XLSTAT: Run Repeated Measures ANOVA Using Mixed Models in Excel

 

Dataset for Example

The (attached) 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 4 different times:

0: pre-test

1: one month post-test

3: 3 months follow-up

6: 6 months follow-up

The dependent variable is a depression score.

We have performed a 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 two fixed factors (time and group) and one interaction factor (time*group).

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 thus the covariance matrix of e is not diagonal.

XLSTAT uses the mixed models theory to treat repeated measures ANOVA and this raises some differences.

Some supplementary options are available like the choice between many covariance structures for the covariance matrix of the error term. For this example we will use the compound symmetry structure.

Please consult the XLSTAT help for more details on covariance structures.

 

Data for Example

Data should have a particular shape when using the mixed model methodology for repeated measures ANOVA. All the measures have to be on the same column with:

• a factor called repeated factor which indicates what repetition is associated to this measure

• a factor called subject factor which indicates what subject is associated to this measure.

In our example, each patient will appear in 4 different lines. The data has to be in the following form:

If your data is in a different form, one column for each measure, you should transform it in order to obtain the preceding structure.

 

Setting Up

• Open XLSTAT.

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

• Once you've clicked on the button, the repeated measures ANOVA dialog box appears. 

In the General tab:

• Select the data from the Excel sheet.

• Choose "dv" as the dependent variable (or variable to model). The goal is to assess the effect of group, time, and their interaction on the variability of the depression score.

• Select "Repeated" and "Subject" as explanatory variables.

• Select "One column for all repetitions"

• Since the column titles are included, check the Variable labels option.

In the Options tab:

• Activate the Interactions option and set the maximum level of interaction to 2.

• Leave the constraint at a1 = 0 to assume the control group has the standard effect on the score. This is required for theoretical reasons but does not affect model results.

• Select the Compound Symmetry structure as the covariance structure (this is the default setting).

• Once you click the OK button, a dialog box appears to define the model factors.

In the Factors and Interactions dialog box:

• Select "time", "group", and "time*group" as fixed effects.

• Select "time" as the repeated factor.

• Select "subj" as the subject factor.

Note: A factor cannot be the subject factor and a fixed effect at he same time. Repeated and subject factors have to be different and both qualitative.

 

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The first results displayed by XLSTAT are the goodness of fit coefficients.

Model parameters are obtained using the restricted maximum likelihood (REML) method and will be different as when a classical ANOVA model is applied. All indexes are used to compare models with different covariance structures.

Covariance parameters are displayed on the next table with their associated Z test. We can see that both parameters are significant.

Model implied covariance and correlation matrix can as well be displayed.

The null model likelihood ratio test compares the model obtained with the specified covariance structure and the model obtained with the classical diagonal covariance structure. The obtained p-value can be used to assess the significance of the model fit.

We can see that choosing the compound symmetry structure has a positive effect on the model fit.

The type III tests of fixed effects show that all factors have significant effects on the depression score. These tests are very important; they replace the Type III SS of the classical ANOVA model.

From the results displayed in the Type III table, we can see that the "Time" variable is the one that has the highest impact on the model. When we look at the model parameters (see below), we can see that time 1, 3 and 6 have a negative impact on the depression score. Patients are less depressed as time passes. Being in the treatment group has also a negative impact on the depression score.

The table depicted above can be used to analyze the impact of the explanatory variables on the depression score and/or to predict the average yield in a situation not yet covered by the experiment, such as the time 6 with the treatment group. In this particular example, the average depression score would be 107.1, given the fact that the influence of the interaction is unknown.

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

Some other output can be useful and are available in XLSTAT like residuals, residuals charts, least square means charts…