## Confidence Intervals for Parameters

There are two methods of computing confidence intervals for the
regression parameters.
One is based
on the profile likelihood function, and the other is based on the
asymptotic normality of the parameter estimators. The latter is
not as time-consuming as the former, since it does not involve
an iterative scheme; however, it is not thought to be as accurate
as the former, especially with small sample size.
You use the CLPARMS= option to request confidence intervals for the
parameters.

*Likelihood Ratio-Based Confidence Intervals*

The likelihood ratio-based confidence
interval is also known as the profile likelihood confidence
interval. The construction of this interval is derived
from the asymptotic distribution of the generalized likelihood
ratio test (Venzon and Moolgavkar 1988).
Suppose that the parameter vector
is and you want to compute a confidence interval for .The profile likelihood function for is defined as

where is the set of all with the *j*th
element fixed at ,and is the log likelihood function for .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.
Let ,where is the percentile
of the chi-square distribution with one degree of freedom.
A % confidence interval for is

The endpoints of the confidence interval are found by solving
numerically for values of that satisfy equality
in the preceding relation.
To obtain an iterative algorithm for computing the confidence
limits, the log likelihood function in a neighborhood of is
approximated by the quadratic function

where is the gradient vector and
is the Hessian matrix.
The increment for the next iteration is obtained by
solving the likelihood equations

where is the Lagrange multiplier, **e**_{j} is the *j*th
unit vector, and is an unknown
constant. The solution is

By substituting this into the equation ,you can estimate as

The upper confidence limit for is computed by starting
at the maximum likelihood estimate of and iterating
with positive values of until convergence
is attained. The process
is repeated for the lower confidence limit using negative values
of .Convergence is controlled by value specified with the PLCONV=
option in the MODEL statement (the default value
of is 1E**-**4).
Convergence is declared on the current iteration if the following
two conditions are satisfied:

and

*Wald Confidence Intervals*

Wald confidence intervals are sometimes called the normal
confidence intervals. They are based
on the asymptotic normality
of the parameter estimators.
The % Wald confidence interval
for is given by

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

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