that the dependent variables are not correlated?

Thanks for any help.. and sorry for the long message 🙂

July 16, 2020 at 10:58 am

Hello. Thank you for such helpful information on your website! Please may I ask; for my own research, I’ve run an intervention testing three DVs (e.g. depression, anxiety, resilience) at two time points (before/after) for two groups (Intervention and control). So, a 2*(2) design, but with 3 DVs that are correlated. I am hoping before intervention, no difference between groups across all three scales, but after the intervention, I would hope for significant changes for intervention group, across all three scales, compared with control and pre-test. Do you know if a MANOVA is appropriate, and if so, do you know how best to run this (for example, in SPSS) – examples seem to only include between-participants? Many thanks.

July 9, 2020 at 3:35 pm

I’m trying to wrap my brain around this with the context of my research. The research question I am currently working on is: What is the impact of gender and the implementation of a freshman academy on academic achievement?

To answer this, I have eight years of student data (some pre-academy and some post-academy) that shows their cohort year, gender, GPA, and number of credits earned.

I’m a bit confused on what to use to answer my research question. Do I run a two-way ANOVA? Or since GPA seemingly also could impact credits earned, and vice-versa, should I look at a two-way MANOVA?

Thanks much, A student with a brain about to explode!

June 24, 2020 at 8:38 am

I am running two one-way MANOVAs for my final year project (and a further two-way with the data from both studies combined).

The first one-way was fine, but the second i’m not so confident about. The Pearson’s correlation came out at .808 when I tested for multicollinearity. I thought this was fine but I may have confused myself a little.

I know the DVs are meant to be correlated, but is this value too high? I have been trying to get some info but even in my textbook (Field, 2013) I can only find ‘acceptable ranges’ in relation to regression models. These ranges also vary between <5 and <10.

I am looking for some clarity in where my Pearson's value should fall between.

June 27, 2020 at 4:16 pm

Multicollinearity only refers to correlation between the independent variables and not the dependent variables. I don’t believe there is an upper limit to the correlation between DVs in MANOVA. You should be fine!

June 19, 2020 at 3:35 pm

I need some help with a data analysis question. I am completing a research course and I am a bit stuck. I am sure the answer to this question will seem very obvious to you. Sadly, it has me a bit confused.

My research will examine stress levels and behavioural responses to unexpected positive stimuli and unexpected negative stimuli across different socio-economic circumstances.

Do I use MANOVA?

Variable include:

Socio-economic status Stress levels behavioural responses to (Surprise: Unexpected positive stimuli and Blindside: unexpected negative stimuli)

Think of a group of guys (differing SES) walking into a parking lot. they all discover parking tickets on their cars. They all react differently. I am trying to understand the factors that caused these different reactions. My hypothesis is that SES, Stress and excitability are contributing factors.

Can you help me out?

Actually, just writing this to you helped! LOL.

I look forward to hearing your reply.

June 20, 2020 at 9:36 pm

Ha! If more people answered their own questions while typing it would make my job easier! 🙂

If your response variables are correlated, then it probably makes sense to use MANOVA. If they’re not correlated, using regular ANOVA (separate models for each DV) is probably fine. That’s the general rule of thumb. In your case, if responses to expected and unexpected stimuli are correlated (positive or negative), definitely consider MANOVA! Just look at the correlation between those variables to decide. I’m guessing yes, but I’m not in your field!

I hope that helps! Best of luck with your analysis!

June 19, 2020 at 2:04 am

Thank you for this blog and a chance to ask questions, I am new to statistics, I have two questions for you.

I have one IV with two groups. Methods of contact: Telephone verses Secured Message

I want to see the affect on my two DVs: Response rate (number of responses from zero divided by number of attempts) and readmissions (number of days from zero to readmission).

First question: Would a MANOVA be a good choice or a two-tail hypothesis be a better choice?

Second question: Statistics wise, would I fair better if I separated my IVs (have two IVs) opposed to one IV with two groups?

June 16, 2020 at 1:46 pm

Thank you for your prompt reply! The test selector actually chose the Hotelling’s T^2, not the MANOVA.

June 16, 2020 at 2:37 pm

Right, and Hotelling’s T^2 is the test statistic that MANOVA uses when you have two groups. So, it is in essence telling you to use MANOVA.

June 16, 2020 at 12:55 pm

Can you explain when to use a Hotelling’s T2 versus MANOVA? I am studying the impact of one dichotomous independent variable (defendant gender: male or female) on three correlated measures of juror decision making which are all continuous scale. I am sampling from jury-eligible individuals who live in one state. I used the statistical test selector function that Laerd Statistics offers, and it came up with the Hotelling’s T2, but it seems to me a MANOVA would do the trick.

June 16, 2020 at 1:42 pm

Hotelling’s T^2 is a generalized form of the t-statistic that allows it to be used for multivariate tests. T-tests use the t-value to calculated the p-value for univariate tests. MANOVA uses Hotelling’s T^2 (and other test statistics) to calculate the p-value for multivariate tests like MANOVA. It looks like MANOVA uses Hotelling’s T2 when there are two groups, which makes sense because you’re assessing males and females.

So, that selector function is basically telling you to use MANOVA and it chose the test statistic for two groups.

Best of luck with your analysis!

June 14, 2020 at 2:36 pm

What do you do if there is not enough of a relationship b/w the DVs? For example if the Pearson’s r is .168? Thank You!

June 15, 2020 at 3:35 pm

If the correlation is very low, you can use regular ANOVA without losing any power. Just perform a separate ANOVA for each DV.

June 13, 2020 at 5:41 pm

I am running a MANOVA with one categorical IV and two continuous DVs. I was running the analysis and my Bartlett’s Test of Sphericity was not significant (p = 0.097) which I’ve never dealt with before. I’m not sure what to do next now that the factorial analysis cannot be done. I’ve heard getting more participants could help but I cannot do that due to time restrictions. I only have 52 participants. Are there other things I could do?

June 10, 2020 at 4:27 pm

June 10, 2020 at 4:07 pm

Thank you for your reply. I think I should apply repeated measures MANOVA in this design, Can you help me in how to conduct this test in SPSS (steps). I have found many SPSS tutorials, but I got confused. thank you

June 10, 2020 at 4:21 pm

Hi, sorry, I don’t have SPSS.

June 8, 2020 at 7:12 pm

Hi Jim, Also I have another question, If I have a design of three groups (2 study and one control) and I want to see their effects on 7 DV (that are correlated). the DV are measured at 3 time intervals (pre, post and follow-up. in such case, you recommend me to use MANOVA ? THANK YOU IN ADVANCE Best

June 10, 2020 at 12:16 pm

Hi Nevein, yes, that sounds like a good approach.

June 8, 2020 at 7:05 pm

Hi Jim, Thank you for this informative post. I’m making a study with two IV (2 groups; study and control groups), and I have 3 DV that I measure at two time intervals (Pre and post). the IV are correlated. in this design, Would you recommend me to use MANOVA in analyzing my data or would you recommend a different approach ?

June 7, 2020 at 9:58 am

Hi. Thanks for explaining the concept so well.

May i ask what should be done if there is multicollinearity in the DV?

June 8, 2020 at 3:16 pm

Hi Aarti, multicollinearity among independent variables just means that they are correlated. For MANOVA, you actually want correlated *dependent* variables because that increases the power of the tests, as I discuss in the post.

May 25, 2020 at 9:56 pm

This post has been extremely helpful in understanding what data analysis i need for my thesis, however, can you help clarify if im on the right track for using MANOVA. I have three IV’s (persisters, remitters, and controls) and three DV’s (Tests – Flanker, Stroop, Parent Report) with data being assessed at 8 years old and then 18 years old, would MANOVA suit this sort of analysis?

