Bayesian mixture modelling with ranked set samples.

We consider the Bayesian estimation of the parameters of a finite mixture model from independent order statistics arising from imperfect ranked set sampling designs. As a cost-effective method, ranked set sampling enables us to incorporate easily attainable characteristics, as ranking information, into data collection and Bayesian estimation. To handle the special structure of the ranked set samples, we develop a Bayesian estimation approach exploiting the Expectation-Maximization (EM) algorithm in estimating the ranking parameters and Metropolis within Gibbs Sampling to estimate the parameters of the underlying mixture model. Our findings show that the proposed RSS-based Bayesian estimation method outperforms the commonly used Bayesian counterpart using simple random sampling. The developed method is finally applied to estimate the bone disorder status of women aged 50 and older.

Another look at regression analysis using ranked set samples with application to an osteoporosis study.

Statistical learning with ranked set samples has shown promising results in estimating various population parameters. Despite the vast literature on rank-based statistical learning methodologies, very little effort has been devoted to studying regression analysis with such samples. A pressing issue is how to incorporate the rank information of ranked set samples into the analysis. We propose two methodologies based on a weighted least squares approach and multilevel modeling to better incorporate the rank information of such samples into the estimation and prediction processes of regression-type models. Our approaches reveal significant improvements in both estimation and prediction problems over already existing methods in the literature and the corresponding ones with simple random samples. We study the robustness of our methods with respect to the misspecification of the distribution of the error terms. Also, we show that rank-based regression models can effectively predict simple random test data by assigning ranks to them a posteriori using judgment poststratification. Theoretical results are augmented with simulations and an osteoporosis study based on a real data set from the Bone Mineral Density (BMD) program of Manitoba to estimate the BMD level of patients using easy to obtain covariates.

Judgment post-stratification in finite mixture modeling: An example in estimating the prevalence of osteoporosis.

Judgment post-stratification is used to supplement observations taken from finite mixture models with additional easy to obtain rank information and incorporate it in the estimation of model parameters. To do this, sampled units are post-stratified on ranks by randomly selecting comparison sets for each unit from the underlying population and assigning ranks to them using available auxiliary information or judgment ranking. This results in a set of independent order statistics from the underlying model, where the number of units in each rank class is random. We consider cases where one or more rankers with different ranking abilities are used to provide judgment ranks. The judgment ranks are then combined to produce a strength of agreement measure for each observation. This strength measure is implemented in the maximum likelihood estimation of model parameters via a suitable expectation maximization algorithm. Simulation studies are conducted to evaluate the performance of the estimators with or without the extra rank information. Results are applied to bone mineral density data from the third National Health and Nutrition Examination Survey to estimate the prevalence of osteoporosis in adult women aged 50 and over.

Bayesian sample size calculation for estimation of the difference between two binomial proportions.

In this study, we discuss a decision theoretic or fully Bayesian approach to the sample size question in clinical trials with binary responses. Data are assumed to come from two binomial distributions. A Dirichlet distribution is assumed to describe prior knowledge of the two success probabilities p1 and p2. The parameter of interest is p = p1 - p2. The optimal size of the trial is obtained by maximising the expected net benefit function. The methodology presented in this article extends previous work by the assumption of dependent prior distributions for p1 and p2.

The choice of sample size: a mixed Bayesian / frequentist approach.

Sample size computations are largely based on frequentist or classical methods. In the Bayesian approach the prior information on the unknown parameters is taken into account. In this work we consider a fully Bayesian approach to the sample size determination problem which was introduced by Grundy et al. and developed by Lindley. This approach treats the problem as a decision problem and employs a utility function to find the optimal sample size of a trial. Furthermore, we assume that a regulatory authority, which is deciding on whether or not to grant a licence to a new treatment, uses a frequentist approach. We then find the optimal sample size for the trial by maximising the expected net benefit, which is the expected benefit of subsequent use of the new treatment minus the cost of the trial.