The LOGISTIC Procedure

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, because 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 CLPARM= 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 chi squared distribution of the generalized likelihood ratio test (Venzon and Moolgavkar 1988). Suppose that the parameter vector is bold-italic beta equals left-parenthesis beta 0 comma beta 1 comma ellipsis comma beta Subscript s Baseline right-parenthesis prime and you want to compute a confidence interval for beta Subscript j. The profile-likelihood function for beta Subscript j Baseline equals gamma is defined as

l Subscript j Superscript asterisk Baseline left-parenthesis gamma right-parenthesis equals max Underscript bold-italic beta element-of script upper B Subscript j Baseline left-parenthesis gamma right-parenthesis Endscripts l left-parenthesis bold-italic beta right-parenthesis

where script upper B Subscript j Baseline left-parenthesis gamma right-parenthesis is the set of all bold-italic beta with the jth element fixed at gamma, and l left-parenthesis bold-italic beta right-parenthesis is the log-likelihood function for bold-italic beta. (The penalized log-likelihood function is used when you specify the FIRTH option in the MODEL statement.) If l Subscript max Baseline equals l left-parenthesis ModifyingAbove bold-italic beta With caret right-parenthesis is the log likelihood evaluated at the maximum likelihood estimate ModifyingAbove bold-italic beta With caret, then 2 left-parenthesis l Subscript max Baseline minus l Subscript j Superscript asterisk Baseline left-parenthesis beta Subscript j Baseline right-parenthesis right-parenthesis has a limiting chi-square distribution with one degree of freedom if beta Subscript j is the true parameter value. Let l 0 equals l Subscript max Baseline minus 0.5 chi 1 squared left-parenthesis 1 minus alpha right-parenthesis, where chi 1 squared left-parenthesis 1 minus alpha right-parenthesis is the 100 left-parenthesis 1 minus alpha right-parenthesisth percentile of the chi-square distribution with one degree of freedom. A 100 left-parenthesis 1 minus alpha right-parenthesis% confidence interval for beta Subscript j is

StartSet gamma colon l Subscript j Superscript asterisk Baseline left-parenthesis gamma right-parenthesis greater-than-or-equal-to l 0 EndSet

The endpoints of the confidence interval are found by solving numerically for values of beta Subscript j 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 bold-italic beta is approximated by the quadratic function

l overTilde left-parenthesis bold-italic beta plus bold-italic delta right-parenthesis equals l left-parenthesis bold-italic beta right-parenthesis plus bold-italic delta prime bold g plus one-half bold-italic delta prime bold upper V bold-italic delta

where bold g equals bold g left-parenthesis bold-italic beta right-parenthesis is the gradient vector and bold upper V equals bold upper V left-parenthesis bold-italic beta right-parenthesis is the Hessian matrix. The increment bold-italic delta for the next iteration is obtained by solving the likelihood equations

StartFraction d Over d bold-italic delta EndFraction StartSet l overTilde left-parenthesis bold-italic beta plus bold-italic delta right-parenthesis plus lamda left-parenthesis bold e prime Subscript j Baseline bold-italic delta minus gamma right-parenthesis EndSet equals bold 0

where lamda is the Lagrange multiplier, bold e Subscript j is the jth unit vector, and gamma is an unknown constant. The solution is

bold-italic delta equals minus bold upper V Superscript negative 1 Baseline left-parenthesis bold g plus lamda bold e Subscript j Baseline right-parenthesis

By substituting this bold-italic delta into the equation l overTilde left-parenthesis bold-italic beta plus bold-italic delta right-parenthesis equals l 0, you can estimate lamda as

lamda equals plus-or-minus left-parenthesis StartFraction 2 left-parenthesis l 0 minus l left-parenthesis bold-italic beta right-parenthesis plus one-half bold g prime bold upper V Superscript negative 1 Baseline bold g right-parenthesis Over bold e prime Subscript j Baseline bold upper V Superscript negative 1 Baseline bold e Subscript j Baseline EndFraction right-parenthesis Superscript one-half

The upper confidence limit for beta Subscript j is computed by starting at the maximum likelihood estimate of bold-italic beta and iterating with positive values of lamda until convergence is attained. The process is repeated for the lower confidence limit by using negative values of lamda.

Convergence is controlled by the value epsilon specified with the PLCONV= option in the MODEL statement (the default value of epsilon is 1E–4). Convergence is declared on the current iteration if the following two conditions are satisfied:

StartAbsoluteValue l left-parenthesis bold-italic beta right-parenthesis minus l 0 EndAbsoluteValue less-than-or-equal-to epsilon

and

left-parenthesis bold g plus lamda bold e Subscript j Baseline right-parenthesis prime bold upper V Superscript negative 1 Baseline left-parenthesis bold g plus lamda bold e Subscript j Baseline right-parenthesis less-than-or-equal-to epsilon

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 100 left-parenthesis 1 minus alpha right-parenthesis% Wald confidence interval for beta Subscript j is given by

ModifyingAbove beta With caret Subscript j Baseline plus-or-minus z Subscript 1 minus alpha slash 2 Baseline ModifyingAbove sigma With caret Subscript j

where z Subscript p is the 100pth percentile of the standard normal distribution, ModifyingAbove beta With caret Subscript j is the maximum likelihood estimate of beta Subscript j, and ModifyingAbove sigma With caret Subscript j is the standard error estimate of ModifyingAbove beta With caret Subscript j.

Last updated: December 09, 2022