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PubMed1988

Mono- through hexanucleotide composition of the sense strand of yeast DNA: a Markov chain analysis.

Here we compare several methods for predicting oligonucleotide frequencies in 392 kb of yeast DNA. As in previous work on E. coli, a relatively simple equation based on tetranucleotide frequencies can be used in predicting the frequencies of longer oligonucleotides. For example, the mean of observed/expected abundances of 4,096 hexamers was 1.00 with a sample standard deviation of .18. This simple predictor arises by considering each base on the sense strand of yeast to depend only on the three bases 5' to it (a 3rd order Markov chain) and is more accurate in estimating oligonucleotide frequencies than other statistical methods examined. This equation is useful in predicting restriction enzyme fragment sizes, selecting restriction enzymes that cut preferentially in coding vs noncoding regions, and in constructing detailed physical maps of whole genomes. When ranked highest to lowest abundance, the observed frequencies of oligomers of a given length (up to 6 bases) are closely tracked by the predicted abundances of a 3rd or 4th order Markov chain. These ordered abundance curves have a power curve shape with a broad linear range with a sharp break at the top end of the curve. There is also a strong disparity between the most and least abundant oligomer with for example a 79-fold variation between the most and least abundant hexamer. The curves reveal a strong dependence of oligomer frequencies on base composition. Unlike E. Coli, there is no sharp downturn at the low end of the curves and hence, no class of oligomers rare relative to other oligomers of the same length.

Base Composition↗
PubMed2002

Bayesian internal dosimetry calculations using Markov Chain Monte Carlo.

A new numerical method for solving the inverse problem of internal dosimetry is described. The new method uses Markov Chain Monte Carlo and the Metropolis algorithm. Multiple intake amounts, biokinetic types, and times of intake are determined from bioassay data by integrating over the Bayesian posterior distribution. The method appears definitive, but its application requires a large amount of computing time.

Algorithms↗
PubMed1999

Markov chain Monte Carlo linkage analysis of a complex qualitative phenotype.

We tested a new computer program, LOKI, that implements a reversible jump Markov chain Monte Carlo (MCMC) technique for segregation and linkage analysis. Our objective was to determine whether this software, designed for use with continuously distributed phenotypes, has any efficacy when applied to the discrete disease states of the simulated data from the Mordor data from GAW Problem 1. Although we were able to identify the genomic location for two of the three quantitative trait loci by repeated application of the software, the MCMC sampler experienced significant mixing problems indicating that the method, as currently formulated in LOKI, was not suitable for the discrete phenotypes in this data set.

Chromosome Mapping↗
PubMed1999

A Bayesian Markov chain Monte Carlo approach to map disease genes in simulated GAW11 data.

A Bayesian method for multipoint mapping of disease genes based on Markov chain Monte Carlo algorithms was applied to the simulated GAW11 data (Study 2). The method is based on repeated Gibbs and more general Metropolis-Hastings steps. For simplicity we assumed a single disease locus model with two alleles. A normal distribution for the underlying latent variable of the qualitative phenotype was assumed. Based on a single replicate of the data no clear evidence of any of the genes underlying the simulated disease was found. However, when three replicates were combined the method was able to locate the locus C correctly on chromosome 3.

Algorithms↗
PubMed2006

[Prediction of protein solvent accessibility with Markov chain model].

Residues in protein sequences can be classified into two (exposed / buried) or three (exposed/intermediate/buried) states according to their relative solvent accessibility. Markov chain model (MCM) had been adopted for statistical modeling and prediction. Different orders of MCM and classification thresholds were explored to find the best parameters. Prediction results for two different data sets and different cut-off thresholds were evaluated and compared with some existing methods, such as neural network, information theory and support vector machine. The best prediction accuracies achieved by the MCM method were 78.9% for the two-state prediction problem and 67.7% for the three-state prediction problem, respectively. A comprehensive comparison for all these results shows that the prediction accuracy and the correlative coefficient of the MCM method are better than or comparable to those obtained by the other prediction methods. At the same time, the advantage of this method is the lower computation complexity and better time-consuming performance.

Algorithms↗
PubMed2002

A note on estimating the posterior density of a quantitative trait locus from a Markov chain Monte Carlo sample.

We provide an overview of the use of kernel smoothing to summarize the quantitative trait locus posterior distribution from a Markov chain Monte Carlo sample. More traditional distributional summary statistics based on the histogram depend both on the bin width and on the sideway shift of the bin grid used. These factors influence both the overall mapping accuracy and the estimated location of the mode of the distribution. Replacing the histogram by kernel smoothing helps to alleviate these problems. Using simulated data, we performed numerical comparisons between the two approaches. The results clearly illustrate the superiority of the kernel method. The kernel approach is particularly efficient when one needs to point out the best putative quantitative trait locus position on the marker map. In such situations, the smoothness of the posterior estimate is especially important because rough posterior estimates easily produce biased mode estimates. Different kernel implementations are available from Rolf Nevanlinna Institute's web page (http://www.rni.helsinki.fi/;fjh).

