Natural history of cervical intraepithelial neoplasia: a meta-analysis

OBJECTIVE: To determine the probabilities of transition of stages in the cervical cancer by conducting a meta-studies on the topic. STUDY DESIGN: We identified health states of interest in the natural history of cervical precancer, identified all possible papers that could meet selection criteria, developed relevance and acceptability criteria for inclusion, then thoroughly reviewed the selected studies. To determine the transition probability data we used a random effects model. We determined probabilities for 4 health state transitions. The 6-month mean predictive transition probability (95% confidence intervals with "prediction interval" in parentheses) for high grade squamous intraepithelial lesions (HSIL) to cancer was 0.0037 (0.00004, 0.03386), for low grade squamous intraepithelial lesions (LSIL) to HSIL was 0.0362 (0.00055, 0.23220), for HSIL to LSIL was 0.0282 (0.00027, 0.35782), and for LSIL to normal was 0.0740 (0.00119, 0.42672). CONCLUSION: The transition probabilities between cervical cancer health states for 6-month intervals are small; however, the cumulative risk of cervical cancer is significant. Markers to identify the cervical precursors that will lead to the transition to cervical cancer are needed.     

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.     

Restricted transition probabilities and their applications to some problems in the dynamics of biological populations

In this paper a theory of a class of restricted transition probabilities is developed and applied to a problem in the dynamics of biological populations under the assumption that the underlying stochastic process is a continuous time parameter Markov chain with stationary transition probabilities. The paper is divided into three parts. Part one contains sufficient background from the theory of Markov processes to define restricted transition probabilities in a rigorous manner. In addition, some basic concepts in the theory of stochastic processes are interpreted from the biological point of view. Part two is concerned with the problem of finding representations for restricted transition probabilities. Finally, in part three the theory of restricted transition probabilities is applied to the problem of finding and analyzing some properties of the distribution function of the maximum size attained by the population in a finite time interval for a rather wide class of Markov processes. Some other applications of restricted transition probabilities to other problems in the dynamics of biological populations are also suggested. These applications will be discussed more fully in a companion paper. The research reported in this paper was supported by the United States Atomic Energy Commission, Division of Biology and Medicine Project AT(45-1)-1729.     

From complex spatial dynamics to simple Markov chain models: do predators and prey leave footprints?

In this paper we present a concept for using presence–absence data to recover information on the population dynamics of predator–prey systems. We use a highly complex and spatially explicit simulation model of a predator–prey mite system to generate simple presence–absence data: the number of patches with both prey and predators, with prey only, with predators only, and with neither species, along with the number of patches that change from one state to another in each time step. The average number of patches in the four states, as well as the average transition probabilities from one state to another, are then depicted in a state transition diagram, constituting the "footprints" of the underlying population dynamics. We investigate to what extent changes in the population processes modeled in the complex simulation (i.e. the predator's functional response and the dispersal rates of both species) are reflected by different footprints
The transition probabilities can be used to forecast the expected fate of a system given its current state. However, the transition probabilities in the modeled system depend on the number of patches in each state. We develop a model for the dependence of transition probabilities on state variables, and combine this information in a Markov chain transition matrix model. Finally, we use this extended model to predict the long-term dynamics of the system and to reveal its asymptotic steady state properties.     

A class of latent Markov models for capture-recapture data allowing for time, heterogeneity, and behavior effects

We propose an extension of the latent class model for the analysis of capture-recapture data which allows us to take into account the effect of a capture on the behavior of a subject with respect to future captures. The approach is based on the assumption that the variable indexing the latent class of a subject follows a Markov chain with transition probabilities depending on the previous capture history. Several constraints are allowed on these transition probabilities and on the parameters of the conditional distribution of the capture configuration given the latent process. We also allow for the presence of discrete explanatory variables, which may affect the parameters of the latent process. To estimate the resulting models, we rely on the conditional maximum likelihood approach and for this aim we outline an EM algorithm. We also give some simple rules for point and interval estimation of the population size. The approach is illustrated by applying it to two data sets concerning small mammal populations.     

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.     

A compact statistical model of the song syntax in Bengalese finch

Songs of many songbird species consist of variable sequences of a finite number of syllables. A common approach for characterizing the syntax of these complex syllable sequences is to use transition probabilities between the syllables. This is equivalent to the Markov model, in which each syllable is associated with one state, and the transition probabilities between the states do not depend on the state transition history. Here we analyze the song syntax in Bengalese finch. We show that the Markov model fails to capture the statistical properties of the syllable sequences. Instead, a state transition model that accurately describes the statistics of the syllable sequences includes adaptation of the self-transition probabilities when states are revisited consecutively, and allows associations of more than one state to a given syllable. Such a model does not increase the model complexity significantly. Mathematically, the model is a partially observable Markov model with adaptation (POMMA). The success of the POMMA supports the branching chain network model of how syntax is controlled within the premotor song nucleus HVC, but also suggests that adaptation and many-to-one mapping from the syllable-encoding chain networks in HVC to syllables should be included in the network model.     

