The GENMOD Procedure

F Statistics

Suppose that $D_0$ is the deviance resulting from fitting a generalized linear model and that $D_1$ is the deviance from fitting a submodel. Then, under appropriate regularity conditions, the asymptotic distribution of $(D_1-D_0)/\phi $ is chi-square with r degrees of freedom, where r is the difference in the number of parameters between the two models and $\phi $ is the dispersion parameter. If $\phi $ is unknown, and $\hat{\phi }$ is an estimate of $\phi $ based on the deviance or Pearson’s chi-square divided by degrees of freedom, then, under regularity conditions, $(n-p)\hat{\phi }/\phi $ has an asymptotic chi-square distribution with $n-p$ degrees of freedom. Here, n is the number of observations and p is the number of parameters in the model that is used to estimate $\phi $. Thus, the asymptotic distribution of

\[ F = \frac{D_1-D_0}{r \hat{\phi }} \]

is the F distribution with r and $n-p$ degrees of freedom, assuming that $(D_1-D_0)/\phi $ and $(n-p)\hat{\phi }/\phi $ are approximately independent.

This F statistic is computed for the Type 1 analysis, Type 3 analysis, and hypothesis tests specified in CONTRAST statements when the dispersion parameter is estimated by either the deviance or Pearson’s chi-square divided by degrees of freedom, as specified by the DSCALE or PSCALE option in the MODEL statement. In the case of a Type 1 analysis, model 0 is the higher-order model obtained by including one additional effect in model 1. For a Type 3 analysis and hypothesis tests, model 0 is the full specified model and model 1 is the submodel obtained from constraining the Type III contrast or the user-specified contrast to be 0.