Dynamic Conditional Density Models for Multivariate Longitudinal Data
Open AccessLongitudinal studies allow to evaluate and make statistical inferences of the relationships between risk factors and the development of disease or death over some period of time. However, when the outcome variables of interest have more than two dimensions, the density estimation and the computation are much more complicated and time-consuming. Here, we propose a simple and easily applicable but more flexible method to estimate the dynamic conditional density models of a two-dimensional outcome variable based on other risk factors for multivariate longitudinal data. Previous studies were all based on either conditional mean or conditional distribution modeling schemes. To address the limitations of all previous work, our improved approach extends the univariate time-varying models to bivariate time-varying models by incorporating both conditional mean and conditional distribution modeling structures. Our study is motivated by the attempt of assessing joint densities of clinical variables in two real word studies, the Framingham Heart Study (FHS) and the National Growth and Health Study (NGHS). Based on the previous work on the nonparametric time-varying modeling under appropriate statistical hypotheses, we extend the models by incorporating dynamic joint density distributions of a two-dimensional response variable. Unlike the other direct estimation methods for the multivariate longitudinal data, we construct dynamic models through the conditional modeling approach.Our conditional approach provides us very useful insights in both the FHS and NGHS data: by only making relaxed assumptions: assuming both X|Y and Y|X are conditional normally distributed, given values of age and other health status covariates, e.g. the cholesterol (Chol) or the diastolic blood pressure (DBP), the exact dynamic joint density functions of a two-dimensional response variable, e.g. the bivariate cardiovascular risk factor: (SBP, BMI) over time can be easily generated. For example, given the mean values of cholesterol, we can estimate the joint density function of SBP and BMI for subjects at age 60. Hence, the exact cumulative densities of SBP $<$ 250 and BMI $<$ 30 at different ages also can be calculated. The thesis is organized as follows. In Chapter 1, we introduce the motivation; review methods commonly used in analyzing multivariate longitudinal data; the available smoothing methods to study time-varying models; the smoothing framework: two-step smoothing for building the conditional models and the conditional specified modeling for cross-sectional data. In Chapter 2, we look at the special cases of the joint models with exponential and normal conditionals and review their theoretical derivations. In Chapter 3, we propose our approach of estimating dynamic conditional specified joint density models, following similar framework in previous chapters. In Chapter 4, we apply our methods on the Framingham Heart Study and the National Growth and Health Study. In Chapter 5, we explore the feasibility of our approach using simulated data, under three different simulation scenarios. In Chapter 6 we will present the consistency properties of all our estimators at each step. And finally, we will make and discussion about our research and the future work.
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