On continuity correction for RSS-structured cluster randomized designs with binary outcomes

Correction for continuity is commonly used to improve the inference for binary data when the event of interest is rare or the sample size is small. A standard approach to reduce the bias in logit estimation is to add a small constant to both event and nonevent counts. The 0.5 adjustment is known as a correction rendering the estimation unbiased up to the order of $$K^{-1}$$ K - 1 , where K is the size of a simple random sample. However, for general designs beyond simple random sampling, the bias in estimating the logit is no longer zero in order $$K^{-1}$$ K - 1 . In this paper, we derive the formula of the correction factor that makes the first-order term of the bias vanish for general designs. We then apply it to estimate the logit when data are from ranked set sampling (RSS) embedded in a cluster randomized design (CRD). An RSS-structured CRD (RSS-CRD), introduced by Wang et al. (J Am Stat Assoc 111: 1576–1590, 2016), is a new two-stage design for more efficient estimation of treatment effect. We propose two methods to estimate the correction factors derived for RSS-CRDs. We numerically compare the proposed methods to those with the default factor 0.5 in terms of bias and mean squared error for estimating the treatment effect, and finally make recommendations to practitioners.