1 2 9 4 P. C. AUSTIN subjects. We then assumed that the following logistic regression model related the probability of treatment to these 10 baseline covariates: logit( pi,treat) = 0,treat + L X1,i + L X2,i + L X3,i + M X4,i + M X5,i + M X6,i + H X7,i + H X8,i + H X9,i + VH X10,i We then generated a treatment status indicator (Zi ) for each subject from a Bernoulli distribution with subject-speciﬁc probability equal to pi,treat. Those subjects with Zi = 1 denote the treated subjects in whom the ATT is deﬁned. We assumed that the following logistic regression model related the probability of the outcome to these covariates and an indicator variable (Z) denoting treatment: logit( pi,outcome) = 0,outcome + Zi + L X1,i + L X2,i + L X3,i + M X4,i + M X5,i + M X6,i + H X7,i + H X8,i + H X9,i + VH X10,i In the above regression model, pi,outcome denotes the probability of the outcome for the i th subject and denotes the log-odds ratio relating treatment to the outcome. We then generated subject-speciﬁc outcomes from a Bernoulli distribution with probability pi,outcome. The regression coefﬁcients for the baseline covariates in the above two regression models were set as follows: L = log(1.1), M = log(1.25), H = log(1.5) and VH = log(2). These are intended to reﬂect low, medium, high and very high effect sizes. We ﬁxed the value of 0,treat = −1.344090 so that approximately 25 per cent of the subjects would be treated. We ﬁxed the value of 0,outcome = −1.098537 so that the probability of the event occurring in the population if all subjects were untreated would be approximately 0.29. To induce a risk difference of 0, was set to be 0. For risk differences of −0.02, −0.05, −0.10 and −0.15, the required value of equaled log(0.90619), log(0.7795362), log(0.6001387) and log(0.45292), respectively. The reader is referred elsewhere for a more detailed explanation of how these values of were determined [12]. The above scenario assumed that the 10 covariates (X1 − X10) were all independently distributed standard normal random variables. We also examined four additional covariate scenarios. In the second covariate scenario, the 10 covariates were from a multivariate normal distribution such that the mean and variance of each random variable were equal to 0 and 1, respectively, while the correlation between pairs of random variables was equal to 0.25. In the third covariate scenario, the ﬁrst ﬁve covariates (X1 − X5) were assumed to be independent Bernoulli random variables with parameter 0.5, while the last ﬁve covariates (X6 − X10) were assumed to be independent standard normal random variables. In the fourth covariate scenario, the ﬁrst nine covariates were assumed to be independent Bernoulli random variables with parameter 0.5, while the tenth covariate was a standard normal random variable. In the ﬁfth covariate scenario, all the 10 covariates (X1 − X10) were all independent Bernoulli random variables with parameter 0.5. The values of 0,treat, 0,outcome and were modiﬁed in order to preserve the proportion of treated subjects, the marginal probability of the outcome, and the required treatment effect. We refer to the ﬁve covariate scenarios as the independent normal covariates scenario, the correlated normal covariates scenario, the ﬁrst mixed covariates scenario, the second mixed covariates and the independent Bernoulli covariates scenario, respectively. Within each of the ﬁve covariate scenarios and for each of the ﬁve absolute risk reductions, we randomly generated 1825 data sets, each consisting of 10 000 subjects. We refer to the above set of 25 scenarios as the scenarios with a 0.29 outcome probability and a weak treatment-selection model. We also examined three additional sets of ﬁve covariate scenarios. In the next set of ﬁve scenarios, the data-generating process was modiﬁed so that the probability of the outcome if all subjects were untreated was 0.15. We refer to this set of ﬁve 25 scenarios as the scenarios with a 0.15 outcome probability and a weak treatment-selection model. We then modiﬁed these two sets of 25 scenarios by changing the weak treatment-selection model to a strong treatment-selection model. A strong treatment- selection process will induce greater differences in baseline covariates between treated and untreated subjects in the unmatched sample. In these two sets of 25 scenarios, the coefﬁcients for the treatment- selection model and the outcomes model were changed to: L = log(1.5), M = log(1.75), H = log(2), and VH = log(2.5). In the two sets of simulations in which there was a strong treatment-selection model, we observed low percentages of treated subjects successfully matched to untreated subjects in some of the covariate scenarios. Therefore, in these two sets of scenarios, minor modiﬁcations were made to the data-generating process by adding additional untreated subjects to the sample. Ten additional copies of each untreated subject were created within each replication of the Monte Carlo simulations. For these 10 additional subjects, outcomes were generated independently using the same subject-speciﬁc Copyright © 2011 John Wiley & Sons, Ltd. Statist. Med. 2011, 30 1292--1301