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 involved in the analysis save the last one. 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 total variances of the data will change. This change of the total variance component is due to the changed variance between the column

 

and row

 

 

means. The residual variance component is due to the mean differences between the changed row values. For the example,

 

 

 For the example,  .92 = .25 + .50 + .17.

Regression Solution for the Column Component of Variance

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

 

 

 

The column variance component equals .25.

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.

 

 

 

Note that the code for the last subject 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.

Since the predictor variables are not orthogonal, the variance contributions of the predictor variables are not additive. Select the variable X as the initial variable and partial out the redundant variance components of the X₁ and X₂ variables, as

 

 

Multiple regression analysis

 

 

 

partitions variance of the variable Y (.92) into the row variance component (.13 + .37 = .50), into the column variance component (.25) and the residual component of variance (.17). Thus, .92 = .25 + .50 + .17, identical to values  obtained by the conceptual analysis of the same data.

Summary Table for the Variance Components

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

 

 

Summary Table for the Standard Components of Variance

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