Interpreting Mauchly’s test results to assess sphericity assumption in repeated measures analysis

I am having some issues with interpreting sphericity in repeated measures test result in MATLAB.

Take a look at this MATLAB example:

load repeatedmeas

rm = fitrm(between, 'y1-y8 ~ Group*Gender+Age+IQ', 'WithinDesign',within)

[TBL,A,C,D] = ranova(rm)

In SPSS, you can run Mauchly’s test which returns a test result for each term in the model i.e Group*Gender, Age and IQ. This allow you to use the correct p-value for each item in the model.

PART 1) In MATLAB you can do:

rm.mauchly()
% or 
rm.mauchly(C)

which seems to be the same and return only one test results contained in a single table row. Should this single test result be used to assess the sphericity status of all the terms in the model together?

PART 2) Additionally, one can also do this:

[TBL,A,C,D] = ranova(rm, 'WithinModel','w1+w2')
G=rm.mauchly(C)

Now we have multiple Mauchly’s test results. My question is how do I interpret each row of G with respect to the model?

Is this the correct contrast matrix here? I ask because C does not look like pairewise differences between y1,y2…y8, which Maucly's test should be performed on.

In summary is PART 1 sufficient or do I need PART 2?

Any help will be appreciated.

Answer 1

[Shubh Dhyani]

Hi Bethel,

I understand that you are having some issues in interpreting sphericity in repeated measures test result in MATLAB.

PART 1:

1. Mauchly's Test:

Mauchly's test of sphericity is used to assess the assumption of sphericity in a repeated measures ANOVA. This assumption states that the variances of the differences between all combinations of related groups (levels) are equal.

2. Using rm.mauchly() in MATLAB:

When you run rm.mauchly() without specifying any contrast matrix, MATLAB returns the Mauchly's test result for the default contrast matrix, which is usually the identity matrix. This test will assess the sphericity of the within-subjects factor across all its levels.

3. Interpretation of the Default Mauchly's Test:

The default Mauchly's test in MATLAB (as produced by rm.mauchly() or rm.mauchly(C)) assesses the sphericity of the entire within-subjects factor. It doesn't test sphericity for each term in the model separately. So, if the p-value from this test is less than your significance threshold (typically 0.05), you would reject the null hypothesis of sphericity for the entire within-subjects factor.

PART 2:

1. Using a Custom Within-Subject Model:

When you specify a custom within-subject model using ranova(rm, 'WithinModel','w1+w2'), MATLAB returns ANOVA results for each of the within-subject terms (w1 and w2).

2. Using rm.mauchly(C) with a Custom Within-Subject Model:

Here, the test will assess the sphericity of the within-subjects factor for each term specified in the within-subject model. Each row in the returned table corresponds to a term in the within-subject model.

3. Interpreting the Contrast Matrix C:

The contrast matrix C provides contrasts for the within-subject factor. If you haven't specified custom contrasts, MATLAB will use the default contrasts, which may not be pairwise differences.

4. Interpretation of Multiple Mauchly's Test Results:

Each row of the Mauchly's test results corresponds to a term in the within-subject model. If the p-value for a row is less than your significance threshold, you would reject the null hypothesis of sphericity for that specific term.

Conclusion:

PART 1 gives you a general assessment of sphericity for the entire within-subjects factor. It's a broad-level test.

PART 2 allows for more granularity by testing sphericity for each term in the within-subject model.

Which approach to use depends on your research question and the level of detail you need. If you're only interested in the general sphericity of the entire factor, PART 1 is sufficient. If you want to know about sphericity for each term in the within-subject model, you would use PART 2. Often, both tests are performed to give a complete picture of the sphericity assumptions in the data. For more information, you can refer to the following link: https://www.mathworks.com/help/stats/repeatedmeasuresmodel.mauchly.html