Regards, Bailey

May 26, 2020 at 9:52 pm

Yes, if the DVs are correlated, then MANOVA is a good choice. If they’re not correlated, just use regular ANOVA.

May 25, 2020 at 7:16 am

I’m experiencing a similar quandary, however, my IV is continuous (Autism quotient score). I’m assuming from your response that it does not matter that there is only one IV. And MANOVA is a perfectly acceptable tool in these scenarios.

Thanks for your continued assistance.

May 23, 2020 at 6:58 am

Hi Jim , I have a question.

I am currently conducting a final year thesis study with 1 independent variable (Gender) and 3 dependent variables (Perceived stress, anxiety and loneliness), titled Gender differences in Perceived stress, anxiety and Loneliness among International Undergraduates during the Covid-19 pandemic. I am trying to find if there are gender differences on the 3 variables. Can I run a one way independent anova 3 times for this study or a Manova? I am genuinely confused as what to run because I only want to see the differences of gender.

May 24, 2020 at 9:32 pm

The answer is you can do it either way. Often times it’s just simpler to understand regular ANOVA results. You can try one-way ANOVA first with three separate models to see what results you get. However, if that doesn’t work *and* those three DVs are correlated, you can try using MANOVA. MANOVA does have some benefits as I point out in this article but it can be harder to interpret. So, if your ANOVA results for separate models seem to miss something you expect to be there, try MANOVA.

May 22, 2020 at 2:45 pm

Yes, I read your article about two-way ANOVA just now and it definitely cleared my confusion. Thank you so much for all your help. I really appreciate it.

May 22, 2020 at 3:09 pm

You’re very welcome, Alice!

May 22, 2020 at 2:18 pm

Thank you so much for your help. Just one more question again. I came across a lot of Youtube tutorials employing Univariate analysis in SPSS to see the interaction effect of the IV’s on the DV when they have two IVs and one DV, like in my case. Will it be correct if I do the same?

May 22, 2020 at 2:36 pm

If you’re performing ANOVA you’re not performing univariate analysis. Univariate analysis is the simplest type of analyses because you have just one variable. The simplest form of ANOVA requires TWO variables. One-way ANOVA requires one IV and one DV.

Additionally, if you’re talking about interaction effects, you’ll need a minimum of three variables. 1 DV and 2 IVs. An interaction involves at least two IVs. So, if you’re performing univariate analysis or one-way ANOVA, you cannot consider interaction effects. Typically, you’d be using at least two-way ANOVA or multiple regression.

I believe you’re talking about ANOVA because you’re posting in the comment section of an ANOVA post. So, I’ll refer you to my article about two-way ANOVA . In it, I talk about interaction effects amongst other topics. I also have an additional article about interaction effects you might want to read. However, read the two-way ANOVA post first because it covers the number and types of variables you need.

May 21, 2020 at 9:49 pm

Hi Jim, Thank you for this post. You did a great job in explaining the concepts.

If I may ask, I have one question:

If I had a thousand of dependent variables, How could I be able to interpret and draw groups of correlation among them using MANOVA?

A real-world example for that would be the quantification of a thousand proteins from cells in three different conditions (control, treatment 1, treatment 2). Currently I’m using ANOVA to analyze each one, but I’m pretty sure many of them are correlated, involved in connected cell-signaling pathways.

Thank you for your time!

May 22, 2020 at 1:46 pm

Wow, so a thousand dependent variables. That would be quite the knot to untangle! The same principles would apply but the results would be vastly more complex because you’re talking about multivariate relationships between a thousand DVs plus whatever IVs you have. I think it would require a custom solution to be honest that incorporates subject-area knowledge.

May 20, 2020 at 5:22 am

Hi, Thank you for the informative and easily understandable post. Can you please suggest a statistical measure for my data; I have three independent variable and eight dependent variables. The DVs are not correlated, for ex. DV1 mean 1.5 and DV5 mean 350.

Thank you very much

May 20, 2020 at 1:19 am

Hi Jim, Firstly, thank you so much for this useful blog. You are a blessing to many of us who are struggling with statistics. Please give me some advice on the following question. I have two independent variables and one dependent variable and I also want to see whether there is an interaction effect of those two IVs on the DV. Would it be better to employ MANOVA instead of using ANOVA?

May 20, 2020 at 1:22 am

You should use ANOVA because you have only one dependent variable. Because you have two IVs (presumably categorical), you should use two-way ANOVA. Click the link to see where I show examples of two-way ANOVA .

May 16, 2020 at 1:09 pm

Wow! Thank you so much for this, Jim. I’m currently working on my final year thesis and I’ve been confused as to whether to use ANOVA or MANOVA. It’s a lot clearer now. Thanks again!

April 13, 2020 at 12:39 pm

Hi, I am currently conducting a study on the effects of student type (domestic or international) on levels of trait emotional intelligence, mindfulness, emotional distress and perceived stress. I used a MANOVA analysis which found non-significant differences in three of the variables but not one. I would like to analyse the variable with the difference, Would I need to run a post-hoc (which apparently isn’t possible because there are less than three categories) or can I simply say the mean differences as it appears in the estimated marginal means?

April 7, 2020 at 8:49 am

Thank you very much!) This information helps a lot!

April 2, 2020 at 10:11 am

First of all, I hope you are safe in these worrying times of disease.

I want to thank you for your wonderful posts on statistics. They make things much clearer!

And if I may I also wanted to ask a question: you say that to apply MANOVA, dependent variables should be correlated, but I wonder if the sign of correlation matters? I mean should all the DVs be positively correlated, or they could be both positively and negatively correlated? For example, I have 5 DVs, and there are both positive and negative correlations between them, is it still possible to apply MANOVA in this case?

Thank you very much.

Kind regards, Oksana

April 5, 2020 at 7:13 pm

No, the correlation sign doesn’t matter. All that matters is that the correlations among the DVs exists because it provides more information for the model to assess. It’s ok, and even good, to use MANOVA in the situation you describe with a mix of positive and negative correlations.

February 21, 2020 at 4:50 am

Thanks Jim for your to the point reply, sure my dependent variable are correlated so i used MANOVA, Now i have both significant as well as non-significant result for DV, but stuck how to identify which one is significant among the five cultivars for a particular dependent variable.

February 14, 2020 at 5:59 am

Hello Jim, Feeling pretty good to see your posts regarding statistics. I am going to analyze my research data. I had taken 5 different apple cultivars and determined TPC, TFC, mineral contents etc. Now which type anova would be good to analyze the data.

February 20, 2020 at 4:04 pm

I’m glad you like my posts!

It sounds like you can either use one-way ANOVA or MANOVA. It appears that you have one factor: apple cultivar. This factor forms the five groups. If your dependent/response variables are correlated, then I’d consider using MANOVA for the reasons I mention in this post. If they’re not correlated, consider performing one-way ANOVA multiple times for each of the DVs. While performing multiple tests can increase the family-wise error rate, it’ll be easier to interpret.

January 28, 2020 at 6:32 am

Can we use individual ANOVA rather than MANOVA since MANOVA is a considerably more complex design than ANOVA and therefore there can be some ambiguity about which independent variable affects which dependent variable. The dependent variables should be largely uncorrelated for MANOVA. If the dependent variables are highly correlated, there is little advantage in including more than one in the test given the resultant loss in degrees of freedom. Under these circumstances, use of a single ANOVA test would be preferable (French et al., 2008)

http://online.sfsu.edu/efc/classes/biol710/manova/MANOVAnewest.pdf Do you agree?

