Interval estimation of risk difference for data sampled from clusters.

Risk difference (RD) is an important measure in epidemiological studies where the probability of developing a disease for individuals in an exposed group, for example, is compared with that in a control group. There are varying cluster sizes in each group and the binary responses within each cluster cannot be assumed independent. Under the cluster sampling scenario, Lui (Statistical Estimation of Epidemiological Risk. Wiley: CA, 2004; 7-27) discusses four methods for the construction of a confidence interval for the RD. In this paper we introduce two very simple methods. One method is based on an estimator of the variance of a ratio estimator (Sampling Techniques (3rd edn). Wiley: New York, 1977; 30-67) and the other method is based on a sandwich estimator of the variance of the regression estimator using the generalized estimating equations approach of Zeger and Liang (Biometrics 1986; 42:121-130). These two methods are then compared, by simulation, in terms of maintaining nominal coverage probability and average coverage length, with the four methods discussed by Lui (Statistical Estimation of Epidemiological Risk. Wiley: CA, 2004; 7-27). Simulations show at least as good properties of these two methods as those of the others. The method based on an estimate of the variance of a ratio estimator performs best overall. It involves a very simple variance expression and can be implemented with a very few computer codes. Therefore, it can be considered as an easily implementable alternative.

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.

Fisher information matrix of the Dirichlet-multinomial distribution.

In this paper we derive explicit expressions for the elements of the exact Fisher information matrix of the Dirichlet-multinomial distribution. We show that exact calculation is based on the beta-binomial probability function rather than that of the Dirichlet-multinomial and this makes the exact calculation quite easy. The exact results are expected to be useful for the calculation of standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters for data that arise in practice in toxicology and other similar fields. Standard errors of the maximum likelihood estimates of the beta-binomial parameters and those of the Dirichlet-multinomial parameters, based on the exact and the asymptotic Fisher information matrix based on the Dirichlet distribution, are obtained for a set of data from Haseman and Soares (1976), a dataset from Mosimann (1962) and a more recent dataset from Chen, Kodell, Howe and Gaylor (1991). There is substantial difference between the standard errors of the estimates based on the exact Fisher information matrix and those based on the asymptotic Fisher information matrix.

Test of treatment effect in pre-drug and post-drug count data with zero-inflation.

A procedure for testing for treatment effect in data similar to the data on premature ventricular contractions (PVC) is presented. We consider a zero-inflated beta-binomial model. Based on this model, we develop score tests to test for treatment effect in the data in which observations in the form of counts are recorded before and after applying a therapy. Results of a small simulation experiment, to study small sample behaviour of a score test and a likelihood ratio test, are reported and the PVC data are analysed. Both the score and the likelihood ratio tests show good level properties. Either the score tests or the likelihood ratio tests can be used for testing the presence of treatment effect. The score tests, however, may be preferable because they use estimates of the parameters only under the null hypothesis and in the important range pi<0.5 power of the score test statistic S1 is slightly better than the likelihood ratio statistic LR1.