The PHREG Procedure

Using the TEST Statement to Test Linear Hypotheses

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

\[  H_{0}\colon \mb {L}{\bbeta }=\mb {c}  \]

where L is a matrix of coefficients for the linear hypotheses, and c is a vector of constants. The Wald chi-square statistic for testing $H_{0}$ is computed as

\[  \chi ^{2}_{W}=\left( \mb {L}\hat{\bbeta }-\mb {c} \right) ’ \left[ \mb {L}\hat{\mb {V}}(\hat{\bbeta })\mb {L}’ \right] ^{-1} \left( \mb {L}\hat{\bbeta }-\mb {c} \right)  \]

where $\hat{\bV }(\hat{\bbeta })$ 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 $\mb {L}$.

Optimal Weights for the AVERAGE option in the TEST Statement

Let $\bbeta _0=(\beta _{i_1}, \ldots , \beta _{i_ s})’$, where $\{ \beta _{i_1},\ldots ,\beta _{i_ s}\} $ is a subset of s regression coefficients. For any vector $\mb {e}=(e_1, \ldots , e_ s)’$ of length s,

\[  \mb {e}’\hat{\bbeta }_0 \sim N(\mb {e}’\bbeta _0, \mb {e}’\hat{\bV }(\hat{\bbeta _0})\mb {e})  \]

To find $\mb {e}$ such that $\mb {e}’\hat{\bbeta }_0$ has the minimum variance, it is necessary to minimize $\mb {e}’\hat{\bV }(\hat{\bbeta _0})\mb {e} $ subject to $\sum _{i=1}^ k e_ i = 1$. Let $\mb {1}_ s$ be a vector of 1’s of length s. The expression to be minimized is

\[  \mb {e}’\hat{\bV }(\hat{\bbeta }_0) \mb {e} - \lambda (\mb {e}’\mb {1}_ s -1)  \]

where $\lambda $ is the Lagrange multiplier. Differentiating with respect to $\mb {e}$ and $\lambda $, respectively, yields

$\displaystyle  \hat{\bV }(\hat{\bbeta }_0) \mb {e} - \lambda \mb {1}_ s  $
$\displaystyle = $
$\displaystyle  \mb {0}  $
$\displaystyle \mb {e}’ \mb {1}_ s -1  $
$\displaystyle = $
$\displaystyle  0  $

Solving these equations gives

\[  \mb {e}= [\mb {1}_ s’{\hat{\bV }^{-1}(\hat{\bbeta }_0)} \mb {1}_ s]^{-1} {\hat{\bV }^{-1}(\hat{\bbeta }_0)} \mb {1}_ s  \]

This provides a one degree-of-freedom test for testing the null hypothesis $H_0: \beta _{i_1} =\ldots =\beta _{i_ s} =0$ with normal test statistic

\[  Z = \frac{\mb {e}\hat{\bbeta }_0}{\sqrt {\mb {e}’\hat{\bV }(\hat{\bbeta }_0)\mb {e}}}  \]

This test is more sensitive than the multivariate test specified by the TEST statement

Multivariate: test X1, ..., Xs;

where X1, …, Xs are the variables with regression coefficients $\beta _{i_1},\ldots ,\beta _{i_ s}$, respectively.