January 28, 2020 at 11:20 pm

I agree that MANOVA models are more complex and interpreting the results can be more difficult. However, some cases really require it. I show one such case in this blog post where if you don’t use MANOVA, you completely miss the findings! I’d say try using ANOVA and see what you get. However, if you have multiple correlated DVs, there are some occasions where you might really need to use it. And, I disagree with what you right about using MANOVA for uncorrelated DVs. If your DVs are uncorrelated, you have less reason to use MANOVA. If they are correlated, you have more reason. Again, this post indicates one such scenario!

January 6, 2020 at 5:27 pm

Jim, I have a question on how far can we expand the independent and defendant amount. In my case I have 5 different offices types (independent variables) and a scores for 32 different categories (dependent variables). Can a MANOVA test distinguish if there is any significant differences between the office types and how would the data be interpreted? Thanks David

November 12, 2019 at 11:43 am

I’m in the toxicology field and my project involves assessing the effects of 5 different compounds, with various concentrations of each, in wild type and mutant animals. I think in this case I would have 3 independent variables, right? Compound, concentration and animal strain. However, if I’m making all these comparisons, I’m losing power. So what I don’t understand is why a result would be statistically significant in a scenario were I would have, for example, only one compound, one concentration and one strain, but if I have multiple of each, this same result wouldn’t be statistically significant? Hopefully I made myself clear! Thank you for your website, you make statistics much more fun!!

April 9, 2019 at 9:27 pm

Hi Jim, My study includes one independent variable with 3 levels (participation in one of 3 networks) and 3 dependent variables which are likely to be correlated (teacher expectations, teacher self-efficacy, and teacher collective efficacy). I want to determine which networks result in higher scores of these factors. I am having trouble wording my questions though and thinking about what information the MANOVA will tell me. I originally worded it as.. Is there a relationship between network membership and scores of teacher expectations, self-efficacy, and collective efficacy? But, to me this implies more of a correlation, which is not possible because it is not two pieces of ordinal data. Should I word it as differences instead? And is it possible for example that more than one network will show a statistically significant relationship to a DV or multiple DVs? Should I have a research question which addresses the possible correlation of my DVs?

I don’t know if that makes any sense. I know what I’m trying to investigate, but I’m not sure how to word the questions and I also wonder how specific I should make my hypotheses. For example, I suspect one network will be statistically higher in teacher expectations and possibly collective efficacy. But, I also see the potential for another one to be statistically higher in collective efficacy in regard to another.

Is it possible to stick with the relationship question and then have a hypothesis that predicts a relationship between more than one IV and the same DV?

January 5, 2019 at 10:55 am

Thanks for the informative post. I am wondering if only Roy’s largest root is significant out of the 4 MANOVA tests, would you still take the result as significant? And how to interpret a quadratic interaction between my within and between subject variables? I am testing my participants’ memory on emotional faces, which have been categorized into negative, neutral and positive faces for data analysis. They are also the within subject DVs, while I got 2 clinical populations and 3 treatment conditions as my between subject variables. If MANOVA doesn’t work, what other tests would you recommend? I am interested in the interaction between the between subject variables.

Thanks, Desiree

December 7, 2018 at 10:15 am

thank you for an informative post! I am currently designing a grant proposal and have always struggled with stats so I was wondering if you could please help me out. I want to be looking at the differences of autistic and neurotypical adults’ performance on reading facial and body language expressions – there will therefore be two populations (autistics vs. non-autistic) and 3 conditions (one where only faces are shown (2 types of emotions will be displayed)), one where only bodies are shown (again, 2 types of emotions will be displayed), one where faces and bodies are both shown (either displaying congruent or incongruent emotions)). All participants will be completing all the conditions. Previous papers that have looked into this used repeated measures ANOVA, however what I want to do is have all participants complete a couple additional questionnaires that will be looking into a condition that is often associated with autism (alexithymia) – it is therefore very possible that my populations will look something like neurotypical adults with low score for alexithymia, neurotypical adults with high score for alexithymia vs. autistic patients with low score for alexithymia, and autistic patients with high score for alexithymia. The presumption is that autistic patients with high score for alexithymia will perform worst, while neurotypical adults with low alexithymia score will perform best. The aim of the project is to determine to what extend do autism and alexithymia contribute to the impairment of expression recognition (individually and combined) – is it still okay to use ANOVA or would MANOVA be better? Or am I meant to be running a regression analysis? Any post-hoc tests that should be considered?

Thank you very much for your help!

December 3, 2018 at 8:16 pm

Dear Jim, thank you very much, your reply was very helpful!

I have now performed the MANOVA analyses and have found that one of my group variables, variable A, (described in my first post) best discriminates between brain measures (the outcome measures), and would like to follow up on this with a post-hoc test. Variable A consists of group 1, 2, 3 and 4 and I would now like to know if one of these groups is driving this effect. Could you please tell me if this is possible to test and what would be the best way to do this? I have read online that ANOVA is sometimes used to follow up, but I am not very interested to see which groups are associated with brain measures, I just want to know which groups differ across all brain measures.

Thank you so much in advance!

Best, Laura

December 4, 2018 at 10:53 am

It sounds like you need to perform a post-hoc test to determine which groups are significantly different from the others for variable A. Post hoc tests a common follow up for both MANOVA and ANOVA. MANOVA and ANOVA tell you that there is a significant effect while the post hoc tests help you map out the nature of those effects–which groups are significantly different from the others.

I hope this helps!

December 2, 2018 at 9:50 am

Thank you for a useful and informative post.

I was wondering if you could help me with a question.

I am conducting a study looking into the effectiveness of a post-therapy relapse-prevention intervention in maintaining treatment grains after therapy. Questionnaires measuring treatment gains quantitatively (e.g. mood) are taken at three time points: pre-therapy (T1), on completion of the therapy (T2) and 2 months after the therapy (T3).

I am comparing a control group completing therapy but without relapse-intervention with a group who receive the relapse-prevention intervention between T2 and T3. In order to test the effectiveness of the intervention, I want to see if the IV (intervention group or control group) predicts the DV (change in outcomes between T2 and T3).However, the DVs are correlated AND I will need to account for the change in outcomes from T1 to T2 (a covariate) and so I thought an MANCOVA would be appropriate.

So I have 2 IVs and 4 DVs.

However I have a much smaller sample size that I had first intended (n=15 per group – total n=30). so my question is is this too small a sample size to conduct a MANCOVA with? If so, what would be a better option?

I hope this makes sense.

Many thanks in advance!

November 19, 2018 at 2:54 am

I’m doing a study with 1 IV and 3 DVs. In theory, my 3 DVs should be at least weakly correlated, although my results do not show this. 2 of them are very strongly correlated (r is very close to +1), while the third is not correlated at all with either of them. Can I still use a MANOVA? or should I do 3 separate ANOVAs instead? If I were to do the ANOVAs separately, is there a way in which I reduce the Type I error?

Thank you in advance.

November 19, 2018 at 9:40 am

It still sounds like MANOVA would be helpful given the correlations. Although, you probably should investigate why the correlations are so high between a pair and non-existent with the third variable when that doesn’t match your theory. It almost sounds like two variables practically measure the same thing and third isn’t related at all. It’s not necessarily wrong, but worth investigating because they don’t match your expectations. You should check to be sure that you’re measuring what you think you’re measuring, and that there were no measurement/data entry errors.

October 24, 2018 at 9:40 am

HI, thanks for your kindly help. I have a doubt. I have a IV with 2 groups (control/experimental), and one DV (sensory processing). However, the DV has 4 main scales. The way in which I measured was through a questionnaire that doesn’t have a single scores, but 4 scores per each scale. So, is it appropriate using a MANOVA with my DV plus 4 IV??

