Prior Distributions (prior + distribution)

Distribution by Scientific Domains

Kinds of Prior Distributions

  • informative prior distribution

  • Selected Abstracts


    Pulak Ghosh
    Summary This paper considers the use of Dirichlet process prior distributions in the statistical analysis of network data. Dirichlet process prior distributions have the advantages of avoiding the parametric specifications for distributions, which are rarely known, and of facilitating a clustering effect, which is often applicable to network nodes. The approach is highlighted for two network models and is conveniently implemented using WinBUGS software. [source]

    Bayesian regression with multivariate linear splines

    C. C. Holmes
    We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data. [source]

    Decision Theory Applied to an Instrumental Variables Model

    ECONOMETRICA, Issue 3 2007
    Gary Chamberlain
    This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (,, ,, ,), where ,is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter ,and a parameter vector ,, which is normalized to be a point on the (k,1)-dimensional unit sphere; ,measures the strength of the association between the endogenous variables and the instrumental variables, and ,is a measure of direction. A prior distribution is introduced for the IV model. The parameters ,, ,, and ,are treated as independent random variables. The distribution for ,is uniform on the unit circle; the distribution for ,is uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for ,is left general. It turns out that given any positive value for ,, the Bayes estimator of ,does not depend on ,; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. The same general concepts are applied to obtain confidence intervals. The prior distribution is used in two ways. The first way is to integrate out the nuisance parameter ,in the IV model. This gives an integrated likelihood function with two scalar parameters, ,and ,. Inverting a likelihood ratio test, based on the integrated likelihood function, provides a confidence interval for ,. This lacks finite sample optimality, but invariance arguments show that the risk function depends only on ,and not on ,or ,. The second approach to confidence sets aims for finite sample optimality by setting up a loss function that trades off coverage against the length of the interval. The automatic uniform priors are used for ,and ,, but a prior is also needed for the scalar ,, and no guidance is offered on this choice. The Bayes rule is a highest posterior density set. Invariance arguments show that the risk function depends only on ,and not on ,or ,. The optimality result combines average risk and maximum risk. The confidence set minimizes the average,with respect to the prior distribution for ,,of the maximum risk, where the maximization is with respect to ,and ,. [source]

    Psychometric Properties of IRT Proficiency Estimates

    Michael J. Kolen
    Psychometric properties of item response theory proficiency estimates are considered in this paper. Proficiency estimators based on summed scores and pattern scores include non-Bayes maximum likelihood and test characteristic curve estimators and Bayesian estimators. The psychometric properties investigated include reliability, conditional standard errors of measurement, and score distributions. Four real-data examples include (a) effects of choice of estimator on score distributions and percent proficient, (b) effects of the prior distribution on score distributions and percent proficient, (c) effects of test length on score distributions and percent proficient, and (d) effects of proficiency estimator on growth-related statistics for a vertical scale. The examples illustrate that the choice of estimator influences score distributions and the assignment of examinee to proficiency levels. In particular, for the examples studied, the choice of Bayes versus non-Bayes estimators had a more serious practical effect than the choice of summed versus pattern scoring. [source]


    Alessandro Barbarino
    This article provides a microfoundation for the rise in optimism that seems to precede market crashes. Small, young markets are more likely to experience stock-price run-ups and crashes. We use a Zeira,Rob type of model in which demand size is uncertain. Optimism then grows rationally if traders' prior distribution over market size has a decreasing hazard. Such prior beliefs are appropriate if most new markets are duds and only a few reach a large size. The crash occurs when capacity outstrips demand. As an illustration, for the period 1971,2001 we fit the model to the Telecom sector. [source]

    Prior knowledge processing for initial state of Kalman filter

    E. Suzdaleva
    Abstract The paper deals with a specification of the prior distribution of the initial state for Kalman filter. The subjective prior knowledge, used in state estimation, can be highly uncertain. In practice, incorporation of prior knowledge contributes to a good start of the filter. The present paper proposes a methodology for selection of the initial state distribution, which enables eliciting of prior knowledge from the available expert information. The proposed methodology is based on the use of the conjugate prior distribution for models belonging to the exponential family. The normal state-space model is used for demonstrating the methodology. The paper covers processing of the prior knowledge for state estimation, available in the form of simulated data. Practical experiments demonstrate the processing of prior knowledge from the urban traffic control area, which is the main application of the research. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Bayesian Hypothesis Testing: a Reference Approach

    José M. Bernardo
    Summary For any probability model M={p(x|,, ,), ,,,, ,,,} assumed to describe the probabilistic behaviour of data x,X, it is argued that testing whether or not the available data are compatible with the hypothesis H0={,=,0} is best considered as a formal decision problem on whether to use (a0), or not to use (a0), the simpler probability model (or null model) M0={p(x|,0, ,), ,,,}, where the loss difference L(a0, ,, ,) ,L(a0, ,, ,) is proportional to the amount of information ,(,0, ,), which would be lost if the simplified model M0 were used as a proxy for the assumed model M. For any prior distribution ,(,, ,), the appropriate normative solution is obtained by rejecting the null model M0 whenever the corresponding posterior expectation ,,,(,0, ,, ,),(,, ,|x)d,d, is sufficiently large. Specification of a subjective prior is always difficult, and often polemical, in scientific communication. Information theory may be used to specify a prior, the reference prior, which only depends on the assumed model M, and mathematically describes a situation where no prior information is available about the quantity of interest. The reference posterior expectation, d(,0, x) =,,,(,|x)d,, of the amount of information ,(,0, ,, ,) which could be lost if the null model were used, provides an attractive nonnegative test function, the intrinsic statistic, which is invariant under reparametrization. The intrinsic statistic d(,0, x) is measured in units of information, and it is easily calibrated (for any sample size and any dimensionality) in terms of some average log-likelihood ratios. The corresponding Bayes decision rule, the Bayesian reference criterion (BRC), indicates that the null model M0 should only be rejected if the posterior expected loss of information from using the simplified model M0 is too large or, equivalently, if the associated expected average log-likelihood ratio is large enough. The BRC criterion provides a general reference Bayesian solution to hypothesis testing which does not assume a probability mass concentrated on M0 and, hence, it is immune to Lindley's paradox. The theory is illustrated within the context of multivariate normal data, where it is shown to avoid Rao's paradox on the inconsistency between univariate and multivariate frequentist hypothesis testing. Résumé Pour un modèle probabiliste M={p(x|,, ,) ,,,, ,,,} censé décrire le comportement probabiliste de données x,X, nous soutenons que tester si les données sont compatibles avec une hypothèse H0={,=,0 doit être considéré comme un problème décisionnel concernant l'usage du modèle M0={p(x|,0, ,) ,,,}, avec une fonction de coût qui mesure la quantité d'information qui peut être perdue si le modèle simplifiéM0 est utilisé comme approximation du véritable modèle M. Le coût moyen, calculé par rapport à une loi a priori de référence idoine fournit une statistique de test pertinente, la statistique intrinsèque d(,0, x), invariante par reparamétrisation. La statistique intrinsèque d(,0, x) est mesurée en unités d'information, et sa calibrage, qui est independante de la taille de léchantillon et de la dimension du paramètre, ne dépend pas de sa distribution à l'échantillonage. La règle de Bayes correspondante, le critère de Bayes de référence (BRC), indique que H0 doit seulement êetre rejeté si le coût a posteriori moyen de la perte d'information à utiliser le modèle simplifiéM0 est trop grande. Le critère BRC fournit une solution bayésienne générale et objective pour les tests d'hypothèses précises qui ne réclame pas une masse de Dirac concentrée sur M0. Par conséquent, elle échappe au paradoxe de Lindley. Cette théorie est illustrée dans le contexte de variables normales multivariées, et on montre qu'elle évite le paradoxe de Rao sur l'inconsistence existant entre tests univariés et multivariés. [source]

    Models for potentially biased evidence in meta-analysis using empirically based priors

    N. J. Welton
    Summary., We present models for the combined analysis of evidence from randomized controlled trials categorized as being at either low or high risk of bias due to a flaw in their conduct. We formulate a bias model that incorporates between-study and between-meta-analysis heterogeneity in bias, and uncertainty in overall mean bias. We obtain algebraic expressions for the posterior distribution of the bias-adjusted treatment effect, which provide limiting values for the information that can be obtained from studies at high risk of bias. The parameters of the bias model can be estimated from collections of previously published meta-analyses. We explore alternative models for such data, and alternative methods for introducing prior information on the bias parameters into a new meta-analysis. Results from an illustrative example show that the bias-adjusted treatment effect estimates are sensitive to the way in which the meta-epidemiological data are modelled, but that using point estimates for bias parameters provides an adequate approximation to using a full joint prior distribution. A sensitivity analysis shows that the gain in precision from including studies at high risk of bias is likely to be low, however numerous or large their size, and that little is gained by incorporating such studies, unless the information from studies at low risk of bias is limited. We discuss approaches that might increase the value of including studies at high risk of bias, and the acceptability of the methods in the evaluation of health care interventions. [source]

    Combining evidence on air pollution and daily mortality from the 20 largest US cities: a hierarchical modelling strategy

