Ranked set sampling in finite populations with bivariate responses: An application to an osteoporosis study

The majority of the research on rank-based sampling designs in finite populations has been concerned with univariate situations. In this article, we study design-based estimation using a bivariate ranked set sampling (BIRSS) for finite populations when we have bivariate response variables. We derive the first and second-order inclusion probabilities associated with a BIRSS design. We show that the size of a BIRSS sample is random and propose using a conditional Poisson sampling (CPS) design to rectify this problem. We then use calculated inclusion probabilities to obtain design-based estimators of correlation coefficients between the bone mineral density (BMD) levels at the baseline and followup of a longitudinal BMD study in the province of Manitoba in Canada. We also study the problem of estimating the parameters of a regression model between the followup BMD and easy to obtain auxiliary information from the underlying population. Finally, we study the problem of classifying patients as those with or without osteoporosis using BIRSS and various CPS designs. We show that BIRSS designs are very flexible and can be used to obtain more efficient design-based estimators in sample surveys when dealing with response variables that are hard to measure or expensive to obtain.