Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models

Daniel O. Scharfstein, Andrea Rotnitzky, James M. Robins

Research output: Contribution to journalArticlepeer-review

626 Scopus citations


Consider a study whose design calls for the study subjects to be followed from enrollment (time t = 0) to time t = T, at which point a primary endpoint of interest Y is to be measured. The design of the study also calls for measurements on a vector Vt) of covariates to be made at one or more times t during the interval [0, T). We are interested in making inferences about the marginal mean μ0 of Y when some subjects drop out of the study at random times Q prior to the common fixed end of follow-up time T. The purpose of this article is to show how to make inferences about μ0 when the continuous drop-out time Q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables. In particular, we consider two models for the conditional hazard of drop-out given (V(T), Y), where V(t) denotes the history of the process Vt) through time t, t ∈ [0, T). In the first model, we assume that λQ(t|V(T), Y) exp(α0Y), where α0 is a scalar parameter and λ0(t|V(t)) is an unrestricted positive function of t and the process V(t). When the process Vt) is high dimensional, estimation in this model is not feasible with moderate sample sizes, due to the curse of dimensionality. For such situations, we consider a second model that imposes the additional restriction that λ0(t|V(t)) = λ0(t) exp(γ′0(t)), where λ0t) is an unspecified baseline hazard function, W(t) = w(t, V(t)), w(·,·) is a known function that maps (t, V(t)) to Rq, and γ0 is a q × 1 unknown parameter vector. When α0 ≠ 0, then drop-out is nonignorable. On account of identifiability problems, joint estimation of the mean μ0 of Y and the selection bias parameter α0 may be difficult or impossible. Therefore, we propose regarding the selection bias parameter α0 as known, rather than estimating it from the data. We then perform a sensitivity analysis to see how inference about α0 changes as we vary α0 over a plausible range of values. We apply our approach to the analysis of ACTG 175, an AIDS clinical trial.

Original languageEnglish (US)
Pages (from-to)1096-1120
Number of pages25
JournalJournal of the American Statistical Association
Issue number448
StatePublished - Dec 1 1999


  • Augmented inverse probability of censoring weighted estimators
  • Cox proportional hazards model
  • Identification; Missing data
  • Noncompliance; Nonparametric methods
  • Randomized trials
  • Sensitivity analysis
  • Time-dependent covariates

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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