A two-state Markov mixture model for a time series of epileptic seizure counts.

This paper discusses a model for a time series of epileptic seizure counts in which the mean of a Poisson distribution changes according to an underlying two-state Markov chain. The EM algorithm (Dempster, Laird, and Rubin, 1977, Journal of the Royal Statistical Society, Series B 39, 1-38) is used to compute maximum likelihood estimators for the parameters of this two-state mixture model and extensions are made allowing for nonstationarity. The model is illustrated using daily seizure counts for patients with intractable epilepsy and results are compared with a simple Poisson distribution and Poisson regressions. Some simulation results are also presented to demonstrate the feasibility of this model.     

A weighted partial likelihood approach for zero‐truncated models

Zero‐truncated data arises in various disciplines where counts are observed but the zero count category cannot be observed during sampling. Maximum likelihood estimation can be used to model these data; however, due to its nonstandard form it cannot be easily implemented using well‐known software packages, and additional programming is often required. Motivated by the Rao–Blackwell theorem, we develop a weighted partial likelihood approach to estimate model parameters for zero‐truncated binomial and Poisson data. The resulting estimating function is equivalent to a weighted score function for standard count data models, and allows for applying readily available software. We evaluate the efficiency for this new approach and show that it performs almost as well as maximum likelihood estimation. The weighted partial likelihood approach is then extended to regression modelling and variable selection. We examine the performance of the proposed methods through simulation and present two case studies using real data.     

Markov models for covariate dependence of binary sequences

Suppose that a heterogeneous group of individuals is followed over time and that each individual can be in state 0 or state 1 at each time point. The sequence of states is assumed to follow a binary Markov chain. In this paper we model the transition probabilities for the 0 to 0 and 1 to 0 transitions by two logistic regressions, thus showing how the covariates relate to changes in state. With p covariates, there are 2(p + 1) parameters including intercepts, which we estimate by maximum likelihood. We show how to use transition probability estimates to test hypotheses about the probability of occupying state 0 at time i (i = 2, ..., T) and the equilibrium probability of state 0. These probabilities depend on the covariates. A recursive algorithm is suggested to estimate regression coefficients when some responses are missing. Extensions of the basic model which allow time-dependent covariates and nonstationary or second-order Markov chains are presented. An example shows the model applied to a study of the psychological impact of breast cancer in which women did or did not manifest distress at four time points in the year following surgery.     

Zero‐Inflated Negative Binomial Mixed Regression Modeling of Over‐Dispersed Count Data with Extra Zeros

In many biometrical applications, the count data encountered often contain extra zeros relative to the Poisson distribution. Zero‐inflated Poisson regression models are useful for analyzing such data, but parameter estimates may be seriously biased if the nonzero observations are over‐dispersed and simultaneously correlated due to the sampling design or the data collection procedure. In this paper, a zero‐inflated negative binomial mixed regression model is presented to analyze a set of pancreas disorder length of stay (LOS) data that comprised mainly same‐day separations. Random effects are introduced to account for inter‐hospital variations and the dependency of clustered LOS observations. Parameter estimation is achieved by maximizing an appropriate log‐likelihood function using an EM algorithm. Alternative modeling strategies, namely the finite mixture of Poisson distributions and the non‐parametric maximum likelihood approach, are also considered. The determination of pertinent covariates would assist hospital administrators and clinicians to manage LOS and expenditures efficiently.     

Parameter estimation and goodness-of-fit in log binomial regression

An estimate of the risk, adjusted for confounders, can be obtained from a fitted logistic regression model, but it substantially over-estimates when the outcome is not rare. The log binomial model, binomial errors and log link, is increasingly being used for this purpose. However this model's performance, goodness of fit tests and case-wise diagnostics have not been studied. Extensive simulations are used to compare the performance of the log binomial, a logistic regression based method proposed by Schouten et al. (1993) and a Poisson regression approach proposed by Zou (2004) and Carter, Lipsitz, and Tilley (2005). Log binomial regression resulted in "failure" rates (non-convergence, out-of-bounds predicted probabilities) as high as 59%. Estimates by the method of Schouten et al. (1993) produced fitted log binomial probabilities greater than unity in up to 19% of samples to which a log binomial model had been successfully fit and in up to 78% of samples when the log binomial model fit failed. Similar percentages were observed for the Poisson regression approach. Coefficient and standard error estimates from the three models were similar. Rejection rates for goodness of fit tests for log binomial fit were around 5%. Power of goodness of fit tests was modest when an incorrect logistic regression model was fit. Examples demonstrate the use of the methods. Uncritical use of the log binomial regression model is not recommended.     

