Exact and approximate unconditional confidence intervals for proportion difference in the presence of incomplete data

Abstract

Confidence interval (CI) construction with respect to proportion/rate difference for paired binary data has become a standard procedure in many clinical trials and medical studies. When the sample size is small and incomplete data are present, asymptotic CIs may be dubious and exact CIs are not yet available. In this article, we propose exact and approximate unconditional test-based methods for constructing CI for proportion/rate difference in the presence of incomplete paired binary data. Approaches based on one- and two-sided Wald's tests will be considered. Unlike asymptotic CI estimators, exact unconditional CI estimators always guarantee their coverage probabilities at or above the pre-specified confidence level. Our empirical studies further show that (i) approximate unconditional CI estimators usually yield shorter expected confidence width (ECW) with their coverage probabilities being well controlled around the pre-specified confidence level; and (ii) the ECWs of the unconditional two-sided-test-based CI estimators are generally narrower than those of the unconditional one-sided-test-based CI estimators. Moreover, ECWs of asymptotic CIs may not necessarily be narrower than those of two-sided-based exact unconditional CIs. Two real examples will be used to illustrate our methodologies.

Corrected profile likelihood confidence interval for binomial paired incomplete data

“…Using the point estimate of the parameter of interest δ and corresponding standard error ( se ), 100(1 − α ) % Wald CI is constructed by where Z α ∕ 2 is the appropriate quantile from standard normal distribution. Because the Wald method along with the likelihood from Tang et al is used to construct this interval, we name it as TW. The second method we introduce here is very similar to the first method.…”

“…b ı Z˛= 2 se, b ı C Z˛= 2 se/ where Z˛= 2 is the appropriate quantile from standard normal distribution. Because the Wald method along with the likelihood from Tang et al [19] is used to construct this interval, we name it as TW. The second method we introduce here is very similar to the first method.…”

“…The second method we introduce here is very similar to the first method. Considering the likelihood proposed by Tang et al [19], we implement Agresti and Min [17] correction for the multinomial part and Agresti and Caffo [16] modification to the binomial parts of the likelihood. Without the constant terms the corrected likelihood takes the form P.n 11 , n 12 , n 21 , w 1 , w 2 jı, p 21 , p 22 / / .p 21 C ı/ .n 12 C0.5/ p 21 .n 21 C0.5/ p 22 .n 22 C0.5/ .1 2p 21 p 22 ı/ .n 11 C0.5/…”

On Confidence Interval Construction for Establishing Equivalence of Two Binary-Outcome Treatments in Matched-Pair Studies in the Presence of Incomplete Data

“…However, all the aforementioned work were confined to matched-pair data without incomplete data. Tang et al [17] proposed the exact and approximate unconditional confidence intervals for proportion difference in the presence of incomplete data. However, these methods could be computationally intensive for moderate to large sample sizes and simple explicit formulas are impossible.…”

Missing value imputation in longitudinal measures of alcohol consumption

“…In longitudinal research, attrition contributes an additional source of error to the known selection biases that occur in cross-sectional research (Twisk and de Vente, 2002). Causes of attrition in medical research often relate to drop-outs being more ill than those with complete data (Fairclough et al, 1998;Kaciroti et al, 2009;Gunnes et al, 2009;Tang et al, 2009;Paddock and Ebener, 2009).…”