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.

## 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.

feafeafeaf says

DVs aren’t supposed to be correlated

Jim Frost says

Hi, you’re thinking about

independentvariables. 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

dependentvariables. However, MANOVA is not only designed for that but there are benefits when they are correlated.Sophia says

Hi Jim,

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!

Best,

Sophia

Jim Frost says

Hi Sophia,

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!

Aba Merci says

Hi Jim,

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

Jim Frost says

Hi Aba,

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!

I hope this helps!

Joe Smith says

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!

Joe Smith says

correlated rather

Jim Frost says

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.

Tan says

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?

Thank you very much

Gemma says

Hi there,

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!

Jim Frost says

Hi Gemma,

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.

I hope this helps!

Esther says

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?

Jim Frost says

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.

I hope this helps!

Omotayo says

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.

Jim Frost says

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.

Jim

Omotayo says

Thanks for your kind response.

Pamela Marcum says

Hi Jim,

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.

Jim Frost says

Hi Pamela,

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!

Jim Frost says

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.

Sarah says

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

Jim Frost says

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.

Pamela Marcum says

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!

laura says

Hi Jim,

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

Jim Frost says

Hi Laura,

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.

I hope this helps!

Minie Bulay says

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?

Jim Frost says

Hi Minie,

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.

I hope this helps!

Meysam says

Hi Jim,

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?

Thank you

Ifedayo Adu says

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.

Jim Frost says

Hi Adu,

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!