Multiple Regression within the Repeated Measures Design

So far we were able to solve the repeated measures design only within the frameworks of a repeated measures t-test, or by using the Microsoft Excel spreadsheet for the double classification analysis of variance design. The multiple regression solution of the repeated measures design approach is quite exacting. A separate coding vector has to be constructed for all subjects but the last one involved in the analysis. These coding vectors are not orthogonal. They are filled with 0s save few 1s marking the location of the subject within the different conditions of the experiment. To compare these different approaches to the repeated measures design, let us first review its idealized model.

Idealized Model of the Repeated Measures Design

This model makes an assumption that, initially, the means and variances of all components of the model are the same.

Following some type of intervention,

the means and variances will change.

Column Variance Component

This change of the total variance component is due to the changed variance between the column means (treatments). The column variance equals .25 as shown below. 

Row Variance Component

The performance of each subject over different experimental conditions can be computed as the row mean. The row means are 
 

Residual Variance Component

The residual variance component is due to the mean differences between the changed row values. For the example, the mean of the difference score for each row is computed as

 
The residual variance equals .17. 

Total Variance Component

The total variance of the criterion variable Y is .92.

Note that .92 (total variance) = .25 (column variance) + .50 (row variance) + .17 (residual variance). Convert the true variance components to standard variance components. The equation becomes 1 = .27 + .55 + .18.

Regression Solution for the Column Component of Variance

The column component of variance can be obtained by using the coded regression analysis as

In a coded regression analysis, the predictor X represents group membership. The results of the regression analysis are displayed below. 

Note that the vector of the predicted values consists of group means and the variance due to different group means equals .25. The value is identical with the column variance obtained earlier.

Approximately 27 percent (.25/.92=.27) of variance in the criterion variable Y can be explained by the experimental conditions (group membership). 

Multiple Regression Solution for the Row Component of Variance

To obtain the row variance component, one has to use the multiple regression analysis where the group membership is coded by codes specifying which scores in different conditions of the experiment were obtained from the same subject, as shown below. 

Use the dummy coding method to code the subjects. The coding vector X1 represents Allen who took part in both conditions. The coding vector X2 represents Becky who also took part in both conditions. Note that the coding vector for the last subject, Cathy, which would have been [0 0 1 0 0 1] have not been included, as it is redundant. You might wish to experiment with including a code for the last subject only to observe that inclusion of this code does not make any difference, since the information about subject-group memberships is fully conveyed by k-1 coding vectors.

The variables involved in the multiple regression analysis are correlated as

Successive Partialing

Since the predictor variables are not orthogonal, the variance contributions of the predictor variables are not additive. Using successive partialing and including the predictor variable X first changes the predictors as

 

 

 

changes the matrix of inter- and cross-correlations as

 

 

changes the matrix of squared inter- and cross-correlations (coefficients of determination) as

 

The primary predictor variable is X. The 1.x subscript trailing the second predictor variable signifies that X1 was residualized on the variables X.  The 2.x1 subscript trailing the third predictor variable signifies that X2 was residualized on the variables X and X1.

 

After successive partialing, the three predictors are orthogonal and the variance contributions due to predictor variables are additive.

Variance Contributions

The standard variance component for the columns (experimental conditions) is .27 and for the rows (subjects) is .55 (.1364 + .4091= .5455). 

 

Cascading the Coefficient of Multiple Determination

Alternatively, we may compute the variance contributions by cascading the coefficient of multiple determination.

1. Variance contribution due to treatments

Enter the predictor variable X into the regression equation. The resulting coefficient of determination is equal to .27. Approximately 27 percent of the variance in Y can be explained by the experimental treatments.

 

 

2. Add subject codes.

Next, enter the subject codes, X1 and X2, into the regression equation.  The resulting coefficient of multiple determination is increased to .818.

 

 

3. Variance contribution due to subjects

Individual differences account for additional 55 percent of the variance in Y. It is computed as .82 - .27. 

4. Variance not accounted for

About 18 percent of the variance in Y can not be explained. It is computed as 1 - .55 - .27.

Summary Table for the Standard Components of Variance

For the example, the standard variance components obtained by the coded multiple regression can be summarized as