October 24, 2018 at 4:13 pm

If those four scales are correlated to some extent, then it sounds like you’d get some real benefits out of MANOVA. If they’re not correlated, you can just run separate ANOVA analyses for each scale. Actually, because there are just two groups, you can use t-tests if they’re not correlated. But, yes, if the four scales are correlated, definitely consider MANOVA! MANOVA provides benefits when the DVs are correlated.

August 21, 2018 at 2:35 pm

Thanks so much for the post! I have a question. Can we use MANOVA when we have more than one independent variables? I have a design in which I am testing 4 dependent variables and I have 4 independent variables also. I have used lmer in R so far for each of the 4 dependent variables looking at the influence of the independent variables. Do you think I should use MANOVA? My dependent variables are correlated.

August 24, 2018 at 2:14 am

Yes, it sounds like MANOVA is a good possible analysis to try with your dataset. You can use more than one independent variable. And, it provides the most benefits when the dependent variables are correlated.

August 21, 2018 at 1:15 pm

Hi Jim – your blog post and the follow-up questions and your replies are very helpful and much appreciated. Would you please advise on the following questions – is MANOVA most appropriate here and what should the sample size be per group? We have 3 groups who will be measured once pre-operatively and 4 times post-operatively. There are 13 measurements, which can be grouped as patient-reported outcomes (5 measures; means) and clinical outcomes (8 measures, means). We want to compare pre- and post-op differences by group and also compare post-operative measures between groups. We expect there to be some correlations among the measures in both the patient-reported outcomes and the clinical outcomes. We also have 10 measures of patient characteristics (e.g., age, gender, BMI, etc.; 4 means, 6 categorical) that we want to compare by group. I hope I’ve provided enough information for you to advise on this. Thanks!

August 7, 2018 at 1:58 am

Hello Jim, Good day Jim. I find your articles on statistics quite interesting. However, I need you to help me with a statistical issue. Please may I know why a researcher would decide not to introduce a moderating variable in a model. Thanks for your anticipated assistance.

From, Adu Emmanuel.

August 7, 2018 at 3:38 pm

Moderation effects are included in models as interaction effects. Read my post about interaction effects for more information.

The only legitimate reasons that I can think of for why analysts would not include an interaction effect in the model are either because the effect is not statistically significant or because theory very strongly suggests that the interaction effect is not appropriate for the model.

If interaction effects are significant and appropriate for the subject area, you risk extreme interpretation problems by not including them in the analysis. You can read about that in the post I link to!

July 24, 2018 at 10:12 am

Thank you for this helpful information about MANOVA, I have a question, In one of the study I have done for my PhD, I have two independent variables, each categorical with two levels, and I measured 3 dependent variables which are correlated. I also have a covariate. Q1: First of all the data significantly violated the assumption of homogeneity of covariance (p<.00000), considering this can I use the Mancova test (my sample size in each of four groups is about 95, in total 380)?

Q2: When I do a separate ANCOVAs on each dependent variable I have significant result for both IVs, however, when I use MANCOVA the effect of one of my IV becomes non-significant, which test should I report in my thesis?

July 15, 2018 at 3:56 pm

hello Jim, I find your blog very useful especially for those struggling in this realm of statistics, like me. I just want to ask, the way of framing questions in MANOVA, like when my IVs are Sex and Study Habits, and the DVs are the Learning gains, GPA and NAT scores? how will I state if MANOVA applies for this?

July 16, 2018 at 1:05 am

If your DVs are correlated, then MANOVA becomes a more powerful analysis because it can use the correlation between the DVs to increase the statistical power. It can also detect multivariate effects that ANOVA can’t, which I demonstrate in the blog post. If your DVs are not correlated, then MANOVA doesn’t really provide additional benefits compared to ANOVA.

July 12, 2018 at 7:45 am

Thank you for this very informative post! I hope you could give me some advice regarding the following issue: I am trying to examine which of two group variables best map to underlying brain measures. I have two independent (group) variables (say A and B) and around 30 dependent variables. I would like to examine whether group variable A (group 1,2 or 3) or B (group i,ii,iii) best discriminate groups based on these brain measures. I was thinking of running MANOVA twice, once with variable A and once with variable B, but is there a way to test whether these results are signficantly different so we can conclude for instance that group variable A best maps to underlying brain measures?

Thanks very much in advance! Best, Laura

July 12, 2018 at 1:44 pm

It makes me happy to hear that you found the post to be helpful!

MANOVA will help you out if those 30 dependent variables are correlated amongst themselves. With so many dependent variables, you will have to be aware of sample size concerns. I’ve written about how to ensure that your linear model has a sufficiently large sample size based on the independent variables and other terms in the model. I don’t know how the number of dependent variables factors in, but it’s something to keep in mind.

As for determining which categorical variable is better, you can try including both in the model at the same time. This approach can give you some valuable head-to-head comparison information. You might also want to read my post about identifying the best independent variable . I write about this in the context of regression models, but it’ll give you some good ideas to consider and other things that you should disregard.

June 19, 2018 at 3:44 pm

Wow, thanks Jim for taking the time to write such a wonderfully detailed and helpful reply, and for providing links to your other posts for much-needed background reading. Can’t express enough how appreciative I am of your willingness to share such hard-earned knowledge with others!

June 17, 2018 at 2:03 pm

Hi Jim, Thank you for this blog. Do you by chance have a reference for you last section “When MANOVA Provides Benefits”?

June 18, 2018 at 11:34 am

Hi Sarah, I don’t have a specific reference but these are commonly understood benefits of MANOVA. I’d imagine that any textbook that covers this analysis will confirm them.

June 14, 2018 at 5:03 pm

First of all, thanks so much for this extremely informative and comprehensive blog. I’m thinking that MANOVA might be exactly the tool that I need for a problem that I am currently grappling with, but an additional “twist” in my data might actually invalidate the use of MANOVA. So here goes (I am an astronomer): I have a plot of gas mass normalized by luminosity (y axis) versus luminosity (x axis) for a large control sample of galaxies. This plot is a nonlinear relation in which dimmer galaxies generally have higher normalized gas masses. Overlayed on this plot is a (smaller) sample of galaxies that is the focus of my research. If I squint and look sideways at the plot, I think that my sample trends towards having somewhat larger normalized gas mass as compared to the control objects with similar luminosity.

I’d like quantify the statistical significance of any gas mass enhancement in the small/study sample. Here’s the problem: the smaller set of data mostly bunches up at the low-luminosity end and as a result does not cover the full range of X values (luminosity) represented by the control sample and therefore does a poor job of sampling that relationship. Because the smaller set of data are at the low-luminosity end, they are expected to have higher normalized gas mass (that’s what the relationship defined by the control sample shows), so the fact that the study sample objects are generally gas-rich is not what is surprising. What WOULD be interesting is if those galaxies have MORE gas mass than what would be predicted by the relationship for their given luminosity.

My concern is that the smaller sample being so heavily skewed towards one end of the X-axis and not “adequately” sampling along the same range of X values as the control sample will invalidate any MANOVA application (?) On the other hand, there is nothing about the example you provide above that would have excluded the possibility that “satisfaction” would have been skewed to one of the extremes for one of the exam methods. In that example, however, the sample sizes being compared are equal (or nearly so), unlike my situation.

