Multivariate Analysis of Variance (MANOVA)

In the univariate case, we extend the results of two-sample hypothesis testing of the means using the t-test to more than two random variables using analysis of variance (ANOVA). In the multivariate case we will now extend the results of two-sample hypothesis testing of the means using Hotelling’s T2 test to more than two random vectors using multivariate analysis of variance (MANOVA).

ANOVA is an analysis that deals with only one dependent variable. MANOVA extends ANOVA when multiple dependent variables need to be analyzed. It is especially useful when these dependent variables are correlated.

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61 Responses to Multivariate Analysis of Variance (MANOVA)

  1. Yvonne says:

    I am trying to figure out which analysis to use. I have 1 DV (dichotomous) ,3 IVs (test scores, which are all ratio), and three covariates (gender, actual age, and past offenses-categorical). I know MANCOVA will not work, as I only have one DV. I thought an ANCOVA would work, but does not because I do not meet the first three assumptions. Please help.

    • Charles says:

      Yvonne,
      Sorry but I don’t understand your scenario. You say that you would use ANCOVA except that the assumptions are not met, but let’s put this issue aside for a moment. To use ANCOVA you would have 3 categorical independent variables (which could be dichotomous) and one dependent variable which would be numeric (and not dichotomous).
      Please describe the situation better.
      Charles

  2. Maria says:

    Dear Charles,
    Could you please help me with the interpration of the following?

    1. If one, in order to have a Wilcoxon test with N=12, gives G-power the values “effect size = 0,8, a=0,05, power=0,8, it means that: they have a probablity of 80% to find statistical significant differences that really exist, only if these differences are big enough (large size effect). Do I get it right or not?

    2. I can have N=12 with two ways:
    (a) 1-tail: “effect size = 0,8, a=0,05, power=0,8”
    (b) 2-tails: “effect size = 0,95, a=0,05, power=0,8”
    What does 1-tail/2-tails mean and more importantly which might be ‘better/more acceptable’ or at least ‘less bad’: (a) or (b)?

    Thank you very much.

  3. JAIRO A. CLAVIJO says:

    Your work with Real Statistics seems to me excellent, monumental and very useful. I would like suggest you to include individual tests for comparing the means of each variable in the MANOVA. Thanks!

  4. Maria says:

    Dear Charles,
    G-power gives N=12 for a Wilcoxon, if I give these values:
    – Tail = 1
    – effect size=0,8
    – a = 0,05
    – power = 0,8
    What do you think about it? Could it be acceptable to procced this way or it does not have any meaning at all?

    Just grateful for your help,
    Maria

    • Charles says:

      Maria,
      If these are the values reported, then you need to make sure that a one-tailed test is appropriate and you only need to detect such a very large effect.
      Charles

  5. Maria says:

    Dear Charles,
    i used the G-power you recommended and I found out that the minimum sample needed for MANOVA in the case of having questionnaires with 6 dependent variables (=5 sub-scores of Likert questions groups+1 total score) and a unique independent variable (= a didactic intervention) is 42. (The values I gave to the G-power were alpha= 0.05, power=9.80, large size effect f=0.40. Since I am not familiar with this procedure, these values came from http://www.statisticssolutions.com/manova-2-levels-and-2-dependent-variables/).
    Moreover, I tried to estimate the minimun sample for Wilcoxon signed-ranked test, but G-power requires to provide values for parameters I don’t really know.

    So, i need to ask you this: is there any test one can do with just 12 pre-questionnaires and 12 post-questionnaires with Likert-type questions, in order to have a clue about whether their didadctic intervention had any effect on students’ scores?

    Thank you very much.

    • Charles says:

      Dear Maria,
      I am not surprised that you will need to have more than 12 elements to perform MANOVA with any power. It is not likely that another test which is suitable will require a sample as small as 12.
      Regarding the minimum sample for Wilcoxon signed-ranked test, to get at least some idea of the sample size requirement use G*Power (or the Real Statistics Power and Sample Size data analysis tool) for the equivalent paired t test. It shouldn’t be a lot different from the sample size requirement for the Wilcoxon test.
      Charles

      • Maria says:

        Thank you very much, Charles.
        In order to use G-power for estimatating the minimun sample for Wilcoxon, I need to give info about tails (options: 1, 2) and parent distribution (options: normal, laplace, logistic, minARE), but I don’t really know what to choose. So, even if I keep the “default values” for the other parameters (effect size=0.5, a=0.05, power=0.95), I don’t know what to do with tails and parent distribution. Could you please give me a clue? Thanks again.

