Sum of squares due to factors

Using matrix least squares techniques (see Section 5.2), the chosen linear model may be fit to the data to obtain a set of parameter estimates, B, from which predicted values of response, fu, may be obtained. It is convenient to define a matrix of estimated responses, Y. 

Some of the variation of the responses about their mean is caused by variation of the factors. The effect of the factors as they appear in the model can be measured by the differences between the predicted responses (p1( ) and the mean response (j ). For this purpose, it is convenient to define a matrix of factor contributions, F. 

This matrix may be used to calculate still another useful sum of squares, the sum of squares due to the factors as they appear in the model, SSfact, sometimes called the sum of squares due to regression. 

For models containing a / 0 term, the sum of squares due to the factors has p — 1 degrees of freedom associated with it, where p is the number of parameters in the model. For models that do not contain a /S0 term, SStact has p degrees of freedom. 

We have already defined the matrix of residuals, R, in Section 5.2. It may be obtained using matrix techniques as 

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