If MANOVA is definitely not the tool for my situation, would a better approach be to just truncate the control set so that the same x values are covered between both samples, and then apply a 1-dimensional test to the Y-axis parameter ? (A steeply-rising “knee” in the gas mass to luminosity relationship in the mid-range of the x-axis would be mostly clipped out if a truncation to the control sample was performed. Losing the information of this underlying relationship in any statistical comparisons between these 2 data sets could be an undesirable consequence?). Thanks for any advice you might have, particularly with regards to the robustness of the MANOVA test to such a situation in which one sample is seriously skewed towards one end of the x-axis relative to the other, and has significantly fewer data points than the other.

June 15, 2018 at 12:07 am

First, I just want to let you know that I LOVE astronomy. I totally don’t have any official education in it but I love to absorb as much of it as possible! In a parallel universe I’m an astronomer or maybe a physicist!

I think what you need to do is actually fit a regression model and include an sample indicator variable and an interaction term. The gas mass normalized by luminosity would be your dependent variable and luminosity would be an independent variable. You’d also need to include an indicator variable that identifies whether the data are from the control sample or your sample (you’d need to put all the data in one data sheet). Then include a two-way interaction term (luminosity*sample indicator). You should also include the indicator variable as a main effect/independent variable. Collectively the indicator variable and interaction term will tell you whether the relationship between the independent and dependent variable is different between your sample and the control sample. The main effect for the indicator variable will tell you if the relationship is shifted up or down on the X-Y scatterplot. The interaction effect tells you whether the slope is different. Use the p-values for these terms to determine whether each one is statistically significant.

I *think* that is what you need to do based on your description. I’ve written a couple of posts that describe this process that you should read: Comparing Regression Lines Understanding Interaction Effects

There is one possible complication that I can see with your scenario. You mentioned that it is a nonlinear relationship. The way forward depends on whether that is nonlinear in the common meaning of the word (i.e., the line isn’t straight) or in the exact statistical sense. The process I describe above works for linear models. But linear models can fit curves using a variety of methods. However, if it’s truly nonlinear in the statistical sense of the word, you’ll need to use an entirely different and unfortunately much more complex methodology. I’ve written two posts that should help you answer that question. The first defines the differences between linear and nonlinear in the world of statistics. The second walks you through the different ways of fitting curves using an example dataset. The Difference Between Linear and Nonlinear Regression Curve Fitting Using Linear and Nonlinear Regression

Hopefully a linear model will fit because that’s a lot more straightforward to work with! Oh, and about your sample being skewed towards one end of the range, I don’t think that’s a problem. You should be able to determine whether the same relationship applies to both samples or not. One issue I see is if the two samples don’t overlap. In that case, if the relationship is different, it might be hard to determine whether the difference is due to whatever the defining characteristic of your sample is or just because you’re in a different range of the data. Sometimes the relationship can change for the same type of data as you move along the range of data.

At any rate, I hope this helps!

June 15, 2018 at 1:03 am

Hi again, one more thought in addition to my previous reply. Given that there is curvature in your data, it probably is a problem that your data just covers a portion of the range. You won’t be able to use the same model because if it’s just a portion it won’t have the same curvature most likely. You might need to truncate the control sample data to fit the same range of data that your sample covers. I’m not sure how many data points the sample data has in that range or if there might be other subject related concerns for that approach. But, it would let you compare the models for the same range of data and if the curvature was different within that range, it would be meaningful. Whereas if you compared your range to the full range, the difference wouldn’t be meaningful.

May 2, 2018 at 4:35 pm

Thanks for your kind response.

May 1, 2018 at 5:36 pm

Hi Jim, I need more clarifications, your comment on this subject seems to be tending towards my needs. However, my question is, between ANOVA and Manova, which of them is suitable for an hypothesis stating that, there is no deference between male and female in terms of impact of Staff development policies, practices on job performance. Thanks for your response in anticipation.

May 1, 2018 at 11:06 pm

Hi Omotayo,

If gender is your independent variable and staff development policies and practices on job performance are your dependent variables, and those two DVs are continuous variables that are correlated, I’d recommend MANOVA.

April 7, 2018 at 1:37 am

Hi Jim! Thank you for your informative article, the concepts are so much clearer to me now.

However, I have a question.What if I have 4 independent variable but only 1 dependent variable? Would ANOVA be able to process that? Or should I employ MANOVA?

April 7, 2018 at 4:12 pm

Hi Esther, It makes me happy to hear that my blogs have helped clarify things for you!

MANOVA requires multiple dependent variables. It is particularly useful when those dependent variables are correlated.

When you have only one dependent variable, you have to use an ANOVA procedure. With some ANOVA procedures, such as General Linear Model, you can have multiple independent variables.

March 20, 2018 at 11:39 am

So what would be the difference between a 2-way ANOVA and a MANOVA? When would you use them and why? Is one more powerful?

Thanks so much!

March 20, 2018 at 11:58 am

There are several differences. For one thing, 2-way ANOVA can handle two independent variables (IV) and only one dependent variable (DV). MANOVA can handle 1 or more IVs and 1 or more DVs. The real key advantage of MANOVA is how it handles multiple DVs at the same time. This provides MANOVA with more power when those DVs are correlated.

Use MANOVA when you have multiple DVs that are correlated. As this post shows, it can detect multivariate patterns in the DVs that ANOVA is simply unable to detect at all. Plus, it is more powerful when those DVs are correlated.

When you have only one DV, use some form of regular ANOVA, which includes 2-way ANOVA.

February 28, 2018 at 1:46 am

Hi Jim, I have a few questions: IV = independent variable DV = dependent variable

1. Suppose I conducted 2 x 2 x 2 factorial design using Manova. My hypotheses cover the direct effect, two-way interaction, and three-way interaction.

I found that based on the direct effect, IV1, IV2, IV3 are significantly related to DV1 and IV1 is the most important factor. Furthermore, interaction effect of X2 and X3 also significantly related to DV1. When writing the discussion part in paper, I provide that reasons why IV1, IV2, IV3 are significantly related to DV1, then do I need to explain IV1 is the most important factor? If not, why? Finally, I discuss the interaction effect.

2. I understand that SPSS will run Manova and then run Anova automatically. The reason is that if the program run Anova in several times, the type I error will increase. Am I correct?

February 14, 2018 at 12:32 am

Dumb question: What test or tests does one run in SPSS to find out if the dependent variables are related, to “gain the benefits of MANOVA” as you say. Thank you sir!

February 14, 2018 at 12:33 am

correlated rather

February 14, 2018 at 1:36 am

Hi Joe, of course there is no such thing as a dumb question! All you need to determine is if they are correlated. So a simple Pearson’s correlation. Or, you can even use regression analysis. Nothing fancy! As long as there is some sort of relationship between the dependent variables, MANOVA is beneficial.

February 10, 2018 at 2:22 am

Hi Jim, Could please explain why computing a variance of several numbers is like analyzing their differences

February 10, 2018 at 5:00 pm

To calculate the variance (which is the square of the standard deviation), you take the difference between each individual value and the mean, square those differences, add all of those squared differences together, and then divide by the number of observations. If you want the standard deviation, which is easier to interpret, you need to take the square root of that.

So, you’re really analyzing the differences between the individual observations and the mean. A larger variance (or standard deviation) indicates that the differences between the individual data points and the mean tends to be larger (the data points tend to fall further from the mean–they’re more dispersed). Smaller values of the variance/standard deviation indicate that the differences are smaller. The data points are clustered more tightly around the mean. That’s how differences come into play with the variance and standard deviation!

December 6, 2017 at 9:20 am

Thanks for writing this very informative post! I am in my final year of Applied Psychology and am currently in the process of completing my final year project. My study is investigating whether a difference exists among the eating patterns and behaviours of college students of the different years (i.e. 1st-4th year college students). My independent variable would be college year and my dependent variables are: (i) eating patterns, (ii) emotional eating and (iii) attitudes towards healthy eating. Would you recommend me to use a MANOVA in analyzing my data or would you recommend a different approach to analyses? I am confused on what approach to use!

