Longitudinal categorical data analysis: A continuous -time Markov chain approach

In this dissertation, we propose a continuous-time Markov chain model to examine the longitudinal data that have three categories in the outcome variable. The advantage of this model is that it permits a different number of measurements for each subject and the duration between two consecutive time points of measurements can be irregular. Using the maximum likelihood principle, we can estimate the transition probability between two time points. By using the information provided by the independent variables, this model can also estimate the transition probability for each subject. The Monte Carlo simulation method will be used to investigate the goodness of model fitting compared with that obtained from other models. A public health example will be used to demonstrate the application of this method. ^

Estimating parameters in Markov models for longitudinal studies with missing data or surrogate outcomes

The discrete-time Markov chain is commonly used in describing changes of health states for chronic diseases in a longitudinal study. Statistical inferences on comparing treatment effects or on finding determinants of disease progression usually require estimation of transition probabilities. In many situations when the outcome data have some missing observations or the variable of interest (called a latent variable) can not be measured directly, the estimation of transition probabilities becomes more complicated. In the latter case, a surrogate variable that is easier to access and can gauge the characteristics of the latent one is usually used for data analysis. ^ This dissertation research proposes methods to analyze longitudinal data (1) that have categorical outcome with missing observations or (2) that use complete or incomplete surrogate observations to analyze the categorical latent outcome. For (1), different missing mechanisms were considered for empirical studies using methods that include EM algorithm, Monte Carlo EM and a procedure that is not a data augmentation method. For (2), the hidden Markov model with the forward-backward procedure was applied for parameter estimation. This method was also extended to cover the computation of standard errors. The proposed methods were demonstrated by the Schizophrenia example. The relevance of public health, the strength and limitations, and possible future research were also discussed. ^

Logistic regression with Markov chains as covariates

The tobacco-specific nitrosamine 4-(methylnitrosamino)-1-(3-pyridyl)-1-butanone (NNK) is an obvious carcinogen for lung cancer. Since CBMN (Cytokinesis-blocked micronucleus) has been found to be extremely sensitive to NNK-induced genetic damage, it is a potential important factor to predict the lung cancer risk. However, the association between lung cancer and NNK-induced genetic damage measured by CBMN assay has not been rigorously examined. ^ This research develops a methodology to model the chromosomal changes under NNK-induced genetic damage in a logistic regression framework in order to predict the occurrence of lung cancer. Since these chromosomal changes were usually not observed very long due to laboratory cost and time, a resampling technique was applied to generate the Markov chain of the normal and the damaged cell for each individual. A joint likelihood between the resampled Markov chains and the logistic regression model including transition probabilities of this chain as covariates was established. The Maximum likelihood estimation was applied to carry on the statistical test for comparison. The ability of this approach to increase discriminating power to predict lung cancer was compared to a baseline "non-genetic" model. ^ Our method offered an option to understand the association between the dynamic cell information and lung cancer. Our study indicated the extent of DNA damage/non-damage using the CBMN assay provides critical information that impacts public health studies of lung cancer risk. This novel statistical method could simultaneously estimate the process of DNA damage/non-damage and its relationship with lung cancer for each individual.^

The physical map of {\it Escherichia coli\/} K-12 strain MG1655: Construction, comparison with other strains and implications for mechanisms controlling oligonucleotide frequency

