Andrea Rotnitzky

Estimation of Regression Coefficients When Some Regressors Are Not Always Observed

Abstract In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...

Analysis of semiparametric regression models for repeated outcomes in the presence of missing data

Abstract We propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data. The proposed estimators do not require full specification of the likelihood. They can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random. These estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable. The likelihood-based parametric G-computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance. But because of possible model misspecification, the parametric G-computation algorithm estimator, in contrast with the proposed w...

The Prevention and Treatment of Missing Data in Clinical Trials

Missing data in clinical trials can have a major effect on the validity of the inferences that can be drawn from the trial. This article reviews methods for preventing missing data and, failing that, dealing with data that are missing.

Adjusting for Nonignorable Drop-Out Using Semiparametric Nonresponse Models

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 (V(T), Y), where V(t) denotes the history of the process V t) through time t, t ∈ [0, T). In the first model, we assume that λQ(t|V(T), Y) exp(α0 Y), where α0 is a scalar paramet...

Semiparametric Efficiency in Multivariate Regression Models with Missing Data

Abstract We consider the efficiency bound for the estimation of the parameters of semiparametric models defined solely by restrictions on the means of a vector of correlated outcomes, Y, when the data on Y are missing at random. We show that the semiparametric variance bound is the asymptotic variance of the optimal estimator in a class of inverse probability of censoring weighted estimators and that this bound is unchanged if the data are missing completely at random. For this case we study the asymptotic performance of the generalized estimating equations (GEE) estimators of mean parameters and show that the optimal GEE estimator is inefficient except for special cases. The optimal weighted estimator depends on unknown population quantities. But for monotone missing data, we propose an adaptive estimator whose asymptotic variance can achieve the bound.

Recovery of Information and Adjustment for Dependent Censoring Using Surrogate Markers

A class of tests and estimators for the parameters of the Cox proportional hazards model, the accelerated failure time model, and a model for the effect of treatment on the mean of a response variable of interest are proposed that use surrogate marker data to recover information lost due to independent censoring and to adjust for bias due to dependent censoring in randomized clinical trials. We construct an adaptive test that (i) is asymptotically distribution free under the null hypothesis of no treatment effect on survival, (ii) incorporates surrogate marker data, and (iii) is guaranteed to be locally more powerful than the ordinary log-rank test against proportional hazards alternatives when the baseline failure time distribution is Weibull. The proposed test is shown to outperform the log-rank test in a series of simulation experiments. We also prove the optimal estimator within our class is semiparametric efficient by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model with data missing at random, and then deriving a representation for the efficient score in this more general problem.

Semiparametric Regression for Repeated Outcomes with Nonignorable Nonresponse

Abstract We consider inference about the parameter β* indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values; that is, nonresponse is nonignorable. We propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal (CAN) for estimating β* when the response probabilities can be parametrically modeled and a CAN estimator exists. The proposed estimators do not require full specification of a parametric likelihood, and their computation does not require numerical integration. Our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse. We show that our class essentially consists of all CAN estimators of β*. We also show that the asymptotic variance of the optimal estimator in our ...

Sensitivity Analysis for Selection bias and unmeasured Confounding in missing Data and Causal inference models

In both observational and randomized studies, subjects commonly drop out of the study (i.e., become censored) before end of follow-up. If, conditional on the history of the observed data up to t, the hazard of dropping out of the study (i.e., censoring) at time t does not depend on the possibly unobserved data subsequent to t, we say drop-out is ignorable or explainable (Rubin, 1976). On the other hand, if the hazard of drop-out depends on the possibly unobserved future, we say drop-out is non-ignorable or, equivalently, that there is selection bias on unobservables. Neither the existence of selection bias on unobservables nor its magnitude is identifiable from the joint distribution of the observables. In view of this fact, we argue that the data analyst should conduct a “sensitivity analysis” to quantify how one’s inference concerning an outcome of interest varies as a function of the magnitude of non-identifiable selection bias.

Estimation and extrapolation of optimal treatment and testing strategies

We review recent developments in the estimation of an optimal treatment strategy or regime from longitudinal data collected in an observational study. We also propose novel methods for using the data obtained from an observational database in one health-care system to determine the optimal treatment regime for biologically similar subjects in a second health-care system when, for cultural, logistical, or financial reasons, the two health-care systems differ (and will continue to differ) in the frequency of, and reasons for, both laboratory tests and physician visits. Finally, we propose a novel method for estimating the optimal timing of expensive and/or painful diagnostic or prognostic tests. Diagnostic or prognostic tests are only useful in so far as they help a physician to determine the optimal dosing strategy, by providing information on both the current health state and the prognosis of a patient because, in contrast to drug therapies, these tests have no direct causal effect on disease progression. Our new method explicitly incorporates this no direct effect restriction.

Likelihood-based inference with singular information matrix

1Department of Biostatistics, Harvard School of Public Health, 655 Huntington Avenue, Boston MA 02115, USA. *E-mail: andrea@hsph.harvard.edu 2Nuffield College, Oxford OX1 1NF, UK 3Centro Nazionale Universitario di Calcolo Elettronico, Consiglio Nazionale delle Ricerche, Via Santa Maria 36, I-56126 Pisa, Italy 4Department of Epidemiology, Harvard School of Public Health, 677 Huntington Avenue, Boston MA 02115, USA