Propensity Score Methods. The estimation of propensity scores and their use in removing bias when treatment assignment is binary has been well established in the literature (Xxxxxxxxx & Xxxxx 1983, Xxxxx & Xxxxxx 1996). In these cases, the defi- nition of the propensity score is the probability of receiving a treatment con- ditional on a set of observed variables X, b(X) = P (Z = 1|X). However, when the treatment is non-binary, the probability of treatment may either be difficult to quantify or is not the best balancing score. Two options for flexible propensity scores that have been proposed in recent literature include the generalized propensity score (GPS) of Hirano and Imbens (2004) and the P-Function of Imai and Van Dyk (2004). Both methods are applicable to a wide variety of treatment assignment models, including non-parametric and semi-parametric models. Generalized Propensity Scores GPS is defined as R = r(Z, X) where r(z, x) = fZ|X(z|x). In words, the GPS is the density function for treatment assignment evaluated at the observed treatment and covariates. In practice, it may be suitable to reduce Rˆi to the linear predictor of a regression model, for ease of computation and analysis. Hirano and Imbens (2004) propose that the GPS satisfies both the proper- ties of a balancing score as well as weak unconfoundedness, the conditional independence of potential outcomes and a single value of treatment given the GPS evaluated at that treatment and observed covariates. This allows the au- thors to state further that the causal quantities of interests are unbiased when conditional on the GPS. Specifically, the dose response function of treatment is defined as E[Y (t)] = Er(z,X){E[Y (t)|z, r(z, X)]}. In addition Rˆi could be used for matching, stratification, or weighting, though these applications are not described in detail by Hirano and Imbens and have been relatively less explored. In cases such as our motivating data, time to a specific treatment such as PEG, TZ, is of interest and a Xxx Proportional Hazards (PH) model for TZ, with or without time-dependent covariates, may be utilized for estimating the Generalized Propensity Score. While the formal definition implies that the GPS would be the density function of time to treatment evaluated at a time of interest t, r(t, x) = fTZ|X(t|x) = hTZ (t|x)STZ (t|x) where hTZ (t|x) is the hazard function and STZ (t|x) is the survival function for TZ, evaluation of hTZ (t|x) is sufficient for the GPS. The time of interest can be baseline (...
