# 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.

### 13 Responses to MANOVA Effect Size

1. Rory says:

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

• Charles says:

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

• Laura says:

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

• Charles says:

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

• Laura says:

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

2. Rory says:

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?

3. Rory says:

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

4. Dave says:

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

• Charles says:

Dave,
From what I can see in the SPSS documentation, s = min(k,m-1), not b.
Are you looking at the following webpage or some other?
http://www.ibm.com/support/knowledgecenter/SSLVMB_21.0.0/com.ibm.spss.statistics.help/alg_manova_ancova_effect-size_ci_multivariate.htm
Charles

• Dave says:

I am sorry, I may not have been clear in my initial question. What I meant to say was that whereas you show effect size computed as

partial η^2 = 1 – Λ^(1/b) ,

SPSS uses

(partial?) η^2 = 1 – Λ^(1/s) .

I.e., SPSS uses “s” in the equation where you use “b”. I actually discovered this while trying to replicate SPSS output by hand, but I eventually found the documentation here: http://www.ibm.com/support/knowledgecenter/SSLVMB_21.0.0/com.ibm.spss.statistics.help/alg_manova_ancova_effect-size_stat.htm.

Is this an error in the SPSS algorithm? Or might it be the difference between computing η^2 (which is what is shown in the SPSS documentation) and computing partial η^2? However, both should be the same for the 1-way design with which I was working. Any insight would be appreciated!
Best,
Dave

• Charles says:

Dave,
From the referenced webpage, it certainly seems that you are correct. I believe that I used the following, which gives a different result.
http://www.csun.edu/~ata20315/psy524/docs/Psy524%20lecture%2011%20MANOVA2.ppt
Now, I am confused as to which is correct.
Charles

• David Corey says:

Charles,
I am glad to see that I am not the only one confused. However, do note that, as you say in the webpage, computing partial η^2 as 1-Λ^(1/b) is in fact equivalent to the first formula you show, in which partial η^2 is computed from F and df. Do you happen to remember where you saw the latter formula applied to MANOVA? If it holds, then partial η^2=1-Λ^(1/b) must be the correct formulation (in which case, SPSS computation of partial η^2 would be incorrect).
Dave