BK-plot of bias from an unobserved binary covariate among subjects not missing outcome. The upper diagonal line is the probability of outcome among subjects not missing outcome in randomization group Z = 1. The lower diagonal line is the probability of outcome among subjects not missing outcome in randomization group group Z = 0. For subjects in group 0, the fraction with X = 1 is φ0s and the probability of outcome is indicated by point A. For subjects in group 1, the fraction with X = 1 is φ1s and the probability of outcome is indicated by point B. The true treatment effect Δs is the difference between the diagonal lines. The apparent treatment effect Δs is the vertical distance between points A and B, which equals Δ + ψsεs, where εs = φ1s - φ0s and ψs = α1s - α0s = the slope of each diagonal line. 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.
Baker and Freedman BMC Medical Research Methodology 2003 3:8 doi:10.1186/1471-2288-3-8