Modeling and Estimation of Patterns of Relationship Formation and Dissolution

Abstract

This dissertation describes and develops methods for modeling sexual partnership formation and termination using retrospectively collected survey data. Such methods are required to produce information necessary to model propagation of sexually transmitted diseases and the impact of interventions on such processes which are being used both to design and to monitor HIV combination prevention studies. Sexual history data are commonly obtained through surveys that collect information on relationships that are ongoing during a fixed time window. This sampling mechanism leads to incomplete sexual history data and duration data that are left truncated and right censored.

In Chapter 1, we describe a common sampling scheme for collecting sexual partnership data, discuss a key assumption required for unbiased estimation, and provide the conditions under which the nonparametric maximum likelihood estimator of the relationship duration distribution is unique and consistent. We also investigate the conditions required for the consistency of the regression coefficient from a Cox proportional hazards model that apply even when the distribution of duration is not completely identifiable due to restrictions on the support of the truncation distribution. Lastly, we will provide some illustrative examples on estimating distribution of most recent partnerships and present spline regression results based on sexual history data from Botswana.

In Chapter 2, we present a Markov framework for modeling and estimation of partnership transition probabilities for sexual history data collected under a retrospective sampling scheme. We propose a stochastic expectation maximization algorithm (stEM) coupled with rejection-sampling scheme in order to estimate transition probabilities from a state of celibacy to monogamy and to concurrency (or vice versa). This approach accommodates the retrospective sampling scheme from which sexual partnership data is obtained and utilizes all available information from the sexual history data. In particular, this paper will address maximum likelihood estimation via stEM when our observed data includes information on the number of certain types of transitions without specifying the sojourn time in the states. For example, with regards to partnership data, the total life time number of partnerships (or number of partnerships within a fixed window of time) may be known even though the sojourn time of each of the partnerships in the different states may not be known. In the process of estimating transition rates, we incorporate such information by using rejection sampling. Simulation results showing the performance of the stEM will be presented. We also provide an application example based on partnership data collected from South Africa.

In Chapter 3, we extend the Markov model presented in Chapter 2 so that the sexual history process can be fully characterized. This approach combines a Markov model and a logistic regression framework. The Markov model states we consider include celibacy, monogamy and concurrency; the logistic regression model classifies the pattern of concurrency, which can be either transitional (older partnership ends first) or embedded (new ends first). By using both types of models we can fully characterize the processes of interest. Estimation of model parameters is based on a stochastic expectation maximization algorithm (stEM) coupled with rejection-sampling scheme. Strategies based on statistics that arise naturally from the estimation procedure itself stEM are used to validate model assumptions. The method is illustrated using sexual history data collected from South Africa. Simulation results are used to demonstrate properties of the estimation methods.