    Francesca Dominici
    Reports over the last decade of association between levels of particles in outdoor air and daily mortality counts have raised concern that air pollution shortens life, even at concentrations within current regulatory limits. Criticisms of these reports have focused on the statistical techniques that are used to estimate the pollution,mortality relationship and the inconsistency in findings between cities. We have developed analytical methods that address these concerns and combine evidence from multiple locations to gain a unified analysis of the data. The paper presents log-linear regression analyses of daily time series data from the largest 20 US cities and introduces hierarchical regression models for combining estimates of the pollution,mortality relationship across cities. We illustrate this method by focusing on mortality effects of PM10 (particulate matter less than 10 ,m in aerodynamic diameter) and by performing univariate and bivariate analyses with PM10 and ozone (O3) level. In the first stage of the hierarchical model, we estimate the relative mortality rate associated with PM10 for each of the 20 cities by using semiparametric log-linear models. The second stage of the model describes between-city variation in the true relative rates as a function of selected city-specific covariates. We also fit two variations of a spatial model with the goal of exploring the spatial correlation of the pollutant-specific coefficients among cities. Finally, to explore the results of considering the two pollutants jointly, we fit and compare univariate and bivariate models. All posterior distributions from the second stage are estimated by using Markov chain Monte Carlo techniques. In univariate analyses using concurrent day pollution values to predict mortality, we find that an increase of 10 ,g m -3 in PM10 on average in the USA is associated with a 0.48% increase in mortality (95% interval: 0.05, 0.92). With adjustment for the O3 level the PM10 -coefficient is slightly higher. The results are largely insensitive to the specific choice of vague but proper prior distribution. The models and estimation methods are general and can be used for any number of locations and pollutant measurements and have potential applications to other environmental agents. [source]

    Measurement error modelling with an approximate instrumental variable

    Paul Gustafson
    Summary., Consider using regression modelling to relate an exposure (predictor) variable to a disease outcome (response) variable. If the exposure variable is measured with error, but this error is ignored in the analysis, then misleading inferences can result. This problem is well known and has spawned a large literature on methods which adjust for measurement error in predictor variables. One theme is that the requisite assumptions about the nature of the measurement error can be stronger than what is actually known in many practical situations. In particular, the assumptions that are required to yield a model which is formally identified from the observable data can be quite strong. The paper deals with one particular strategy for measurement error modelling, namely that of seeking an instrumental variable, i.e. a covariate S which is associated with exposure and conditionally independent of the outcome given exposure. If these two conditions hold exactly, then we call S an exact instrumental variable, and an identified model results. However, the second is not checkable empirically, since the actual exposure is unobserved. In practice then, investigators typically seek a covariate which is plausibly thought to satisfy it. We study inferences which acknowledge the approximate nature of this assumption. In particular, we consider Bayesian inference with a prior distribution that posits that S is probably close to conditionally independent of outcome given exposure. We refer to this as an approximate instrumental variable assumption. Although the approximate instrumental variable assumption is more realistic for most applications, concern arises that a non-identified model may result. Thus the paper contrasts inferences arising from the approximate instrumental variable assumption with their exact instrumental variable counterparts, with particular emphasis on the benefit of basing inferences on a more realistic model versus the cost of basing inferences on a non-identified model. [source]

    The Bayesian choice of crop variety and fertilizer dose

    Chris M Theobald
    Recent contributions to the theory of optimizing fertilizer doses in agricultural crop production have introduced Bayesian ideas to incorporate information on crop yield from several environments and on soil nutrients from a soil test, but they have not used a fully Bayesian formulation. We present such a formulation and demonstrate how the resulting Bayes decision procedure can be evaluated in practice by using Markov chain Monte Carlo methods. The approach incorporates expert knowledge of the crop and of regional and local soil conditions and allows a choice of crop variety as well as of fertilizer level. Alternative dose,response functions are expressed in terms of a common interpretable set of parameters to facilitate model comparisons and the specification of prior distributions. The approach is illustrated with a set of yield data from spring barley nitrogen,response trials and is found to be robust to changes in the dose,response function and the prior distribution for indigenous soil nitrogen. [source]

    Fractional Bayesian Lag Length Inference in Multivariate Autoregressive Processes

    Mattias Villani
    The posterior distribution of the number of lags in a multivariate autoregression is derived under an improper prior for the model parameters. The fractional Bayes approach is used to handle the indeterminacy in the model selection caused by the improper prior. An asymptotic equivalence between the fractional approach and the Schwarz Bayesian Criterion (SBC) is proved. Several priors and three loss functions are entertained in a simulation study which focuses on the choice of lag length. The fractional Bayes approach performs very well compared to the three most widely used information criteria, and it seems to be reasonably robust to changes in the prior distribution for the lag length, especially under the zero-one loss. [source]

    European Mathematical Genetics Meeting, Heidelberg, Germany, 12th,13th April 2007

