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Table 1 Cell probabilities in a generic stratum s

From: A simple method for analyzing data from a randomized trial with a missing binary outcome

randomization group unobserved covariate probability of outcome given group, unobserved covariate, s, not missing probabilitity of unobserved covariate given group, s, not missing probability of outcome given group, s not missing
Z X pr(Y = 1|z, s, x, R = 1) pr(x|z, s, R = 1) pr(Y = 1|z, s, R = 1)
   = pr(Y = 1|z, s, x) if MAR   
1 0 α0s + Δ s (1 - φ1s )  
    0s + Δ s ) (1 - φ1s ) + (α1s + Δ s ) φ1s
  1 α1s + Δ s φ1s  
0 0 α0s (1 - φ0s )  
     α0s (1 - φ0s ) + α1s φ0s
  1 α1s φ0s  
difference between randomization groups: Δ s + ψ s ε s , where ε s = φ1s - φ0s , ψ s = α1s - α0s
  1. Under missing at random (MAR), the probabilities in the third column are the same for subjects not missing outcome as for all subjects, so Δ s represents the true treatment effect, which is the same for both levels of x. Because the distribution of x is different among subjects not missing outcome in each randomization group, the apparent treatment effect is the difference in weighted averages over x in the last column, namely, Δ s + ψ s ε s . To bound the overall bias Σ s ψ s ε s pr (S = s), we specify an upper bound for ε s based only on the fraction missing and a plausible value for the maximum of ψ s based on the estimates of ψ s if an observed covariate were missing.