        • Charles says:

          Maria,
          Generally you should choose the 2 tailed test. I would choose the normal distribution, but see how much the sample size changes if you try the other values. As I said in my previous response, I would find out the sample size required for the paired t test and assume that the sample size required is probably similar to that value.
          Charles

          • Maria says:

            Thank you very much, Charles. You are always very helpful.

          • Maria says:

            Dear Charles,
            Could you please tell me what you think about the following? Is it ok to procceed ths way?

            1. For Wilcoxon with 2 tails, parent distribution=normal, effect size=0,5, alpha=0,05, power (1-β error prob) = 0,3, G-power gives N=11. Is it ok to lower power so much? (0,95 became 0,3 in order to lower the sample at a convenient size).

            2. For t-test with 2 tails, effect size=0,5, alpha=0,05, power= 0,8, G-power gives a N=34. If I lower the power to 0,3, then N=11 as well.

            Thank you very much.

          • Charles says:

            Maria,
            Power of 30% is quite bad. It means that you have a high possibility of a type II error.
            The best you can do is to increase the effect size, which means that you test will only find very large effects (not great either).
            Charles

  6. Maria says:

    Dear Charles,
    What is the minimum number of questiionnaires one should have in order to perform MANOVA? Can they do the test with 12 pre/post questionnaires in case they have (a) one categorical independent variable (didactic intervention) with 2 occasions of measumeremnt (before and after), and (b) six dependent variables (the scores of the 5 sub-scales of the questionnaire that includes Likert-questions and the total score)?

    Thank you very much for your time,
    Maria

    • Charles says:

      Maria,
      I have not yet added the sample size requirements for ANOVA to the Real Statistics Resource Pack, but you can use the excellent free tool G*Power to calculate the sample size requirements for MANOVA. See the website
      http://www.gpower.hhu.de/en.html
      Charles

      • Maria says:

        Thank you very much, Charles. And something more. In the case I described before, namely if someone has 12 pre and 12 post- questionnaires with 6 dependent variables (=5 sub-scores of Likert questions groups+1 total score) and the unique independent variable is a didactic intervention, would it be a good idea to perform a Wilcoxon test (a) for each sub-score before and afer, (b) for the total score before and after, instead of performing MANOVA?
        Thank you very much once more.

        • Charles says:

          Maria,
          If there is a fair amount of correlation between the dependent variables, then MANOVA is generally preferred, but if there is little correlation then there isn’t any reason to do MANOVA. In this case you could perform a paired t test or if the assumptions for a t test are not met, then Wilcoxon’s signed ranks test.
          One other thing, you look at the following webpage since you seem to have a repeated measures test:
          Multivariate Repeated Measures
          Charles

  7. Juliana says:

    Hi Charles,

    I know I have to use multivariate regression with my 1 dependent variables and 4 independent variables. All variables are correlated with the minimum number 0.3. Some says i need to use discriminate function analysis to analyze it completely. But i dont even familiar with it. I read that manova is the reverse of it. Can u advise me wether i can benefit manova from it? Thank you

    • Charles says:

      Hi Juliana,
      If you only have 1 dependent variable then you don’t need to use MANOVA. The correct tool depends on the specific questions you want to answer or hypotheses you want to test.
      Charles

      • Juliana says:

        Thanks Charles,

        I need to create model/equation from it. Do you think multiple regression is enough? Adjusted R 0.834, F and P value less than 0.05. It looks ok right? ????????. Thank you in advance sir for your kind help.

        • Charles says:

          Juliana,
          I don’t have enough information to say for sure, but multiple regression may indeed be the way to go.
          Charles

  8. Learner says:

    Hello Professor !

    My objective is to study the impact of gender and income group (2 categorical predictors) on 5 continuous variables (factors obtained by PCA).

    I am using two-way MANOVA (using SPSS) for the same.

    I obtained statistical significance (p-value alpha.

    Interpretation:
    Factors differ significantly for different income group, while there is no significant difference between Factor for male and female.

    Am I using the right test?
    Is my interpretation correct?
    Please suggest. Thanks

    • Learner says:

      I realized that some how I missed some text in my earlier comment:

      I obtained statistical significance (p-value alpha).

