XLSTAT: Repeated measures ANOVA in Excel

 

Background

The difference between classical ANOVA and repeated measures ANOVA  is that measures on the same patient at different times are not supposed to be independent and, thus, the covariance matrix of the errors is not diagonal.

 

Dataset for Example

The data correspond to an experiment in which a treatment for depression is studied. Two groups of patients (1: control / 2: treatment) have been followed at five different times (0: pre-test, 1: one month post-test, 3: 3 months follow-up and 6: 6 months follow-up). The dependant variable is a depression score.

We have performed repeated measures ANOVA in order to determine the effect of the treatment and the effect of time on the depression score. The repeated measures ANOVA model is the same as the classical ANOVA model with interactions:

We have one fixed factor (group), one subjet factor (the patients) and the repetition. The interaction between the group and the repetition is included in the model.

We use the least squares estimation method (LS) to estimate the model. The variance-covariance matrix is supposed to be spherical.

This method is very simple: First, a classical ANOVA is performed on each time measured. Results are the same as in a classical analysis of variance. Then, results based on the covariance matrix and on the repetition, factor are given.

You can use other form of covariance matrix with XLSTAT using mixed models. Please refer to the following tutorial for an example.

Data structure

Data can be presented using two formats. The classical one is one column per repetition. That means that for each repeated measure of the dependent variable there will be one column.

 

Setting Up

• Open XLSTAT.

• Click on the Modeling data menu and select the ANOVA feature:

• Once the button is clicked, the repeated measures ANOVA dialog box appears.

In the General tab:

• Select the data from the Excel sheet.

• Select "dv0-dv1-dv3-dv6" as dependent variable (or variable to model). The goal is to determine the effect of the group on the variability of the depression score.

• Since the column titles were selected, select the Variable labels option.

 

In the Options tab:

• Select LS as the estimation method (Least Squares).

• Leave the constraint option at a1 = 0, assuming the control group has the standard effect on the score.

• Applying a constraint is required for theoretical reasons, but it does not affect the results (e.g., goodness of fit). The constraint only changes how the model is expressed, not its performance.

• After clicking the OK button, a dialog box appears to select the factors to include in the model.

In the Factors and interactions dialog box:

• Select Group as fixed effect.

• The repeated factor is repetition.

• The subject factor is subject.

Note:

• These factors are automatically generated by XLSTAT.
• A factor cannot be the subject or repeated factor and a fixed effect at the same time.

• Once you have clicked on the OK button, the computation starts.

• The results will then be displayed.

 

Interpreting the Results

The first results displayed by XLSTAT are the basic statistics associated to the dependent variable.

For each measure, an ANOVA is performed. Results associated to time 0, 1, 3 and 6 are displayed. For more details, you can check the  tutorial on one-way ANOVA.

The analysis of variance table pre-test (dv0) is:

The analysis of variance table one month post-test (dv1) is:

The analysis of variance table 3 months follow-up (dv3) is:

The analysis of variance table 6 months follow-up (dv0) is:

We can see that the group has an effect significantly greater than 0 on the depression score after 1 month of treatment.

Once the four analyses have been performed, some additional outputs related to the repeated design are displayed.

The first table is very important and helps to validate the sphericity of the covariance matrix of the errors. This test is called Mauchly’s test.

Given that the p-value is smaller than 0.05 (alpha) we can reject the null hypothesis and accept the alternative hypothesis that the the covariance matrix is not spherical. The Greenhouse-Geisser and the Huynt-Feldt corrections have been applied here to adjust the p-value. Each of these corrections estimate a statistic called espilon. The closer it is to one, the more spherical the covariance matrix is.

 

The two following tables can now be analyzed. First, the tests on the inter-subject effects which show the effect of the group variable on the whole dataset without taking into account the repetitions (or measures). We see that the group has a significant impact on the depression score. Then, the tests on the intra-subject effects show the impact of time (of the different measures) on the dependent variable. It can be useful to look at the interaction terms between repetition and the explanatory factors. We see that the repetition factor has a significant impact on the depression score; the interaction has also a significant impact.

This study has shown that both time and treatment have a significant impact on the depression score.

Some other outputs can be useful and are available in XLSTAT: residuals, residuals charts, least square means charts, multiple means comparisons...