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Unformatted text preview: Lecture 11: Transition models for repeated categorical data Laurent Briollais & Rafiq Chowdhury Samuel Lunenfeld Research Institute, Mount Sinai Hospital Dalla Lana School of Public Health, UofT December 5, 2011 1 CHL 5210 Fall 2011 Categorical data analysis Lecture #11 2 Previous lecture • Logit model with random intercept for matched pairs • The logisticnormal model • The Generalized Linear Mixed Model (GLMM) • Discussion conditional vs. marginal approaches CHL 5210 Fall 2011 Categorical data analysis Lecture #11 3 Outline • Generalities • Markov Chains • Firstorder Markov model with explanatory variables • Secondorder Markov model with explanatory variables • Multistate Markov models • Examples in SAS • Concluding remarks • The project CHL 5210 Fall 2011 Categorical data analysis Lecture #11 4 1 Generalities • Book: A.Islam, R. Chowdhury, S. Huda. Markov models with covariate dependence for repeated measures . Nova Science Publishers, Inc. New York. 2009. • Papers: G. Bonney. Regressive logistic models for familial disease and other binary traits . Biometrics , 1986, 42: 611625. G. Bonney. Logistic regression for dependent binary observa tions . Biometrics , 1987, 43:951973. • Transition models are special cases of logit or loglinear models for re peated categorical observations. When the repeated observations can be assumed to follow a Markov chain, an appropriate Markov chain model can be fit. • The firstorder Markov property is satisfied when the prediction of a fu ture response only depends on the response immediately prior to it, and no other responses occurring before that previous response. • Estimation of Markov chain models can be done via MLE or iteratively reweighted least squares (IRLS). • A SAS/IML program is available to fit these models (written by Rafiq Chowdhury). CHL 5210 Fall 2011 Categorical data analysis Lecture #11 5 2 Markov Chains • Markov chain models are now very popular in various disciplines (time series, disease progression, stock market, meteorology, ...) • Recent approaches can deal with covariate dependence of transition prob abilities = transition models. • Discretetime Markov chains are discretetime stochastic processes with a discrete state space. That means that the random variable (potentially) changes states at discrete time points (say, every hour), and the states come from a set of discrete (many times, also finite) possible states. So, if the state of the random variable at time t is represented by y t , then the stochastic process is represented by { y , y 1 , y 2 , ... } . • For firstorder Markov chains, the conditional distribution of y t +1 given { y , y 1 , y 2 , ..., y t } is identical to the distribution of y t +1 given only y t . In other words, this means that in order to predict the state at the next time point, we only need to consider the current state. Any states prior to the current state add nothing to predictability of the next state. Thisto the current state add nothing to predictability of the next state....
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This note was uploaded on 02/23/2012 for the course CHL 5210H taught by Professor Leisun during the Fall '11 term at University of Toronto.
 Fall '11
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