The PHREG Procedure

Using the TEST Statement to Test Linear Hypotheses

Subsections:

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

\begin{eqnarray*} \hat{\bV }(\hat{\bbeta }_0) \mb{e} - \lambda \mb{1}_ s & =& \mb{0} \\ \mb{e}’ \mb{1}_ s -1 & =& 0 \end{eqnarray*}

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.