    Article first published online: 28 MAY 200
    Saurabh Ghosh 11 Indian Statistical Institute, Kolkata, India High correlations between two quantitative traits may be either due to common genetic factors or common environmental factors or a combination of both. In this study, we develop statistical methods to extract the contribution of a common QTL to the total correlation between the components of a bivariate phenotype. Using data on bivariate phenotypes and marker genotypes for sib-pairs, we propose a test for linkage between a common QTL and a marker locus based on the conditional cross-sib trait correlations (trait 1 of sib 1 , trait 2 of sib 2 and conversely) given the identity-by-descent sharing at the marker locus. The null hypothesis cannot be rejected unless there exists a common QTL. We use Monte-Carlo simulations to evaluate the performance of the proposed test under different trait parameters and quantitative trait distributions. An application of the method is illustrated using data on two alcohol-related phenotypes from the Collaborative Study On The Genetics Of Alcoholism project. Rémi Kazma 1 , Catherine Bonaïti-Pellié 1 , Emmanuelle Génin 12 INSERM UMR-S535 and Université Paris Sud, Villejuif, 94817, France Keywords: Gene-environment interaction, sibling recurrence risk, exposure correlation Gene-environment interactions may play important roles in complex disease susceptibility but their detection is often difficult. Here we show how gene-environment interactions can be detected by investigating the degree of familial aggregation according to the exposure of the probands. In case of gene-environment interaction, the distribution of genotypes of affected individuals, and consequently the risk in relatives, depends on their exposure. We developed a test comparing the risks in sibs according to the proband exposure. To evaluate the properties of this new test, we derived the formulas for calculating the expected risks in sibs according to the exposure of probands for various values of exposure frequency, relative risk due to exposure alone, frequencies of latent susceptibility genotypes, genetic relative risks and interaction coefficients. We find that the ratio of risks when the proband is exposed and not exposed is a good indicator of the interaction effect. We evaluate the power of the test for various sample sizes of affected individuals. We conclude that this test is valuable for diseases with moderate familial aggregation, only when the role of the exposure has been clearly evidenced. Since a correlation for exposure among sibs might lead to a difference in risks among sibs in the different proband exposure strata, we also add an exposure correlation coefficient in the model. Interestingly, we find that when this correlation is correctly accounted for, the power of the test is not decreased and might even be significantly increased. Andrea Callegaro 1 , Hans J.C. Van Houwelingen 1 , Jeanine Houwing-Duistermaat 13 Dept. of Medical Statistics and Bioinformatics, Leiden University Medical Center, The Netherlands Keywords: Survival analysis, age at onset, score test, linkage analysis Non parametric linkage (NPL) analysis compares the identical by descent (IBD) sharing in sibling pairs to the expected IBD sharing under the hypothesis of no linkage. Often information is available on the marginal cumulative hazards (for example breast cancer incidence curves). Our aim is to extend the NPL methods by taking into account the age at onset of selected sibling pairs using these known marginal hazards. Li and Zhong (2002) proposed a (retrospective) likelihood ratio test based on an additive frailty model for genetic linkage analysis. From their model we derive a score statistic for selected samples which turns out to be a weighed NPL method. The weights depend on the marginal cumulative hazards and on the frailty parameter. A second approach is based on a simple gamma shared frailty model. Here, we simply test whether the score function of the frailty parameter depends on the excess IBD. We compare the performance of these methods using simulated data. Céline Bellenguez 1 , Carole Ober 2 , Catherine Bourgain 14 INSERM U535 and University Paris Sud, Villejuif, France 5 Department of Human Genetics, The University of Chicago, USA Keywords: Linkage analysis, linkage disequilibrium, high density SNP data Compared with microsatellite markers, high density SNP maps should be more informative for linkage analyses. However, because they are much closer, SNPs present important linkage disequilibrium (LD), which biases classical nonparametric multipoint analyses. This problem is even stronger in population isolates where LD extends over larger regions with a more stochastic pattern. We investigate the issue of linkage analysis with a 500K SNP map in a large and inbred 1840-member Hutterite pedigree, phenotyped for asthma. Using an efficient pedigree breaking strategy, we first identified linked regions with a 5cM microsatellite map, on which we focused to evaluate the SNP map. The only method that models LD in the NPL analysis is limited in both the pedigree size and the number of markers (Abecasis and Wigginton, 2005) and therefore could not be used. Instead, we studied methods that identify sets of SNPs with maximum linkage information content in our pedigree and no LD-driven bias. Both algorithms that directly remove pairs of SNPs in high LD and clustering methods were evaluated. Null simulations were performed to control that Zlr calculated with the SNP sets were not falsely inflated. Preliminary results suggest that although LD is strong in such populations, linkage information content slightly better than that of microsatellite maps can be extracted from dense SNP maps, provided that a careful marker selection is conducted. In particular, we show that the specific LD pattern requires considering LD between a wide range of marker pairs rather than only in predefined blocks. Peter Van Loo 1,2,3 , Stein Aerts 1,2 , Diether Lambrechts 4,5 , Bernard Thienpont 2 , Sunit Maity 4,5 , Bert Coessens 3 , Frederik De Smet 4,5 , Leon-Charles Tranchevent 3 , Bart De Moor 2 , Koen Devriendt 3 , Peter Marynen 1,2 , Bassem Hassan 1,2 , Peter Carmeliet 4,5 , Yves Moreau 36 Department of Molecular and Developmental Genetics, VIB, Belgium 7 Department of Human Genetics, University of Leuven, Belgium 8 Bioinformatics group, Department of Electrical Engineering, University of Leuven, Belgium 9 Department of Transgene Technology and Gene Therapy, VIB, Belgium 10 Center for Transgene Technology and Gene Therapy, University of Leuven, Belgium Keywords: Bioinformatics, gene prioritization, data fusion The identification of genes involved in health and disease remains a formidable challenge. Here, we describe a novel bioinformatics method to prioritize candidate genes underlying pathways or diseases, based on their similarity to genes known to be involved in these processes. It is freely accessible as an interactive software tool, ENDEAVOUR, at Unlike previous methods, ENDEAVOUR generates distinct prioritizations from multiple heterogeneous data sources, which are then integrated, or fused, into one global ranking using order statistics. ENDEAVOUR prioritizes candidate genes in a three-step process. First, information about a disease or pathway is gathered from a set of known "training" genes by consulting multiple data sources. Next, the candidate genes are ranked based on similarity with the training properties obtained in the first step, resulting in one prioritized list for each data source. Finally, ENDEAVOUR fuses each of these rankings into a single global ranking, providing an overall prioritization of the candidate genes. Validation of ENDEAVOUR revealed it was able to efficiently prioritize 627 genes in disease data sets and 76 genes in biological pathway sets, identify candidates of 16 mono- or polygenic diseases, and discover regulatory genes of myeloid differentiation. Furthermore, the approach identified YPEL1 as a novel gene involved in craniofacial development from a 2-Mb chromosomal region, deleted in some patients with DiGeorge-like birth defects. Finally, we are currently evaluating a pipeline combining array-CGH, ENDEAVOUR and in vivo validation in zebrafish to identify novel genes involved in congenital heart defects. Mark Broom 1 , Graeme Ruxton 2 , Rebecca Kilner 311 Mathematics Dept., University of Sussex, UK 12 Division of Environmental and Evolutionary Biology, University of Glasgow, UK 13 Department of Zoology, University of Cambridge, UK Keywords: Evolutionarily stable strategy, parasitism, asymmetric game Brood parasites chicks vary in the harm that they do to their companions in the nest. In this presentation we use game-theoretic methods to model this variation. Our model considers hosts which potentially abandon single nestlings and instead choose to re-allocate their reproductive effort to future breeding, irrespective of whether the abandoned chick is the host's young or a brood parasite's. The parasite chick must decide whether or not to kill host young by balancing the benefits from reduced competition in the nest against the risk of desertion by host parents. The model predicts that three different types of evolutionarily stable strategies can exist. (1) Hosts routinely rear depleted broods, the brood parasite always kills host young and the host never then abandons the nest. (2) When adult survival after deserting single offspring is very high, hosts always abandon broods of a single nestling and the parasite never kills host offspring, effectively holding them as hostages to prevent nest desertion. (3) Intermediate strategies, in which parasites sometimes kill their nest-mates and host parents sometimes desert nests that contain only a single chick, can also be evolutionarily stable. We provide quantitative descriptions of how the values given to ecological and behavioral parameters of the host-parasite system influence the likelihood of each strategy and compare our results with real host-brood parasite associations in nature. Martin Harrison 114 Mathematics Dept, University of Sussex, UK Keywords: Brood parasitism, games, host, parasite The interaction between hosts and parasites in bird populations has been studied extensively. Game theoretical methods have been used to model this interaction previously, but this has not been studied extensively taking into account the sequential nature of this game. We consider a model allowing the host and parasite to make a number of decisions, which depend on a number of natural factors. The host lays an egg, a parasite bird will arrive at the nest with a certain probability and then chooses to destroy a number of the host eggs and lay one of it's own. With some destruction occurring, either natural or through the actions of the parasite, the host chooses to continue, eject an egg (hoping to eject the parasite) or abandon the nest. Once the eggs have hatched the game then falls to the parasite chick versus the host. The chick chooses to destroy or eject a number of eggs. The final decision is made by the host, choosing whether to raise or abandon the chicks that are in the nest. We consider various natural parameters and probabilities which influence these decisions. We then use this model to look at real-world situations of the interactions of the Reed Warbler and two different parasites, the Common Cuckoo and the Brown-Headed Cowbird. These two parasites have different methods in the way that they parasitize the nests of their hosts. The hosts in turn have a different reaction to these parasites. Arne Jochens 1 , Amke Caliebe 2 , Uwe Roesler 1 , Michael Krawczak 215 Mathematical Seminar, University of Kiel, Germany 16 Institute of Medical Informatics and Statistics, University of Kiel, Germany Keywords: Stepwise mutation model, microsatellite, recursion equation, temporal behaviour We consider the stepwise mutation model which occurs, e.g., in microsatellite loci. Let X(t,i) denote the allelic state of individual i at time t. We compute expectation, variance and covariance of X(t,i), i=1,,,N, and provide a recursion equation for P(X(t,i)=z). Because the variance of X(t,i) goes to infinity as t grows, for the description of the temporal behaviour, we regard the scaled process X(t,i)-X(t,1). The results furnish a better understanding of the behaviour of the stepwise mutation model and may in future be used to derive tests for neutrality under this model. Paul O'Reilly 1 , Ewan Birney 2 , David Balding 117 Statistical Genetics, Department of Epidemiology and Public Health, Imperial, College London, UK 18 European Bioinformatics Institute, EMBL, Cambridge, UK Keywords: Positive selection, Recombination rate, LD, Genome-wide, Natural Selection In recent years efforts to develop population genetics methods that estimate rates of recombination and levels of natural selection in the human genome have intensified. However, since the two processes have an intimately related impact on genetic variation their inference is vulnerable to confounding. Genomic regions subject to recent selection are likely to have a relatively recent common ancestor and consequently less opportunity for historical recombinations that are detectable in contemporary populations. Here we show that selection can reduce the population-based recombination rate estimate substantially. In genome-wide studies for detecting selection we observe a tendency to highlight loci that are subject to low levels of recombination. We find that the outlier approach commonly adopted in such studies may have low power unless variable recombination is accounted for. We introduce a new genome-wide method for detecting selection that exploits the sensitivity to recent selection of methods for estimating recombination rates, while accounting for variable recombination using pedigree data. Through simulations we demonstrate the high power of the Ped/Pop approach to discriminate between neutral and adaptive evolution, particularly in the context of choosing outliers from a genome-wide distribution. Although methods have been developed showing good power to detect selection ,in action', the corresponding window of opportunity is small. In contrast, the power of the Ped/Pop method is maintained for many generations after the fixation of an advantageous variant Sarah Griffiths 1 , Frank Dudbridge 120 MRC Biostatistics Unit, Cambridge, UK Keywords: Genetic association, multimarker tag, haplotype, likelihood analysis In association studies it is generally too expensive to genotype all variants in all subjects. We can exploit linkage disequilibrium between SNPs to select a subset that captures the variation in a training data set obtained either through direct resequencing or a public resource such as the HapMap. These ,tag SNPs' are then genotyped in the whole sample. Multimarker tagging is a more aggressive adaptation of pairwise tagging that allows for combinations of two or more tag SNPs to predict an untyped SNP. Here we describe a new method for directly testing the association of an untyped SNP using a multimarker tag. Previously, other investigators have suggested testing a specific tag haplotype, or performing a weighted analysis using weights derived from the training data. However these approaches do not properly account for the imperfect correlation between the tag haplotype and the untyped SNP. Here we describe a straightforward approach to testing untyped SNPs using a missing-data likelihood analysis, including the tag markers as nuisance parameters. The training data is stacked on top of the main body of genotype data so there is information on how the tag markers predict the genotype of the untyped SNP. The uncertainty in this prediction is automatically taken into account in the likelihood analysis. This approach yields more power and also a more accurate prediction of the odds ratio of the untyped SNP. Anke Schulz 1 , Christine Fischer 2 , Jenny Chang-Claude 1 , Lars Beckmann 121 Division of Cancer Epidemiology, German Cancer Research Center (DKFZ) Heidelberg, Germany 22 Institute of Human Genetics, University of Heidelberg, Germany Keywords: Haplotype, haplotype sharing, entropy, Mantel statistics, marker selection We previously introduced a new method to map genes involved in complex diseases, using haplotype sharing-based Mantel statistics to correlate genetic and phenotypic similarity. Although the Mantel statistic is powerful in