December 6, 2017 at 11:42 am

If the dependent variables are correlated, then you gain benefits by using MANOVA rather than ANOVA. However, if they are not correlated, ANOVA might be just fine. I’m not an expert in that field, but it seems like your dependent variables might be correlated.

I hope this helps. Best of luck with your analysis!

October 1, 2017 at 2:08 am

DVs aren’t supposed to be correlated

October 1, 2017 at 1:45 pm

Hi, you’re thinking about independent variables. When independent variables are correlated it is known as multicollinearity, and it can be a problem. I’ve written a post about multicollinearity in case you are interested.

Most analyses aren’t designed to handle multiple dependent variables. However, MANOVA is not only designed for that but there are benefits when they are correlated.

Comments and Questions Cancel reply

How to Perform MANOVA in R

What is manova.

MANOVA is useful when there are multiple dependent variables and they are inter-correlated. While analyzing the differences among groups, MANOVA reduces the type I error which can be inflated by performing separate univariate ANOVA for each dependent variable.

Assumptions of MANOVA

Manova hypotheses, one-way (one factor) manova in r, manova example dataset.

For MANOVA, the dataset should contain more observations per group in independent variables than a number of the dependent variables. It is particularly important for testing the Homogeneity of the variance-covariance matrices using Box’s M-test.

Summary statistics and visualization of dataset

Perform one-way manova, post-hoc test.

The MANOVA results suggest that there are statistically significant ( p < 0.001) differences between plant varieties, but it does not tell which groups are different from each other. To know which groups are significantly different, the post-hoc test needs to carry out.

Test MANOVA assumptions

Assumptions of multivariate normality.

Note: As per Multivariate Central Limit Theorem, if the sample size is large (say n > 20) for each combination of the independent and dependent variable, we can assume the assumptions of multivariate normality.

If the sample size is large (say n > 50), the visual approaches such as QQ-plot and histogram will be better for assessing the normality assumption. Read more here

Here both Skewness and Kurtosis p value should be > 0.05 for concluding the multivariate normality.

Homogeneity of the variance-covariance matrices

Multivariate outliers, linearity assumption, multicollinearity assumption, enhance your skills with courses on statistics and r, you may also enjoy, calculate coverage from bam file, python: why vif return inf value, find max and min sequence length in fasta, get non-overlapping portion between two regions in bedtools.

Conduct and Interpret a One-Way MANOVA

What is the One-Way MANOVA?

MANOVA is short for M ultivariate AN alysis O f Va riance.  The main purpose of a one-way ANOVA is to test if two or more groups differ from each other significantly in one or more characteristics.  A factorial ANOVA compares means across two or more variables.  Again, a one-way ANOVA has one independent variable that splits the sample into two or more groups whereas the factorial ANOVA has two or more independent variables that split the sample in four or more groups.  A MANOVA now has two or more independent variables and two or more dependent variables.

For some statisticians the MANOVA doesn’t only compare differences in mean scores between multiple groups but also assumes a cause effect relationship whereby one or more independent, controlled variables (the factors) cause the significant difference of one or more characteristics.  The factors sort the data points into one of the groups causing the difference in the mean value of the groups.

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A research team wants to test the user acceptance with a new online travel booking tool.  The team conducts a study where they assign 30 randomly chosen people into 3 groups.  The first group needs to book their travel through an automated online-portal; the second group books over the phone via a hotline; the third group sends a request via the online-portal and receives a call back.  The team measures user acceptance as the behavioral intention to use the system, they do measure the latent construct behavioral intention with 3 variables – ease of use, perceived usefulness, effort to use.

In the example, some statisticians argue that the MANOVA can only find the differences in the behavioral intention to use the system.  However, some statisticians argue that you can establish a causal relationship between the channel they used and the behavioral intention for future use.  It is generally assumed that the MANOVA is an ‘ analysis of dependencies’ .  It is referred to as such because it proves an assumed cause-effect relationship between two or more independent variables and two or more dependent variables.  In more statistical terms, it tests the effect of one or more independent variables on one or more dependent variables.

Other things you may want to try:

When faced with a question similar to the one in our example you could also try to run a 3 factorial ANOVAs , testing the influence of the three independent variables (the three channels) on each of the three dependent variables (ease of use, perceived usefulness, effort to use) individually.  However running multiple factorial ANOVAs does not account for the full variability in all three dependent variables and thus the test has lesser power than the MANOVA.

Another thing you might want to try is running a factor analysis on the three dependent variables and then running a factorial ANOVA.  The factor analysis reduces the variance within the three dependent variables to one factor, thus this procedure does have lesser power than the MANOVA. A third approach would be to conduct a discriminant analysis and switch the dependent and independent variables.  That is the discriminant analysis uses the three groups (online, phone, call back) as the dependent variable and identifies the significantly discriminating variables from the list of continuous-level variables (ease of use, perceived usefulness, effort to use).

Mathematically, the MANOVA is fully equivalent to the discriminant analysis.  The difference consists of a switching of the independent and dependent variables.  Both the MANOVA and the discriminant analysis are a series of canonical regressions.  The MANOVA is therefore the best test use when conducting experiments with latent variables.  This is due to the fact that it only requires a nominal scale for the independent variables which typically represent the treatment.  This includes multiple continuous-level independent variables – which typically measure one latent (not directly observable) construct.

The MANOVA is much like the one-way ANOVA and the factorial ANOVA in that the one-way ANOVA has exactly one independent and one dependent variable.  The factorial ANOVAs can have one or more independent variables but always has only one dependent variable.  On the other hand the MANOVA can have two or more dependent variables.

The following table helps to quickly identify the right analysis of variance to choose in different scenarios.

Examples of typical questions that are answered by the MANOVA are as follows:

• Medicine – Does a drug work? Does the average life expectancy, perceived pain, and level of side-effects significantly differ between the three experimental groups that got the drug versus the established product, versus the control—and within each of the groups two subgroups for a high dose versus a low dose?
• Sociology – Are rich people living in the country side happier?  Do they enjoy their lives more and have a more positive outlook on their futures? Do different income classes report a significantly different satisfaction, enjoyment and outlook on their lives? Does the area in which they live (suburbia/city/rural) affect their happiness and positive outlook?
• Management Studies – Which brands from the BCG matrix do have a higher customer loyalty, brand appeal, customer satisfaction? The BCG matrix measures brands in a brand portfolio with their business growth rate (high/low) and their market share (high/low).  To which brand are customers more loyal, more attracted, and more satisfied with? Stars, cash cows, dogs, or question marks?

The One-Way MANOVA in SPSS

Our research question for the one-way MANOVA in SPSS is as follows:

Do gender and the outcome of the final exam influence the standardized test scores of math, reading, and writing?

The research question indicates that this analysis has multiple independent variables (exam and gender) and multiple dependent variables (math, reading, and writing test scores).  We will skip the check for multivariate normality of the dependent variables; the sample we are going to look at has some violations of the assumption set forth by the MANOVA.

The MANOVA can be found in SPSS in Analyze/General Linear Model/Multivariate… , which opens the dialog for multivariate GLM procedure (that is GLM with more than one dependent variable).  The multivariate GLM model is used to specify the MANOVAs.

To answer our research question we need to specify a full-factorial model that includes the test scores for math, reading, and writing as dependent variable.  Plus the independent variables gender and exam, which represent a fixed factor in our research design.