      So for income group, what kind of post-hoc test I should apply? [If my approach and interpretation is correct]

    • Charles says:

      Learner,
      Since you have multiple dependent variables and multiple categorical independent variables, MANOVA seems to be a reasonable test to use.
      Without seeing your results, I cannot comment on whether your interpretation is correct.
      Charles

      • Learner says:

        Thanks a ton for your response.

        I was just worried about applying 2-way MANOVA as my response variables are uncorrelated (as, I’ve obtained them from PCA).
        Is this a problem? If yes, What are the alternatives for multiple response and predictor variables. Please suggest.

        I realized that there is some typo in my text again [I’ve written the results obtained by me]. I’ll try to learn and ask again, in case of any difficulty regarding interpretation.

        • Charles says:

          Learner,
          If the dependent variables are uncorrelated, then there really isn’t any point in using MANOVA. You might as well use separate ANOVAs (and/or their follow-up tests).
          Charles

  9. Irsalan says:

    Thank you for your article. Ho would I do poat hoc tests such as tukey tests on a Manova.

    • Charles says:

      Irsalan,

      I show two types of follow up tests on the website: ANOVA and contrasts. The contrasts case is described on the webpage
      MANOVA Follow up using Contrasts

      I believe that you can modify the contrasts approach to create Tukey’s HSD test in the same manner as you do as a follow up to ANOVA.

      Charles

  10. PVP says:

    Dear Sir,
    I am trying to analyze and use post hoc test to find if there is a significant differences in the performances of four different multi class( three different classes) classifiers for single data set. From the literature, I understand an ANOVA followed by post will differentiate the classifiers. Which ANOVA should I use? Kindly help me.
    ?

    • Charles says:

      I assume that you are referring to follow up tests after MANOVA.

      You can read more about ANOVA follow-up on the webpage
      http://www.real-statistics.com/multivariate-statistics/multivariate-analysis-of-variance-manova/manova-follow-up-anova/

      but often it is best to use a different sort of post hoc test. I give an example of this on the following webpage:
      http://www.real-statistics.com/multivariate-statistics/multivariate-analysis-of-variance-manova/manova-follow-up-contrasts/

      There are other options that I don’t (yet) discuss on the website.

      Charles

      • PVP says:

        Dear Sir,
        Thank you very much for the reply.

        • PVP says:

          Dear Sir,
          I repeat my question once again. I am comparing the performance of three different classifiers on a single data set and i have derived 4 performance metrics for each classifier(accuracy,precision,recall and specificity). I want to analyze is there a significant difference between the classifiers. From the literature, I understand an ANOVA followed by post hoc test will differentiate the classifiers levels. Also i understand that i have to perform analysis with repeated measures ANOVA. Kindly help to find a conclusion as to which ANOVA should i perform- One way, Two way or the MANOVA(suggested reply for my previous query). Your suggestion is very important.

          • Learner says:

            @PVP,

            Your response variables are four performance measures, say, accuracy,precision,recall and specificity (all are continuous).
            Explanatory variable is: Classifier (three types csay, LDA, QDA & KNN)

            It’s the problem of multivariate regression, in particular one-way MANOVA. Make sure to check for the assumptions before applying MANOVA.

          • Charles says:

            PVP,
            I see that Learner responded to your question. Do you now have the information you need?
            Charles

  11. Selvan says:

    Kindly suggest me best statistical method based on the following details.

    I want to analyze distribution of mosquitoes in six different locations based on the physico-chemical parameters, climatic variabilities, spatio and temporal variabilities.

  12. Kayleigh says:

    Hi Charles,

    I want to compare change in tree volume with change in elevation. I also want to test if tree species is a factor, and if maybe a certain species is affected more at a specific elevation. I have further (possibly co-variates) i want to add in, but for now i want to run that analysis. Help please!

    Thanks

    • Charles says:

      Based on your description, you only need to use a two factor Anova. One factor is Species and the other factor is Elevation (with levels 0-1,000 meters, 1001-2000 meters, etc.). The dependent variable is tree Volume. If you have other covariates, these become additional factors

      You could turn this into a regression problem based on Species as categorical independent variables, Elevation independent variable and Volume as the dependent variable.

      Charles

      • Kayleigh says:

        Thank you!