narrowing down candidate regions, the precise localization of a gene is hampered in genomic regions where linkage disequilibrium is so high that neighboring markers are found to be significant at similar magnitude and we are not able to discriminate between them. Here, we present a new approach to localize susceptibility genes by combining haplotype sharing-based Mantel statistics with an iterative entropy-based marker selection algorithm. For each marker at which the Mantel statistic is evaluated, the algorithm selects a subset of surrounding markers. The subset is chosen to maximize multilocus linkage disequilibrium, which is measured by the normalized entropy difference introduced by Nothnagel et al. (2002). We evaluated the algorithm with respect to type I error and power. Its ability to localize the disease variant was compared to the localization (i) without marker selection and (ii) considering haplotype block structure. Case-control samples were simulated from a set of 18 haplotypes, consisting of 15 SNPs in two haplotype blocks. The new algorithm gave correct type I error and yielded similar power to detect the disease locus compared to the alternative approaches. The neighboring markers were clearly less often significant than the causal locus, and also less often significant compared to the alternative approaches. Thus the new algorithm improved the precision of the localization of susceptibility genes. Mark M. Iles 123 Section of Epidemiology and Biostatistics, LIMM, University of Leeds, UK Keywords: tSNP, tagging, association, HapMap Tagging SNPs (tSNPs) are commonly used to capture genetic diversity cost-effectively. However, it is important that the efficacy of tSNPs is correctly estimated, otherwise coverage may be insufficient. If the pilot sample from which tSNPs are chosen is too small or the initial marker map too sparse, tSNP efficacy may be overestimated. An existing estimation method based on bootstrapping goes some way to correct for insufficient sample size and overfitting, but does not completely solve the problem. We describe a novel method, based on exclusion of haplotypes, that improves on the bootstrap approach. Using simulated data, the extent of the sample size problem is investigated and the performance of the bootstrap and the novel method are compared. We incorporate an existing method adjusting for marker density by ,SNP-dropping'. We find that insufficient sample size can cause large overestimates in tSNP efficacy, even with as many as 100 individuals, and the problem worsens as the region studied increases in size. Both the bootstrap and novel method correct much of this overestimate, with our novel method consistently outperforming the bootstrap method. We conclude that a combination of insufficient sample size and overfitting may lead to overestimation of tSNP efficacy and underpowering of studies based on tSNPs. Our novel approach corrects for much of this bias and is superior to the previous method. Sample sizes larger than previously suggested may still be required for accurate estimation of tSNP efficacy. This has obvious ramifications for the selection of tSNPs from HapMap data. Claudio Verzilli 1 , Juliet Chapman 1 , Aroon Hingorani 2 , Juan Pablo-Casas 1 , Tina Shah 2 , Liam Smeeth 1 , John Whittaker 124 Department of Epidemiology and Population Health, London School of Hygiene and Tropical Medicine, UK 25 Division of Medicine, University College London, UK Keywords: Meta-analysis, Genetic association studies We present a Bayesian hierarchical model for the meta-analysis of candidate gene studies with a continuous outcome. Such studies often report results from association tests for different, possibly study-specific and non-overlapping markers (typically SNPs) in the same genetic region. Meta analyses of the results at each marker in isolation are seldom appropriate as they ignore the correlation that may exist between markers due to linkage disequlibrium (LD) and cannot assess the relative importance of variants at each marker. Also such marker-wise meta analyses are restricted to only those studies that have typed the marker in question, with a potential loss of power. A better strategy is one which incorporates information about the LD between markers so that any combined estimate of the effect of each variant is corrected for the effect of other variants, as in multiple regression. Here we develop a Bayesian hierarchical linear regression that models the observed genotype group means and uses pairwise LD measurements between markers as prior information to make posterior inference on adjusted effects. The approach is applied to the meta analysis of 24 studies assessing the effect of 7 variants in the C-reactive protein (CRP) gene region on plasma CRP levels, an inflammatory biomarker shown in observational studies to be positively associated with cardiovascular disease. Cathryn M. Lewis 1 , Christopher G. Mathew 1 , Theresa M. Marteau 226 Dept. of Medical and Molecular Genetics, King's College London, UK 27 Department of Psychology, King's College London, UK Keywords: Risk, genetics, CARD15, smoking, model Recently progress has been made in identifying mutations that confer susceptibility to complex diseases, with the potential to use these mutations in determining disease risk. We developed methods to estimate disease risk based on genotype relative risks (for a gene G), exposure to an environmental factor (E), and family history (with recurrence risk ,R for a relative of type R). ,R must be partitioned into the risk due to G (which is modelled independently) and the residual risk. The risk model was then applied to Crohn's disease (CD), a severe gastrointestinal disease for which smoking increases disease risk approximately 2-fold, and mutations in CARD15 confer increased risks of 2.25 (for carriers of a single mutation) and 9.3 (for carriers of two mutations). CARD15 accounts for only a small proportion of the genetic component of CD, with a gene-specific ,S, CARD15 of 1.16, from a total sibling relative risk of ,S= 27. CD risks were estimated for high-risk individuals who are siblings of a CD case, and who also smoke. The CD risk to such individuals who carry two CARD15 mutations is approximately 0.34, and for those carrying a single CARD15 mutation the risk is 0.08, compared to a population prevalence of approximately 0.001. These results imply that complex disease genes may be valuable in estimating with greater precision than has hitherto been possible disease risks in specific, easily identified subgroups of the population with a view to prevention. Yurii Aulchenko 128 Department of Epidemiology & Biostatistics, Erasmus Medical Centre Rotterdam, The Netherlands Keywords: Compression, information, bzip2, genome-wide SNP data, statistical genetics With advances in molecular technology, studies accessing millions of genetic polymorphisms in thousands of study subjects will soon become common. Such studies generate large amounts of data, whose effective storage and management is a challenge to the modern statistical genetics. Standard file compression utilities, such as Zip, Gzip and Bzip2, may be helpful to minimise the storage requirements. Less obvious is the fact that the data compression techniques may be also used in the analysis of genetic data. It is known that the efficiency of a particular compression algorithm depends on the probability structure of the data. In this work, we compared different standard and customised tools using the data from human HapMap project. Secondly, we investigate the potential uses of data compression techniques for the analysis of linkage, association and linkage disequilibrium Suzanne Leal 1 , Bingshan Li 129 Department of Molecular and Human Genetics, Baylor College of Medicine, Houston, USA Keywords: Consanguineous pedigrees, missing genotype data Missing genotype data can increase false-positive evidence for linkage when either parametric or nonparametric analysis is carried out ignoring intermarker linkage disequilibrium (LD). Previously it was demonstrated by Huang et al (2005) that no bias occurs in this situation for affected sib-pairs with unrelated parents when either both parents are genotyped or genotype data is available for two additional unaffected siblings when parental genotypes are missing. However, this is not the case for consanguineous pedigrees, where missing genotype data for any pedigree member within a consanguinity loop can increase false-positive evidence of linkage. The false-positive evidence for linkage is further increased when cryptic consanguinity is present. The amount of false-positive evidence for linkage is highly dependent on which family members are genotyped. When parental genotype data is available, the false-positive evidence for linkage is usually not as strong as when parental genotype data is unavailable. Which family members will aid in the reduction of false-positive evidence of linkage is highly dependent on which other family members are genotyped. For a pedigree with an affected proband whose first-cousin parents have been genotyped, further reduction in the false-positive evidence of linkage can be obtained by including genotype data from additional affected siblings of the proband or genotype data from the proband's sibling-grandparents. When parental genotypes are not available, false-positive evidence for linkage can be reduced by including in the analysis genotype data from either unaffected siblings of the proband or the proband's married-in-grandparents. Najaf Amin 1 , Yurii Aulchenko 130 Department of Epidemiology & Biostatistics, Erasmus Medical Centre Rotterdam, The Netherlands Keywords: Genomic Control, pedigree structure, quantitative traits The Genomic Control (GC) method was originally developed to control for population stratification and cryptic relatedness in association studies. This method assumes that the effect of population substructure on the test statistics is essentially constant across the genome, and therefore unassociated markers can be used to estimate the effect of confounding onto the test statistic. The properties of GC method were extensively investigated for different stratification scenarios, and compared to alternative methods, such as the transmission-disequilibrium test. The potential of this method to correct not for occasional cryptic relations, but for regular pedigree structure, however, was not investigated before. In this work we investigate the potential of the GC method for pedigree-based association analysis of quantitative traits. The power and type one error of the method was compared to standard methods, such as the measured genotype (MG) approach and quantitative trait transmission-disequilibrium test. In human pedigrees, with trait heritability varying from 30 to 80%, the power of MG and GC approach was always higher than that of TDT. GC had correct type 1 error and its power was close to that of MG under moderate heritability (30%), but decreased with higher heritability. William Astle 1 , Chris Holmes 2 , David Balding 131 Department of Epidemiology and Public Health, Imperial College London, UK 32 Department of Statistics, University of Oxford, UK Keywords: Population structure, association studies, genetic epidemiology, statistical genetics In the analysis of population association studies, Genomic Control (Devlin & Roeder, 1999) (GC) adjusts the Armitage test statistic to correct the type I error for the effects of population substructure, but its power is often sub-optimal. Turbo Genomic Control (TGC) generalises GC to incorporate co-variation of relatedness and phenotype, retaining control over type I error while improving power. TGC is similar to the method of Yu et al. (2006), but we extend it to binary (case-control) in addition to quantitative phenotypes, we implement improved estimation of relatedness coefficients, and we derive an explicit statistic that generalizes the Armitage test statistic and is fast to compute. TGC also has similarities to EIGENSTRAT (Price et al., 2006) which is a new method based on principle components analysis. The problems of population structure(Clayton et al., 2005) and cryptic relatedness (Voight & Pritchard, 2005) are essentially the same: if patterns of shared ancestry differ between cases and controls, whether distant (coancestry) or recent (cryptic relatedness), false positives can arise and power can be diminished. With large numbers of widely-spaced genetic markers, coancestry can now be measured accurately for each pair of individuals via patterns of allele-sharing. Instead of modelling subpopulations, we work instead with a coancestry coefficient for each pair of individuals in the study. We explain the relationships between TGC, GC and EIGENSTRAT. We present simulation studies and real data analyses to illustrate the power advantage of TGC in a range of scenarios incorporating both substructure and cryptic relatedness. References Clayton, D. al. (2005) Population structure, differential bias and genomic control in a large-scale case-control association study. Nature Genetics37(11) November 2005. Devlin, B. & Roeder, K. (1999) Genomic control for association studies. Biometics55(4) December 1999. Price, A. al. (2006) Principal components analysis corrects for stratification in genome-wide association studies. Nature Genetics38(8) (August 2006). Voight, B. J. & Pritchard, J. K. (2005) Confounding from cryptic relatedness in case-control association studies. Public Library of Science Genetics1(3) September 2005. Yu, al. (2006) A unified mixed-model method for association mapping that accounts for multiple levels of relatedness. Nature Genetics38(2) February 2006. Hervé Perdry 1 , Marie-Claude Babron 1 , Françoise Clerget-Darpoux 133 INSERM U535 and Univ. Paris Sud, UMR-S 535, Villejuif, France Keywords: Modifier genes, case-parents trios, ordered transmission disequilibrium test A modifying locus is a polymorphic locus, distinct from the disease locus, which leads to differences in the disease phenotype, either by modifying the penetrance of the disease allele, or by modifying the expression of the disease. The effect of such a locus is a clinical heterogeneity that can be reflected by the values of an appropriate covariate, such as the age of onset, or the severity of the disease. We designed the Ordered Transmission Disequilibrium Test (OTDT) to test for a relation between the clinical heterogeneity, expressed by the covariate, and marker genotypes of a candidate gene. The method applies to trio families with one affected child and his parents. Each family member is genotyped at a bi-allelic marker M of a candidate gene. To each of the families is associated a covariate value; the families are ordered on the values of this covariate. As the TDT (Spielman et al. 1993), the OTDT is based on the observation of the transmission rate T of a given allele at M. The OTDT aims to find a critical value of the covariate which separates the sample of families in two subsamples in which the transmission rates are significantly different. We investigate the power of the method by simulations under various genetic models and covariate distributions. Acknowledgments H Perdry is funded by ARSEP. Pascal Croiseau 1 , Heather Cordell 2 , Emmanuelle Génin 134 INSERM U535 and University Paris Sud, UMR-S535, Villejuif, France 35 Institute of Human Genetics, Newcastle University, UK Keywords: Association, missing data, conditionnal logistic regression Missing data is an important problem in association studies. Several methods used to test for association need that individuals be genotyped at the full set of markers. Individuals with missing data need to be excluded from the analysis. This could involve an important decrease in sample size and a loss of information. If the disease susceptibility locus (DSL) is poorly typed, it is also possible that a marker in linkage disequilibrium gives a stronger association signal than the DSL. One may then falsely conclude that the marker is more likely to be the DSL. We recently developed a Multiple Imputation method to infer missing data on case-parent trios Starting from the observed data, a few number of complete data sets are generated by Markov-Chain Monte Carlo approach. These complete datasets are analysed using standard statistical package and the results are combined as described in Little & Rubin (2002). Here we report the results of simulations performed to examine, for different