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 = 1z, s, x, R = 1) 
pr(xz, s, R = 1) 
pr(Y = 1z, s, R = 1) 
= pr(Y = 1z, s, x) if MAR 



1 
0 
α_{0s }+ Δ_{s} 
(1  φ_{1s}) 

(α_{0s }+ Δ_{s}) (1  φ_{1s}) + (α_{1s }+ Δ_{s}) φ_{1s} 

1 
α_{1s }+ Δ_{s} 
φ_{1s} 



0 
0 
α_{0s} 
(1  φ_{0s}) 

α_{0s}(1  φ_{0s}) + α_{1s }φ_{0s} 

1 
α_{1s} 
φ_{0s} 



difference between randomization groups: 
Δ_{s }+ ψ_{s}ε_{s}, where ε_{s }= φ_{1s } φ_{0s}, ψ_{s }= α_{1s } α_{0s} 



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. 

Baker and Freedman BMC Medical Research Methodology 2003 3:8 doi:10.1186/1471228838 