A quantitative approach for polymerase chain reactions based on a hidden Markov model

Polymerase chain reaction (PCR) is a major DNA amplification technology from molecular biology. The quantitative analysis of PCR aims at determining the initial amount of the DNA molecules from the observation of typically several PCR amplifications curves. The mainstream observation scheme of the DNA amplification during PCR involves fluorescence intensity measurements. Under the classical assumption that the measured fluorescence intensity is proportional to the amount of present DNA molecules, and under the assumption that these measurements are corrupted by an additive Gaussian noise, we analyze a single amplification curve using a hidden Markov model(HMM). The unknown parameters of the HMM may be separated into two parts. On the one hand, the parameters from the amplification process are the initial number of the DNA molecules and the replication efficiency, which is the probability of one molecule to be duplicated. On the other hand, the parameters from the observational scheme are the scale parameter allowing to convert the fluorescence intensity into the number of DNA molecules and the mean and variance characterizing the Gaussian noise. We use the maximum likelihood estimation procedure to infer the unknown parameters of the model from the exponential phase of a single amplification curve, the main parameter of interest for quantitative PCR being the initial amount of the DNA molecules. An illustrative example is provided. This research was financed by the Swedish foundation for Strategic Research through the Gothenburg Mathematical Modelling Centre.     

The analysis of hospital infection data using hidden Markov models

Surveillance data for communicable nosocomial pathogens usually consist of short time series of low-numbered counts of infected patients. These often show overdispersion and autocorrelation. To date, almost all analyses of such data have ignored the communicable nature of the organisms and have used methods appropriate only for independent outcomes. Inferences that depend on such analyses cannot be considered reliable when patient-to-patient transmission is important. We propose a new method for analysing these data based on a mechanistic model of the epidemic process. Since important nosocomial pathogens are often carried asymptomatically with overt infection developing in only a proportion of patients, the epidemic process is usually only partially observed by routine surveillance data. We therefore develop a 'structured' hidden Markov model where the underlying Markov chain is generated by a simple transmission model. We apply both structured and standard (unstructured) hidden Markov models to time series for three important pathogens. We find that both methods can offer marked improvements over currently used approaches when nosocomial spread is important. Compared to the standard hidden Markov model, the new approach is more parsimonious, is more biologically plausible, and allows key epidemiological parameters to be estimated.     

A general model for detecting genetic determinants underlying longitudinal traits with unequally spaced measurements and nonstationary covariance structure

A mixture model for determining quantitative trait loci (QTL) affecting growth trajectories has been proposed in the literature. In this article, we extend this model to a more general situation in which longitudinal traits for each subject are measured at unequally spaced time intervals, different subjects have different measurement patterns, and the residual correlation within subjects is nonstationary. We derive an EM-simplex hybrid algorithm to estimate the allele frequencies, Hardy-Weinberg disequilibrium, and linkage disequilibrium between QTL in the original population and parameters contained in the growth equation and in the covariance structure. A worked example of head circumference growth in 145 children is used to validate our extended model. A simulation study is performed to examine the statistical properties of the parameter estimation obtained from this example. Finally, we discuss the implications and extensions of our model for detecting QTL that affect growth trajectories.     

Stochastic model for analysis of longitudinal data on aging and mortality

Aging-related changes in a human organism follow dynamic regularities, which contribute to the observed age patterns of incidence and mortality curves. An organism's 'optimal' (normal) physiological state changes with age, affecting the values of risks of disease and death. The resistance to stresses, as well as adaptive capacity, declines with age. An exposure to improper environment results in persisting deviation of individuals' physiological (and biological) indices from their normal state (due to allostatic adaptation), which, in turn, increases chances of disease and death. Despite numerous studies investigating these effects, there is no conceptual framework, which would allow for putting all these findings together, and analyze longitudinal data taking all these dynamic connections into account. In this paper we suggest such a framework, using a new version of stochastic process model of aging and mortality. Using this model, we elaborated a statistical method for analyses of longitudinal data on aging, health and longevity and tested it using different simulated data sets. The results show that the model may characterize complicated interplay among different components of aging-related changes in humans and that the model parameters are identifiable from the data.     