Quasi Likelihood/Moment Method for Generalized and Restricted Generalized Poisson Regression Models and Its Application

This paper reviews the generalized Poisson regression model, the restricted generalized Poisson regression model and the mixed Poisson regression (negative binomial regression and Poisson inverse Gaussian regression) models which can be used for regression analysis of counts. The aim of this study is to demonstrate the quasi likelihood/moment method, which is used for estimation of the parameters of mixed Poisson regression models, also applicable to obtain the estimates of the parameters of the generalized Poisson regression and the restricted generalized Poisson regression models. Besides, at the end of this study an application related to this method for zoological data is given.     

A general framework for constraint approaches to adjusted risk differences

The risk difference is an intelligible measure for comparing disease incidence in two exposure or treatment groups. Despite its convenience in interpretation, it is less prevalent in epidemiological and clinical areas where regression models are required in order to adjust for confounding. One major barrier to its popularity is that standard linear binomial or Poisson regression models can provide estimated probabilities out of the range of (0,1), resulting in possible convergence issues. For estimating adjusted risk differences, we propose a general framework covering various constraint approaches based on binomial and Poisson regression models. The proposed methods span the areas of ordinary least squares, maximum likelihood estimation, and Bayesian inference. Compared to existing approaches, our methods prevent estimates and confidence intervals of predicted probabilities from falling out of the valid range. Through extensive simulation studies, we demonstrate that the proposed methods solve the issue of having estimates or confidence limits of predicted probabilities out of (0,1), while offering performance comparable to its alternative in terms of the bias, variability, and coverage rates in point and interval estimation of the risk difference. An application study is performed using data from the Prospective Registry Evaluating Myocardial Infarction: Event and Recovery (PREMIER) study.     

A hidden Markov model approach to neuron firing patterns.

Analysis and characterization of neuronal discharge patterns are of interest to neurophysiologists and neuropharmacologists. In this paper we present a hidden Markov model approach to modeling single neuron electrical activity. Basically the model assumes that each interspike interval corresponds to one of several possible states of the neuron. Fitting the model to experimental series of interspike intervals by maximum likelihood allows estimation of the number of possible underlying neuron states, the probability density functions of interspike intervals corresponding to each state, and the transition probabilities between states. We present an application to the analysis of recordings of a locus coeruleus neuron under three pharmacological conditions. The model distinguishes two states during halothane anesthesia and during recovery from halothane anesthesia, and four states after administration of clonidine. The transition probabilities yield additional insights into the mechanisms of neuron firing.     

Bayesian semiparametric intensity estimation for inhomogeneous spatial point processes

In this work we propose a fully Bayesian semiparametric method to estimate the intensity of an inhomogeneous spatial point process. The basic idea is to first convert intensity estimation into a Poisson regression setting via binning data points on a regular grid, and then model the log intensity semiparametrically using an adaptive version of Gaussian Markov random fields to smooth the corresponding counts. The inference is carried by an efficient Markov chain Monte Carlo simulation algorithm. Compared to existing methods for intensity estimation, for example, parametric modeling and kernel smoothing, the proposed estimator not only provides inference regarding the dependence of the intensity function on possible covariates, but also uses information from the data to adaptively determine the amount of smoothing at the local level. The effectiveness of using our method is demonstrated through simulation studies and an application to a rainforest dataset.     

Evaluation and Estimation of Various Markov Models with Applications to Membrane Channel Kinetics

Hidden Markov modelling is a powerful and efficient digital signal processing strategy for extracting the maximum likelihood model from a finite length sample of noisy data. Assuming the number of states in the model is known, then the state levels, transition probabilities, initial state distribution and the noise variance can be estimated. We investigate the applicability of this technique in membrane channel kinetics not only as a parameter estimator, but also as an aid to discriminating between various model types according to their statistical likelihood. We survey three representative classes of channel dynamics, namely: aggregated Markov models, semi-Markov models (with asymptotically convergent transition probabilities), and coupled Markov models; reformulating each within a discrete-time hidden Markov model framework. We then provide numerical evidence of the effectiveness of the procedure using simulated channel data and hence show that the correct model, as well as the model parameters, can be discerned. We also demonstrate that the model likelihood can be used to indicate the approximate number of states in the model.     