patterns of missing data, how often the true DSL gives the highest association score among different loci in LD. We found that multiple imputation usually correctly detect the DSL site even if the percentage of missing data is high. This is not the case for the naïve approach that consists in discarding trios with missing data. In conclusion, Multiple imputation presents the advantage of being easy to use and flexible and is therefore a promising tool in the search for DSL involved in complex diseases. Salma Kotti 1 , Heike Bickeböller 2 , Françoise Clerget-Darpoux 136 University Paris Sud, UMR-S535, Villejuif, France 37 Department of Genetic Epidemiology, Medical School, University of Göttingen, Germany Keywords: Genotype relative risk, internal controls, Family based analyses Family based analyses using internal controls are very popular both for detecting the effect of a genetic factor and for estimating the relative disease risk on the corresponding genotypes. Two different procedures are often applied to reconstitute internal controls. The first one considers one pseudocontrol genotype formed by the parental non-transmitted alleles called also 1:1 matching of alleles, while the second corresponds to three pseudocontrols corresponding to all genotypes formed by the parental alleles except the one of the case (1:3 matching). Many studies have compared between the two procedures in terms of the power and have concluded that the difference depends on the underlying genetic model and the allele frequencies. However, the estimation of the Genotype Relative Risk (GRR) under the two procedures has not been studied. Based on the fact that on the 1:1 matching, the control group is composed of the alleles untransmitted to the affected child and on the 1:3 matching, the control group is composed amongst alleles already transmitted to the affected child, we expect a difference on the GRR estimation. In fact, we suspect that the second procedure leads to biased estimation of the GRRs. We will analytically derive the GRR estimators for the 1:1 and 1:3 matching and will present the results at the meeting. Family based analyses using internal controls are very popular both for detecting the effect of a genetic factor and for estimating the relative disease risk on the corresponding genotypes. Two different procedures are often applied to reconstitute internal controls. The first one considers one pseudocontrol genotype formed by the parental non-transmitted alleles called also 1:1 matching of alleles, while the second corresponds to three pseudocontrols corresponding to all genotypes formed by the parental alleles except the one of the case (1:3 matching). Many studies have compared between the two procedures in terms of the power and have concluded that the difference depends on the underlying genetic model and the allele frequencies. However, the estimation of the Genotype Relative Risk (GRR) under the two procedures has not been studied. Based on the fact that on the 1:1 matching, the control group is composed of the alleles untransmitted to the affected child and on the 1:3 matching, the control group is composed amongst alleles already transmitted to the affected child, we expect a difference on the GRR estimation. In fact, we suspect that the second procedure leads to biased estimation of the GRR. We will analytically derive the GRR estimator for the 1:1 and 1:3 matching and will present the results at the meeting. Luigi Palla 1 , David Siegmund 239 Department of Mathematics,Free University Amsterdam, The Netherlands 40 Department of Statistics, Stanford University, California, USA Keywords: TDT, assortative mating, inbreeding, statistical power A substantial amount of Assortative Mating (AM) is often recorded on physical and psychological, dichotomous as well as quantitative traits that are supposed to have a multifactorial genetic component. In particular AM has the effect of increasing the genetic variance, even more than inbreeding because it acts across loci beside within loci, when the trait has a multifactorial origin. Under the assumption of a polygenic model for AM dating back to Wright (1921) and refined by Crow and Felsenstein (1968,1982), the effect of assortative mating on the power to detect genetic association in the Transmission Disequilibrium Test (TDT) is explored as parameters, such as the effective number of genes and the allelic frequency vary. The power is reflected by the non centrality parameter of the TDT and is expressed as a function of the number of trios, the relative risk of the heterozygous genotype and the allele frequency (Siegmund and Yakir, 2007). The noncentrality parameter of the relevant score statistic is updated considering the effect of AM which is expressed in terms of an ,effective' inbreeding coefficient. In particular, for dichotomous traits it is apparent that the higher the number of genes involved in the trait, the lower the loss in power due to AM. Finally an attempt is made to extend this relation to the Q-TDT (Rabinowitz, 1997), which involves considering the effect of AM also on the phenotypic variance of the trait of interest, under the assumption that AM affects only its additive genetic component. References Crow, & Felsenstein, (1968). The effect of assortative mating on the genetic composition of a population. Eugen.Quart.15, 87,97. Rabinowitz,, 1997. A Transmission Disequilibrium Test for Quantitative Trait Loci. Human Heredity47, 342,350. Siegmund, & Yakir, (2007) Statistics of gene mapping, Springer. Wright, (1921). System of mating.III. Assortative mating based on somatic resemblance. Genetics6, 144,161. Jérémie Nsengimana 1 , Ben D Brown 2 , Alistair S Hall 2 , Jenny H Barrett 141 Leeds Institute of Molecular Medicine, University of Leeds, UK 42 Leeds Institute for Genetics, Health and Therapeutics, University of Leeds, UK Keywords: Inflammatory genes, haplotype, coronary artery disease Genetic Risk of Acute Coronary Events (GRACE) is an initiative to collect cases of coronary artery disease (CAD) and their unaffected siblings in the UK and to use them to map genetic variants increasing disease risk. The aim of the present study was to test the association between CAD and 51 single nucleotide polymorphisms (SNPs) and their haplotypes from 35 inflammatory genes. Genotype data were available for 1154 persons affected before age 66 (including 48% before age 50) and their 1545 unaffected siblings (891 discordant families). Each SNP was tested for association to CAD, and haplotypes within genes or gene clusters were tested using FBAT (Rabinowitz & Laird, 2000). For the most significant results, genetic effect size was estimated using conditional logistic regression (CLR) within STATA adjusting for other risk factors. Haplotypes were assigned using HAPLORE (Zhang et al., 2005), which considers all parental mating types consistent with offspring genotypes and assigns them a probability of occurence. This probability was used in CLR to weight the haplotypes. In the single SNP analysis, several SNPs showed some evidence for association, including one SNP in the interleukin-1A gene. Analysing haplotypes in the interleukin-1 gene cluster, a common 3-SNP haplotype was found to increase the risk of CAD (P = 0.009). In an additive genetic model adjusting for covariates the odds ratio (OR) for this haplotype is 1.56 (95% CI: 1.16-2.10, p = 0.004) for early-onset CAD (before age 50). This study illustrates the utility of haplotype analysis in family-based association studies to investigate candidate genes. References Rabinowitz, D. & Laird, N. M. (2000) Hum Hered50, 211,223. Zhang, K., Sun, F. & Zhao, H. (2005) Bioinformatics21, 90,103. Andrea Foulkes 1 , Recai Yucel 1 , Xiaohong Li 143 Division of Biostatistics, University of Massachusetts, USA Keywords: Haploytpe, high-dimensional, mixed modeling The explosion of molecular level information coupled with large epidemiological studies presents an exciting opportunity to uncover the genetic underpinnings of complex diseases; however, several analytical challenges remain to be addressed. Characterizing the components to complex diseases inevitably requires consideration of synergies across multiple genetic loci and environmental and demographic factors. In addition, it is critical to capture information on allelic phase, that is whether alleles within a gene are in cis (on the same chromosome) or in trans (on different chromosomes.) In associations studies of unrelated individuals, this alignment of alleles within a chromosomal copy is generally not observed. We address the potential ambiguity in allelic phase in this high dimensional data setting using mixed effects models. Both a semi-parametric and fully likelihood-based approach to estimation are considered to account for missingness in cluster identifiers. In the first case, we apply a multiple imputation procedure coupled with a first stage expectation maximization algorithm for parameter estimation. A bootstrap approach is employed to assess sensitivity to variability induced by parameter estimation. Secondly, a fully likelihood-based approach using an expectation conditional maximization algorithm is described. Notably, these models allow for characterizing high-order gene-gene interactions while providing a flexible statistical framework to account for the confounding or mediating role of person specific covariates. The proposed method is applied to data arising from a cohort of human immunodeficiency virus type-1 (HIV-1) infected individuals at risk for therapy associated dyslipidemia. Simulation studies demonstrate reasonable power and control of family-wise type 1 error rates. Vivien Marquard 1 , Lars Beckmann 1 , Jenny Chang-Claude 144 Division of Cancer Epidemiology, German Cancer Research Center (DKFZ) Heidelberg, Germany Keywords: Genotyping errors, type I error, haplotype-based association methods It has been shown in several simulation studies that genotyping errors may have a great impact on the type I error of statistical methods used in genetic association analysis of complex diseases. Our aim was to investigate type I error rates in a case-control study, when differential and non-differential genotyping errors were introduced in realistic scenarios. We simulated case-control data sets, where individual genotypes were drawn from a haplotype distribution of 18 haplotypes with 15 markers in the APM1 gene. Genotyping errors were introduced following the unrestricted and symmetric with 0 edges error models described by Heid et al. (2006). In six scenarios, errors resulted from changes of one allele to another with predefined probabilities of 1%, 2.5% or 10%, respectively. A multiple number of errors per haplotype was possible and could vary between 0 and 15, the number of markers investigated. We examined three association methods: Mantel statistics using haplotype-sharing; a haplotype-specific score test; and Armitage trend test for single markers. The type I error rates were not influenced for any of all the three methods for a genotyping error rate of less than 1%. For higher error rates and differential errors, the type I error of the Mantel statistic was only slightly and of the Armitage trend test moderately increased. The type I error rates of the score test were highly increased. The type I error rates were correct for all three methods for non-differential errors. Further investigations will be carried out with different frequencies of differential error rates and focus on power. Arne Neumann 1 , Dörthe Malzahn 1 , Martina Müller 2 , Heike Bickeböller 145 Department of Genetic Epidemiology, Medical School, University of Göttingen, Germany 46 GSF-National Research Center for Environment and Health, Neuherberg & IBE-Institute of Epidemiology, Ludwig-Maximilians University München, Germany Keywords: Interaction, longitudinal, nonparametric Longitudinal data show the time dependent course of phenotypic traits. In this contribution, we consider longitudinal cohort studies and investigate the association between two candidate genes and a dependent quantitative longitudinal phenotype. The set-up defines a factorial design which allows us to test simultaneously for the overall gene effect of the loci as well as for possible gene-gene and gene time interaction. The latter would induce genetically based time-profile differences in the longitudinal phenotype. We adopt a non-parametric statistical test to genetic epidemiological cohort studies and investigate its performance by simulation studies. The statistical test was originally developed for longitudinal clinical studies (Brunner, Munzel, Puri, 1999 J Multivariate Anal 70:286-317). It is non-parametric in the sense that no assumptions are made about the underlying distribution of the quantitative phenotype. Longitudinal observations belonging to the same individual can be arbitrarily dependent on one another for the different time points whereas trait observations of different individuals are independent. The two loci are assumed to be statistically independent. Our simulations show that the nonparametric test is comparable with ANOVA in terms of power of detecting gene-gene and gene-time interaction in an ANOVA favourable setting. Rebecca Hein 1 , Lars Beckmann 1 , Jenny Chang-Claude 147 Division of Cancer Epidemiology, German Cancer Research Center (DKFZ) Heidelberg, Germany Keywords: Indirect association studies, interaction effects, linkage disequilibrium, marker allele frequency Association studies accounting for gene-environment interactions (GxE) may be useful for detecting genetic effects and identifying important environmental effect modifiers. Current technology facilitates very dense marker spacing in genetic association studies; however, the true disease variant(s) may not be genotyped. In this situation, an association between a gene and a phenotype may still be detectable, using genetic markers associated with the true disease variant(s) (indirect association). Zondervan and Cardon [2004] showed that the odds ratios (OR) of markers which are associated with the disease variant depend highly on the linkage disequilibrium (LD) between the variant and the markers, and whether the allele frequencies match and thereby influence the sample size needed to detect genetic association. We examined the influence of LD and allele frequencies on the sample size needed to detect GxE in indirect association studies, and provide tables for sample size estimation. For discordant allele frequencies and incomplete LD, sample sizes can be unfeasibly large. The influence of both factors is stronger for disease loci with small rather than moderate to high disease allele frequencies. A decline in D' of e.g. 5% has less impact on sample size than increasing the difference in allele frequencies by the same percentage. Assuming 80% power, large interaction effects can be detected using smaller sample sizes than those needed for the detection of main effects. The detection of interaction effects involving rare alleles may not be possible. Focussing only on marker density can be a limited strategy in indirect association studies for GxE. Cyril Dalmasso 1 , Emmanuelle Génin 2 , Catherine Bourgain 2 , Philippe Broët 148 JE 2492 , Univ. Paris-Sud, France 49 INSERM UMR-S 535 and University Paris Sud, Villejuif, France Keywords: Linkage analysis, Multiple testing, False Discovery Rate, Mixture model In the context of genome-wide linkage analyses, where a large number of statistical tests are simultaneously performed, the False Discovery Rate (FDR) that is defined as the expected proportion of false discoveries among all discoveries is nowadays widely used for taking into account the multiple testing problem. Other related criteria have been considered such as the local False Discovery Rate (lFDR) that is a variant of the FDR giving to each test its own measure of significance. The lFDR is defined as the posterior probability that a null hypothesis is true. Most of the proposed methods for estimating the lFDR or the FDR rely on distributional assumption under the null hypothesis. However, in observational studies, the empirical null distribution may be very different from the theoretical one. In this work, we propose a mixture model based approach that provides estimates of the lFDR and the FDR in the context of large-scale variance component linkage analyses. In particular, this approach allows estimating the empirical null distribution, this latter being a