Confidence Intervals for Parameters

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, 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 $\chi^2$ distribution of the generalized likelihood ratio test (Venzon and Moolgavkar 1988). Suppose that the parameter vector is ${\beta}= (\beta_{0},\beta_{1}, ... ,\beta_{s})'$ and you want to compute a confidence interval for $\beta_{j}$ .The profile likelihood function for $\beta_{j}=\theta$ is defined as

$l_j^*(\theta) = \max_{{\beta}\in {\cal B}_j(\theta)} l({\beta})$

where ${\cal B}_j(\theta)$ is the set of all ${\beta}$ with the jth element fixed at $\theta$ ,and $l({\beta})$ is the log likelihood function for ${\beta}$ .If $l_{\max} = l(\hat{{\beta}})$ is the log likelihood evaluated at the maximum likelihood estimate $\hat{{\beta}}$ , then $2( l_{\max} - l_j^{*}(\beta_{j} ))$ has a limiting chi-square distribution with one degree of freedom if $\beta_{j}$ is the true parameter value. Let $l_0=l_{\max} - .5\chi^2_{1-\alpha,1}$ ,where $\chi^2_{1-\alpha,1}$ is the $100(1-\alpha)$ percentile of the chi-square distribution with one degree of freedom. A $100(1-\alpha)$ % confidence interval for $\beta_{j}$ is

$\{\theta : l_j^*(\theta) \geq l_{0} \}$

The endpoints of the confidence interval are found by solving numerically for values of $\beta_{j}$ 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 ${\beta}$ is approximated by the quadratic function

$\tilde{l}({\beta}+ {\delta}) = l({\beta}) + {\delta}'g+ \frac{1}2{\delta}' V{\delta}$

where $g=g({\beta})$ is the gradient vector and $V=V({\beta})$ is the Hessian matrix. The increment ${\delta}$ for the next iteration is obtained by solving the likelihood equations

$\frac{d}{d{\delta}}\{ \tilde{l}({\beta}+ {\delta}) + \lambda( e_j'{\delta}- \theta)\} = 0$

where $\lambda$ is the Lagrange multiplier, e_j is the jth unit vector, and $\theta$ is an unknown constant. The solution is

${\delta}= -V^{-1}(g+ \lambda e_j)$

By substituting this ${\delta}$ into the equation $\tilde{l}({\beta}+ {\delta}) = l_0$ ,you can estimate $\lambda$ as

$\lambda = +- \biggl(\frac{2(l_0 - l({\beta}) + \frac{1}2{g}'V^{-1}g)} {e_j'V^{-1}e_j}\biggr)^{ \frac{1}2}$

The upper confidence limit for $\beta_{j}$ is computed by starting at the maximum likelihood estimate of ${\beta}$ and iterating with positive values of $\lambda$ until convergence is attained. The process is repeated for the lower confidence limit using negative values of $\lambda$ .

Convergence is controlled by 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:

$| l({\beta})-l_{0}| \leq \epsilon$

and

$({g} + \lambda{e_j})'{V}^{-1}({g} + \lambda{e_j}) \leq \epsilon$

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 $100(1-\alpha)$ % Wald confidence interval for $\beta_{j}$ is given by

$\hat{\beta}_j +- z_{1-\alpha/2}\hat{\sigma}_j$

where z_p is the 100pth percentile of the standard normal distribution, $\hat{\beta}_j$ is the maximum likelihood estimate of $\beta_{j}$ , and $\hat{\sigma}_j$ is the standard error estimate of $\hat{\beta}_j$ .

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