An illness-death process with time-dependent covariates

A general model for the illness-death stochastic process with covariates has been developed for the analysis of survival data. This model incorporates important baseline and time-dependent covariates in order to make an appropriate adjustment for the transition and survival probabilities. The follow-up period is subdivided into small intervals and a constant hazard is assumed for each interval. An approximation formula is derived to estimate the transition parameters when the exact transition time is unknown. The method developed is illustrated with data from a study on the prevention of the recurrence of a myocardial infarction and subsequent mortality, the Beta-Blocker Heart Attack Trial (BHAT). This method provides an analytical approach with which the effectiveness of the treatment can be compared between the placebo and propranolol treatment groups with respect to fatal and nonfatal events simultaneously.     

A compartmental model for the study of diurnal rhythms in cell proliferation

A mathematical model taking into account the observed diurnal variations in cell kinetics is presented. Its principle is to divide each phase of the cell cycle into a definite number of compartments and to assume time-dependent probabilities of transition from one compartment to the following; general properties of the model are derived.The particular case where the only time-dependent transition probabilities are those corresponding to the G₁ phase is studied. A characterization of the joint percentages of S and M cells variations is given. The application of the model to interpretation of published experimental data obtained in hamster cheek pouch epithelium is given.     

Estimation of Bayesian confidence intervals for the product of conditional survival probabilities

A model is derived to estimate the survival probability of a time interval when censorings occur. The time interval is divided into partial intervals in order to obtain the conditional survival probabilities, each of which is a parameter of a Binomial distributed random variable. To allow for the dependence between the events in the different intervals these parameters are transformed. Corresponding a priori density functions are formulated regarding both the Bayesian uniform distribution and the special model. The a posteriori density function is derived for the product of the conditional survival probabilities, and formulae for the BAYE sian confidence interval and the expectation are given. Lower and upper bounds for the confidence interval and the expectation are derived. Some examples are given to compare the results with other methods.     

Bias in Markov models of disease

We examine bias in Markov models of diseases, including both chronic and infectious diseases. We consider two common types of Markov disease models: ones where disease progression changes by severity of disease, and ones where progression of disease changes in time or by age. We find sufficient conditions for bias to exist in models with aggregated transition probabilities when compared to models with state/time dependent transition probabilities. We also find that when aggregating data to compute transition probabilities, bias increases with the degree of data aggregation. We illustrate by examining bias in Markov models of Hepatitis C, Alzheimer’s disease, and lung cancer using medical data and find that the bias is significant depending on the method used to aggregate the data. A key implication is that by not incorporating state/time dependent transition probabilities, studies that use Markov models of diseases may be significantly overestimating or underestimating disease progression. This could potentially result in incorrect recommendations from cost-effectiveness studies and incorrect disease burden forecasts.     

Relation of transition probabilities in the Leslie model to those from experimental cumulative distributions

We explore the relationship between transition probabilities in the Leslie model and those derived from experimental cumulative distributions. The nature of the two kinds of probabilities are discussed, and a formula derived for converting from one to the other. A numerical example is given to illustrate the differences.     

transition priors for protein hidden Markov models: an empirical study towards maximum discrimination.

Insertions and deletions in a profile hidden Markov model (HMM) are modeled by transition probabilities between insert, delete and match states. These are estimated by combining observed data and prior probabilities. The transition prior probabilities can be defined either ad hoc or by maximum likelihood (ML) estimation. We show that the choice of transition prior greatly affects the HMM's ability to discriminate between true and false hits. HMM discrimination was measured using the HMMER 2.2 package applied to 373 families from Pfam. We measured the discrimination between true members and noise sequences employing various ML transition priors and also systematically scanned the parameter space of ad hoc transition priors. Our results indicate that ML priors produce far from optimal discrimination, and we present an empirically derived prior that considerably decreases the number of misclassifications compared to ML. Most of the difference stems from the probabilities for exiting a delete state. The ML prior, which is unaware of noise sequences, estimates a delete-to-delete probability that is relatively high and does not penalize noise sequences enough for optimal discrimination.     

A random time interval approach for analysing the impact of a possible intermediate event on a terminal event

We consider the impact of a possible intermediate event on a terminal event in an illness-death model with states 'initial', 'intermediate' and 'terminal'. One aim is to unambiguously describe the occurrence of the intermediate event in terms of the observable data, the problem being that the intermediate event may not occur. We propose to consider a random time interval, whose length is the time spent in the intermediate state. We derive an estimator of the joint distribution of the left and right limit of the random time interval from the Aalen-Johansen estimator of the matrix of transition probabilities and study its asymptotic properties. We apply our approach to hospital infection data. Estimating the distribution of the random time interval will usually be only a first step of an analysis. We illustrate this by analysing change in length of hospital stay following an infection and derive the large sample properties of the respective estimator.     

