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# Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models

Daniel O. Scharfstein, Andrea Rotnitzky and James M. Robins
Journal of the American Statistical Association
Vol. 94, No. 448 (Dec., 1999), pp. 1096-1120
DOI: 10.2307/2669923
Stable URL: http://www.jstor.org/stable/2669923
Page Count: 25
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## Abstract

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 V(t) 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 (V̄(T), Y), where V̄(t) denotes the history of the process V(t) through time t, t ∈ [0, T). In the first model, we assume that $\lambda_Q(t\mid\bar V(T), Y) = \lambda_0(t\mid\bar V(t))$ exp(α0 Y), where α0 is a scalar parameter and $\lambda_0(t\mid\bar V(t))$ is an unrestricted positive function of t and the process $\bar V(t)$. When the process $\bar V(t)$ 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 $\lambda_0(t\mid\bar V(t)) = \lambda_0(t)$ exp(γ'0 W(t)), where λ0(t) is an unspecified baseline hazard function, W(t) = w(t, V̄(t)), w(·, ·) is a known function that maps (t, V̄(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 μ0 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.

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