Table 1.
Cell probabilities in a generic stratum s
| 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 + Δs | (1 - φ1s) | |
| (α0s + Δs) (1 - φ1s) + (α1s + Δs) φ1s | ||||
| 1 | α1s + Δs | φ1s | ||
| 0 | 0 | α0s | (1 - φ0s) | |
| α0s(1 - φ0s) + α1s φ0s | ||||
| 1 | α1s | φ0s | ||
| difference between randomization groups: | Δs + ψsεs, where εs = φ1s - φ0s, ψs = α1s - α0s | |||
Under missing at random (MAR), the probabilities in the third column are the same for subjects not missing outcome as for all subjects, so Δs 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, Δs + ψsεs. To bound the overall bias Σsψsεspr (S = s), we specify an upper bound for εs based only on the fraction missing and a plausible value for the maximum of ψs based on the estimates of ψs if an observed covariate were missing.