Time-Varying GPS and Propensity Process Clause Samples

Time-Varying GPS and Propensity Process. Inclusion of time-varying covariates in the ▇▇▇ PH model allows for the evaluation of the GPS at the time of treatment and the entire Propensity Process from the baseline to the time of treatment, and hence the removal of selection bias due to unbalanced time-varying covariates. The model, pre- sented in equation (3.4), incorporates both the covariates that are measured at baseline and remain static over time (X1), as well as those that have repeated measurements over time (X2(t)). hT V (t|X(t)) = h0(t)exp .βT X1 + β X2(t)Σ . (3.4) ^ ^ We denote the parameter estimates by βT V,1 and βT V,2. We use model (3.4) with time-dependent covariates to define the estimated GPS and the estimated Propensity Process for matching. For subject i, the estimated GPS is defined as the the linear predictor of this model evaluated at the time of treatment, ^ ^ i. e., GPSIP,i = ▇^▇ ▇,▇▇▇ + ▇^▇ ▇,▇▇▇(▇▇,▇) and the estimated Propensity Process ˆ PPi = {θi(t) = βT V,1X1 + βT V,2X2(t); t ∈ [0, TZ,i]}, is the full functional pro- cess from baseline until time of treatment. Computation of these propensity scores is not straightforward. First, the time-dependent covariates X2(·) are often not measured at time of treatment for treated individuals. Secondly, untreated individuals within the risk set are unlikely to have covariate mea- surements at the exact time of treatment for their potential treated match. Thus, within the matching process, all individuals are subject to interpolation of the covariate value at a specific time by a model that estimates the covariate process over time. To estimate or interpolate the covariate process, we model each time- dependent covariate using a mixed model with both fixed and random effects of the time variable. This allows a predictive curve to be fit for the covariate process of each individual from baseline until the end of follow-up, thereby providing values for each covariate over the continuous support of the follow up time. For individual i (i = 1, . . . , n) at times tij (j = 1, . . . , mi), the model for covariate k (k = 1, . . . , p2) in X2,i is given in equation (3.5). X(k)(tij) = µ(k)(tij) + ν(k)(tij) + s(k), (3.5) 2,i i ii where µ(k)(·) is the population average nonparametric curve (i.e., the fixed effect), ν(k)(·) is the subject-specific curve (i.e., the random effect), and s(k) are independent random errors. To estimate µ(k)(·) and ν(k)(·), we can use the approach of basis expansion, that is, write µ(k)(·) = b(·)T γ(k) and ν(k)(·) =...