The SURVEYPHREG Procedure

Contrasts

For a testable hypothesis $H_{0} \colon \mb{L} \bbeta = 0 $, the Wald F statistic is computed as

\[ F_{\mbox{Wald}} = \frac{(\mb{L}^* \hat{\bbeta })' ({\mb{L}^*}' \widehat{\mb{V}} \mb{L}^* )^{-1} (\mb{L}^* \hat{\bbeta }) }{\mbox{rank}(\mb{L}) } \]

where $\mb{L}$ is a contrast vector or matrix that you specify, ${\bbeta }$ is the vector of regression parameters, $\hat{\bbeta }$ is the estimated regression coefficients, $\widehat{\mb{V}}$ is the estimated covariance matrix of $\hat{\bbeta }$, rank($\mb{L}$) is the rank of $\mb{L}$, and $\mb{L}^*$ is a matrix such that

  • $\mb{L}^*$ has the same number of columns as $\mb{L}$

  • $\mb{L}^*$ has full row rank

  • the rank of $\mb{L}^*$ equals the rank of the $\mb{L}$ matrix

  • all rows of $\mb{L}^*$ are estimable functions

  • the Wald F statistic computed by using the $\mb{L}^*$ matrix is equivalent to the Wald F statistic computed by using the $\mb{L}$ matrix with any row deleted that is a linear combination of previous rows

If $\mb{L}$ is a full-rank matrix and all rows of $\mb{L}$ are estimable functions, then $\mb{L}^*$ is the same as $\mb{L}$. It is possible that $\mb{L}$ matrix cannot be constructed for a given set of linear contrasts, in which case the contrasts are not testable.

If the DF=NONE option in the MODEL statement is specified, then the procedure performs a chi-square significance test.