The SURVEYREG Procedure



Testing Effects

For each effect in the model, PROC SURVEYREG computes an $\mb{L}$ matrix such that every element of $\mb{L}\bbeta $ is estimable; the $\mb{L}$ matrix has the maximum possible rank that is associated with the effect. To test the effect, the procedure uses the Wald F statistic for the hypothesis $H_0\colon \mb{L} \bbeta = 0$. The Wald F statistic equals

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

with numerator degrees of freedom equal to $\mr{rank}(\mb{L}’\widehat{\mb{V}}\mb{L})$.

In the Taylor series method, the denominator degrees of freedom is equal to the number of clusters minus the number of strata (unless you specify the denominator degrees of freedom with the DF= option in the MODEL statement). For details about denominator degrees of freedom in replication methods, see the section Denominator Degrees of Freedom. It is possible that the $\mb{L}$ matrix cannot be constructed for an effect, in which case that effect is not testable. For more information about how the matrix $\mb{L}$ is constructed, see the discussion in Chapter 15: The Four Types of Estimable Functions.

You can use the TEST statement to perform F tests that test Type I, Type II, or Type III hypotheses. For details about the syntax of the TEST statement, see the section TEST Statement in Chapter 19: Shared Concepts and Topics.


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.