Bayes Theorem↗
PubMed2005

Markov chains of infinite order and asymptotic satisfaction of balance: application to the adaptive integration method.

Adaptive Monte Carlo methods can be viewed as implementations of Markov chains with infinite memory. We derive a general condition for the convergence of a Monte Carlo method whose history dependence is contained within the simulated density distribution. In convergent cases, our result implies that the balance condition need only be satisfied asymptotically. As an example, we show that the adaptive integration method converges.

Algorithms↗
PubMed2003

Binary N-step Markov chains and long-range correlated systems.

A theory of systems with long-range correlations based on the consideration of binary N-step Markov chains is developed. In our model, the conditional probability that the ith symbol in the chain equals zero (or unity) is a linear function of the number of unities among the preceding N symbols. The correlation and distribution functions as well as the variance of number of symbols in the words of arbitrary length L are obtained analytically and numerically. If the persistent correlations are not extremely strong, the variance is shown to be nonlinearly dependent on L. A self-similarity of the studied stochastic process is revealed. The applicability of the developed theory to the coarse-grained written and DNA texts is discussed.

Bacillus subtilis↗
PubMed2001

A piecewise-homogeneous Markov chain process of lung transplantation.

BACKGROUND: Markov and semi-Markov models are increasingly used in clinical and public health epidemiology to represent disease processes. We present a Markov model of events following lung transplantation as a case study in clinical epidemiology. METHODS: A five-state discrete-time Markov model with two-way transitions between acute event states is applied to the analysis of 356 lung transplant patients. A two-state continuous time Markov model for chronic disease onset is fitted. Values of transition parameters are estimated by maximum likelihood using numerical methods. RESULTS: Accurate estimates of acute and chonic event rates, and survival probabilities are calculated from transition probabilities. Costs attributed to different acute and chronic states are calculated. CONCLUSIONS: Transition models provide a useful and flexible representation of acute and chronic events and can be used to explore the economic impact of changes in therapy.

Algorithms↗
PubMed1996

An exact McNemar test for paired binary Markov chains.

A straightforward extension of the McNemar test for paired binary data yields an exact test for the equality of the limiting marginal distributions for bivariate binary Markov chains. The exact distribution of the test statistics under the null hypothesis of equal marginals depends on the classical cell occupancy statistics for the Bose-Einstein model. Exact p-values are computed for the one-sided test, and the mean and variance of the test statistic are found. The power of the Markov-McNemar test is found to be close to the power of the classical McNemar test for independent paired observations when the independence assumption holds. The method is applied to the comparison of ribosomal DNA sequences.

Animals↗
PubMed2003

Exploring heterogeneity in tumour data using Markov chain Monte Carlo.

We describe a Bayesian approach to incorporate between-individual heterogeneity associated with parameters of complicated biological models. We emphasize the use of the Markov chain Monte Carlo (MCMC) method in this context and demonstrate the implementation and use of MCMC by analysis of simulated overdispersed Poisson counts and by analysis of an experimental data set on preneoplastic liver lesions (their number and sizes) in the presence of heterogeneity. These examples show that MCMC-based estimates, derived from the posterior distribution with uniform priors, may agree well with maximum likelihood estimates (if available). However, with heterogeneous parameters, maximum likelihood estimates can be difficult to obtain, involving many integrations. In this case, the MCMC method offers substantial computational advantages.

Animals↗
PubMed1993

Markov chain modelling of bioassay toxicity procedures.

Many bioassay procedures to test a new substance for toxicity, like some phase I clinical trials, can be modelled as first-order Markov chains with absorbing states. We provide a general framework for such procedures. We show how one may calculate under any dose response curve assumption final absorption probabilities, and a number of total average costs such as time required until trial completion, number of subjects treated, number of subjects exhibiting unacceptable levels of toxicity, and amount of medication administered. One may use the above information associated with the trial for an a priori comparison between competing designs under consideration. We provide an example of such a comparison.

Biological Assay↗
PubMed2001

Markov chain Monte Carlo linkage analysis: effect of bin width on the probability of linkage.

