On the Semiparametric Efficiency of the Scott-Wild Estimator under Choice-Based and Two-Phase Sampling

Suppose that for each of a number of subjects, we measure a response y and a vector of covariates x, in order to estimate the parameters β of a regression model which describes the conditional distribution of y given x. If we have sampled directly from the conditional distribution, or even the joint distribution, we can estimate β without knowledge of the distribution of the covariates. In the case of a discrete response, which takes one of J values y1, . . . , yJ , say, we often estimate β using a case-control sample, where we sample from the conditional distribution of X given Y = yj . This is particularly advantageous if some of the values yj occur with low probability. In case-control sampling, the likelihood involves the distribution of the covariates, which may be quite complex, and direct parametric modelling of this distribution may be too difficult. To get around this problem, the covariate distribution can be treated non-parametrically. In a series of papers, (Scott and Wild 1986, 1997, 2001, Wild 1991) Scott and Wild have developed an estimation technique which yields a semi-parametric estimate of β. They dealt with the unknown distribution of the covariates by profiling it out of the likelihood, and derived a set of estimating equations whose solution is the semi-parametric estimator of β. This technique also works well for more general sampling schemes, for example for two-phase outcome-dependent stratified sampling. Here, the sample space is partitioned into S disjoint strata which are defined completely by the values of the response and possibly some of the covariates. In the first phase of sampling, a prospective sample of size N is taken from the joint distribution of x and y, but only the stratum the individual belongs to is observed. In the second phase, for s = 1, . . . , S, a sample of size n 1 is selected from the n (s) 0 individuals in stratum s who were selected in the first phase, and the rest of the covariates are measured. Such a sampling scheme can reduce the cost of studies by confining the measurement of expensive variables to the