MANOVA Effect Size

As for ANOVA, the partial eta-squared η2 can be used as a measure of effect size for MANOVA.

This statistic is calculated by

partial η2 = \frac{df_1 \cdot F}{df_1 \cdot F + df_2}

which is equivalent to the following, where b and s are as in Property 4 and 5 of Manova Basic Concepts.

Wilks Lambda: 1 – Λ1/b

Hotelling-Lawley Trace: \frac{T_0^2/s}{T_0^2/s+1}

Pillai-Bartlett Trace: V/s

For Example 1 of Manova Basic Concepts, these values were calculated by the Real Statistics Manova data analysis tool in range L6:L8 in Figure 2 of Real Statistics Support for Manova.

35 thoughts on “MANOVA Effect Size”

  1. Hi Charles,
    I am using g power to calculate my sample size for a MANOVA with two groups and two dependent variables.
    I have used alpha .05, power .80 and effect size .14.
    Is this correct? Sorry I am useless with statistics, any help is much appreciated thank you!

    Reply

    • Katie,
      Alpha of .05 and power of .80 are commonly used, although power = .90 may be better.
      As far as the effect size is concerned, this is trickier. This really depends on what hypothesis you are trying to test and your expectation as to the size of the effect that you are trying to detect. The smaller the effect, the larger the sample size needs to be. If you make the effect size unrealistically high then the sample size might be easier to obtain (because it is smaller), but you may not be able to detect the effect size that you really need.
      Charles

      Reply

  2. hy charles…i want to learn abaout effect size on manova..but i can used it..can i ask formula table calculator to find Effect size using table excel ..thank you

    Reply

  4. Hi Charles
    I found your site now… can you tell me if in a MANOVA of three independent factors the sum of the effect sizes has to be equal to one, I have 16 dependent variables… thaks!

    Reply

  5. Hi,

    Does anyone know a reference for the partial eta squared effect sizes being small , medium, large (.01/.06/.14 respectively).

    For example did Cohen (1988) state this as many other effect sizes use this paper as a reference?
    Cohen, J. (1988). Statistical power analysis for the behavioral sciences. Abingdon. England: Routledge.

    Thanks

    Reply

      • Great, thanks Charles!

        I have performed a MANOVA so will be taking partial eta squared values as my effect size. My post-hoc analysis is individual ANOVA in which the table is also presenting partial eta squared, is it correct for me to take partial eta squared as an effective size for the ANOVA’s too and would the same values apply as to what classifies small medium large (.01, .06, .14)?

        Thanks!

        Reply

  6. Hi Charles,

    What are the “small medium large” values for the partial eta-square effect size?

    Thanks in advance!

    Regards,

    Hugo.

    Reply

  7. Hello Charles,
    I am not a statistician and maybe my question is basic but I would like to use data on studies using MANOVA for a Forest Plot.
    I will be using partial eta-squared η2 for the effect size, can you confirm with me what will then be the standard error to be used?
    Many thanks for your website.

    Reply

  8. Hi Charles. I’ve just discovered your website. Thanks for all the great information.

    One question about effect size. In the second equation above (part. eta^2=1-lambda^b), SPSS seems to use min(k, m-1) in place of the formula for “b” shown in your page, MANOVA Basic Concepts.

    Are you familiar with SPSS’s method for computing part. eta^2?

    Thanks!
    Dave

    Reply

  9. sorry it should look like this

    within——-between——-statistic——-value———-interpretation
    intercept—-intercept——————————————none
    intercept——age——————————————–effect of age alone
    frequency—intercept—————————————effect of frequency alone
    frequency—–age——————————————-effect of age*freq interaction
    etc etc

    Reply

    • When using the MANOVA_POWER function for a ONE WAY MANOVA:
      1. What is iter ? ? does it mean iterations =1 if not a repeated measures experiment?
      2. What is “prec”
      3. For effect size, I can’t find definitions of Pillai’s V or ttype
      4. When using the Power and Sample Size data analysis tool for a ONE WAY MANOVA, the drop down menu lists “Sum Count” as 1000. What is “Sum Count” and when should I change the default entry of 1000?

