Improved Horvitz–Thompson Estimation of Model Parameters from Two-phase Stratified Samples: Applications in Epidemiology

Abstract

The case-cohort study involves two-phase sampling: simple random sampling from an infinite super-population at phase one and stratified random sampling from a finite cohort at phase two. Standard analyses of case-cohort data involve solution of inverse probability weighted (IPW) estimating equations, with weights determined by the known phase two sampling fractions. The variance of parameter estimates in (semi)parametric models, including the Cox model, is the sum of two terms: (i) the model based variance of the usual estimates that would be calculated if full data were available for the entire cohort; and (ii) the design based variance from IPW estimation of the unknown cohort total of the efficient influence function (IF) contributions. This second variance component may be reduced by adjusting the sampling weights, either by calibration to known cohort totals of auxiliary variables correlated with the IF contributions or by their estimation using these same auxiliary variables. Both adjustment methods are implemented in the R survey package. We derive the limit laws of coefficients estimated using adjusted weights. The asymptotic results suggest practical methods for construction of auxiliary variables that are evaluated by simulation of case-cohort samples from the National Wilms Tumor Study and by log-linear modeling of case-cohort data from the Atherosclerosis Risk in Communities Study. Although not semiparametric efficient, estimators based on adjusted weights may come close to achieving full efficiency within the class of augmented IPW estimators.

HIV Protective Efficacy and Correlates of Tenofovir Blood Concentrations in a Clinical Trial of PrEP for HIV Prevention

“…Information from covariates in all active arm participants was used to predict plasma drug concentrations from pill count coverage, age, and occupation. These models, through influence functions, were used to adjust the sampling for imbalances in the cohort selected for plasma testing 15,16. Estimates of HIV protective effect could be affected by potential confounding with factors associated with adherence and HIV risk, therefore the analysis was conducted both with drug levels as the only covariate and also adjusted for covariates found to correlate with adherence (gender, age, and any sex with study partner7).…”

Calibration weighted estimation of semiparametric transformation models for two‐phase sampling

“…The intuition behind weight calibration is to perturb the inverse probability weights just a little, but enough so that the weighted sum of certain variables, termed calibration variables or adjustment variables , matches the corresponding unweighted sums for all subjects in the cohort. We choose the calibration variables to be the asymptotic linear expansion of $\hat{\theta }$, namely ${\left[\partial \left\{{\sum }_{ij}{\widehat{e}}_{ij}\left(\theta \right)\right\}∕\partial \theta \right]}^{-1}{\hat{e}}_{ij}\left(\theta \right)$, which corresponds to the efficient influence function under a maximum likelihood estimating procedure used by Breslow et al [2]. Such calibration variables are not known for the whole cohort, but can be approximated if there exist phase I variables that are correlated with the missing phase II covariates.…”

“…Such calibration variables are not known for the whole cohort, but can be approximated if there exist phase I variables that are correlated with the missing phase II covariates. Specifically, following Breslow et al [2, 1] and Kulich and Lin [22], we propose the following three-step procedure.…”

“…Instead of calibrating the calibration variables to their overall totals as we have done, one could also calibrate to the subtotals within each pair of strata combinations. Following the argument of Breslow et al [2] (Section 3), this may be more efficient in theory. But as the number of samples for estimating each λ decreases, Breslow et al reported that finite sample performance may actually decrease.…”

“…For analyzing time to event data, Breslow et al [1, 2] developed improved IPW methods for Cox regression models [9]. Our research is motivated by the need to go beyond the Cox model when analyzing data from vaccine clinical trials.…”