The LOGISTIC Procedure

Testing Linear Hypotheses about the Regression Coefficients

Linear hypotheses for $\bbeta $ are expressed in matrix form as

\[  H_0\colon \bL \bbeta = \mb {c}  \]

where $\bL $ is a matrix of coefficients for the linear hypotheses, and $\mb {c}$ is a vector of constants. The vector of regression coefficients $\bbeta $ includes slope parameters as well as intercept parameters. The Wald chi-square statistic for testing $H_0$ is computed as

\[  \chi ^2_{W} = (\bL {\widehat{\bbeta }} - \mb {c})’ [{\bL \widehat{\bV }({\widehat{\bbeta }})\bL ’}]^{-1} (\bL {\widehat{\bbeta }} - \mb {c})  \]

where $\widehat{\bV }({\widehat{\bbeta }})$ is the estimated covariance matrix. Under $H_0$, $\chi ^2_{W}$ has an asymptotic chi-square distribution with r degrees of freedom, where r is the rank of $\bL $.