The SURVEYREG Procedure

Contrasts

You can use the CONTRAST statement to perform custom hypothesis tests. If the hypothesis is testable in the univariate case, the Wald F statistic for $H_{0}: \mb {L} \bbeta = 0$ is computed as

$F_{\mbox{Wald}} = \frac{(\mb {L}_{\mbox{Full}}\hat{\bbeta }) (\mb {L}_{\mbox{Full}}\widehat{\mb {V}} \mb {L}_{\mbox{Full}})^{-1} (\mb {L}_{\mbox{Full}}\hat{\bbeta }) }{\mbox{rank}(\mb {L})}$

where $\mb {L}$ is the contrast vector or matrix you specify, ${\bbeta }$ is the vector of regression parameters, $\hat{\bbeta }=\mb {(X’WX)^-X’WY}$ , $\widehat{\mb {V}}$ is the estimated covariance matrix of $\hat{\bbeta }$ , rank( $\mb {L}$ ) is the rank of $\mb {L}$ , and $\mb {L_\mr {Full}}$ is a matrix such that

$\mb {L_\mr {Full}}$ has the same number of columns as $\mb {L}$
$\mb {L_\mr {Full}}$ has full row rank
the rank of $\mb {L_\mr {Full}}$ equals the rank of the $\mb {L}$ matrix
all rows of $\mb {L_\mr {Full}}$ are estimable functions
the Wald F statistic computed using the $\mb {L_\mr {Full}}$ 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 $\mb {L_\mr {Full}}$ is the same as $\mb {L}$ . It is possible that $\mb {L_\mr {Full}}$ matrix cannot be constructed for contrasts in a CONTRAST statement, in which case the contrasts are not testable.