The dialog box Post Hoc tests is used to conduct a separate comparison between factor levels, this is useful if the MANOVA includes factors have more than two factor levels.  In our case we select two factors and each has only two factor levels (male/female and pass/fail).  The MANOVA’s F-test will test the null hypothesis that all means are the same.  It does not indicate which of the means in our design are different.  In order to find this information, post hoc tests need to be conducted as part of our MANOVA.  In order to compare different groups (i.e., factor levels) we select the Student-Newman-Keuls test (or short S-N-K), which pools the groups that do not differ significantly from each other, thereby improving the reliability of the post hoc comparison by increasing the sample size used in the comparison.  Additionally, it is simple to interpret.

The Options… Dialog allows us to add descriptive statistics, the Levene Test and the practical significance to the output.  Also we might want to add the pairwise t-tests to compare the marginal means of the main and interaction effects.  The Bonferroni adjustment corrects the degrees of freedom to account for multiple pairwise tests.

The Contrast… dialog in the GLM procedure model gives us the option to group multiple groups into one and test the average mean of the two groups against our third group.  Please note that the contrast is not always the mean of the pooled groups! Because Contrast = (mean first group + mean second group)/2.  It is only equal to the pooled mean if the groups are of equal size.  In our example we do without contrasts.

Lastly the Plots… dialog allows us to add profile plots for the main and interaction effects to our MANOVA.  However it is easier to create the marginal means plots that are typically reported in academic journals in Excel.

Interpret the key results for General MANOVA

In this topic, step 1: test the equality of means from all the responses, step 2: determine which response means have the largest differences for each factor, step 3: assess the differences between group means, step 4: assess the univariate results to examine individual responses, step 5: determine whether your model meets the assumptions of the analysis.

• If a main effect is significant, the level means for the factor are significantly different from each other across all responses in your model.
• If an interaction term is significant, the effects of each factor are different at each level of the other factors across all responses in your model. For this reason, you should not analyze the individual effects of terms involved in significant higher-order interactions.

MANOVA Tests for Method

MANOVA Tests for Plant

MANOVA Tests for Method*Plant

Key Results: P

The p-values for the production method are statistically significant at the 0.10 significance level. The p-values for the manufacturing plant are not significant at the 0.10 significance level for any of the tests. The p-values for the interaction between plant and method are statistically significant at the 0.10 significance level. Because the interaction is statistically significant, the effect of the method depends on the plant.

Use the eigen analysis to assess how the response means differ between the levels of the different model terms. You should focus on the eigenvectors that correspond to high eigenvalues. To display the eigen analysis, go to Stat > ANOVA > General MANOVA > Results and select Eigen analysis under Display of Results .

EIGEN Analysis for Method

Key Result: Eigenvalue, Eigenvector

In these results, the first eigenvalue for method (0.5848) is greater than the second eigenvalue (0.00000). Therefore, you should put higher importance on the first eigenvector. The first eigenvector for method is 0.144062, -0.003968. The highest absolute value within this vector is for the usability rating. This suggests that the means for usability have the largest difference between the factor levels for method. This information is helpful for assessing the means table.

Use the Means table to understand the statistically significant differences between the factor levels in your data. The mean of each group provides an estimate of each population mean. Look for differences between group means for terms that are statistically significant.

For main effects, the table displays the groups within each factor and their means. For interaction effects, the table displays all possible combinations of the groups. If an interaction term is statistically significant, do not interpret the main effects without considering the interaction effects.

To display the means, go to Stat > ANOVA > General MANOVA > Results , select Univariate analysis of variance , and enter the terms in Display least squares means corresponding to the terms .

Least Squares Means for Responses

Key Result: Mean

In these results, the Means table shows how the mean usability and quality ratings varies by method, plant, and the method*plant interaction. Method and the interaction term are statistically significant at the 0.10 level. The table shows that method 1 and method 2 are associated with mean usability ratings of 4.819 and 6.212 respectively. The difference between these means is larger than the difference between the corresponding means for quality rating. This confirms the interpretation of the eigen analysis.

However, because the Method*Plant interaction term is also statistically significant, do not interpret the main effects without considering the interaction effects. For example, the table for the interaction term shows that with method 1, plant C is associated with the highest usability rating and the lowest quality rating. However, with method 2, plant A is associated with the highest usability rating and a quality rating that is nearly equal to the highest quality rating.

When you perform General MANOVA , you can choose to calculate the univariate statistics to examine the individual responses. The univariate results can provide a more intuitive understanding of the relationships in your data. However, the univariate results can differ from the multivariate results.

To display the univariate results, go to Stat > ANOVA > General MANOVA > Results and select Univariate analysis of variance under Display of Results .

• If a categorical factor is significant, you can conclude that not all the level means are equal.
• If an interaction term is significant, the relationship between a factor and the response depends on the other factors in the term. In this case, you should not interpret the main effects without considering the interaction effect.
• If a covariate is statistically significant, you can conclude that changes in the value of the covariate are associated with changes in the mean response value.
• If a polynomial term is significant, you can conclude that the data contain curvature.

Analysis of Variance for Usability Rating, using Adjusted SS for Tests

Analysis of Variance for Quality Rating, using Adjusted SS for Tests

In these results, the p-value for the main effect of method and the method*plant interaction effect are statistically significant at the 0.10 level in the model for usability rating. The main effects of both method and plant are statistically significant in the model for quality rating. You can conclude that changes in these variables are associated with changes in the response variables.

Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results.

When you perform General MANOVA , Minitab displays residual plots for all response variables that are in your model. You must determine whether the residual plots for all response variables indicate that the model meets the assumptions.

For more information on how to handle patterns in the residual plots, go to Residual plots for General MANOVA and click the name of the residual plot in the list at the top of the page.

Residuals versus fits plot

Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.

Residuals versus order plot

Normal probability plot of the residuals

Use the normal probability plot of the residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.

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Statistical Methods and Data Analytics

One-Way MANOVA | SPSS Annotated Output

This page shows an example of multivariate analysis of variance (manova) in SPSS with footnotes explaining the output. The data used in this example are from the following experiment.

A researcher randomly assigns 33 subjects to one of three groups. The first group receives technical dietary information interactively from an on-line website. Group 2 receives the same information from a nurse practitioner, while group 3 receives the information from a video tape made by the same nurse practitioner. Each subject then made three ratings: difficulty, usefulness, and importance of the information in the presentation. The researcher looks at three different ratings of the presentation (difficulty, usefulness and importance) to determine if there is a difference in the modes of presentation. In particular, the researcher is interested in whether the interactive website is superior because that is the most cost-effective way of delivering the information. In the dataset, the ratings are presented in the variables useful , difficulty and importance . The variable group indicates the group to which a subject was assigned.

We are interested in how the variability in the three ratings can be explained by a subject’s group.  Group is a categorical variable with three possible values: 1, 2 or 3.  Because we have multiple dependent variables that cannot be combined, we will choose to use manova.  Our null hypothesis in this analysis is that a subject’s group has no effect on any of the three different ratings, and we can test this hypothesis on the dataset, manova.sav .

We can start by examining the three outcome variables.

Next, we can enter our manova command. In SPSS, manova can be conducted through the generalized linear model function, GLM. In the manova command, we first list the outcome variables, then indicate any categorical factors after “by” and any covariates after “with”.  Here, group is a categorical factor.  We must also indicate the lowest and highest values found in group . We are also asking SPSS to print the eigenvalues generated. These will be useful in seeing how the test statistics are calculated.

