Here we will see how to deal with the sphericity problem by using SPSS instead of StatView. 

 

But first, to review that we have a problem in StatView, use the file 3-level.svd, which has one factor with 3 levels organized as a compact variable to do a RM ANOVA in StatView.  The summary table pasted here shows a significant (p<.05) main effect of this factor:

 

 

BUT: This is in fact Max & Onghena’s sample data, and as we know, 3 levels of a factor means a possible sphericity violation (violation of “equality of variances of differences between all treatment pairs”).  So, again reviewing, we can test for such a violation in StatView, which says it’s OK (there’s no significant correlation: 

 

 

BUT: in fact these data do have a sphericity problem – Max &Onghena constructed this dataset precisely to come out significant by an uncorrected RM ANOVA, but not significant by one that corrects the df for correlations – so here we can see that this test in StatView is not doing what it’s supposed to.  This underscores M&O’s claim that tests for sphericity violations themselves aren’t reliable.(Possibly the reason this test fails is the small amount of data).

 

Finally we can use StatView to run a MANOVA and look at Wilks’ lambda: the result is not close to significant:

 

 

This is too conservative.  Had this result been significant, we would have been satisfied that it was not because of overestimating the df; but since it is not, we still don’t know where we stand.  But we have exhausted the analysis possibilities in StatView.  At this point we have no choice but to leave StatView, so as to get a valid estimate of the problem, and correct the analysis procedure accordingly. 

 

We want corrected df:  epsilon describes the extent of the violation, and epsilon-hat is the estimate of that parameter; when it < 1, it is used directly to adjust the df (e.g. if uncorrected df = 4 and epsilon = .75, then corrected df = 3).  The basic method is due to Box (1954), so is called the Box adjustment.  Huynh & Feldt (in the 70s) proposed a variation that is supposed to correct for some bias in calculating epsilon, and this is what we use.  (Geisser and Greenhouse 1958 provided a formerly popular rule-of-thumb procedure which is overly conservative but simple.  The most extreme, and computationally simplest, correction is the lower bound.  But nowadays since a computer is doing all the work for us, there’s no reason to use a simpler but less exact method.)

 

How to do RM ANOVA in SPSS (with corrected df):

1. Data set up – like an expanded compact file (one row per subject, one column per condition) but with only one row of column labels and no subject label column – see file 3-level.xls (can compare this with .svd version).  Have this as an Excel file. If you don’t already have an earlier Excel version, first “expand” the compact variable in StatView before saving in Excel format, then in Excel collapse label rows and then save again.

 

2. Open your Excel file in SPSS, checking Read variable names.  (Do not try to open a StatView file – there is an import wizard in SPSS, but it wants text files).  You are in the Data Editor; if there is a problem with your file you will be jumped to the Viewer.  Just try again.

 

3. Choose Analyze – GLM – repeated measures.  (Note that just as in StatView, RM is used for pure RM, and for mixed designs.)

 

4. Whereas in StatView you first set up the file in a way that makes explicit the structure of the data (the compact variable), in SPSS you do this as part of launching the analysis – name your factors and say how many levels in each, then define them as conditions for the analysis:

 

type a factor name and its number of levels, click ADD

repeat until all factors are named

click DEFINE (brings up new window)

on left, see all your column labels

on right, see your declared data structure

send column labels over into data structure

clicking OK runs the analysis

 

5. Results: the first box echoes data structure so you can confirm you did it right:

 

Within-Subjects Factors

Measure: MEASURE_1 
 

FACTOR1	Dependent Variable
1
COND1
2
COND2
3
COND3

 

then the MANOVA results as from StatView:

 

Multivariate Tests
 

Effect		Value	F	Hypothesis df	Error df	Sig.
FACTOR1
Pillai's Trace	.615	2.401	2.000	3.000	.238
Wilks' Lambda	.385	2.401	2.000	3.000	.238
Hotelling's Trace	1.601	2.401	2.000	3.000	.238
Roy's Largest Root	1.601	2.401	2.000	3.000	.238

aExact statistic

bDesign: InterceptWithin Subjects Design: FACTOR1

 

then a different test of sphericity (still useless):

 

Mauchly's Test of Sphericity

Measure: MEASURE_1 
 

	Mauchly's W	Approx. Chi-Square	df	Sig.	Epsilon
Within Subjects Effect
				Greenhouse-Geisser	Huynh-Feldt	Lower-bound
FACTOR1
.424	2.578	2	.273	.634	.795	.500

Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.

aMay be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of Within-Subjects Effects table.

bDesign: InterceptWithin Subjects Design: FACTOR1

 

then the crucial bit: “tests of within-subjects effects”: without (“sphericity assumed”) vs. with correction (“Huynh-Feldt”) = significant vs. not

 

Tests of Within-Subjects Effects

Measure: MEASURE_1 
 

Source		Type III Sum of Squares	df	Mean Square	F	Sig.
FACTOR1
Sphericity Assumed	928.533	2	464.267	4.725	.044
Greenhouse-Geisser	928.533	1.269	731.910	4.725	.077
Huynh-Feldt	928.533	1.590	583.935	4.725	.060
Lower-bound	928.533	1.000	928.533	4.725	.095
Error(FACTOR1)
Sphericity Assumed	786.133	8	98.267
Greenhouse-Geisser	786.133	5.075	154.916
Huynh-Feldt	786.133	6.361	123.596
Lower-bound	786.133	4.000	196.533

 

Note everything shown here that we want can be done in StatView EXCEPT the crucial, corrected-df, analyses.

 

This same example is shown on a better-looking page from our statistics consultant at OAC.

 

To practice on your own: use the file M&O-2.xls, which has 2 factors each with 3 levels.  It is already set up to have only one row of labels.  This is the second sample dataset used by Max&Onghena, and the results of the analysis can also be seen on the OAC demonstration page.  To analyze 2 factors, add the second factor after the first, then define them both.

Repeated from the earlier section on Repeated Measures: One reason students sometimes avoid Repeated Measures analyses is that there is no automatic option for post-hoc tests.  See Hays section 13.25 (p. 579-583) about using Scheffe and Tukey HSD procedures or Bonferroni t-tests for post-hoc testing of within-subject factors - including the use of the corrected df; see Winer p. 529 about using  a factorial 1-way ANOVA for testing simple effects (a comparison of levels of one factor to a single level of another factor).