Efficient tests for one sample correlated binary data with applications

Abstract

Four testing procedures are considered for testing the response rate of one sample correlated binary data with a cluster size of one or two, which often occurs in otolaryngologic and ophthalmologic studies. Although an asymptotic approach is often used for statistical inference, it is criticized for unsatisfactory type I error control in small sample settings. An alternative to the asymptotic approach is an unconditional approach. The first unconditional approach is the one based on estimation, also known as parametric bootstrap (Lee and Young in Stat Probab Lett 71(2):143–153, 2005). The other two unconditional approaches considered in this article are an approach based on maximization (Basu in J Am Stat Assoc 72(358):355–366, 1977), and an approach based on estimation and maximization (Lloyd in Biometrics 64(3):716–723, 2008a). These two unconditional approaches guarantee the test size and are generally more reliable than the asymptotic approach. We compare these four approaches in conjunction with a test proposed by Lee and Dubin (Stat Med 13(12):1241–1252, 1994) and a likelihood ratio test derived in this article, in regards to type I error rate and power for sample sizes from small to medium. An example from an otolaryngologic study is provided to illustrate the various testing procedures. The unconditional approach based on estimation and maximization using the test in Lee and Dubin (Stat Med 13(12):1241–1252, 1994) is preferable due to the power advantageous.

Appendix

Likelihood ratio test statistic $\mathbf{T}_\mathbf{LR}$. The log-likelihood is expressed as

$$\begin{aligned} l(\pi , \rho | \mathbf{{N_1}},\mathbf{{N_2}})&= log(\frac{(n-n_1)!}{N_{22}! N_{21}! N_{20}!})+N_{11}\log (\pi )+N_{10}\log (1-\pi )\\&+N_{22}\log [\pi ^2+\rho \pi (1-\pi )]+N_{21}\log [2(1-\rho )(1-\pi )]\\&+N_{20}\log [(1-\pi )^2+\rho \pi (1-\pi )]. \end{aligned}$$

Differentiating $l(\pi , \rho )$ with respect to $(\pi , \rho )$ yields the score function

$$\begin{aligned} \frac{\partial l}{\partial \pi }&= \frac{N_{11}+2N_{22}}{\pi } - \frac{N_{10}+N_{21}+2N_{20}}{1-\pi } - \frac{N_{22} \rho }{\pi (\pi +\rho -\pi \rho )}\\&+\frac{N_{20}\rho }{(1-\pi )(1-\pi +\pi \rho )},\\ \frac{\partial l}{\partial \rho }&= -\frac{N_{21}}{1-\rho }+\frac{N_{22}(1-\pi )}{\pi +\rho -\pi \rho } + \frac{N_{20}\pi }{1-\pi +\pi \rho }. \end{aligned}$$

The unrestricted MLE of $(\pi , \rho )$, denoted by $(\hat{\pi }, \hat{\rho })$ is the solution to the following equations which can be obtained by Fishing-Score method,

$$\begin{aligned} \frac{\partial l}{\partial \pi } = 0 \quad \hbox { and }\quad \frac{\partial l}{\partial \rho } = 0. \end{aligned}$$

After a lengthy algebra calculation, the $\hat{\rho }$ can be derived as a solution of a third-order polynomial

$$\begin{aligned} a \rho ^3 + b\rho ^2 + c\rho +d = 0, \end{aligned}$$

where

$$\begin{aligned} a&= N_{10}(N_{11}-N_{21})n_2,\\ b&= N_{10}^2N_{21}\!+\!N_{10}^2N_{22} +N_{10}N_{11}N_{21} +N_{10}N_{20}N_{21}+2N_{10}N_{20}N_{22} +3N_{10}N_{21}N_{22}\\&+2N_{10}N_{22}^2+N_{11}^2N_{20}+N_{11}^2N_{21}+2N_{11}N_{20}^2 +N_{11}N_{20}N_{21}+2N_{11}N_{20}N_{22}\\&-N_{11}N_{21}^2+N_{11}N_{21}N_{22}-2N_{20}^2N_{21}-2N_{20}N_{21}^2 -2N_{20}N_{21}N_{22}-N_{21}^2N_{22},\\ c&= N_{10}^2N_{21}-N_{10}N_{11}N_{20}+N_{10}N_{11}N_{21} -N_{10}N_{11}N_{22}+4N_{10}N_{20}N_{21}\\&+2N_{10}N_{20}N_{22}+N_{10}N_{21}^2+2N_{10}N_{21}N_{22} -2N_{10}N_{22}^2+N_{11}^2N_{21}-2N_{11}N_{20}^2\\&+N_{11}N_{20}N_{21}+2N_{11}N_{20}N_{22}+3N_{11}N_{21}N_{22} +4N_{20}^2N_{21}+4N_{20}^2N_{22}\\&+2N_{20}N_{21}^2+6N_{20}N_{21}N_{22}+4N_{20}N_{22}^2 +2N_{21}N_{22}^2,\\ d&= -N_{10}^2N_{22}+N_{10}N_{11}N_{21} -4N_{10}N_{20}N_{22}-N_{11}^2N_{20}+2N_{11}N_{20}N_{21}\\&-4N_{11}N_{20}N_{22}+N_{11}N_{21}^2-4N_{20}^2N_{22} -4N_{20}N_{22}^2+N_{21}^2N_{22}, \end{aligned}$$

and

$$\begin{aligned} \hat{\pi }=\frac{(N_{21}+2N_{22}-n_2\hat{\rho })\pm \sqrt{(\hat{\rho }n_2)^2+2\hat{\rho }n_2N_{21}-4N_{20}N_{22}+N_{21}^2}}{2n_2(1-\hat{\rho })}. \end{aligned}$$

We then compute the log-likelihoods for the solutions in the parameters’ space, and the parameter with the largest value is the solution. Another method may be used to derive the LR test (Evans and Forcina 2013).

Under null hypothesis $H_0: \pi =\pi _0$, the MLE of $\rho$ is given by

$$\begin{aligned} \hat{\rho }_{H_0}=-\frac{N_{21}+N_{22}-\pi _0(N_{20}+2N_{21}+3N_{22})+2n_2\pi _0^2\pm \sqrt{f}}{2\pi _0(1-\pi _0)n_2}, \end{aligned}$$

where $f= (4N_{21}n_2+(N_{20}-N_{22})^2)\pi _0^2 + 2(N_{20}(N_{21}+2N_{22})-n_2(2N_{21}+N_{22}))\pi _0+(N_{21}+N_{22})^2$ and only keep the solution in the parameter space. When both $\hat{\rho }_{H_0}$ are in the parameter space, the one with the larger null log likelihood is the solution.