# Inference about Population Treatment Effects

## No Censoring

First suppose there is no right censoring, such that T is observed for all trial participants. A traditional (i.e., unweighted) approach to estimating treatment effects is a difference in outcome means between the two randomized arms of the trial. Let i = 1, ..., n + m index the trial and cohort participants. The within-trial estimator is defined as

where here and in the sequel = ^^”+”*. If trial participants are assumed

to constitute a random sample from the target population, it is straightforward to show Д _{T} is a consistent and asymptotically normal estimator of Д. By contrast, if we are not willing to assume trial participants are a random sample from the target population, then Д _{T} is no longer guaranteed to be consistent.

Below we consider two estimators of Д that do not assume trial participants are a random sample from the target population. Both estimators utilize sampling scores. Following Cole and Stuart (2010), assume a logistic regression model for the sampling scores such that *P(S* = 1| Z = z) = {1 + exp(-zp)}^{-1}, where p is a p x 1 vector of coefficient parameters. Let ^{p} denote the weighted maximum likelihood estimator of p, where each trial participant has weight П^{—1} = 1 and each individual in the cohort has weight П^{-}. ^{1}= *m/(N -* n), where N is the size of the target population (Scott and Wild 1986). Because the cohort is assumed to be a representative sample of the target, we inflate the size of the cohort to that of the target population. This allows for consistent estimation of the sampling scores. Let *P(S* = 1| Z = z) = w(z, p), w. = w(Zj, p), and w. = w(Z_{i;} p). The IPSW estimator (Buchanan et al. 2015; Cole and Stuart 2010) of the PATE is

Another approach for estimating the PATE uses stratification based on the sampling scores (O’Muircheartaigh and Hedges 2013; Tipton 2013; Tipton et al. 2014) and is computed in the following steps. First, p is estimated using a logistic regression model as described above and the estimated sampling scores *wi* are computed. These estimated sampling scores are used to form *L* strata. The difference of sample means within each stratum is computed among those in the trial. The PATE is then estimated as a weighted sum of the differences of sample means across strata. The stratum specific weights used in computing this weighted average equal estimates of the proportion of individuals in the target population within the stratum. Specifically, let n be the number of individuals in the trial in stratum l and m_{l} be the number of individuals in the cohort in stratum l. Let S_{il} = 1 denote trial participation for individual i in stratum l for i =1, ..., *(n _{l}* + m

_{l}) and l = 1, ..., L (and

*S*= 0 otherwise). If

_{il}*S*= 1, then let X

_{il}_{il}and T

_{il}denote the treatment assignment and outcome for individual i in stratum l; otherwise, if S

_{il}= 0, then let X

_{il}= T

_{il}= 0. The sampling score stratified estimator is defined as

where *w _{l}* =

*N*where

_{l}/N*N*= П=+

_{l}^{т}П

^{-1}and n

_{S l}is the weight for individual i

in stratum l.