key quantity for any simultaneous inference procedure. The proposed method is applied on a real dataset. Arief Gusnanto 1 , Frank Dudbridge 150 MRC Biostatistics Unit, Cambridge UK Keywords: Significance, genome-wide, association, permutation, multiplicity Genome-wide association scans have introduced statistical challenges, mainly in the multiplicity of thousands of tests. The question of what constitutes a significant finding remains somewhat unresolved. Permutation testing is very time-consuming, whereas Bayesian arguments struggle to distinguish direct from indirect association. It seems attractive to summarise the multiplicity in a simple form that allows users to avoid time-consuming permutations. A standard significance level would facilitate reporting of results and reduce the need for permutation tests. This is potentially important because current scans do not have full coverage of the whole genome, and yet, the implicit multiplicity is genome-wide. We discuss some proposed summaries, with reference to the empirical null distribution of the multiple tests, approximated through a large number of random permutations. Using genome-wide data from the Wellcome Trust Case-Control Consortium, we use a sub-sampling approach with increasing density to estimate the nominal p-value to obtain family-wise significance of 5%. The results indicate that the significance level is converging to about 1e-7 as the marker spacing becomes infinitely dense. We considered the concept of an effective number of independent tests, and showed that when used in a Bonferroni correction, the number varies with the overall significance level, but is roughly constant in the region of interest. We compared several estimators of the effective number of tests, and showed that in the region of significance of interest, Patterson's eigenvalue based estimator gives approximately the right family-wise error rate. Michael Nothnagel 1 , Amke Caliebe 1 , Michael Krawczak 151 Institute of Medical Informatics and Statistics, University Clinic Schleswig-Holstein, University of Kiel, Germany Keywords: Association scans, Bayesian framework, posterior odds, genetic risk, multiplicative model Whole-genome association scans have been suggested to be a cost-efficient way to survey genetic variation and to map genetic disease factors. We used a Bayesian framework to investigate the posterior odds of a genuine association under multiplicative disease models. We demonstrate that the p value alone is not a sufficient means to evaluate the findings in association studies. We suggest that likelihood ratios should accompany p values in association reports. We argue, that, given the reported results of whole-genome scans, more associations should have been successfully replicated if the consistently made assumptions about considerable genetic risks were correct. We conclude that it is very likely that the vast majority of relative genetic risks are only of the order of 1.2 or lower. Clive Hoggart 1 , Maria De Iorio 1 , John Whittakker 2 , David Balding 152 Department of Epidemiology and Public Health, Imperial College London, UK 53 Department of Epidemiology and Public Health, London School of Hygiene and Tropical Medicine, UK Keywords: Genome-wide association analyses, shrinkage priors, Lasso Testing one SNP at a time does not fully realise the potential of genome-wide association studies to identify multiple causal variants of small effect, which is a plausible scenario for many complex diseases. Moreover, many simulation studies assume a single causal variant and so more complex realities are ignored. Analysing large numbers of variants simultaneously is now becoming feasible, thanks to developments in Bayesian stochastic search methods. We pose the problem of SNP selection as variable selection in a regression model. In contrast to single SNP tests this approach simultaneously models the effect of all SNPs. SNPs are selected by a Bayesian interpretation of the lasso (Tibshirani, 1996); the maximum a posterior (MAP) estimate of the regression coefficients, which have been given independent, double exponential prior distributions. The double exponential distribution is an example of a shrinkage prior, MAP estimates with shrinkage priors can be zero, thus all SNPs with non zero regression coefficients are selected. In addition to the commonly-used double exponential (Laplace) prior, we also implement the normal exponential gamma prior distribution. We show that use of the Laplace prior improves SNP selection in comparison with single -SNP tests, and that the normal exponential gamma prior leads to a further improvement. Our method is fast and can handle very large numbers of SNPs: we demonstrate its performance using both simulated and real genome-wide data sets with 500 K SNPs, which can be analysed in 2 hours on a desktop workstation. Mickael Guedj 1,2 , Jerome Wojcik 2 , Gregory Nuel 154 Laboratoire Statistique et Génome, Université d'Evry, Evry France 55 Serono Pharmaceutical Research Institute, Plan-les-Ouates, Switzerland Keywords: Local Replication, Local Score, Association In gene-mapping, replication of initial findings has been put forwards as the approach of choice for filtering false-positives from true signals for underlying loci. In practice, such replications are however too poorly observed. Besides the statistical and technical-related factors (lack of power, multiple-testing, stratification, quality control,) inconsistent conclusions obtained from independent populations might result from real biological differences. In particular, the high degree of variation in the strength of LD among populations of different origins is a major challenge to the discovery of genes. Seeking for Local Replications (defined as the presence of a signal of association in a same genomic region among populations) instead of strict replications (same locus, same risk allele) may lead to more reliable results. Recently, a multi-markers approach based on the Local Score statistic has been proposed as a simple and efficient way to select candidate genomic regions at the first stage of genome-wide association studies. Here we propose an extension of this approach adapted to replicated association studies. Based on simulations, this method appears promising. In particular it outperforms classical simple-marker strategies to detect modest-effect genes. Additionally it constitutes, to our knowledge, a first framework dedicated to the detection of such Local Replications. Juliet Chapman 1 , Claudio Verzilli 1 , John Whittaker 156 Department of Epidemiology and Public Health, London School of Hygiene and Tropical Medicine, UK Keywords: FDR, Association studies, Bayesian model selection As genomewide association studies become commonplace there is debate as to how such studies might be analysed and what we might hope to gain from the data. It is clear that standard single locus approaches are limited in that they do not adjust for the effects of other loci and problematic since it is not obvious how to adjust for multiple comparisons. False discovery rates have been suggested, but it is unclear how well these will cope with highly correlated genetic data. We consider the validity of standard false discovery rates in large scale association studies. We also show that a Bayesian procedure has advantages in detecting causal loci amongst a large number of dependant SNPs and investigate properties of a Bayesian FDR. Peter Kraft 157 Harvard School of Public Health, Boston USA Keywords: Gene-environment interaction, genome-wide association scans Appropriately analyzed two-stage designs,where a subset of available subjects are genotyped on a genome-wide panel of markers at the first stage and then a much smaller subset of the most promising markers are genotyped on the remaining subjects,can have nearly as much power as a single-stage study where all subjects are genotyped on the genome-wide panel yet can be much less expensive. Typically, the "most promising" markers are selected based on evidence for a marginal association between genotypes and disease. Subsequently, the few markers found to be associated with disease at the end of the second stage are interrogated for evidence of gene-environment interaction, mainly to understand their impact on disease etiology and public health impact. However, this approach may miss variants which have a sizeable effect restricted to one exposure stratum and therefore only a modest marginal effect. We have proposed to use information on the joint effects of genes and a discrete list of environmental exposures at the initial screening stage to select promising markers for the second stage [Kraft et al Hum Hered 2007]. This approach optimizes power to detect variants that have a sizeable marginal effect and variants that have a small marginal effect but a sizeable effect in a stratum defined by an environmental exposure. As an example, I discuss a proposed genome-wide association scan for Type II diabetes susceptibility variants based in several large nested case-control studies. Beate Glaser 1 , Peter Holmans 158 Biostatistics and Bioinformatics Unit, Cardiff University, School of Medicine, Heath Park, Cardiff, UK Keywords: Combined case-control and trios analysis, Power, False-positive rate, Simulation, Association studies The statistical power of genetic association studies can be enhanced by combining the analysis of case-control with parent-offspring trio samples. Various combined analysis techniques have been recently developed; as yet, there have been no comparisons of their power. This work was performed with the aim of identifying the most powerful method among available combined techniques including test statistics developed by Kazeem and Farrall (2005), Nagelkerke and colleagues (2004) and Dudbridge (2006), as well as a simple combination of ,2-statistics from single samples. Simulation studies were performed to investigate their power under different additive, multiplicative, dominant and recessive disease models. False-positive rates were determined by studying the type I error rates under null models including models with unequal allele frequencies between the single case-control and trios samples. We identified three techniques with equivalent power and false-positive rates, which included modifications of the three main approaches: 1) the unmodified combined Odds ratio estimate by Kazeem & Farrall (2005), 2) a modified approach of the combined risk ratio estimate by Nagelkerke & colleagues (2004) and 3) a modified technique for a combined risk ratio estimate by Dudbridge (2006). Our work highlights the importance of studies investigating test performance criteria of novel methods, as they will help users to select the optimal approach within a range of available analysis techniques. David Almorza 1 , M.V. Kandus 2 , Juan Carlos Salerno 2 , Rafael Boggio 359 Facultad de Ciencias del Trabajo, University of Cádiz, Spain 60 Instituto de Genética IGEAF, Buenos Aires, Argentina 61 Universidad Nacional de La Plata, Buenos Aires, Argentina Keywords: Principal component analysis, maize, ear weight, inbred lines The objective of this work was to evaluate the relationship among different traits of the ear of maize inbred lines and to group genotypes according to its performance. Ten inbred lines developed at IGEAF (INTA Castelar) and five public inbred lines as checks were used. A field trial was carried out in Castelar, Buenos Aires (34° 36' S , 58° 39' W) using a complete randomize design with three replications. At harvest, individual weight (P.E.), diameter (D.E.), row number (N.H.) and length (L.E.) of the ear were assessed. A principal component analysis, PCA, (Infostat 2005) was used, and the variability of the data was depicted with a biplot. Principal components 1 and 2 (CP1 and CP2) explained 90% of the data variability. CP1 was correlated with P.E., L.E. and D.E., meanwhile CP2 was correlated with N.H. We found that individual weight (P.E.) was more correlated with diameter of the ear (D.E.) than with length (L.E). Five groups of inbred lines were distinguished: with high P.E. and mean N.H. (04-70, 04-73, 04-101 and MO17), with high P.E. but less N.H. (04-61 and B14), with mean P.E. and N.H. (B73, 04-123 and 04-96), with high N.H. but less P.E. (LP109, 04-8, 04-91 and 04-76) and with low P.E. and low N.H. (LP521 and 04-104). The use of PCA showed which variables had more incidence in ear weight and how is the correlation among them. Moreover, the different groups found with this analysis allow the evaluation of inbred lines by several traits simultaneously. Sven Knüppel 1 , Anja Bauerfeind 1 , Klaus Rohde 162 Department of Bioinformatics, MDC Berlin, Germany Keywords: Haplotypes, association studies, case-control, nuclear families The area of gene chip technology provides a plethora of phase-unknown SNP genotypes in order to find significant association to some genetic trait. To circumvent possibly low information content of a single SNP one groups successive SNPs and estimates haplotypes. Haplotype estimation, however, may reveal ambiguous haplotype pairs and bias the application of statistical methods. Zaykin et al. (Hum Hered, 53:79-91, 2002) proposed the construction of a design matrix to take this ambiguity into account. Here we present a set of functions written for the Statistical package R, which carries out haplotype estimation on the basis of the EM-algorithm for individuals (case-control) or nuclear families. The construction of a design matrix on basis of estimated haplotypes or haplotype pairs allows application of standard methods for association studies (linear, logistic regression), as well as statistical methods as haplotype sharing statistics and TDT-Test. Applications of these methods to genome-wide association screens will be demonstrated. Manuela Zucknick 1 , Chris Holmes 2 , Sylvia Richardson 163 Department of Epidemiology and Public Health, Imperial College London, UK 64 Department of Statistics, Oxford Center for Gene Function, University of Oxford, UK Keywords: Bayesian, variable selection, MCMC, large p, small n, structured dependence In large-scale genomic applications vast numbers of markers or genes are scanned to find a few candidates which are linked to a particular phenotype. Statistically, this is a variable selection problem in the "large p, small n" situation where many more variables than samples are available. An additional feature is the complex dependence structure which is often observed among the markers/genes due to linkage disequilibrium or their joint involvement in biological processes. Bayesian variable selection methods using indicator variables are well suited to the problem. Binary phenotypes like disease status are common and both Bayesian probit and logistic regression can be applied in this context. We argue that logistic regression models are both easier to tune and to interpret than probit models and implement the approach by Holmes & Held (2006). Because the model space is vast, MCMC methods are used as stochastic search algorithms with the aim to quickly find regions of high posterior probability. In a trade-off between fast-updating but slow-moving single-gene Metropolis-Hastings samplers and computationally expensive full Gibbs sampling, we propose to employ the dependence structure among the genes/markers to help decide which variables to update together. Also, parallel tempering methods are used to aid bold moves and help avoid getting trapped in local optima. Mixing and convergence of the resulting Markov chains are evaluated and compared to standard samplers in both a simulation study and in an application to a gene expression data set. Reference Holmes, C. C. & Held, L. (2006) Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis1, 145,168. Dawn Teare 165 MMGE, University of Sheffield, UK Keywords: CNP, family-based analysis, MCMC Evidence is accumulating that segmental copy number polymorphisms (CNPs) may represent a significant portion of human genetic variation. These highly polymorphic systems require handling as phenotypes rather than co-dominant markers, placing new demands on family-based analyses. We present an integrated approach to meet these challenges in the form of a graphical model, where the underlying discrete CNP phenotype is inferred from the (single or replicate) quantitative measure within the analysis, whilst assuming an allele based system segregating through the pedigree. [source]


