Secondary outcome analysis for data from an outcome-dependent sampling design

Abstract

Outcome-dependent sampling (ODS) scheme is a cost-effective way to conduct a study. For a study with continuous primary outcome, an ODS scheme can be implemented where the expensive exposure is only measured on a simple random sample and supplemental samples selected from 2 tails of the primary outcome variable. With the tremendous cost invested in collecting the primary exposure information, investigators often would like to use the available data to study the relationship between a secondary outcome and the obtained exposure variable. This is referred as secondary analysis. Secondary analysis in ODS designs can be tricky, as the ODS sample is not a random sample from the general population. In this article, we use the inverse probability weighted and augmented inverse probability weighted estimating equations to analyze the secondary outcome for data obtained from the ODS design. We do not make any parametric assumptions on the primary and secondary outcome and only specify the form of the regression mean models, thus allow an arbitrary error distribution. Our approach is robust to second- and higher-order moment misspecification. It also leads to more precise estimates of the parameters by effectively using all the available participants. Through simulation studies, we show that the proposed estimator is consistent and asymptotically normal. Data from the Collaborative Perinatal Project are analyzed to illustrate our method.

Keywords: biased sampling; estimating equation; missing data; secondary analysis; semiparametric estimation; validation sample.

FIGURE 1

Sample relative efficiencies (SREs) comparing ${\widehat{\xi }}_{\mathit{AIPW}}$ and ${\widehat{\xi }}_{\mathit{SRS}}$ to ${\widehat{\xi }}_{\mathit{IPW}}$ in terms of estimating γ₁, under various combinations of simple random sample (SRS) and supplemental samples. The SRE is defined as $SR{E}_{\mathit{AIPW}:\mathit{IPW}}=\mathit{var}\left({\widehat{\xi }}_{\mathit{IPW}}\right)/\mathit{var}\left({\widehat{\xi }}_{\mathit{AIPW}}\right)$. The X-axis is the fraction of SRS in the validation sample: n₀/n_V

FIGURE 2

Sample relative efficiencies (SREs) comparing ${\widehat{\xi }}_{\mathit{AIPW}}$ and ${\widehat{\xi }}_{\mathit{SRS}}$ to ${\widehat{\xi }}_{\mathit{IPW}}$ in terms of estimating γ₁, under different values of ρ. The sample relative efficiency is defined as $SR{E}_{\mathit{AIPW}:\mathit{IPW}}=\mathit{var}\left({\widehat{\xi }}_{\mathit{IPW}}\right)/\mathit{var}\left({\widehat{\xi }}_{\mathit{AIPW}}\right)$. AIPW, augmented IPW; IPW, inverse probability weighted