Dynamic Alignment Models for Neural Coding

Recently, there have been remarkable advances in modeling the relationships between the sensory environment, neuronal responses, and behavior. However, most models cannot encompass variable stimulus-response relationships such as varying response latencies and state or context dependence of the neural code. Here, we consider response modeling as a dynamic alignment problem and model stimulus and response jointly by a mixed pair hidden Markov model (MPH). In MPHs, multiple stimulus-response relationships (e.g., receptive fields) are represented by different states or groups of states in a Markov chain. Each stimulus-response relationship features temporal flexibility, allowing modeling of variable response latencies, including noisy ones. We derive algorithms for learning of MPH parameters and for inference of spike response probabilities. We show that some linear-nonlinear Poisson cascade (LNP) models are a special case of MPHs. We demonstrate the efficiency and usefulness of MPHs in simulations of both jittered and switching spike responses to white noise and natural stimuli. Furthermore, we apply MPHs to extracellular single and multi-unit data recorded in cortical brain areas of singing birds to showcase a novel method for estimating response lag distributions. MPHs allow simultaneous estimation of receptive fields, latency statistics, and hidden state dynamics and so can help to uncover complex stimulus response relationships that are subject to variable timing and involve diverse neural codes.     

Comparison of Bayesian methods for flexible modeling of spatial risk surfaces in disease mapping

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)‐based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF‐based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.     

Marginalized zero‐inflated Poisson models with missing covariates

Unlike zero‐inflated Poisson regression, marginalized zero‐inflated Poisson (MZIP) models for counts with excess zeros provide estimates with direct interpretations for the overall effects of covariates on the marginal mean. In the presence of missing covariates, MZIP and many other count data models are ordinarily fitted using complete case analysis methods due to lack of appropriate statistical methods and software. This article presents an estimation method for MZIP models with missing covariates. The method, which is applicable to other missing data problems, is illustrated and compared with complete case analysis by using simulations and dental data on the caries preventive effects of a school‐based fluoride mouthrinse program.     

Quasi-likelihood estimation for relative risk regression models

For a prospective randomized clinical trial with two groups, the relative risk can be used as a measure of treatment effect and is directly interpretable as the ratio of success probabilities in the new treatment group versus the placebo group. For a prospective study with many covariates and a binary outcome (success or failure), relative risk regression may be of interest. If we model the log of the success probability as a linear function of covariates, the regression coefficients are log-relative risks. However, using such a log-linear model with a Bernoulli likelihood can lead to convergence problems in the Newton-Raphson algorithm. This is likely to occur when the success probabilities are close to one. A constrained likelihood method proposed by Wacholder (1986, American Journal of Epidemiology 123, 174-184), also has convergence problems. We propose a quasi-likelihood method of moments technique in which we naively assume the Bernoulli outcome is Poisson, with the mean (success probability) following a log-linear model. We use the Poisson maximum likelihood equations to estimate the regression coefficients without constraints. Using method of moment ideas, one can show that the estimates using the Poisson likelihood will be consistent and asymptotically normal. We apply these methods to a double-blinded randomized trial in primary biliary cirrhosis of the liver (Markus et al., 1989, New England Journal of Medicine 320, 1709-1713).     

Semiparametric analysis of zero-inflated count data

Medical and public health research often involve the analysis of count data that exhibit a substantially large proportion of zeros, such as the number of heart attacks and the number of days of missed primary activities in a given period. A zero-inflated Poisson regression model, which hypothesizes a two-point heterogeneity in the population characterized by a binary random effect, is generally used to model such data. Subjects are broadly categorized into the low-risk group leading to structural zero counts and high-risk (or normal) group so that the counts can be modeled by a Poisson regression model. The main aim is to identify the explanatory variables that have significant effects on (i) the probability that the subject is from the low-risk group by means of a logistic regression formulation; and (ii) the magnitude of the counts, given that the subject is from the high-risk group by means of a Poisson regression where the effects of the covariates are assumed to be linearly related to the natural logarithm of the mean of the counts. In this article we consider a semiparametric zero-inflated Poisson regression model that postulates a possibly nonlinear relationship between the natural logarithm of the mean of the counts and a particular covariate. A sieve maximum likelihood estimation method is proposed. Asymptotic properties of the proposed sieve maximum likelihood estimators are discussed. Under some mild conditions, the estimators are shown to be asymptotically efficient and normally distributed. Simulation studies were carried out to investigate the performance of the proposed method. For illustration purpose, the method is applied to a data set from a public health survey conducted in Indonesia where the variable of interest is the number of days of missed primary activities due to illness in a 4-week period.     

