Recently proposed double-robust estimators for a population mean from incomplete data and for a finite number of counterfactual means can have much higher efficiency than the usual double-robust estimators under misspecification of the outcome model. In this paper, we derive a new class of double-robust estimators for the parameters of regression models with incomplete cross-sectional or longitudinal data, and of marginal structural mean models for cross-sectional data with similar efficiency properties. Unlike the recent proposals, our estimators solve outcome regression estimating equations. In a simulation study, the new estimator shows improvements in variance relative to the standard double-robust estimator that are in agreement with those suggested by asymptotic theory.

Summary: We study a class of parameters with the so-called mixed bias property. For parameters with this property, the bias of the semiparametric efficient one-step estimator is equal to the mean of the product of the estimation errors of two nuisance functions. In nonparametric models, parameters with the mixed bias property admit so-called rate doubly robust estimators, i.e., estimators that are consistent and asymptotically normal when one succeeds in estimating both nuisance functions at sufficiently fast rates, with the possibility of trading off slower rates of convergence for the estimator of one of the nuisance functions against faster rates for the estimator of the other nuisance function. We show that the class of parameters with the mixed bias property strictly includes two recently studied classes of parameters which, in turn, include many parameters of interest in causal inference. We characterize the form of parameters with the mixed bias property and of their influence functions. Furthermore, we derive two functional loss functions, each being minimized at one of the two nuisance functions. These loss functions can be used to derive loss-based penalized estimators of the nuisance functions.

We consider the doubly robust estimation of the parameters in a semiparametric conditional odds ratio model. Our estimators are consistent and asymptotically normal in a union model that assumes either of two variation independent baseline functions is correctly modelled but not necessarily both. Furthermore, when either outcome has finite support, our estimators are semiparametric efficient in the union model at the intersection submodel where both nuisance functions models are correct. For general outcomes, we obtain doubly robust estimators that are nearly efficient at the intersection submodel. Our methods are easy to implement as they do not require the use of the alternating conditional expectations algorithm of Chen (2007).

We propose a new class of models for making inference about the mean of a vector of repeated outcomes when the outcome vector is incompletely observed in some study units and missingness is nonmonotone. Each model in our class is indexed by a set of unidentified selection-bias functions which quantify the residual association of the outcome at each occasion t and the probability that this outcome is missing after adjusting for variables observed prior to time t and for the past nonresponse pattern. In particular, selection-bias functions equal to zero encode the investigator's a priori belief that nonresponse of the next outcome does not depend on that outcome after adjusting for the observed past. We call this assumption sequential explainability. Since each model in our class is nonparametric, it fits the data perfectly well. As such, our models are ideal for conducting sensitivity analyses aimed at evaluating the impact that different degrees of departure from sequential explainability have on inference about the marginal means of interest. Although the marginal means are identified under each of our models, their estimation is not feasible in practice because it requires the auxiliary estimation of conditional expectations and probabilities given high-dimensional variables. We henceforth discuss the estimation of the marginal means under each model in our class assuming, additionally, that at each occasion either one of the following two models holds: a parametric model for the conditional probability of nonresponse given current outcomes and past recorded data or a parametric model for the conditional mean of the outcome on the nonrespondents given the past recorded data. We call the resulting procedure 2^T-multiply robust as it protects at each of the T time points against misspecification of one of these two working models, although not against simultaneous misspecification of both. We extend our proposed class of models and estimators to incorporate data configurations which include baseline covariates and a parametric model for the conditional mean of the vector of repeated outcomes given the baseline covariates.

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 class attains the semiparametric variance bound for the model. When the model for nonresponse is richly parameterized, joint estimation of the regression parameter β* and the nonresponse model parameter τ* which encodes the magnitude of nonignorable selection bias, may be difficult or impossible. Therefore we propose regarding the selection bias parameter τ* as known, rather than estimating it from the data. We then perform a sensitivity analysis that examines how inference concerning the regression parameter β* changes as we vary τ* over a range of plausible values. We apply our approach to the analysis of ACTG Trial 002, an AIDS clinical trial.