On Parametrization, Robustness and Sensitivity Analysis in a Marginal Structural Cox Proportional Hazards Model for Point Exposure

In this paper, some new statistical methods are proposed, for making inferences about the parameter indexing a Cox proportional hazards marginal structural model for point exposure. Under the key assumption that unmeasured confounding is absent, we propose a new class of closed-form estimators that are doubly robust in the sense that they remain consistent and asymptotically normal for the e¤ect of treatment provided the marginal structural model is correctly speci…ed and, at least one of the following holds: (i) a model for the treatment assignment mechanism is correctly speci…ed or, (ii) a model for part of the observed data likelihood not involving the treatment assignment mechanism is correctly speci…ed. In order to ensure that condition (ii) provides a genuine opportunity for valid inference, we propose a new parametrization of the observed data law, that is congenial with the marginal proportional hazards assumption. In addition, because the assumption of no unmeasured confounding can seldom be established with certainty with observational data, a second contribution of the current paper is to propose a general framework for estimation without the assumption of no unmeasured confounding. For this purpose, a sensitivity analysis technique is developed, that allows an investigator to assess, under model (i), the extent to which unmeasured confounding may alter inferences about causal e¤ects. The current article concerns the development of improved statistical methods for making inferences about the parameter indexing a Cox proportional hazards marginal structural model for point exposure. Under the key assumption that unmeasured confounding is absent, inverse-probability-of treatment-weighted estimation and augmented inverse-probability-of treatment-weighted estimation have previously been described, to obtain consistent and asymptotically normal estimators for this model. Unfortunately, estimators obtained using these previous methods rely on the crucial assumption that the treatment mechanism is consistently estimated. Furthermore, the assumption that there is no unmeasured confounding may be inappropriate, if the investigator fails to collect at least one key risk factor which predicts treatment assignment. In observational studies, investigators tend to collect and adjust for a large number of confounders, precisely to minimize the presence of unmeasured confounding. As a result, whether succesful in their e¤ort to reduce unmeasured confounding or not, the curse of dimensionality implies that for good …nite sample performance, parametric or semi-parametric models must be used to estimate the treatment mechanism. In the event that this latter model is incorrect, the corresponding inferences are likely to be 1 Hosted by The Berkeley Electronic Press severely biased even when all confounders are observed. As a remedy, here we develop a new class of estimators that are doubly robust when confounding is absent. In a marginal structural model, an estimator is doubly robust if it remains consistent and asymptotically normal for the e¤ect of treatment provided that there is no unmeasured confounding, the marginal structural model is correctly speci…ed and, at least one of the following holds; i) the model for the treatment assignment mechanismM1 is correctly speci…ed or, ii) the model for part of the observed data likelihood not involving the treatment assignment mechanismM2 is correctly speci…ed. Doubly robust estimation extends inverse-probability-of treatment weighted estimation of marginal structural models and o¤ers at least two major advantages over the latter which we emphasize. Firstly, as stated above, doubly robust estimation is more robust to model misspeci…cation in the estimated weights used in inverse-probability weighting, and thus, an estimator that is doubly robust, is consistent and asymptotically normal under many more laws than one that is not. Secondly, doubly robust estimation can lead to more e¢ cient estimation than inverse-probabilityof-treatment-weighting. Existing literature on the theory of double robustness is too rich to summarize here; but see Sharfstein, Rotnitzky and Robins (1999), Robins (2000), Robins and Rotnizky (2001), van der Laan and Robins (2003) and Tsiatis (2006). van der Laan and Robins (2003). Yu and van der Laan (2003), and Tchetgen Tchetgen (2006) previously considered doubly robust methods for the Cox proportional hazards model in a more general setting which allowed for time-varying exposure and time-varying confounding. The current setting di¤ers in two crucial ways. First, because studies with a point exposure play a pivotal role in several …elds, including epidemiology, economics, biostatistics, political science and other social sciences, here we focus on the setting of a point exposure. The time-varying setting will be addressed elsewhere. Second, whereas all previous methods used a pooled logistic regression approximation to the Cox regression model, no such approximation is needed here, and inferences are obtained for a structural regression model in which the proportional hazards assumption is exact. Lastly, our proposed estimators are closed form, which is generally not the case in the time-varying setting, even under a pooled logistic regression approach. We emphasize that the proposed approach is new and does not immediately follow from available results on doubly robust estimation of pooled logistic regression. Whilst the assumption of no unmeasured confounding may be enforced in an experimental context, mainly by randomizing treatment; there is seldom a guaranty that this assumption holds in an observational study. In addition, because this latter assumption is empirically untestable from nonexperimental data, a second contribution of the current paper is to propose a general framework for estimation without the assumption of no unmeasured confounding. For this purpose, a new sensitivity analysis technique is developed, that allows an investigator to assess, under model M1; and thus under the assumption that the treatment process is consistently estimated given the observed data, the extent to which unmeasured confounding may alter inferences about causal e¤ects. We emphasize that to the best of our knowledge, there currently exist no sensitivity analysis methodology for unmeasured counfounding under a marginal Cox regression model, therefore the current paper aims to directly address this important gap in the causal inference literature.