PROC PHREG: Testing Linear Hypotheses about Regression Coefficients :: SAS/STAT(R) 9.2 User's Guide, Second Edition

The PHREG Procedure

Testing Linear Hypotheses about Regression Coefficients

Linear hypotheses for $\text{[math]}$ are expressed in matrix form as

$\text{[math]}$

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 $\text{[math]}$ is computed as

$\text{[math]}$

where $\text{[math]}$ is the estimated covariance matrix. Under $\text{[math]}$ , $\text{[math]}$ has an asymptotic chi-square distribution with r degrees of freedom, where r is the rank of $\text{[math]}$ .

Optimal Weights for the AVERAGE option in the TEST Statement

Let $\text{[math]}$ , where $\text{[math]}$ is a subset of $\text{[math]}$ regression coefficients. For any vector $\text{[math]}$ of length $\text{[math]}$ ,

$\text{[math]}$

To find $\text{[math]}$ such that $\text{[math]}$ has the minimum variance, it is necessary to minimize $\text{[math]}$ subject to $\text{[math]}$ . Let $\text{[math]}$ be a vector of 1’s of length $\text{[math]}$ . The expression to be minimized is

$\text{[math]}$

where $\text{[math]}$ is the Lagrange multipler. Differentiating with respect to $\text{[math]}$ and $\text{[math]}$ , respectively, yields

	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$
	$\text{[math]}$	$\text{[math]}$	$\text{[math]}$

Solving these equations gives

$\text{[math]}$

This provides a one degree-of-freedom test for testing the null hypothesis $\text{[math]}$ with normal test statistic

$\text{[math]}$

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

   Multivariate: test X1, ..., Xs;

where X $\text{[math]}$ , ..., X $\text{[math]}$ are the variables with regression coefficients $\text{[math]}$ , respectively.

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