Confidence intervals for the difference in the success rates of two treatments in the analysis of correlated binary responses.

In clinical studies, we often compare the success rates of two treatment groups where post-treatment responses of subjects within clusters are usually correlated. To estimate the difference between the success rates, interval estimation procedures that do not account for this intraclass correlation are likely inappropriate. To address this issue, we propose three interval procedures by direct extensions of recently proposed methods for independent binary data based on the concepts of design effect and effective sample size used in sample surveys. Each of them is then evaluated with four competing variance estimates. We also extend three existing methods recommended for complex survey data using different weighting schemes required for those three existing methods. An extensive simulation study is conducted for the purposes of evaluating and comparing the performance of the proposed methods in terms of coverage and expected width. The interval estimation procedures are illustrated using three examples in clinical and social science studies. Our analytic arguments and numerical studies suggest that the methods proposed in this work may be useful in clustered data analyses.

Confidence intervals for the common intraclass correlation in the analysis of clustered binary responses.

In cluster randomized trials, it is often of interest to estimate the common intraclass correlation at the design stage for sample size and power calculations, which are greatly affected by the value of a common intraclass correlation. In this article, we construct confidence intervals (CIs) for the common intraclass correlation coefficient of several treatment groups. We consider the profile likelihood (PL)-based approach using the beta-binomial models and the approach based on the concept of generalized pivots using the ANOVA estimator and its asymptotic variance. We compare both approaches with a number of large sample procedures as well as both parametric and nonparametric bootstrap procedures in terms of coverage and expected CI length through a simulation study, and illustrate the methodology with two examples from biomedical fields. The results support the use of the PL-based CI as it holds the preassigned confidence level very well and overall gives a very competitive length.

A Comparison of Some Approximate Confidence Intervals for a Single Proportion for Clustered Binary Outcome Data.

Interval estimation of the proportion parameter in the analysis of binary outcome data arising in cluster studies is often an important problem in many biomedical applications. In this paper, we propose two approaches based on the profile likelihood and Wilson score. We compare them with two existing methods recommended for complex survey data and some other methods that are simple extensions of well-known methods such as the likelihood, the generalized estimating equation of Zeger and Liang and the ratio estimator approach of Rao and Scott. An extensive simulation study is conducted for a variety of parameter combinations for the purposes of evaluating and comparing the performance of these methods in terms of coverage and expected lengths. Applications to biomedical data are used to illustrate the proposed methods.

Weighted profile likelihood-based confidence interval for the difference between two proportions with paired binomial data.

Inference on the difference between two binomial proportions in the paired binomial setting is often an important problem in many biomedical investigations. Tang et al. (2010, Statistics in Medicine) discussed six methods to construct confidence intervals (henceforth, we abbreviate it as CI) for the difference between two proportions in paired binomial setting using method of variance estimates recovery. In this article, we propose weighted profile likelihood-based CIs for the difference between proportions of a paired binomial distribution. However, instead of the usual likelihood, we use weighted likelihood that is essentially making adjustments to the cell frequencies of a 2 × 2 table in the spirit of Agresti and Min (2005, Statistics in Medicine). We then conduct numerical studies to compare the performances of the proposed CIs with that of Tang et al. and Agresti and Min in terms of coverage probabilities and expected lengths. Our numerical study clearly indicates that the weighted profile likelihood-based intervals and Jeffreys interval (cf. Tang et al.) are superior in terms of achieving the nominal level, and in terms of expected lengths, they are competitive. Finally, we illustrate the use of the proposed CIs with real-life examples.

Semiparametric approach for non-monotone missing covariates in a parametric regression model.

Missing covariate data often arise in biomedical studies, and analysis of such data that ignores subjects with incomplete information may lead to inefficient and possibly biased estimates. A great deal of attention has been paid to handling a single missing covariate or a monotone pattern of missing data when the missingness mechanism is missing at random. In this article, we propose a semiparametric method for handling non-monotone patterns of missing data. The proposed method relies on the assumption that the missingness mechanism of a variable does not depend on the missing variable itself but may depend on the other missing variables. This mechanism is somewhat less general than the completely non-ignorable mechanism but is sometimes more flexible than the missing at random mechanism where the missingness mechansim is allowed to depend only on the completely observed variables. The proposed approach is robust to misspecification of the distribution of the missing covariates, and the proposed mechanism helps to nullify (or reduce) the problems due to non-identifiability that result from the non-ignorable missingness mechanism. The asymptotic properties of the proposed estimator are derived. Finite sample performance is assessed through simulation studies. Finally, for the purpose of illustration we analyze an endometrial cancer dataset and a hip fracture dataset.

Inference concerning a common dispersion of several treatment groups in the analysis of over/underdispersed count data.

Over/underdispersed count data arise in many biostatistical practices in which a number of different treatment groups are compared in an experiment. In the analysis of several treatment groups of such count data, a very common statistical inference problem is to test whether these data come from the same population. The usual practice for testing homogeneity of several treatment groups in terms of means and dispersions is first to test the equality of dispersions and then to test the equality of the means based on the result of the test for equality of dispersions. Previous studies reported test procedures for testing the homogeneity of the means of several treatment groups with an assumption of equal or unequal dispersions. This article develops test procedures for testing the validity of the equal or unequal dispersions assumption of several treatment groups in the analysis of over/underdispersed count data. We consider the C(α) test based on the maximum likelihood (ML) method using the negative binomial model as well as the three other C(α) tests based on the method of moments, extended quasi-likelihood, and double extended quasi-likelihood using the models specified by the first two moments of counts. Monte Carlo simulations are then used to study the comparative behavior of these C(α) tests along with the likelihood ratio test in terms of size and power. The simulation results demonstrate that all four statistics hold the nominal level reasonably well in most of the data situations studied here, and the C(α) test based on ML shows some edge in power over the other three C(α) tests. Finally, applications to biostatistical practices are analyzed.

