Faculty Advisor

Balgobin Nandram

Faculty Advisor

Bogdan Vernescu


We analyze data on bone mineral density (BMD) and age for white females age 20+ in the third National Health and Nutrition Examination Survey. For the sample the age of each individual is known, but some individuals did not have their BMD measured, mainly because they did not show up in the mobile examination centers. We have data from 35 counties, the small areas. We use two types of models to analyze the data. In the ignorable nonresponse model, BMD does not depend on whether an individual responds or not. In the nonignorable nonresponse model, BMD is related to whether he/she responds. We incorporate this relationship in our model by using a Bayesian approach. We further divide these two types of models into continuous and categorical data models. Our nonignorable nonresponse models have one important feature: They are ``close' to the ignorable nonresponse model thereby reducing the effects of the untestable assumptions so common in nonresponse models. In the continuous data models, because the age of all nonrespondents are known and there is a relation between BMD and age, age is used as a covariate. In the categorical data models BMD has three levels (normal, osteopenia, osteoporosis) and age has two levels (younger than 50 years, at least 50 years). Thus, age is a supplemental margin for the $2 imes 3$ categorical table. Our research on the categorical models is much deeper than on the continuous models. Our models are hierarchical, a feature that allows a ``borrowing of strength' across the counties. Individual inference for most of the counties is unreliable because there is large variation. This ``borrowing of strength' is therefore necessary because it permits a substantial reduction in variation. The joint posterior density of the parameters for each model is complex. Thus, we fit each model using Markov chain Monte Carlo methods to obtain samples from the posterior density. These samples are used to make inference about BMD and age, and the relation between BMD and age. For the continuous data models, we show that there is an important relation between BMD and age by using a deviance measure, and we show that the nonignorable nonresponse models are to be preferred. For the categorical data models, we are able to estimate the proportion of individuals in each BMD and age cell of the categorical table, and we can assess the relation between BMD and age using the Bayes factor. A sensitivity analysis shows that there are differences, typically small, in inference that permits different levels of association between BMD and age. A simulation study shows that there is not much difference in inference between the ignorable nonresponse models and the nonignorable nonresponse models. As expected, BMD depends on age and this inference can be obtained for some small counties. For the data we use, there are virtually no young individuals with osteoporosis. The nonignorable nonresponse models generalize the ignorable nonresponse models, and therefore, allow broader inference.


Worcester Polytechnic Institute

Degree Name



Mathematical Sciences

Project Type


Date Accepted





missing data, Bayesian, nonresponse, Bone densitometry, Measurement, Surveys