## Confidence Intervals for Parameters

*Likelihood Ratio-Based Confidence Intervals*

PROC GENMOD produces likelihood ratio-based confidence
intervals, also known as profile likelihood confidence intervals,
for parameter estimates for generalized linear models. These
are not computed for GEE models, since there is no likelihood
for this type of model.
Suppose that the parameter vector is
and that you want a confidence interval for .The profile likelihood function for is defined as

where is the vector with the *j*th element fixed at and *l* is the log likelihood function.
If is the log likelihood
evaluated at the maximum likelihood estimate
, then has a
limiting chi-square distribution with one degree of
freedom if is the true parameter value.
A confidence interval for is

where is the percentile
of the chi-square distribution with one degree of freedom.
The endpoints of the confidence interval can be
found by solving numerically for values of that satisfy equality in the preceding relation.
PROC GENMOD solves this by starting at the
maximum likelihood estimate of .The log likelihood function is approximated with a
quadratic surface, for which an exact solution is possible.
The process is iterated until
convergence to an endpoint is attained.
The process is repeated for the other endpoint.
Convergence is controlled by the
CICONV= option in the MODEL statement.
Suppose is the number specified in the CICONV= option.
The default value of is 10^{-4}.
Let the parameter of interest be and
define **r** = **u**_{j}, the unit vector
with a 1 in position *j* and 0s elsewhere.
Convergence is declared on the current iteration
if the following two conditions are satisfied:

where , **s**, and **H** are
the log likelihood, the gradient, and the Hessian
evaluated at the current parameter vector and
is a constant computed by the procedure.
The first condition for convergence means that the log-likelihood
function must be within of the correct
value, and the second condition means that the gradient vector
must be proportional to the restriction vector **r**.
When you request the LRCI option in the MODEL
statement, PROC GENMOD computes profile likelihood
confidence intervals for all parameters in the model,
including the scale parameter, if there is one.
The interval endpoints are displayed in a table as well as
the values of the remaining parameters at the solution.

*Wald Confidence Intervals *

You can request that PROC GENMOD produce Wald
confidence intervals for the parameters.
The (1-)100% Wald confidence interval
for a parameter is defined as

where *z*_{p} is the 100*p*th percentile of the standard normal
distribution, is the parameter estimate, and
is the estimate of its standard error.

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.