Caepalic Shells of Regression Analysis

Regression analysis can be expressed in three basic modes, using standard, deviation, and obtained scores. The most concise mode is that using standard scores.

 

Regression and Multiple Regression Analysis in Standard Scores

 

The multiple regression model, expressed in standard scores, partitions the standard variance of the criterion variable z_y into its predictable and error components.

 

                                                               

 

The multivariate regression model is an extension of the bivariate regression model where

                                                                  

 

and

                                                               

 

In the multivariate regression model the predictable component equals

 

                                                    

 

and the residual component equals

 

                                                               

 

Regression and Multiple Regression Analysis in Deviation Scores

 

The relationships between deviation scores x and y and standard scores z_x and z_y are

 

 

and

 

 

Thus equation

                                                                  

 

Can be written in the deviation scores as

 

                                                                

 

and, solving for the predicted scores,

 

                                                                

 

The geometric equation of a line, expressed in deviation scores is

 

                                                                    

where

 

                                                                  

 

The multivariate regression model, expressed in deviation scores is

 

                                                     

 

Regression and Multiple Regression Analysis in Obtained Scores

 

In the equation

 

 

 

change the deviation scores into the obtained scores as

 

                                                                

and

                                                                

 

Thus,

                                                         

 

Solve for the Y’ as

                                                         

 

The geometric equation of a line, expressed in obtained scores is

 

                                                                

 

Thus the intercept, A, equals

 

                                                              

 

Note that the regression weights of the models expressed in the obtained and in the deviation scores are the same, i.e.,

 

                                                                     

 

The predictable component of the multivariate regression model expressed in the obtained scores is

                                                

 

where the intercept equals

 

                                              

 

The residual component of the regression model expressed in the obtained scores equals

 

                                                               

 

For the example, the first partial regression coefficient equals .32 and the second partial regression coefficient equals .60. In this particular example, the beta and the b weights are the same, since the predictor variables and the criterion variable have the same variances. The intercept can be computed as A = 3.00 - .32(3.00) - .60(3.00) which equals .23.

Example of Multiple Regression Analysis in Standard Scores

Using the data matrix for the example of multiple regression analysis introduced at the beginning of this chapter,

 

 

regression equations for the predicted and error scores can be written, using standard scores, as

 

                                                            

and

                                                               

 

Using the above equations, the multiple regression analysis can be build as

 

                            

 

 

The means of the predicted and error scores sum to the mean of the criterion variable

 

                                                           

 

Since the variables are in the standard form, all the means are zero. The variances of the predicted and the error scores sum to the variance of the criterion variable

 

                                                               

 

Since the criterion variable is in the standard form

 

 

 

The variance of the predicted scores equals the coefficient of multiple determination

 

                                                                   

 

and the variance of the error scores equals the coefficient of multiple alienation

 

 

 

The variances components contributed by the predictor variables do not sum to the variance of the predicted scores since the predictor variables are not orthogonal.

Example of Multiple Regression Analysis in Deviation Scores

For our example, regression equations for the predicted and residual components of the multiple regression analysis can be written, using deviation scores, as

                                                            

 

And

                                                                 

 

Using the above equations, the multiple regression analysis for our example can be build as

                                  

 

The means of the predicted and error scores sum to the mean of the criterion variable

 

                                                            

 

Since the variables are in the form of deviation scores, all the means are zero. The specification equation for partitioning of variance by the multiple regression analysis states that the variances of the predicted and the error scores sum to the variance of the criterion variable

                                                              

 

Dividing its both sides by the variance of the criterion variable can standardize the above equation as

                                                             

 

This equation can be also expressed as

 

 

The coefficient of multiple determination equals

 

 

 

and the coefficient of multiple alienation equals

 

 

 

The variances contributed by the predictors do not sum to the variance of the predicted scores, since the predictor variables are correlated.

Example of Multiple Regression Analysis in Obtained Scores 

For our example, the regression equation in the obtained scores may be written as

 

                                                      

 

The equation, expressing the residual component in the obtained scores is

 

                                                                

 

Using the above equations, the multiple regression analysis can be build as

 

                                    

 

The means of the predicted and error scores sum to the mean of the criterion variable

 

                                                            

 

Since the mean of the error component must be zero, the mean of the predicted scores must equal the mean of the criterion variable

 

                                                                 

Since the variables are in the form of deviation scores, all the means are zero. The specification equation for partitioning of variance by the multiple regression analysis states that the variances of the predicted and the error scores sum to the variance of the criterion variable

                                                              

 

Dividing its both sides by the variance of the criterion variable can standardize the above equation as

                                                             

 

This equation can be also expressed as

 

 

The coefficient of multiple determination equals

 

 

 

and the coefficient of multiple alienation equals

 

 

 

The variances contributed by the predictors do not sum to the variance of the predicted scores, since the predictor variables are not orthogonal.