Manova Output

a. Case summary – This provides counts of the observations to be included in the manova and the counts of observations to be dropped due to missing data or data that falls out-of-range. For example, a record where the value for group is 4, after we have specified that the maximum value for group is 3, would be considered out-of-range.

b. Effect – This indicates the predictor variable in question. In our model, we are looking at the effect of group .

c. Value – This is the test statistic for the given effect and multivariate statistic listed in the prior column.  For each predictor variable, SPSS calculates four test statistics.  All of these test statistics are calculated using the eigenvalues of the model (see superscript m).  See superscripts h, i, j and k for explanations of each of the tests.

d. Approx. F – This is the approximate F statistic for the given effect and test statistic.

e. Hypoth. DF – This is the number of degrees of freedom in the model.

f. Error DF – This is the number of degrees of freedom associated with the model errors.  There are instances in manova when the degrees of freedom may be a non-integer.

g. Sig. of F – This is the p-value associated with the F statistic and the hypothesis and error degrees of freedom of a given effect and test statistic.  The null hypothesis that a given predictor has no effect on either of the outcomes is evaluated with regard to this p-value. For a given alpha level, if the p-value is less than alpha, the null hypothesis is rejected. If not, then we fail to reject the null hypothesis. In this example, we reject the null hypothesis that group has no effect on the three different ratings at alpha level .05 because the p-values are all less than .05.

h. Pillais – This is Pillai’s Trace, one of the four multivariate criteria test statistics used in manova. We can calculate Pillai’s trace using the generated eigenvalues (see superscript m). Divide each eigenvalue by (1 + the eigenvalue), then sum these ratios.  So in this example, you would first calculate 0.89198790/(1+0.89198790) = 0.471455394, 0.00524207/(1+0.00524207) = 0.005214734, and 0/(1+0)=0.  When these are added, we arrive at Pillai’s trace: (0.471455394 + 0.005214734 + 0) = .47667.

i. Hotellings – This is Lawley-Hotelling’s Trace. It is very similar to Pillai’s Trace. It is the sum of the eigenvalues (see superscript m) and is a direct generalization of the F statistic in ANOVA. We can calculate the Hotelling-Lawley Trace by summing the characteristic roots listed in the output: 0.8919879 + 0.00524207 + 0 = 0.89723.

j. Wilks – This is Wilk’s Lambda. This can be interpreted as the proportion of the variance in the outcomes that is not explained by an effect. To calculate Wilks’ Lambda, for each eigenvalue, calculate 1/(1 + the eigenvalue), then find the product of these ratios. So in this example, you would first calculate 1/(1+0.8919879) = 0.5285446, 1/(1+0.00524207) = 0.9947853, and 1/(1+0)=1. Then multiply 0.5285446 * 0.9947853 * 1 = 0.52579.

k. Roys – This is Roy’s Largest Root. Note there are two definitions of Roy’s Largest Root depending on whether you use SPSS MANOVA or SPSS GLM. In MANOVA, the calculation is 0.8919879/(1+0.8919879) =  0.4714544. In GLM, Roy’s Largest root is defined as the largest eigenvalue, which is 0.8919879. Based on this definition, Roy’s Largest Root can behave differently from the other three test statistics.  In instances where the other three are not significant and Roy’s is significant, the effect should be considered not significant. You can reference the following article for a detailed description of this discrepancy:

Kuhfeld, W. F. (1986). A note on Roy’s largest root.  Psychometrika ,  51 (3), 479-481.

l. Note – This indicates that the F statistic for Wilk’s Lambda was calculated exactly. For the other test statistics, the F values are approximate (as indicated by the column heading).

m. Eigenvalues and Canonical Correlations – This section of output provides the eigenvalues from the product of the sum-of-squares matrix of the model and the sum-of-squares matrix of the errors. There is one eigenvalue for each of the three eigenvectors of the product of the model sum of squares matrix and the error sum of squares matrix, a 3×3 matrix. Because only two are listed here, we can assume the third eigenvalue is zero. These values can be used to calculate the four multivariate test statistics.

n. Univariate F-tests – The manova procedure provides both univariate and multivariate output. This section of output provides summarized output from a one-way anova for each of the outcomes in the manova. Each row corresponds to a different one-way anova, one for each dependent variable in the manova.  While the manova tested a single hypothesis, each line in this output corresponds to a test of a different hypothesis.  Generally, if your manova suggests that an effect is significant, you would expect at least one of these one-way anova tests to indicate that the effect is significant on a single outcome.

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8.3 - test statistics for manova.

SAS uses four different test statistics based on the MANOVA table:

\(\Lambda^* = \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\)

Here, the determinant of the error sums of squares and cross-products matrix E is divided by the determinant of the total sum of squares and cross-products matrix T = H + E . If H is large relative to E , then | H + E | will be large relative to | E |. Thus, we will reject the null hypothesis if Wilks lambda is small (close to zero).

\(T^2_0 = trace(\mathbf{HE}^{-1})\)

Here, we are multiplying H by the inverse of E ; then we take the trace of the resulting matrix. If H is large relative to E , then the Hotelling-Lawley trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

\(V = trace(\mathbf{H(H+E)^{-1}})\)

Here, we are multiplying H by the inverse of the total sum of squares and cross products matrix T = H + E . If H is large relative to E , then the Pillai trace will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

Here, we multiply H by the inverse of E and then compute the largest eigenvalue of the resulting matrix. If H is large relative to E , then Roy's root will take a large value. Thus, we will reject the null hypothesis if this test statistic is large.

Recall: The trace of a p x p matrix

\(\mathbf{A} = \left(\begin{array}{cccc}a_{11} & a_{12} & \dots & a_{1p}\\ a_{21} & a_{22} & \dots & a_{2p} \\ \vdots & \vdots & & \vdots \\ a_{p1} & a_{p2} & \dots & a_{pp}\end{array}\right)\)

is equal to

\(trace(\mathbf{A}) = \sum_{i=1}^{p}a_{ii}\)

Statistical tables are not available for the above test statistics. However, each of the above test statistics has an F approximation: The following details the F approximations for Wilks lambda. Details for all four F approximations can be found on the SAS website .

1. Wilks Lambda

\begin{align} \text{Starting with }&& \Lambda^* &= \dfrac{|\mathbf{E}|}{|\mathbf{H+E}|}\\  \text{Let, }&& a &= N-g - \dfrac{p-g+2}{2},\\ &&\text{} b &= \left\{\begin{array}{ll} \sqrt{\frac{p^2(g-1)^2-4}{p^2+(g-1)^2-5}}; &\text{if } p^2 + (g-1)^2-5 > 0\\ 1; & \text{if } p^2 + (g-1)^2-5 \le 0 \end{array}\right. \\ \text{and}&& c &= \dfrac{p(g-1)-2}{2} \\ \text{Then}&& F &= \left(\dfrac{1-\Lambda^{1/b}}{\Lambda^{1/b}}\right)\left(\dfrac{ab-c}{p(g-1)}\right) \overset{\cdot}{\sim} F_{p(g-1), ab-c} \\ \text{Under}&& H_{o} \end{align}

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MANOVA hypothesis

If we fail to reject the null hypothesis of MANOVA at an alpha level, does that mean that all the underlying univariate ANOVA tests will not be significant as well at the same alpha level?

• $\begingroup$ Did you really mean to include "not" in "not be significant"? $\endgroup$ –  whuber ♦ Commented Jun 2, 2015 at 19:20
• $\begingroup$ Yeah, my apology $\endgroup$ –  Tommy Yu Commented Jun 2, 2015 at 19:54

2 Answers 2

Not necessarily. But if you're planning to do each individual ANOVA anyway (presumably adjusting for multiple comparisons), why bother with the MANOVA?

If you fail to reject the null hypothesis of MANOVA at some alpha level, then you can report your result as " There is no statistically significant difference ..." And you do not need any follow up tests.

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