#### Faculty Advisor

Balgobin Nandram

#### Faculty Advisor

Bodgan Vernescu

#### Identifier

etd-050605-155002

#### Abstract

Data from the 1995 National Health Interview Survey (NHIS) indicate that, due to chronic conditions, the onset of activity limitation typically occurs between age 40-70 years (i.e., the proportion of young adults with activity limitation is small and roughly constant with age and then it starts to change, roughly increasing). We use a Bayesian hierarchical model to detect the change point of a positive activity limitation status (ALS) across twelve domains based on race, gender, and education. We have two types of data: weighted and unweighted. We obtain weighted binomial counts using a regression analysis with the sample weights. Given the proportion of individuals in the population with positive ALS, we assume that the number of individuals with positive ALS at each age group has a binomial probability mass function. The proportions across age are different, and have the same beta distribution up to the change point (unknown), and the proportions after the change point have a different beta distribution. We consider two different analyses. The ﬁrst considers each domain individually in its own model and the second considers the twelve domains simultaneously in a single model to“borrow strength" as in small area estimation. It is reasonable to assume that each domain has its own onset.In the ﬁrst analysis, we use the Gibbs sampler to ﬁt the model, and a computation of the marginal likelihoods, using an output analysis from the Gibbs sampler, provides the posterior distribution of the change point. We note that a reversible jump sampler fails in this analysis because it tends to get stuck either age 40 or age 70. In the second analysis, we use the Gibbs sampler to ﬁt only the joint posterior distribution of the twelve change points. This is a difficult problem because the joint density requires the numerical computation of a triple integral at each iteration. The other parameters of the process are obtained using data augmentation by a Metropolis sampler and a Rao-Blackwellization. We found that overall the age of onset is about 50 to 60 years.

#### Publisher

Worcester Polytechnic Institute

#### Degree Name

MS

#### Department

Mathematical Sciences

#### Project Type

Thesis

#### Date Accepted

2005-05-06

#### Copyright Statement

All authors have granted to WPI a nonexclusive royalty-free license to distribute copies of the work. Copyright is held by the author or authors, with all rights reserved, unless otherwise noted. If you have any questions, please contact wpi-etd@wpi.edu.

#### Accessibility

Unrestricted

#### Repository Citation

Bai, Yan, "A Bayesian Approach to Detect the Onset of Activity Limitation Among Adults in NHIS" (2005). *Masters Theses (All Theses, All Years)*. 765.

https://digitalcommons.wpi.edu/etd-theses/765

#### Subjects

Change point, Gibbs sampler, Hierarchical Bayesian model, Reversible jump, Bayesian statistical decision theory, Medical statistics, Health behavior, Age factors