Also known as withinsubjects design, theses tests are used when each subject is measured multiple times. Different treatments may applied to each subject over time, or to groups of subjects in a uniform way. Similar to paired ttests, these tests increase the power of the analysis by accounting for the idiosyncratic differences between subjects.
The following conditions make a study appropriate for repeatedmeasures ANOVA:
Questions which might be suitable for this type of analysis include: Does an experimental diet lead to better test performance of two groups of study animals? Which medium leads to the most proliferation in several cell lines over time? Do subjects improve their balance over time when given a sequence of experimental treatments?
Here we will use a real data set to ask whether different concentrations of a tree bark extract lead to different survival rates of termites. These data can be used to see if the tree bark compound would be suitable for development as an antitermite treatment.
Open Termites.xls (see the Data Appendix). This study has a "mixed design" or "twoway design with one repeated measure" in the terminology of Portney & Watkins, with two treatment levels applied to different blocks of subjects, and many measurements in time for each subject.
Go to Analyze > General Linear Models > Repeated Measures. The first dialog requires you to "define factors". Here we need to make a name two new objects, the WithinSubject Factor Name, which you can name by what is actually being assessed at each measure. In this case, it is the number of termites surviving. There are 13 measures in our data set (they skipped days 3 and 9). Second, you need to type in the Measure Name. This should just be the time units for the repeated measures, which in this case is day. Type in each, making sure to click "Add", then choose Define.
The next dialog shows all the 13 levels of the "survival" factor, named as "day". We want to match these up with the 13 columns of measurements we have. Select day1 to day15, and click the arrorw to move them into the WithinSubjects box. Then move dose into the BetweenSubjects Factors box.
It should now look like this:
In this example, we only have two doses. If we had more levels in this factor, we would want to examine the differences between each category using the Post Hoc dialog (Tukey).
To create a graph of the results, click Plots. Move dose (or whatever betweensubjects factor you have) into the Seperarte Lines box, and survival into the Horizontal Axis box.
Finally, choose Options, and at least click the Estimates of effect size and Homogeneity tests boxes.
Choose Continue, and then OK to run the test.
Before looking at the results, it is necessary to digress briefly to discuss the concept of sphericity.
Sphericity
In other parametric tests, we have been concerned with the normal distribution of data and homogeneity of variances. In a repeatedmeasures design, we are also concerned with equal correlations between the data at different time points; this is known in statistics as sphericity. This assumption considers the covariance between measurements.
If the sphericity assumption is violated, the chance of a Type I error (incorrectly rejecting the null hypothesis of no difference between groups) increases. This is a troubling outcome, and unfortunately difficult to resolve.
Alternatives include multivariate analyses of variance (MANOVA), which do not require sphericity. SPSS runs a MANOVA by default for a repeatedmeasures ANOVA, with the results in the Multivariate Tests table. There is rarely any major difference between them in terms of significance values, but if necessary to choose the appropriate test, consult a specialized text on multivariate statistics (e.g., Manly 2005).
SPSS performs two tests related to sphericity, Box's Test for Equality of Covariance Matrices and Mauchly's Test of Sphericity. Portney & Watkins provide a succinct description of Mauchly's test (p. 447).
If the result of the Mauchly test is significant (p ² 0.05), there is a significant violation of the assumption of sphericity. Therefore, we should correct the degrees of freedom when performing the ANOVA; SPSS does this automatically and notes it in a footnote beneath the Mauchly test table. The correction is called epsilon.
SPSS reports all possible significance values, using the different epsilon corrections. Here are the meanings of each of these:
Returning to the model results, we first see the multivariate analysis of variance tests. These test the effect of the withinsubject factor, survival, as if each measurement were a different variable; that is what makes this a multivariate test. The different flavors of MANOVA are all identical here, showing a significant effect of the day measured; this is not interesting or surprising, since we expect that termites will start dying off in the petri dishes quite naturally.
However, the next set of values, survival x dose, show no effect. This indicates that the survival of termites did not differ depending on the concentration of tree bark extract. This indicates that the tree bark extract would not be useful as an antitermite treatment. But this result should be treated very cautiously, since the multivariate test is less powerful than a repeatedmeasure ANOVA.
Multivariate Tests(b)
Effect 
Value 
F 
Hypothesis df 
Error df 
Sig. 
Partial Eta Squared 

survival 
Pillai's Trace 
.979 
11.535(a) 
12.00 
3.00 
.034 
.979 
Wilks' Lambda 
.021 
11.535(a) 
12.00 
3.00 
.034 
.979 

