Web - based Supplementary Materials for “ A Note on MAR , Identifying Restrictions , Model Comparison , and Sensitivity Analysis in Pattern Mixture Models With and Without Covariates for Incomplete Data

for all j ≥ 2 and k < j. These conditionals are normal distributions since we assume Y |S is multivariate normal. Suppose that there exists j such that ps(yj|yj−1) is not the same for all s ≥ j. Then from (1), p≥j(yj|yj−1) will be a mixture of normals whereas pk(yj|yj−1) will be a normal distribution. Thus, Molenbergh’s condition will not be satisfied, i.e. the MAR constraints do not exist. On the other hand, if for all 1 < j < J , the conditional distributions ps(yj|yj−1) are identical for s ≥ j, then pk(yj|yj−1) and p≥j(yj|yj−1) are both normally distributed and the identification restrictions pk(yj|yj−1) = p≥j(yj|yj−1) will result in MAR. Corollary 1 Proof: Since Y1 is always observed (by assumption), S|Y ' S|Y1 implies that S|Y mis,Y obs ' S|Y obs, where Y mis and Y obs denote the missing and observed data respectively. This shows that MAR holds. On the other hand, MAR implies that