    Pulak Ghosh
    Summary We extend the standard multivariate mixed model by incorporating a smooth time effect and relaxing distributional assumptions. We propose a semiparametric Bayesian approach to multivariate longitudinal data using a mixture of Polya trees prior distribution. Usually, the distribution of random effects in a longitudinal data model is assumed to be Gaussian. However, the normality assumption may be suspect, particularly if the estimated longitudinal trajectory parameters exhibit multi-modality and skewness. In this paper we propose a mixture of Polya trees prior density to address the limitations of the parametric random effects distribution. We illustrate the methodology by analysing data from a recent HIV-AIDS study. [source]

    Optimal Spending Functions for Asymmetric Group Sequential Designs

    Keaven M. Anderson
    Abstract We present optimized group sequential designs where testing of a single parameter , is of interest. We require specification of a loss function and of a prior distribution for ,. For the examples presented, we pre-specify Type I and II error rates and minimize the expected sample size over the prior distribution for ,. Minimizing the square of sample size rather than the sample size is found to produce designs with slightly less aggressive interim stopping rules and smaller maximum sample sizes with essentially identical expected sample size. We compare optimal designs using Hwang-Shih-DeCani and Kim-DeMets spending functions to fully optimized designs not restricted by a spending function family. In the examples selected, we also examine when there might be substantial benefit gained by adding an interim analysis. Finally, we provide specific optimal asymmetric spending function designs that should be generally useful and simply applied when a design with minimal expected sample size is desired. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

