Bayesian Framework for Estimation and Inference. The observed outcome Yi, which is only observed when Si = 1, is assumed to have a normal distribution, fg, within each of the principal strata and with Table 2.1: Individual observed likelihood by observed treatment and survival group Gi Observed Group Zi Si LL LD DL DD O(1,0) 1 0 - - πDL,ifDL,i πDD,ifDD,i O(0,1) 0 1 πLL,ifLL,i - πDL,ifDL,i - O(0,0) 0 0 - πLD,ifLD,i - πDD,ifDD,i g parameters and covariates that differ by strata. Specifically, the outcome dis- tributions are defined as Yg,i ∼ N (X1,gηg, σ2) for g ∈ LL, LD, DL. X1,LL includes the column for intercept, one or both of the treatment variables de- pending on the treatment assignment model considered, and the estimated propensity score corresponding ot the treatment assignment model. The out- come models for the LD and DL strata do not include any treatment covariates as the individuals with an observed outcome in each of these strata are either all treated or all untreated, respectively. Xxxxxxxxx, X0,XX and X1,DL include columns for intercept and propensity score only. Using the stratified distributions and the probability of each principal stratum, the structure of the observed data likelihood for any individual and for all possible combinations of Zi and Si is given in Table 2.1. Each cell value is the likelihood of the observed data if the values of the individuals’ strata are known. Thus the conditional probability of Gi = g given the observed data is the ratio of each cell to the total of that row. Rows O(1, 0) and O(0, 0) are included in this table for a comprehensive understanding of the possible com- binations of treatment and survival, but individuals who fall into these groups do not have outcome data that will contribute to the observed data likelihood since Si = 0 and thus Yi is unobserved. Therefore, individuals in this group will only contribute to the model for the probability of principal strata, which is reflected in the observed data likelihood in the Appendix I. Prior distributions for the specified parameters in the observed data like- lihood should be chosen carefully, with thought to distributions that may be informative, proper, and conjugate where appropriate. For this analysis, con- jugate multivariate normal and inverse-gamma distributions are assigned for g the prior distributions of the different forms of ηg and σ2 respectively. The prior distributions for αg are non-informative and are proportional to 1. De- tails of each prior distribution are provided in Appendix ...
Bayesian Framework for Estimation and Inference. The outcome Yi, which is only observed when Si = 1 and Mi = 0, is as- sumed to have a normal distribution, fg,i, within each of the principal strata and with parameters and covariates that differ by strata. Specifically, the out- come distributions are defined as Yg,i ∼ N (X1,gηg, σ2) for g ∈ LL, LD, DL. X1,LL includes the column for intercept, one or both of the treatment variables depending on the treatment assignment model considered, and the estimated propensity score corresponding to the treatment assignment model. The out- come models for the LD and DL strata do not include any treatment covariates as the individuals with an observed outcome in each of these strata are either Table 4.1: Individual observed likelihood by observed groups if Gi is known Gi Group LL LD DL DD O(1, 1, 0) (1 − φLL,1,i)× (1 − φLD,1,i)× - - πLL,ifLL,i πLD,ifLD,i O(1, 1, 1) φLL,1,iπLL,i φLD,1,iπLD,i - - O(1, 0, −) - - πDL,i πDD,i O(0, 1, 0) (1 − φLL,0,i)× - (1 − φDL,0,i)× - πLL,ifLL,i πDL,ifDL,i O(0, 1, 1) φLL,0,iπLL,i - φDL,0,iπDL,i - O(0, 0, −) - πLD,i - πDD,i all treated or all untreated respectively. Xxxxxxxxx, X0,XX and X1,DL include columns for intercept and propensity score only. Using the stratified outcome distributions, the probability of each princi- pal stratum, and the probability missing outcome the structure of the observed data likelihood for any individual and for all possible combinations of Zi, Si, and Mi is given in Table 4.1. Each cell value is the likelihood of the observed data if the values of the individual strata are known for the above described modeling approach 2 in which a model for the missing mechanism is intro- duced into the framework. If instead we employ the first approach, in which latent ignorability is assumed outright, we can drop all φg,z from the table. The conditional probability of Gi = g given the observed data is the ratio of each cell to the total of that row. Rows O(1, 0, −) and O(0, 0, −) are included in this table for a comprehensive understanding of the possible combinations of treatment and survival, but individuals who fall into these groups do not have outcome data that will contribute to the observed data likelihood since Si = 0. Therefore, individuals in this group will only contribute to the model for the probability of principal strata, which is reflected in the observed data likelihood in Appendix C.1. Prior distributions for the specified parameters in the observed data like- lihood should be chosen carefully, with th...
