Generalized Linear Model (GLM) Hypothesis Testing - Testing the Whole Model

Given the Model SS and the Error SS, one can perform a test that all the regression coefficients for the X variables (b1 through bk) are zero. This test is equivalent to a comparison of the fit of the regression surface defined by the predicted values (computed from the whole model regression equation) to the fit of the regression surface defined solely by the dependent variable mean (computed from the reduced regression equation containing only the intercept). Assuming that X'X is full-rank, the whole model hypothesis mean square ( MSH)

MSH = (Model SS) / k

is an estimate of the variance of the predicted values. The error mean square ( MSE)

s2 = MSE = (Error SS) / (n - k - 1)

is an unbiased estimate of the residual or error variance. The test statistic is

F = MSH / MSE

where F has (k, n - k - 1) degrees of freedom.

If X'X is not full rank, r + 1 is substituted for k, where r is the rank or the number of non-redundant columns of X'X.

Note: in the case of non-intercept models, some multiple regression programs will compute the full model test based on the proportion of variance around 0 (zero) accounted for by the predictors; for more information (see Kvålseth, 1985; OKunade, Chang, and Evans, 1993). STATISTICA GLM will actually compute both values (i.e., based on the residual variance around 0, and around the respective dependent variable means; see also the Between effects options on the Summary tab Results of the dialog).

Whole Model Tests

Six Types of Sums of Squares

Error Terms for Tests

Testing Specific Hypotheses

Testing Hypotheses for Repeated Measures and Dependent Variables