Fitting regression models to case-control data by maximum likelihood

Efficient estimation in multi-phase case-control studies

“…In this paper we present an efficient semiparametric maximum-likelihood solution for multiphase population-based case-control studies that unifies and extends previous work by Prentice & Pyke (1979), Breslow & Cain (1988), Scott & Wild (1997), Breslow & Holubkov (1997), and others. In the main body of the paper we present the results in a way that is intended to give the reader an appreciation of the nature of the problem, the nature of the solution and how it can be implemented.…”

“…The conditional maximum likelihood approach developed by Breslow & Cain (1988) and the semiparametric profile likelihood approach developed by Scott & Wild (1997) are both closely related to a simple prospective scheme in which units are examined sequentially, retained in the sample with known probability r hi if (Y = h, X (1) = x (1) i ) and otherwise discarded. Under this scheme, the probability that Y = h, given that (X (1) = x (1) i , X = x) and that the unit is selected, would be…”

“…This paper unifies and extends the methodology for special cases of multiphase case-control studies given by Prentice and Pyke (1979), Scott and Wild (1997), Breslow and Holubkov (1997), Lawless et al (1999) and to provide a general semiparametric-efficient solution to the problem of analyzing multi-phase case-control studies that collect discrete or continuous covariate information at the last stage of sampling and discrete covariate information at previous phases. The methods are relatively easy to implement, particularly in the case of logistic regression where we have shown how to make relatively minor modifications to the score vector and information matrix for ordinary prospective logistic regression.…”

“…In this paper we discuss the analysis of multi-phase, or multi-stage, case-control studies and present an efficient semiparametric maximum-likelihood approach that unifies and extends earlier work, including the seminal case-control paper by Prentice & Pyke (1979) as well as work by Breslow & Cain (1988), Scott & Wild (1997), Breslow & Holubkov (1997), and others. The theoretical derivations apply to arbitrary binary regression models but we present results for logistic regression and show that the approach can be implemented by including additional intercept terms in the logistic model and then making some simple corrections to the score and information equations from the prospective loglikelihood.…”

“…By adapting the results of Scott & Wild (1997), the profile likelihood estimator, β, can be obtained as follows. Take the likelihood that would be appropriate for fitting the augmented logistic model (3) prospectively, add forcing terms c i (…”