Supplementary Web Material - Biometrics

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N0 = vnc(S 2 0p + S2 1p /r) + 1. (2) For internal pilot studies, we propose a formula that produces restricted designs, i.e., N0 > n0p (Wittes et al., 1999): � N0 = max vnc(S 2 0p + S2 1p /r) + 1, n0p � + n0s(min) , (3) where S 2 zp is the sample variance of group z calculated from the pilot data, nzp is the sample size of the pilot data in group z, and n0s(min) is an arbitrary minimum sample size for the control group second segment. We assume that the pre-specified ratio r is maintained for both stages of the study so that n1p/n0p = n1s(min)/n0s(min) = r. Our adaptation of tSS for internal pilot studies involves deriving group-specific variance estimates. Let S 2 zSS � −1 = (nzs) (nzs − 1)S 2 zs nzpnzs + D Nz 2 � z . The noncompliance SS (NSS) test statistic is then tNSS = (S 2 0SS /N0+S 2 1SS /N1) −1/2 ( ¯ Y1− ¯Y0), compared to tα/2,νNSS , where νNSS = (S2 0SS /N0 + S2 2 1SS /N1) (S 2 0SS /N0) 2 /n0s + (S 2 1SS /N1) 2 /n1s Note that nzs are the degrees of freedom for S 2 zSS and were hence used in νNSS instead of nzs − 1. Our adaptation of tBB involves accounting for the potential bias in two variance estimates for the special case of r = 1, thus n1p = n0p = np, n1s = n0s = ns, n1s(min) = n0s(min) = ns(min), and N1 = N0 = N. We calculate the variance 4 .
S 2 0BB + S2 1BB = S2 0 + S2 1 + 2 np − 1 where the term −2 np−1 (νp−2)vnc (νp − 2)vnc 1{N>np+n s(min)}, (4) is an approximate bound for bias of S 2 0 + S 2 1 based on the proof of theorem 1 in Miller (2005) using the Welch approximation of S 2 0p + S 2 1p to a chi-square random variable. The derivation of (4) is found in the subsection below. The noncompliance BB (NBB) test statistic is then tNBB = (S 2 0BB /N +S2 1BB /N)−1/2 ( ¯ Y1 − ¯ Y0), compared to tα/2,νNBB , where νNBB = (S 2 0BB /N + S2 1BB /N)2 (S 2 0BB /N)2 /(N − 1) + (S 2 1BB /N)2 /(N − 1) . The proposed sample size formulas involve adapting the approximate inflation factor in (1) to handle unequal variances that may be due to noncompliance with selection bias. However, use of the Welch approximation in sample size calculations and both tests in the presence of noncompliance is not entirely accurate. When ρc = 1, the Welch degrees of freedom are used to approximate the distribution of the variance of mean differences, a sum of weighted chi-square random variables, to a chi-square random variable. However, in the presence of noncompliance, the variance of mean differences is a sum of weighted central and noncentral chi-square random variables, where the noncentrality parameters depend on true adherence-subgroup means and compliance group distribution. The source of this inaccuracy is a special case of averaging over a factor causing heterogeneity in the treatment-specific outcome distribution. Even in cases with ρc = 1, other factors not considered in the sample size calculation (e.g., sex, age, etc.) are averaged over that may also cause heterogeneity in the outcome distribution. A closed-form approximate method that is reasonably accurate in accounting for heterogeneity from noncompliance is of practical utility, thus it is of interest to evaluate the performance of the proposed procedures for different levels of noncompliance and selection bias. 5
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Supplementary Web Material - Biometrics

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