In general, the tested hypothesis can be written as
where is an r
1 vector-valued function of the parameters
given by the r expressions that are specified in the TEST statement.
Let be the estimate of the covariance matrix of
. Let
be the unconstrained estimate of
and
be the constrained estimate of
such that
. Let
Using this notation, the test statistics for the three types of tests are computed as follows.
The Wald test statistic is defined as
The Lagrange multiplier test statistic is
where is the vector of Lagrange multipliers from the computation of the restricted estimate
.
The likelihood ratio test statistic is
where represents the constrained estimate of
and
is the concentrated log-likelihood value.
The following statements use the TEST statement to perform a likelihood ratio test:
proc hpqlim; model y = x1 x2 x3; test x1 = 0, x2 * .5 + 2 * x3 = 0 /lr; run;
For more information, see the section Tests on Parameters in SAS/ETS 14.1 User's Guide.