Hotelling's Trace 
46.139 
11.535(a) 
12.00 
3.00 
.034 
.979 

Roy's Largest Root 
46.139 
11.535(a) 
12.00 
3.00 
.034 
.979 

survival * dose 
Pillai's Trace 
.737 
.699(a) 
12.00 
3.00 
.717 
.737 
Wilks' Lambda 
.263 
.699(a) 
12.00 
3.00 
.717 
.737 

Hotelling's Trace 
2.798 
.699(a) 
12.00 
3.00 
.717 
.737 

Roy's Largest Root 
2.798 
.699(a) 
12.00 
3.00 
.717 
.737 
a Exact statistic
b Design: Intercept+dose
Within Subjects Design: survival
Next comes the results for the repeatedmeasures ANOVA. This requires that the covariance matrix of the data have "sphericity", as explained above. These data definitely do not; the covariances differ at different points in the experiment.
Mauchly's Test of Sphericity(b)
Measure: day
Within Subjects Effect 
Mauchly's W 
Approx. ChiSquare 
Df 
Sig. 
Epsilon(a) 

GreenhouseGeisser 
HuynhFeldt 
Lowerbound 

survival 
.000 
233.195 
77 
.000 
.172 
.216 
.083 
Tests the null hypothesis that the error covariance matrix of the orthonormalized transformed dependent variables is proportional to an identity matrix.
a May be used to adjust the degrees of freedom for the averaged tests of significance. Corrected tests are displayed in the Tests of WithinSubjects Effects table.
b Design: Intercept+dose
Within Subjects Design: survival
Therefore, when we look below at the withinsubject effects, we will look at them in the following order:
Tests of WithinSubjects Effects
Measure: day
Source 
Type III Sum of Squares 
df 
Mean Square 
F 
Sig. 
Partial Eta Squared 

survival 
Sphericity Assumed 
7130.356 
12 
594.196 
112.924 
.000 
.890 
GreenhouseGeisser 
7130.356 
2.059 
3463.254 
112.924 
.000 
.890 

HuynhFeldt 
7130.356 
2.591 
2751.769 
112.924 
.000 
.890 

Lowerbound 
7130.356 
1.000 
7130.356 
112.924 
.000 
.890 

survival * dose 
Sphericity Assumed 
561.952 
12 
46.829 
8.900 
.000 
.389 
GreenhouseGeisser 
561.952 
2.059 
272.943 
8.900 
.001 
.389 

HuynhFeldt 
561.952 
2.591 
216.870 
8.900 
.000 
.389 

Lowerbound 
561.952 
1.000 
561.952 
8.900 
.010 
.389 

Error(survival) 
Sphericity Assumed 
884.000 
168 
5.262 

GreenhouseGeisser 
884.000 
28.824 
30.669 

HuynhFeldt 
884.000 
36.277 
24.368 

Lowerbound 
884.000 
14.000 
63.143 
Notice that the Fratios for all of the test are the same for the two groups. Even though the sphericity assumption has not been supported, the corrections applied do not change the final story.
In particular, both "survival" (the day of measurement) and the interaction between day and dose are highly significant explanatory factors of the termite numbers. This differs from the MANOVA results, and since this is a more powerful test, we should focus just on the repeatedmeasures. The tree bark extract does have an effective antitermite compound.
Finally, examine the profile plot. This immediately explains the results: the higher dose of tree bark extract led to significantly lower termite suvival.
Other options
In the Repeated Measures dialog box, if you have multiple explanatory factors, you can choose which interactions to include in the model using the Model option.
This dialog also gives you the option to choose which type of sums of squares to use. This is a complex topic, but essentially, if the cell frequencies in of the betweensubject factors are unbalanced (i.e., the values between the different treatments are unequal), Type IV sums of squares is recommended.
Additinonally, there are other procedures which can accomplish appropriate analysis.