2 9 7 1 P. C. AUSTIN Table II. Coverage and width of empirical 95 per cent conﬁdence intervals and estimation of sampling variances of treatment effects – 0.29 treatment effect, weak treatment-selection model. Coverage of 95 per cent conﬁdence intervals True risk difference Independent Paired Lengths of 95 per cent conﬁdence intervals Independent Paired Ratio of length of independent CI to paired CI Ratio of mean estimated variance of treatment effect to variance of empirical sampling distribution Independent Paired Independent normal covariates 0 0.957 0.938 −0.02 0.963 0.947 −0.05 0.964 0.949 −0.1 0.963 0.95 −0.15 0.961 0.944 Correlated normal covariates 0 0.965 0.946 −0.02 0.975 0.958 −0.05 0.97 0.948 −0.1 0.972 0.951 −0.15 0.969 0.952 Mixture covariate scenario 1 0 0.97 0.948 −0.02 0.964 0.951 −0.05 0.961 0.95 −0.1 0.964 0.95 −0.15 0.962 0.949 Mixture covariate scenario 2 0 0.958 0.95 −0.02 0.958 0.95 −0.05 0.962 0.951 −0.1 0.953 0.942 −0.15 0.955 0.945 Independent Bernoulli covariates 0 0.958 0.952 −0.02 0.953 0.95 −0.05 0.955 0.95 −0.1 0.95 0.946 −0.15 0.959 0.954 0.057 0.057 0.056 0.055 0.053 0.063 0.063 0.063 0.062 0.061 0.056 0.056 0.056 0.054 0.053 0.054 0.054 0.053 0.052 0.05 0.052 0.052 0.051 0.05 0.048 0.053 0.053 0.052 0.051 0.05 0.057 0.057 0.057 0.056 0.055 0.053 0.053 0.052 0.051 0.05 0.052 0.051 0.051 0.05 0.048 0.051 0.05 0.05 0.048 0.047 1.075 1.075 1.077 1.078 1.06 1.105 1.105 1.105 1.107 1.109 1.057 1.057 1.077 1.059 1.06 1.038 1.059 1.039 1.04 1.042 1.02 1.04 1.02 1.042 1.021 1.084 1.076 1.093 1.126 1.138 1.255 1.247 1.24 1.262 1.244 1.131 1.124 1.111 1.122 1.124 1.091 1.093 1.07 1.07 1.085 1.07 1.078 1.077 1.072 1.058 0.944 0.938 0.956 0.99 1.008 1.02 1.016 1.013 1.037 1.032 0.991 0.986 0.977 0.992 1.001 0.988 0.991 0.972 0.977 0.998 1.014 1.023 1.024 1.022 1.014 The ratio between the mean length of conﬁdence intervals obtained using methods for independent samples and the mean length of conﬁdence intervals obtained using methods for paired samples ranged from a low of 1 to a high of 1.197; the median ratio was 1.074 (25th and 75th percentiles: 1.045 and 1.113) across the 100 scenarios. Thus, in half of the 100 scenarios, the estimated conﬁdence intervals were at least 7.4 per cent wider when methods for independent samples were used compared with when methods for paired samples were used. As above, the relative difference between the widths of the conﬁdence intervals was greater when there was a strong treatment-selection process compared with when there was a weak treatment-selection process. The analyses reported in the above two paragraphs suggest that conﬁdence intervals constructed using methods for paired samples tend to have coverage rates that were closer to the advertised rates compared with when methods for independent samples were used. Furthermore, methods for paired samples resulted in estimates with greater precision, since the estimated conﬁdence intervals are narrower compared with when methods for independent samples were used. The square of the ratio between the mean estimated standard error when methods for independent samples were used and the standard deviation of the empirical sampling distribution of the estimated risk differences across the 1825 simulated data sets is reported in the second rightmost column of Tables II–V. A similar ratio obtained when methods for paired samples was used is reported in the rightmost column of Tables II–V. When methods for independent samples were used, this ratio ranged from a low of 0.950 to a high of 1.508; the median ratio was 1.149 (25th and 75th percentiles: 1.085 and 1.250, respectively) across the 100 scenarios. Thus, in 50 per cent of the scenarios, variance estimates obtained using methods for independent samples overestimated the sampling variance of the estimated Copyright © 2011 John Wiley & Sons, Ltd. Statist. Med. 2011, 30 1292--1301