Complete NotI, SfiI, XbaI and BlnI cleavage maps of Escherichia coli K-12 strain MG1655 were constructed. Techniques used included: CHEF pulsed field gel electrophoresis; transposon mutagenesis; fragment hybridization to the ordered $\lambda$ library of Kohara et al.; fragment and cosmid hybridization to Southern blots; correlation of fragments and cleavage sites with EcoMap, a sequence-modified version of the genomic restriction map of Kohara et al.; and correlation of cleavage sites with DNA sequence databases. In all, 105 restriction sites were mapped and correlated with the EcoMap coordinate system.^ NotI, SfiI, XbaI and BlnI restriction patterns of five commonly used E. coli K-12 strains were compared to those of MG1655. The variability between strains, some of which are separated by numerous steps of mutagenic treatment, is readily detectable by pulsed-field gel electrophoresis. A model is presented to account for the difference between the strains on the basis of simple insertions, deletions, and in one case an inversion. Insertions and deletions ranged in size from 1 kb to 86 kb. Several of the larger features have previously been characterized and some of the smaller rearrangements can potentially account for previously reported genetic features of these strains.^ Some aspects of the frequency and distribution of NotI, SfiI, XbaI and BlnI cleavage sites were analyzed using a method based on Markov chain theory. Overlaps of Dam and Dcm methylase sites with XbaI and SfiI cleavage sites were examined. The one XbaI-Dam overlap in the database is in accord with the expected frequency of this overlap. The occurrence of certain types of SfiI-Dcm overlaps are overrepresented. Of the four subtypes of SfiI-Dcm overlap, only one has a partial inhibitory effect on the activity of SfiI. Recognition sites for all four enzymes are rarer than expected based on oligonucleotide frequency data, with this effect being much stronger for XbaI and BlnI than for NotI and SfiI. The latter two enzyme sites are rare mainly due to apparent negative selection against GGCC (both) and CGGCCG (NotI). The former two enzyme sites are rare mainly due to effects of the VSP repair system on certain di-tri- and tetranucleotides, most notably CTAG. Models are proposed to explain several of the anomalies of oligonucleotide distribution in E. coli, and the biological significance of the systems that produce these anomalies is discussed. ^

A comparison of Markov and generalized linear models of hospital mortality

This paper reports a comparison of three modeling strategies for the analysis of hospital mortality in a sample of general medicine inpatients in a Department of Veterans Affairs medical center. Logistic regression, a Markov chain model, and longitudinal logistic regression were evaluated on predictive performance as measured by the c-index and on accuracy of expected numbers of deaths compared to observed. The logistic regression used patient information collected at admission; the Markov model was comprised of two absorbing states for discharge and death and three transient states reflecting increasing severity of illness as measured by laboratory data collected during the hospital stay; longitudinal regression employed Generalized Estimating Equations (GEE) to model covariance structure for the repeated binary outcome. Results showed that the logistic regression predicted hospital mortality as well as the alternative methods but was limited in scope of application. The Markov chain provides insights into how day to day changes of illness severity lead to discharge or death. The longitudinal logistic regression showed that increasing illness trajectory is associated with hospital mortality. The conclusion is reached that for standard applications in modeling hospital mortality, logistic regression is adequate, but for new challenges facing health services research today, alternative methods are equally predictive, practical, and can provide new insights. ^

A multivariate frailty model for disease recurrences and survival

A multivariate frailty hazard model is developed for joint-modeling of three correlated time-to-event outcomes: (1) local recurrence, (2) distant recurrence, and (3) overall survival. The term frailty is introduced to model population heterogeneity. The dependence is modeled by conditioning on a shared frailty that is included in the three hazard functions. Independent variables can be included in the model as covariates. The Markov chain Monte Carlo methods are used to estimate the posterior distributions of model parameters. The algorithm used in present application is the hybrid Metropolis-Hastings algorithm, which simultaneously updates all parameters with evaluations of gradient of log posterior density. The performance of this approach is examined based on simulation studies using Exponential and Weibull distributions. We apply the proposed methods to a study of patients with soft tissue sarcoma, which motivated this research. Our results indicate that patients with chemotherapy had better overall survival with hazard ratio of 0.242 (95% CI: 0.094 - 0.564) and lower risk of distant recurrence with hazard ratio of 0.636 (95% CI: 0.487 - 0.860), but not significantly better in local recurrence with hazard ratio of 0.799 (95% CI: 0.575 - 1.054). The advantages and limitations of the proposed models, and future research directions are discussed. ^