The PHREG Procedure

Using the TEST Statement to Test Linear Hypotheses

Optimal Weights for the AVERAGE option in the TEST Statement

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.