Estimation of heterogeneity variance based on a generalized Q statistic in meta‐analysis of log‐odds‐ratio

Abstract For estimation of heterogeneity variance τ2 in meta‐analysis of log‐odds‐ratio, we derive new mean‐ and median‐unbiased point estimators and new interval estimators based on a generalized Q statistic, QF, in which the weights depend on only the studies' effective sample sizes. We compare them with familiar estimators based on the inverse‐variance‐weights version of Q, QIV. In an extensive simulation, we studied the bias (including median bias) of the point estimators and the coverage (including left and right coverage error) of the confidence intervals. Most estimators add 0.5 to each cell of the 2×2 table when one cell contains a zero count; we include a version that always adds 0.5. The results show that: two of the new point estimators and two of the familiar point estimators are almost unbiased when the total sample size n≥250 and the probability in the Control arm (piC) is 0.1, and when n≥100 and piC is 0.2 or 0.5; for 0.1≤τ2≤1, all estimators have negative bias for small to medium sample sizes, but for larger sample sizes some of the new median‐unbiased estimators are almost median‐unbiased; choices of interval estimators depend on values of parameters, but one of the new estimators is reasonable when piC=0.1 and another, when piC=0.2 or piC=0.5; and lack of balance between left and right coverage errors for small n and/or piC implies that the available approximations for the distributions of QIV and QF are accurate only for larger sample sizes.


S1.3 Kulinskaya-Dollinger confidence interval
Instead of assuming that σ 2 i = σ2 i , Kulinskaya and Dollinger [2015] took into account the sampling variability in the σ2 i .Using theoretical results and simulations, they obtained a corrected formula that approximates the first moment of the null distribution of Q IV for LOR and a formula (as a function of that corrected first moment) for the corresponding second moment.Both corrected moments are smaller than those of the usual chi-square distribution.Because the full details are complicated, we do not show the corrected moments here.Kulinskaya and Dollinger [2015] give some details and include a link to an R program for the calculations.The KD confidence interval is obtained by referring Q(τ 2 ) to a gamma distribution with the corrected two moments.Median bias of

S2 Supplemental Figures
Median bias of Median bias of Median bias of

Figure S6 :
Figure S6: Coverage of 95% confidence intervals for between-study variance of LOR (the "only" versions of PL, QP and FPC; KD; and the model versions of FPU) vs τ 2 , for equal sample sizes n = 20, 40, 100 and 250, p iC = .5,θ = 0 and f = 0.5.11 Figure S7: Miss-left probability of PL, QP, KD, FPC, and FPU 95% confidence intervals for between-study variance of LOR vs τ 2 , for equal sample sizes n = 20, 40, 100 and 250, p iC = .2,θ = 0 and f = 0.5.Solid lines: the "only" versions of PL, QP and FPC; KD; and the model version of FPU.Dashed lines: the "always" version of FPC and the naïve version of FPU.
Figure S8: Miss-left probability of PL, QP, KD, FPC, and FPU 95% confidence intervals for between-study variance of LOR vs τ 2 , for equal sample sizes n = 20, 40, 100 and 250, p iC = .5,θ = 0 and f = 0.5.Solid lines: the "only" versions of PL, QP and FPC; KD; and the model version of FPU.Dashed lines: the "always" version of FPC and the naïve version of FPU.
Figure S9: Miss-right probability of PL, QP, KD, FPC, and FPU 95% confidence intervals for between-study variance of LOR vs τ 2 , for equal sample sizes n = 20, 40, 100 and 250, p iC = .2,θ = 0 and f = 0.5.Solid lines: the "only" versions of PL, QP and FPC; KD; and the model version of FPU.Dashed lines: the "always" version of FPC and the naïve version of FPU.
Figure S10: Miss-right probability of PL, QP, KD, FPC, and FPU 95% confidence intervals for between-study variance of LOR vs τ 2 , for equal sample sizes n = 20, 40, 100 and 250, p iC = .5,θ = 0 and f = 0.5.Solid lines: the "only" versions of PL, QP and FPC; KD; and the model version of FPU.Dashed lines: the "always" version of FPC and the naïve version of FPU.

Table S1 :
Stead et al. (2013)data on the use of physician advice for smoking cessation

Table S2 :
Summary statistics for selected estimators in simulations for Stead et al. example