Efficient estimation in multi-phase case-control studies

Abstract

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.

Adjusting for Non-Response in Population-Based Case-Control Studies

“…Finally, it is particularly noteworthy that even including all of the information we have available in the selection and response models is substantially less efficient than the semi‐parametric maximum likelihood method of Lee et al (2010) for estimating β 3 ; the relative efficiency is 72% using estimating equations derived from CML and worse at only about 57% with estimating equations derived from weighting. Put another way, there is quite substantial information captured via and the selection and response model parameters in the completely observed data, as discussed in the previous subsection, because incorporating considerably improves the efficiency of estimation of β .…”

“…In particular, when the non‐respondents are fully sampled, the situation is formally equivalent to a three‐phase case‐control study in which values of Y and x 1 are obtained from all units in the cohort at the first phase, values of x 2 are obtained for all those units selected for the case‐control study at the second phase, and the remaining components of x are obtained from the respondents, “chosen” with probabilities depending on y , x 1 , and x 2 , at the third phase. In the special case where both x 1 and x 2 can take only a finite number of possible values, efficient estimators for three‐phase studies have been developed by Lee et al (2010).…”

“…After some rearrangement, the estimating equations in Lee et al (2010) defining the efficient estimators can be expressed in a form that has a close relationship with the CML estimator (see Example 2 in ) and uses elements of the estimating equations in the previous section. The CML estimator is defined by setting W ( x i , y i ; β , π 1 , π 2 ) =∂ log f * ( y i | x i ; β , π 1 , π 2 )/∂ β in .…”

“…Efficiency can be increased by considering the finite‐population data as the first‐phase and the study data as the second phase. For further discussion and references see Scott & Wild (2009) and Lee et al (2010). Key early references include White (1982) and Breslow & Cain (1988).…”

Using the Whole Cohort in the Analysis of Case-Control Data

“…Methods to recover some of this information, based on theory and methodology for two and three phase stratified sampling designs, have been shown in a few idealized situations to offer substantially more precise parameter estimates in comparison with standard methods. [5,16] Less is known about their performance in more realistic situations typified by ARIC and WHI.…”