Saddlepoint Approximations of Marginal Densities and Confidence Intervals in the Logistic Regression Measurement Error Model

The aim of this note is to use the saddlepoint approximation by Gatto and Ronchetti (1996) in the logistic regression measurement error model, with medical data sets. We approximate the marginal densities of the estimator and use them for obtaining second-order accurate confidence intervals (with the bias-correction proposed by Efron, 1987). The logistic regression measurement error model relates a binary response variable to predictor variables or covariates which are measured with errors. For example, it is not possible to measure the exact blood glucose level because some random variations due to the accuracy of the measure instrument, the contents of the last meal, or the blood sample preservation affect the accuracy of the measure. When an imperfect measurement (proxy) is used instead of the true predictor, an asymptotically unbiased estimator for the regression coefficients was proposed by Stefanski and Carroll (1985). Saddlepoint approximations lead to accurate approximations of densities and tail probabilities. The introduction of the saddlepoint approximation into statistics goes back to Daniels (1954); for a general reference see Field and Ronchetti (1990); for recent applications see Wang (1993), Ronchetti and Welsh (1994), Li et al. (1995), and Davison, Hinkley, and Wharton (1995). Section 2 summarizes the derivation of the unbiased estimator and the steps leading to the saddlepoint approximation. Numerical examples are presented in Section 3. Comparisons with standard normal approximations and with Monte Carlo bootstrap approximations are also shown. Some final remarks are given in Section 4. The Monte Carlo bootstrap approximations are based on 10,000 resamplings; these approximations do not change if we increase the number of resamplings. In our comparisons we do not consider the Edgeworth approximation, which is often inaccurate in the tails of the distribution (see Field and Ronchetti, 1990).