The current statistical literature on causal inference is mostly concerned with binary or categorical exposures, even though exposures of a quantitative nature are frequently encountered in epidemiologic research. In this article, we review the available methods for estimating the dose-response curve for a quantitative exposure, which include ordinary regression based on an outcome regression model, inverse propensity weighting and stratification based on a propensity function model, and an augmented inverse propensity weighting method that is doubly robust with respect to the two models. We note that an outcome regression model often imposes an implicit constraint on the dose-response curve, and propose a flexible modeling strategy that avoids constraining the dose-response curve. We also propose two new methods: a weighted regression method that combines ordinary regression with inverse propensity weighting and a stratified regression method that combines ordinary regression with stratification. The proposed methods are similar to the augmented inverse propensity weighting method in the sense of double robustness, but easier to implement and more generally applicable. The methods are illustrated with an obstetric example and compared in simulation studies.
For semiparametric survival models with interval censored data and a cure fraction, it is often difficult to derive nonparametric maximum likelihood estimation due to the challenge in maximizing the complex likelihood function. In this paper, we propose a computationally efficient EM algorithm, facilitated by a gamma-poisson data augmentation, for maximum likelihood estimation in a class of generalized odds rate mixture cure (GORMC) models with interval censored data. The gamma-poisson data augmentation greatly simplifies the EM estimation and enhances the convergence speed of the EM algorithm. The empirical properties of the proposed method are examined through extensive simulation studies and compared with numerical maximum likelihood estimates. An R package “GORCure” is developed to implement the proposed method and its use is illustrated by an application to the Aerobic Center Longitudinal Study dataset.
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