      Reply

      • Hello Joel,
        1. The power calculation uses the noncentral F distribution. The values from this distribution can’t be obtained using a simple formula; instead it is calculated using an iterative approach. The more iterations, the more accurate the answer (although at some point additional iterations may not generate any further accuracy). iter = the number of iterations used; where 1000 is the default. This should be more than enough for anything you are apt to do and so you can keep the default. If you have any doubt, simply increase the value of iter to some higher number and see whether the output changes; in the unlikely event that it does change, then you should experiment by using higher and higher values until no change is detected. Note that the more iterations the slower the calculations, and so iter = 1,000 is chosen since it is high enough to get an accurate answer but low enough so that the calculations are fast.
        2. In general, the more iterations, the more accurate (i.e. precise) the result. However, at some point additional iterations don’t add enough precision to make the slower calculation time worthwhile. prec is a parameter that specifies the desired level of precision. You can think of of it as the number of significant decimals in the result (e.g. 1.276 is more precise than 1.3) Essentially in an iterative calculation, the iterations continue until either the maximum number of iterations (i.e. iter) is reached or until the desired level of precision (i.e. prec) is reached, at which point the iteration stops even if the maximum number of iteration has not been reached. The default precision values should be sufficient for all your needs.
        3. ttype is defined at MANOVA Power and Sample Size. Namely, if ttype = 1 then effect size = partial eta-square; if ttype = 1 then effect size = eta-square; if ttype = 3 then effect size = Pillai’s V. Pillai’s V is defined at MANOVA Basic Concepts
        4. Sum Count is the same as iter. I originally used “sum count” since the iteration is really an infinite sum, but after adding a certain number of terms, the sum doesn’t change enough to continue adding more terms. I will shortly change this to Iterations for consistency.
        Charles

        Reply

        • Thanks very much. Your web site is fantastic in both theory and practical examples. The practice files are great learning tools too.
          Joel

          Reply

  10. Correction: the constant (intercept term in the model) is listed as a variable in both the within-subject and the between-subject factor columns in my manova table.

    Within Between Statistic F-value p-value (my interpretation)
    intercept Intercept none
    intercept age effect of age alone
    frequency intercept effect of frequency alone
    frequency age effect of frequency*age
    … …

    Is my interpretation correct?

    Reply

  11. Hello I’m hoping for some clarification on MANOVA when there are both within-subject and between-subject factors. Can I assess the effect size of my within-subject factors separately from my between-subject factors?

    In my experiment, the dependent variable is complex (i.e. bivariate and continuous); there are three independent variables: sample age (this is a between-subject factor, as each sample is only measured at one age), and amplitude and frequency (these are within-subject factors, as each sample is measured at all (3 levels) amplitudes and all (16 levels) frequencies.

    In MATLAB, I have created a table similar to the one in your example on soil types. I have samples measured in triplicate at 7 different ages (3×7=21 rows), and the within-subject factors (3 amplitudes x 16 frequencies x bivariate = 96 columns). I then use fitrm() and perform MANOVA on the repeatedmeasuresmodel object that it produces, which gives a table of the test statistics. In addition to my between-subject factor, sample age, the default is to also add a constant to the tested model, which in the table is listed similarly to sample age as a between subject group. The table has Pillai, Wilks, Hotelling and Roy F and p values for all combinations of my within-subject factors (and their interactions) with the two between subject factors (sample age and “constant”). For example:

    Constant , Frequency , Pillai p
    ….
    Age, Frequency*Strain, Pillai p

    How am I to assess the effect size of my within-subject factors separately from my between-subject factors? Specifically, I want to answer whether sample age has a statistically significant effect on my bivariate data irrespective of amplitude and frequency.

    Many thanks in advance!
    -R

    Reply

    • Rory,
      I plan to add some of the MANOVA capabilities you are asking about in the future, but trying to address your questions now would take too much time and would delay some of the things I am working on now. Sorry, but I will need to come back to your questions at some future time.
      Charles

      Reply

      • Hey Charles,

        I was wondering if you ever followed up on this? Any help would be greatly appreciated re: effect sizes for within or between groups in MANOVA. The study design I have is looking at within groups (within the year 2010 group, with 2 groups of participants [level 1 or 2 of a program]; and within the year 2012 group, with 2 groups of participants [level 1 or 2 of a program]). I am unsure as to whether it needs to examine effect sizes (eta-squared) at the between groups level too, or whether I should just analyse within-groups, obtaining eta-squared for both and comparing. Does that make sense?
        Any help would be appreciated beyond words.

        Cheers,

        Laura

        Reply

        • Laura,
          I added effect size for the MANOVA omnibus test. Regarding the effect size for follow up tests, you need to look at the webpages for the follow up tests: ANOVA follow up and Contrasts follow up. I need to add a Tukey HSD follow up test, but the Contrasts follow should indicate how this would work..
          Charles

          Reply

    • Hey Rory,

      I have a similar situation (looking at within groups and between groups) and was wondering if you found the answer you were after! I’ve been asked to find “whether eta-squared/effect sizes in MANOVA are for within groups or between groups”, or which eta-squared corresponds to which grouping. Does that make sense?

      Please let me know if you found anything – I’ve trawled through articles for hours now and still unsure!

      Cheers,

      Laura

      Reply

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