Continuous time Markov models for binary longitudinal data

Longitudinal data usually consist of a number of short time series. A group of subjects or groups of subjects are followed over time and observations are often taken at unequally spaced time points, and may be at different times for different subjects. When the errors and random effects are Gaussian, the likelihood of these unbalanced linear mixed models can be directly calculated, and nonlinear optimization used to obtain maximum likelihood estimates of the fixed regression coefficients and parameters in the variance components. For binary longitudinal data, a two state, non-homogeneous continuous time Markov process approach is used to model serial correlation within subjects. Formulating the model as a continuous time Markov process allows the observations to be equally or unequally spaced. Fixed and time varying covariates can be included in the model, and the continuous time model allows the estimation of the odds ratio for an exposure variable based on the steady state distribution. Exact likelihoods can be calculated. The initial probability distribution on the first observation on each subject is estimated using logistic regression that can involve covariates, and this estimation is embedded in the overall estimation. These models are applied to an intervention study designed to reduce children's sun exposure.     

One‐Sided Test to Assess Correlation in Linear Logistic Models using Estimating Equations

A score‐type test is proposed for testing the hypothesis of independent binary random variables against positive correlation in linear logistic models with sparse data and cluster specific covariates. The test is developed for univariate and multivariate one‐sided alternatives. The main advantage of using score test is that it requires estimation of the model only under the null hypothesis, that in this case corresponds to the binomial maximum likelihood fit. The score‐type test is developed from a class of estimating equations with block‐diagonal structure in which the coefficients of the linear logistic model are estimated simultaneously with the correlation. The simplicity of the score test is illustrated in two particular examples.     

A New Approach for Handling Longitudinal Count Data with Zero‐Inflation and Overdispersion: Poisson Geometric Process Model

For time series of count data, correlated measurements, clustering as well as excessive zeros occur simultaneously in biomedical applications. Ignoring such effects might contribute to misleading treatment outcomes. A generalized mixture Poisson geometric process (GMPGP) model and a zero‐altered mixture Poisson geometric process (ZMPGP) model are developed from the geometric process model, which was originally developed for modelling positive continuous data and was extended to handle count data. These models are motivated by evaluating the trend development of new tumour counts for bladder cancer patients as well as by identifying useful covariates which affect the count level. The models are implemented using Bayesian method with Markov chain Monte Carlo (MCMC) algorithms and are assessed using deviance information criterion (DIC).     

Stochastic models for heterogeneous DNA sequences

The composition of naturally occurring DNA sequences is often strikingly heterogeneous. In this paper, the DNA sequence is viewed as a stochastic process with local compositional properties determined by the states of a hidden Markov chain. The model used is a discrete-state, discreteoutcome version of a general model for non-stationary time series proposed by Kitagawa (1987). A smoothing algorithm is described which can be used to reconstruct the hidden process and produce graphic displays of the compositional structure of a sequence. The problem of parameter estimation is approached using likelihood methods and an EM algorithm for approximating the maximum likelihood estimate is derived. The methods are applied to sequences from yeast mitochondrial DNA, human and mouse mitochondrial DNAs, a human X chromosomal fragment and the complete genome of bacteriophage lambda.     

Bayesian latent multi‐state modeling for nonequidistant longitudinal electronic health records

Large amounts of longitudinal health records are now available for dynamic monitoring of the underlying processes governing the observations. However, the health status progression across time is not typically observed directly: records are observed only when a subject interacts with the system, yielding irregular and often sparse observations. This suggests that the observed trajectories should be modeled via a latent continuous‐time process potentially as a function of time‐varying covariates. We develop a continuous‐time hidden Markov model to analyze longitudinal data accounting for irregular visits and different types of observations. By employing a specific missing data likelihood formulation, we can construct an efficient computational algorithm. We focus on Bayesian inference for the model: this is facilitated by an expectation‐maximization algorithm and Markov chain Monte Carlo methods. Simulation studies demonstrate that these approaches can be implemented efficiently for large data sets in a fully Bayesian setting. We apply this model to a real cohort where patients suffer from chronic obstructive pulmonary disease with the outcome being the number of drugs taken, using health care utilization indicators and patient characteristics as covariates.