The SURVEYPHREG Procedure

Contrasts

For a testable hypothesis $H_{0} \colon \mb{L} \bbeta = 0 $, you can request different Wald tests by using the DF= option in the MODEL statement.

Let

\[ Q = (\mb{L}^* \hat{\bbeta })’ ({\mb{L}^*}’ \widehat{\mb{V}} \mb{L}^* )^{-1} (\mb{L}^* \hat{\bbeta }) \]

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 $\hat{\bbeta }$, and $\mb{L}^*$ is a matrix such that the following are true:

  • $\mb{L}^*$ 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 that is computed by using the $\mb{L}^*$ matrix is equivalent to the Wald F statistic computed by using the $\mb{L}$ matrix.

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 such an $\mb{L}^*$ matrix cannot be constructed for a given set of linear contrasts, in which case the contrasts are not testable. Let r be the rank of $\mb{L}$. The following table describes the Wald tests available in PROC SURVEYPHREG.

     

Numerator

Denominator

Value of DF=

Test Request

Test Statistic

Degrees of Freedom

Degrees of Freedom

NONE

Chi-square

Q

r

$\infty $

v

Customized F

vQ/rd

r

v

DESIGN

Unadjusted F

Q/r

r

d

DESIGN (v)

Unadjusted F

Q/r

r

v

PARMADJ

Adjusted F

(d–r+1)Q/rd

r

d–r+1

PARMADJ (v)

Adjusted F

(v–r+1)Q/rv

r

v–r+1

DESIGNADJ

Adjusted F

Q/r

r

d