An approximate likelihood for genetic data under a model with recombination and population splitting

We describe a new approximate likelihood for population genetic data under a model in which a single ancestral population has split into two daughter populations. The approximate likelihood is based on the ‘Product of Approximate Conditionals’ likelihood and ‘copying model’ of Li and Stephens [Li, N., Stephens, M., 2003. Modeling linkage disequilibrium and identifying recombination hotspots using single-nucleotide polymorphism data. Genetics 165 (4), 2213–2233]. The approach developed here may be used for efficient approximate likelihood-based analyses of unlinked data. However our copying model also considers the effects of recombination. Hence, a more important application is to loosely-linked haplotype data, for which efficient statistical models explicitly featuring non-equilibrium population structure have so far been unavailable. Thus, in addition to the information in allele frequency differences about the timing of the population split, the method can also extract information from the lengths of haplotypes shared between the populations. There are a number of challenges posed by extracting such information, which makes parameter estimation difficult. We discuss how the approach could be extended to identify haplotypes introduced by migrants.     

A marginalized pattern-mixture model for longitudinal binary data when nonresponse depends on unobserved responses

This paper proposes a method for modeling longitudinal binary data when nonresponse depends on unobserved responses. The proposed method presumes that the target of inference is the marginal distribution of the response at each occasion and its dependence on covariates, and can accommodate both monotone and non-monotone missingness. The approach involves a marginally specified pattern-mixture model that directly parameterizes both the marginal means at each occasion and the dependence of each response on indicators of nonresponse pattern. This formulation readily incorporates a variety of nonresponse processes assumed within a sensitivity analysis. Once identifying restrictions have been made, estimation of model parameters proceeds via solution to a set of modified generalized estimating equations. The proposed method provides an alternative to standard selection and pattern-mixture modeling frameworks, while featuring certain advantages of each. The paper concludes with application of the method to data from a contraceptive clinical trial with substantial dropout.     

A semiparametric model for the analysis of recurrent-event panel data

In many longitudinal studies, interest focuses on the occurrence rate of some phenomenon for the subjects in the study. When the phenomenon is nonterminating and possibly recurring, the result is a recurrent-event data set. Examples include epileptic seizures and recurrent cancers. When the recurring event is detectable only by an expensive or invasive examination, only the number of events occurring between follow-up times may be available. This article presents a semiparametric model for such data, based on a multiplicative intensity model paired with a fully flexible nonparametric baseline intensity function. A random subject-specific effect is included in the intensity model to account for the overdispersion frequently displayed in count data. Estimators are determined from quasi-likelihood estimating functions. Because only first- and second-moment assumptions are required for quasi-likelihood, the method is more robust than those based on the specification of a full parametric likelihood. Consistency of the estimators depends only on the assumption of the proportional intensity model. The semiparametric estimators are shown to be highly efficient compared with the usual parametric estimators. As with semiparametric methods in survival analysis, the method provides useful diagnostics for specific parametric models, including a quasi-score statistic for testing specific baseline intensity functions. The techniques are used to analyze cancer recurrences and a pheromone-based mating disruption experiment in moths. A simulation study confirms that, for many practical situations, the estimators possess appropriate small-sample characteristics.     

A generalized mover-stayer model for panel data

A generalized mover-stayer model is described for conditionally Markov processes under panel observation. Marginally the model represents a mixture of nested continuous-time Markov processes in which sub-models are defined by constraining some transition intensities to zero between two or more states of a full model. A Fisher scoring algorithm is described which facilitates maximum likelihood estimation based only on the first derivatives of the transition probability matrices. The model is fit to data from a smoking prevention study and is shown to provide a significant improvement in fit over a time-homogeneous Markov model. Extensions are developed which facilitate examination of covariate effects on both the transition intensities and the mover-stayer probabilities.