# Mauchly's test of Sphericity not defined. Why?

#### ozaz

##### New Member
Hello,

With regard to Mauchly's test of Sphericity in order to assess validity of F statistic calculated for a one way repeated measures anova:

I understand why Mauchly's test of Sphericity has no meaning when there are are only 2 levels of a repeated measures factor.

However, I notice (using GLM.....repeated measures in SPSS) that Mauchly's test of Sphericity also appears to be undefined (or gives the useless output of Mauchly's W = '.' , p = '.') when the number of levels of a repeated measure is equal to or greater than the number of cases (subjects). In these instances, even though Mauchly's statistic is not calculated, Greenhouse-Geisser, Huynh-Feldt, and Lower-Bound Epsilon values are being calculated.

I would be really happy if someone could provide some insight on why Mauchly's statistic is not calculated in these cases and what should be done to assess sphericity in the absence of Mauchly's statistic.

I have noticed the question asked previously as part of other threads on this and other forums but I have never seen a response.

#### Masteras

##### TS Contributor
This situation perhaps arises due to singular matrices, i think. Anyway, if you cannot check sphericity, report the other three statistics you mentioned.

#### ozaz

##### New Member
So the fact that the statistic is not calculated in these instances does not mean that the assumption of sphericity should be discarded. It is rather a nuance of the maths involved in it's calculation (something I know nothing about). Is that correct?

If I need to resort to Greenhouse-Geisser, Huynh-Feldt, and Lower-Bound Epsilon values to judge sphericity, are there rules of thumb / cut off values that one may use to judge the assumption of sphericity?

#### jamesmartinn

##### Member
The rule of thumb my stats prof suggested was using the Huynh-Feldt rather than the Greenhouse-Geisser correction when elipson lies near or above .75. In the notes she doesn't give a rationale.

Also, IIRC, I don't think SPSS gives Mauchly's Sphericity statistic for repeated measure factors with <2 levels.

#### ozaz

##### New Member
@jamesmartinn,
Thanks for your input.

Actually, I was asking how to use the epsilon value/s to gauge whether correction of the F-statistic is even necessary (given that the Mauchly's statistic which I would normally use for this task is unavailable), rather than how to choose which correction method is more appropriate after it has been decided that correction is necessary.

I do understand that you don't get a sphericity statistic when levels of the repeated measure are not greater than two. However, the scenario I raised in my original post is different. The scenario I describe can be seen when the levels of the repeated measure are greater than two.

#### Masteras

##### TS Contributor
You do not look at the epsilons, you look at the p-values generated. When levels are >2 sphericity should be available. Are you saying that SPSS did not provide you an output in this case?

#### ozaz

##### New Member
Yes, I am talking about instances when levels > 2 (ie when I would expect to see output).

Maybe it is clearer if I provide an example...

Here is an experiment with 5 subjects (s1...s5) and 7 levels of a repeated measure (p1....p7) (data also available in the attachment data.txt)

subject p1 p2 p3 p4 p5 p6 p7
S1 122 129 134 134 132 130 120
S2 137 140 141 141 140 128 126
S3 123 123 130 129 130 130 129
S4 130 138 139 141 142 136 127
S5 126 130 141 143 141 139 133

I run glm....repeated measures. Please see output.pdf for the spss output. In the output for Mauchly's test, notice that Mauchly W = .000, approx chi square = . and p-value value = .

I don't understand why parts of the output for the Mauchly's test seem to be missing or not calculated, and how to assess the assumption of sphericity given that Mauchly's test p value is not there.

#### ozaz

##### New Member
In my original post I noted that Mauchly's test of Sphericity appears to be undefined when the number of levels of a repeated measure is equal to or greater than the number of cases.

Actually that statement is inaccurate. It now looks like I am only seeing this when the number of levels of a repeated measure is greater than the number of cases. However, I'm not completely sure if this is a general phenomena or some peculiarity with my data. Hopefully that will be clarified by someone after looking at my uploads in the previous post.

#### Masteras

##### TS Contributor
the message in the third table says: Cannot produce multivariate test statistics because of insufficient residual degrees
of freedom. that is the problem. the dfs in this case are negative!

#### valex

##### New Member
In my original post I noted that Mauchly's test of Sphericity appears to be undefined when the number of levels of a repeated measure is equal to or greater than the number of cases.

Actually that statement is inaccurate. It now looks like I am only seeing this when the number of levels of a repeated measure is greater than the number of cases. However, I'm not completely sure if this is a general phenomena or some peculiarity with my data. Hopefully that will be clarified by someone after looking at my uploads in the previous post.

Is than this true? When the number of levels of a repeated measure is greater than the number of cases, sphericity cannot be calculated? It seems also my problem, for some interaction effects, SPSS does not give me any value for the sphericity:
I have a factor with 4 level and another with 6 (I have 11 cases), although the degree of freedom are perfectly positive (119) no sphericity is given...
Why?? I don't get it!