Batch effect is one type of variability that is not of primary interest but ubiquitous in sizable genomic experiments. To minimize the impact of batch effects, an ideal experiment design should ensure the even distribution of biological groups and confounding factors across batches. However, due to the practical complications, the availability of the final collection of samples in genomics study might be unbalanced and incomplete, which, without appropriate attention in sample-to-batch allocation, could lead to drastic batch effects. Therefore, it is necessary to develop effective and handy tool to assign collected samples across batches in an appropriate way in order to minimize the impact of batch effects.

In ophthalmologic studies, measurements obtained from both eyes of an individual are often highly correlated. Ignoring the correlation could lead to incorrect inferences. An asymptotic method was proposed by Tang and others (2008) for testing equality of proportions between two groups under Rosner's model. In this article, we investigate three testing procedures for general g ≥ 2 groups. Our simulation results show the score testing procedure usually produces satisfactory type I error control and has reasonable power. The three test procedures get closer when sample size becomes larger. Examples from ophthalmologic studies are used to illustrate our proposed methods.

In ophthalmologic studies, bilateral correlated data often arise when information involving paired organs (e.g., eyes) are measured from each subject. Adjusted chi-square approach for testing the equality of proportions has been proposed in the literature. In this article, we investigate and derive three alter- native testing procedures for the problem. Our simulation results show the score testing procedure usually produces satisfactory type I error control with higher power, and therefore is recommended. Examples from ophthalmologic studies are used to illustrate our proposed methods.