        I have tried to run each of these but I get different error messages. I divided my elevations into three categories and then averaged the volumes for each of the eight tree species for these to create the matrix like in your example. Not all species occured at each level (and not all levels have the same n), therefore I have blanks and my range of data is 0-1 (0 means no change in species volume, 1 complete change due to frost).

        To run it as a regression, how do I say that the species variable will be an additional categorical independent variable?

        • Charles says:

          Kayleigh,

          If I understand correctly, the problem when running MANOVA is that you have some missing data. You have two choices for resolving this:

          (1) Remove any missing data. In order to keep the sample sizes equal, you might have to further remove samples randomly. If you have more than a few missing data elements, this will likely take a way lots of your data, thereby reducing the power of the test.
          (2) impute values for the missing data elements. Various techniques for doing this are described on the Real Statistics website.

          If you are only missing a relatively few data elements then either technique should be acceptable. If you are missing lots of data, then neither approach may be acceptable.

          To use a regression, you can make the species variable an additional categorical independent variable.

          Charles

  13. Manasi says:

    I have a fatigue questionnaire in my experiment with constructs like alertness, concentration difficulty, irritability etc. 15 subjects filled it out at four different times (T1, T2, T3 and T4) on two different days (i.e. Day 1 and Day 2). I am trying to see if there is any difference between the ratings at the four different times. Which ANOVA should I be using? Can you please help.

  14. abdul says:

    sir kindly guide me about MANOVA and multiple regression. is they same or not?

    • Charles says:

      Abdul,
      MANOVA and multiple regression are not the same. MANOVA is like ANOVA (which can have multiple independent variables, but only one dependent variable) but MANOVA can support multiple dependent variables. Multiple regression is related to ANOVA and it too supports multiple independent variables but only one dependent variable. Analogous to MANOVA is multivariate multiple regression.
      Charles

  15. MOHD AMIN WANI says:

    Quite useful for the research purpose.Hope to get benefits in the upcoming future.

  16. Getash says:

    Hello, sir
    I wonder if I could get the right way to calculate multiple regression with two dependent variables. Does MANOVA substitute it?

    • Charles says:

      Getash,
      Multivariate regression is the tool that performs multiple regression with multiple dependent variables. I plan to add this to the Real Statistics Resource Pack and website shortly, probably sometime in January.
      Charles

  17. amk says:

    verily i can not get where Manova is even located in the Ms exceel

    • Charles says:

      Excel does not have a MANOVA function or data analysis tool. If you install the Real Statistics Resource Pack you will be able to get access to various MANOVA functions and the Single Factor MANOVA data analysis tool. Installation is free.
      Charles

  18. Kevin says:

    Do you also have or are planning a function that can do a one-way (or n-way) repeated measures MANOVA? I have a 3 sets (on 1 factor, IV with 3 levels) of bivariate responses (DV) from a group of observers that have repeated each the experiment multiple times, and I would like to check whether the means of the bivariate responses are significantly different.
    I’ve done a two-way manova (IV1=3 sets, IV2=observer) on my data using matlab, as I couldn’t find any repeated measures manova function for matlab. However, I’m not quite sure that this would be the correct way to do things. Doing a one-way manova on the individual means of the observer data was another attempt, but I think I loose valuable information that way, as the intra-observer variability, which is quite large, will not be taken into account in this way.
    When searching the net for a potential solution I came across your website and I really liked that it’s very easy to have all the analysis results in one place and that it’s very easy to use. So I was hoping to be able to do one-way (and perhaps in the future n-way) repeated measures manova.
    Hope you can help out??

    kind regards,
    Kevin

    • Charles says:

      Kevin,
      I plan to provide additional multivariate statistics support in a future release, including additional Manova capabilities, but this won’t be part of the next release. Stay tuned.
      Charles

  19. SUNITA SINGHAL . says:

    Sir, Thank you for providing such a great tool. I am not from mathematics. I want to use function in your toll that is equivalent to multicompare function of MatLab. multicompare function is used to test statistical significance .
    Awaiting your reply.

    • Charles says:

      Sunita,
      The Real Statistics software contains most of the features of this function but in separate tests. In particular, you can look at the Tukey HSD, Tukey-Kramer, Scheffe, Bonferonni and Dunn-Sidak options of the One Factor ANOVA data analysis tool. These tests can also be used with two-factor ANOVA.
      Charles

  20. karan says:

    Kindly suggest me how to perform MANOVA in excel

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