An Inhomogeneous Markov Model to Fit the Migration Process

In this paper we study the migration process considering an inhomogeneous Markov model. This is a certain condition to investigate age-dependent population distributions, where the transition probabilities are not constant. We consider also a death process for a population alive in a region at age t and, as a result of this, combined transition probabilities between the states of the concerning Markov chain. The model has non-stationary distribution for t →∞, because the condition of ergodicity does not hold.     

Sequence of events following adoption of contraception: An exploratory analysis of 1973 fertility history data

Abstract

Fertility histories from the 1973 United States National Survey of Family Growth are analyzed in the context of a model of contraceptive use based on a Semi‐Markov processes. This model provides a means of constructing data‐based estimates of probabilities of pregnancy following initial acceptance of a contraceptive method. The algorithm used to construct these estimates recognizes multiple intervals of contraceptive used prior to the events: pregnancy, marital dissolution, or sterilization.

Estimated probabilities of the events marital dissolution and pregnancy for women seeking to delay pregnancy are presented, as are probabilities of contraceptive sterilization for women seeking to prevent subsequent pregnancy. These estimates are compared to one‐step transition probabilities and directly observed NSFG data on pregnancy, marital dissolution, or contraceptive sterilization in an attempt to judge the validity of the model and to assess biases which may result from its use.     

Empirical models of transitions between coral reef states: effects of region, protection, and environmental change

There has been substantial recent change in coral reef communities. To date, most analyses have focussed on static patterns or changes in single variables such as coral cover. However, little is known about how community-level changes occur at large spatial scales. Here, we develop Markov models of annual changes in coral and macroalgal cover in the Caribbean and Great Barrier Reef (GBR) regions. We analyzed reef surveys from the Caribbean and GBR (1996-2006). We defined a set of reef states distinguished by coral and macroalgal cover, and obtained Bayesian estimates of the annual probabilities of transitions between these states. The Caribbean and GBR had different transition probabilities, and therefore different rates of change in reef condition. This could be due to differences in species composition, management or the nature and extent of disturbances between these regions. We then estimated equilibrium probability distributions for reef states, and coral and macroalgal cover under constant environmental conditions. In both regions, the current distributions are close to equilibrium. In the Caribbean, coral cover is much lower and macroalgal cover is higher at equilibrium than in the GBR. We found no evidence for differences in transition probabilities between the first and second halves of our survey period, or between Caribbean reefs inside and outside marine protected areas. However, our power to detect such differences may have been low. We also examined the effects of altering transition probabilities on the community state equilibrium, along a continuum from unfavourable (e.g., increased sea surface temperature) to favourable (e.g., improved management) conditions. Both regions showed similar qualitative responses, but different patterns of uncertainty. In the Caribbean, uncertainty was greatest about effects of favourable changes, while in the GBR, we are most uncertain about effects of unfavourable changes. Our approach could be extended to provide risk analysis for management decisions.     

Complexity,Predictability and Time Homogeneity of Syntax in the Songs of Cassin’s Vireo (Vireo cassinii)

Many species of animals deliver vocalizations in sequences presumed to be governed by internal rules, though the nature and complexity of these syntactical rules have been investigated in relatively few species. Here I present an investigation into the song syntax of fourteen male Cassin’s Vireos (Vireo cassinii), a species whose song sequences are highly temporally structured. I compare their song sequences to three candidate models of varying levels of complexity–zero-order, first-order and second-order Markov models–and employ novel methods to interpolate between these three models. A variety of analyses, including sequence simulations, Fisher’s exact tests, and model likelihood analyses, showed that the songs of this species are too complex to be described by a zero-order or first-order Markov model. The model that best fit the data was intermediate in complexity between a first- and second-order model, though I also present evidence that some transition probabilities are conditioned on up to three preceding phrases. In addition, sequences were shown to be predictable with more than 54% accuracy overall, and predictability was positively correlated with the rate of song delivery. An assessment of the time homogeneity of syntax showed that transition probabilities between phrase types are largely stable over time, but that there was some evidence for modest changes in syntax within and between breeding seasons, a finding that I interpret to represent changes in breeding stage and social context rather than irreversible, secular shifts in syntax over time. These findings constitute a valuable addition to our understanding of bird song syntax in free-living birds, and will contribute to future attempts to understand the evolutionary importance of bird song syntax in avian communication.     

Sequence alignments and pair hidden Markov models using evolutionary history

This work presents a novel pairwise statistical alignment method based on an explicit evolutionary model of insertions and deletions (indels). Indel events of any length are possible according to a geometric distribution. The geometric distribution parameter, the indel rate, and the evolutionary time are all maximum likelihood estimated from the sequences being aligned. Probability calculations are done using a pair hidden Markov model (HMM) with transition probabilities calculated from the indel parameters. Equations for the transition probabilities make the pair HMM closely approximate the specified indel model. The method provides an optimal alignment, its likelihood, the likelihood of all possible alignments, and the reliability of individual alignment regions. Human alpha and beta-hemoglobin sequences are aligned, as an illustration of the potential utility of this pair HMM approach.