The PHREG Procedure
Using the TEST Statement to Test Linear Hypotheses
Linear hypotheses for 
are expressed in matrix form as
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 
is computed as
where 
is the estimated covariance matrix. Under , 
has an asymptotic chi-square distribution with r degrees of freedom, where r is the rank of 
.
Optimal Weights for the AVERAGE Option in the TEST Statement
Let 
, where 
is a subset of s regression coefficients. For any vector 
of length s,
To find 
such that 
has the minimum variance, it is necessary to minimize 
subject to 
. Let 
be a vector of 1’s of length s. The expression to be minimized is
where 
is the Lagrange multiplier. Differentiating with respect to and , respectively, yields
Solving these equations gives
This provides a one degree-of-freedom test for testing the null hypothesis 
with normal test statistic
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 
, respectively.