We analyzed part of the Genetic Analysis Workshop (GAW) 12 simulated data using Monte Carlo Markov chain (MCMC) methods that are implemented in the computer program Loki. The MCMC method reports the "probability of linkage" (PL) across the chromosomal regions of interest. The point of maximum PL can then be taken as a "location estimate" for the location of the quantitative trait locus (QTL). However, Loki does not provide a formal statistical test of linkage. In this paper, we explore how the bin width used in the calculations affects the max PL and the location estimate. We analyzed age at onset (AO) and quantitative trait number 5, Q5, from 26 replicates of the general simulated data in one region where we knew a major gene, MG5, is located. For each trait, we found the max PL and the corresponding location estimate, using four different bin widths. We found that bin width, as expected, does affect the max PL and the location estimate, and we recommend that users of Loki explore how their results vary with different bin widths.

Chromosome Mapping↗
PubMed1997

Mapping-linked quantitative trait loci using Bayesian analysis and Markov chain Monte Carlo algorithms.

A Bayesian method for mapping linked quantitative trait loci (QTL) using multiple linked genetic markers is presented. Parameter estimation and hypothesis testing was implemented via Markov chain Monte Carlo (MCMC) algorithms. Parameters included were allele frequencies and substitution effects for two biallelic QTL, map positions of the QTL, and markers, allele frequencies of the markers, and polygenic and residual variances. Missing data were polygenic effects and multi-locus marker-QTL genotypes. Three different MCMC schemes for testing the presence of a single or two linked QTL on the chromosome were compared. The first approach includes a model indicator variable representing two unlinked QTL, affecting the trait, one linked and one unlinked QTL, or both QTL linked with the markers. The second approach incorporates an indicator variable for each QTL into the model for phenotype, allowing or not allowing for a substitution effect of a QTL, on phenotype, and the third approach is based on model determination by reversible jump MCMC. Methods were evaluated empirically by analyzing simulated granddaughter designs. All methods identified correctly a second, linked QTL and did not reject the one-QTL model when there was only a single QTL, and no additional or an unlinked QTL.

Algorithms↗
PubMedDate not supplied

Identifiability and convergence issues for Markov chain Monte Carlo fitting of spatial models.

The marked increase in popularity of Bayesian methods in statistical practice over the last decade owes much to the simultaneous development of Markov chain Monte Carlo (MCMC) methods for the evaluation of requisite posterior distributions. However, along with this increase in computing power has come the temptation to fit models larger than the data can readily support, meaning that often the propriety of the posterior distributions for certain parameters depends on the propriety of the associated prior distributions. An important example arises in spatial modelling, wherein separate random effects for capturing unstructured heterogeneity and spatial clustering are of substantive interest, even though only their sum is well identified by the data. Increasing the informative content of the associated prior distributions offers an obvious remedy, but one that hampers parameter interpretability and may also significantly slow the convergence of the MCMC algorithm. In this paper we investigate the relationship among identifiability, Bayesian learning and MCMC convergence rates for a common class of spatial models, in order to provide guidance for prior selection and algorithm tuning. We are able to elucidate the key issues with relatively simple examples, and also illustrate the varying impacts of covariates, outliers and algorithm starting values on the resulting algorithms and posterior distributions.

Algorithms↗
PubMed2006

Acceleration of Markov chain Monte Carlo simulations through sequential updating.

Strict detailed balance is not necessary for Markov chain Monte Carlo simulations to converge to the correct equilibrium distribution. In this work, we propose a new algorithm which only satisfies the weaker balance condition, and it is shown analytically to have better mobility over the phase space than the Metropolis algorithm satisfying strict detailed balance. The new algorithm employs sequential updating and yields better sampling statistics than the Metropolis algorithm with random updating. We illustrate the efficiency of the new algorithm on the two-dimensional Ising model. The algorithm is shown to identify the correct equilibrium distribution and to converge faster than the Metropolis algorithm with strict detailed balance. The main advantages of the new algorithm are its simplicity and the feasibility of parallel implementation through domain decomposition.

Journal Article↗
PubMed1996

Markov chain Monte Carlo methods in biostatistics.

Appropriate models in biostatistics are often quite complicated. Such models are typically most easily fit using Bayesian methods, which can often be implemented using simulation techniques. Markov chain Monte Carlo (MCMC) methods are an important set of tools for such simulations. We give an overview and references of this rapidly emerging technology along with a relatively simple example. MCMC techniques can be viewed as extensions of iterative maximization techniques, but with random jumps rather than maximizations at each step. Special care is needed when implementing iterative maximization procedures rather than closed-form methods, and even more care is needed with iterative simulation procedures: it is substantially more difficult to monitor convergence to a distribution than to a point. The most reliable implementations of MCMC build upon results from simpler models fit using combinations of maximization algorithms and noniterative simulations, so that the user has a rough idea of the location and scale of the posterior distribution of the quantities of interest under the more complicated model. These concerns with implementation, however, should not deter the biostatistician from using MCMC methods, but rather help to ensure wise use of these powerful techniques.

Algorithms↗