Interval estimation of the mean difference in the analysis of over-dispersed count data.

This paper focuses on the development and study of the confidence interval procedures for mean difference between two treatments in the analysis of over-dispersed count data in order to measure the efficacy of the experimental treatment over the standard treatment in clinical trials. In this study, two simple methods are proposed. One is based on a sandwich estimator of the variance of the regression estimator using the generalized estimating equations (GEEs) approach of Zeger and Liang (1986) and the other is based on an estimator of the variance of a ratio estimator (1977). We also develop three other procedures following the procedures studied by Newcombe (1998) and the procedure studied by Beal (1987). As assessed by Monte Carlo simulations, all the procedures have reasonably well coverage properties. Moreover, the interval procedure based on GEEs outperforms other interval procedures in the sense that it maintains the coverage very close to the nominal coverage level and that it has the shortest interval length, a satisfactory location property, and a very simple form, which can be easily implemented in the applied fields. Illustrative applications in the biological studies for these confidence interval procedures are also presented.

Profile likelihood-based confidence interval of the intraclass correlation for binary outcome data sampled from clusters.

The intraclass correlation in binary outcome data sampled from clusters is an important and versatile measure in many biological and biomedical investigations. Properties of the different estimators of the intraclass correlation based on the parametric, semi-parametric, and nonparametric approaches have been studied extensively, mainly in terms of bias and efficiency [see, for example, Ridout et al., Biometrics 1999, 55:137-148; Paul et al., Journal of Statistical Computation and Simulation 2003, 73:507-523; and Lee, Statistical Modelling 2004, 4: 113-126], but little attention has been paid to extending these results to the problem of the confidence intervals. In this article, we generalize the results of the four point estimators by constructing asymptotic confidence intervals obtaining closed-form asymptotic and sandwich variance expressions of those four point estimators. It appears from simulation results that the asymptotic confidence intervals based on these four estimators have serious under-coverage. To remedy this, we introduce the Fisher's z-transformation approach on the intraclass correlation coefficient, the profile likelihood approach based on the beta-binomial model, and the hybrid profile variance approach based on the quadratic estimating equation for constructing the confidence intervals of the intraclass correlation for binary outcome data. As assessed by Monte Carlo simulations, these confidence interval approaches show significant improvement in the coverage probabilities. Moreover, the profile likelihood approach performs quite well by providing coverage levels close to nominal over a wide range of parameter combinations. We provide applications to biological data to illustrate the methods.

Interval estimation of the over-dispersion parameter in the analysis of one-way layout of count data.

The over-dispersion parameter is an important and versatile measure in the analysis of one-way layout of count data in biological studies. For example, it is commonly used as an inverse measure of aggregation in biological count data. Its estimation from finite data sets is a recognized challenge. Many simulation studies have examined the bias and efficiency of different estimators of the over-dispersion parameter for finite data sets (see, for example, Clark and Perry, Biometrics 1989; 45:309-316 and Piegorsch, Biometrics 1990; 46:863-867), but little attention has been paid to the accuracy of the confidence intervals (CIs) of it. In this paper, we first derive asymptotic procedures for the construction of confidence limits for the over-dispersion parameter using four estimators that are specified by only the first two moments of the counts. We also obtain closed-form asymptotic variance formulae for these four estimators. In addition, we consider the asymptotic CI based on the maximum likelihood (ML) estimator using the negative binomial model. It appears from the simulation results that the asymptotic CIs based on these five estimators have coverage below the nominal coverage probability. To remedy this, we also study the properties of the asymptotic CIs based on the restricted estimates of ML, extended quasi-likelihood, and double extended quasi-likelihood by eliminating the nuisance parameter effect using their adjusted profile likelihood and quasi-likelihoods. It is shown that these CIs outperform the competitors by providing coverage levels close to nominal over a wide range of parameter combinations. Two examples to biological count data are presented.

Bias-corrected maximum likelihood estimator of the intraclass correlation parameter for binary data.

A popular model to analyse over/under-dispersed proportions is to assume the extended beta-binomial model with dispersion (intraclass correlation) parameter phi and then to estimate this parameter by maximum likelihood. However, it is well known that maximum likelihood estimate (MLE) may be biased when the sample size n or the total Fisher information is small. In this paper we obtain a bias-corrected maximum likelihood (BCML) estimator of the intraclass correlation parameter and compare it, by simulation, in terms of bias and efficiency, with the MLE, an estimator Q(2) based on optimal quadratic estimating equations of Crowder and recommended by Paul et al. and a double extended quasi-likelihood (DEQL) estimator proposed by Lee. The BCML estimator has superior bias and efficiency properties in most instances. Analyses of a set of toxicological data from Paul and a set of medical data pertaining to chromosomal abnormalities among survivors of the atomic bomb in Hiroshima from Otake and Prentice show, in general, much improvement in standard errors of the BCML estimates over the other three estimates.