    Testing Random Effects in the Linear Mixed Model Using Approximate Bayes Factors

    BIOMETRICS, Issue 2 2009
    Benjamin R. Saville
    Summary Deciding which predictor effects may vary across subjects is a difficult issue. Standard model selection criteria and test procedures are often inappropriate for comparing models with different numbers of random effects due to constraints on the parameter space of the variance components. Testing on the boundary of the parameter space changes the asymptotic distribution of some classical test statistics and causes problems in approximating Bayes factors. We propose a simple approach for testing random effects in the linear mixed model using Bayes factors. We scale each random effect to the residual variance and introduce a parameter that controls the relative contribution of each random effect free of the scale of the data. We integrate out the random effects and the variance components using closed-form solutions. The resulting integrals needed to calculate the Bayes factor are low-dimensional integrals lacking variance components and can be efficiently approximated with Laplace's method. We propose a default prior distribution on the parameter controlling the contribution of each random effect and conduct simulations to show that our method has good properties for model selection problems. Finally, we illustrate our methods on data from a clinical trial of patients with bipolar disorder and on data from an environmental study of water disinfection by-products and male reproductive outcomes. [source]

    A General Class of Pattern Mixture Models for Nonignorable Dropout with Many Possible Dropout Times

    BIOMETRICS, Issue 2 2008
    Jason Roy
    Summary In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women. [source]

    On Smoothing Trends in Population Index Modeling

    BIOMETRICS, Issue 4 2007
    Chiara Mazzetta
    Summary In this article, we consider the U.K. Common Birds Census counts and their use in monitoring bird abundance. We use a state,space modeling approach within a Bayesian framework to describe population level trends over time and contribute to the alert system used by the British Trust for Ornithology. We account for potential overdispersion and excess zero counts by modeling the observation process with a zero-inflated negative binomial, while the system process is described by second-order polynomial growth models. In order to provide a biological motivation for the amount of smoothing applied to the observed series the system variance is related to the demographic characteristics of the species, so as to help the specification of its prior distribution. In particular, the available information on productivity and survival is used to formulate prior expectations on annual percentage changes in the population level and then used to constrain the variance of the system process. We discuss an example of how to interpret alternative choices for the degree of smoothing and how these relate to the classification of species, over time, into conservation lists. [source]

    Bayesian statistics in medical research: an intuitive alternative to conventional data analysis

    AStat, Lyle C. Gurrin BSc (Hons)
    Summary Statistical analysis of both experimental and observational data is central to medical research. Unfortunately, the process of conventional statistical analysis is poorly understood by many medical scientists. This is due, in part, to the counter-intuitive nature of the basic tools of traditional (frequency-based) statistical inference. For example, the proper definition of a conventional 95% confidence interval is quite confusing. It is based upon the imaginary results of a series of hypothetical repetitions of the data generation process and subsequent analysis. Not surprisingly, this formal definition is often ignored and a 95% confidence interval is widely taken to represent a range of values that is associated with a 95% probability of containing the true value of the parameter being estimated. Working within the traditional framework of frequency-based statistics, this interpretation is fundamentally incorrect. It is perfectly valid, however, if one works within the framework of Bayesian statistics and assumes a ,prior distribution' that is uniform on the scale of the main outcome variable. This reflects a limited equivalence between conventional and Bayesian statistics that can be used to facilitate a simple Bayesian interpretation based on the results of a standard analysis. Such inferences provide direct and understandable answers to many important types of question in medical research. For example, they can be used to assist decision making based upon studies with unavoidably low statistical power, where non-significant results are all too often, and wrongly, interpreted as implying ,no effect'. They can also be used to overcome the confusion that can result when statistically significant effects are too small to be clinically relevant. This paper describes the theoretical basis of the Bayesian-based approach and illustrates its application with a practical example that investigates the prevalence of major cardiac defects in a cohort of children born using the assisted reproduction technique known as ICSI (intracytoplasmic sperm injection). [source]

    Data cloning: easy maximum likelihood estimation for complex ecological models using Bayesian Markov chain Monte Carlo methods

    ECOLOGY LETTERS, Issue 7 2007
    Subhash R. Lele
    Abstract We introduce a new statistical computing method, called data cloning, to calculate maximum likelihood estimates and their standard errors for complex ecological models. Although the method uses the Bayesian framework and exploits the computational simplicity of the Markov chain Monte Carlo (MCMC) algorithms, it provides valid frequentist inferences such as the maximum likelihood estimates and their standard errors. The inferences are completely invariant to the choice of the prior distributions and therefore avoid the inherent subjectivity of the Bayesian approach. The data cloning method is easily implemented using standard MCMC software. Data cloning is particularly useful for analysing ecological situations in which hierarchical statistical models, such as state-space models and mixed effects models, are appropriate. We illustrate the method by fitting two nonlinear population dynamics models to data in the presence of process and observation noise. [source]

    Estimating the number of ozone peaks in Mexico City using a non-homogeneous Poisson model

    ENVIRONMETRICS, Issue 5 2008
    Jorge A. Achcar
    Abstract In this paper, we consider the problem of estimating the number of times an air quality standard is exceeded in a given period of time. A non-homogeneous Poisson model is proposed to analyse this issue. The rate at which the Poisson events occur is given by a rate function ,(t), t,,,0. This rate function also depends on some parameters that need to be estimated. Two forms of ,(t), t,,,0 are considered. One of them is of the Weibull form and the other is of the exponentiated-Weibull form. The parameters estimation is made using a Bayesian formulation based on the Gibbs sampling algorithm. The assignation of the prior distributions for the parameters is made in two stages. In the first stage, non-informative prior distributions are considered. Using the information provided by the first stage, more informative prior distributions are used in the second one. The theoretical development is applied to data provided by the monitoring network of Mexico City. The rate function that best fit the data varies according to the region of the city and/or threshold that is considered. In some cases the best fit is the Weibull form and in other cases the best option is the exponentiated-Weibull. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Estimation of immigration rate using integrated population models

    Fitsum Abadi
    Summary 1.,The dynamics of many populations is strongly affected by immigrants. However, estimating and modelling immigration is a real challenge. In the past, several methods have been developed to estimate immigration rate but they either require strong assumptions or combine in a piecewise manner the results from separate analyses. In most methods the effects of covariates cannot be modelled formally. 2.,We developed a Bayesian integrated population model which combines capture,recapture data, population counts and information on reproductive success into a single model that estimates and models immigration rate, while directly assessing the impact of environmental covariates. 3.,We assessed parameter identifiability by comparing posterior distributions of immigration rates under varying priors, and illustrated the application of the model with long term demographic data of a little owl Athene noctua population from Southern Germany. We further assessed the impact of environmental covariates on immigration. 4.,The resulting posterior distributions were insensitive to different prior distributions and dominated by the observed data, indicating that the immigration rate was identifiable. Average yearly immigration into the little owl population was 0·293 (95% credible interval 0·183,0·418), which means that ca 0·3 female per resident female entered the population every year. Immigration rate tended to increase with increasing abundance of voles, the main prey of little owls. 5.Synthesis and applications. The means to estimate and model immigration is an important step towards a better understanding of the dynamics of geographically open populations. The demographic estimates obtained from the developed integrated population model facilitate population diagnoses and can be used to assess population viability. The structural flexibility of the model should constitute a useful tool for wildlife managers and conservation ecologists. [source]

    Can forecasting performance be improved by considering the steady state?

    An application to Swedish inflation, interest rate
    Abstract This paper investigates whether the forecasting performance of Bayesian autoregressive and vector autoregressive models can be improved by incorporating prior beliefs on the steady state of the time series in the system. Traditional methodology is compared to the new framework,in which a mean-adjusted form of the models is employed,by estimating the models on Swedish inflation and interest rate data from 1980 to 2004. Results show that the out-of-sample forecasting ability of the models is practically unchanged for inflation but significantly improved for the interest rate when informative prior distributions on the steady state are provided. The findings in this paper imply that this new methodology could be useful since it allows us to sharpen our forecasts in the presence of potential pitfalls such as near unit root processes and structural breaks, in particular when relying on small samples.,,Copyright © 2008 John Wiley & Sons, Ltd. [source]

    An outlier robust hierarchical Bayes model for forecasting: the case of Hong Kong

    William W. Chow
    Abstract This paper introduces a Bayesian forecasting model that accommodates innovative outliers. The hierarchical specification of prior distributions allows an identification of observations contaminated by these outliers and endogenously determines the hyperparameters of the Minnesota prior. Estimation and prediction are performed using Markov chain Monte Carlo (MCMC) methods. The model forecasts the Hong Kong economy more accurately than the standard V AR and performs in line with other complicated BV AR models. It is also shown that the model is capable of finding most of the outliers in various simulation experiments. Copyright © 2004 John Wiley & Sons, Ltd. [source]

    Bias modelling in evidence synthesis

    Rebecca M. Turner
    Summary., Policy decisions often require synthesis of evidence from multiple sources, and the source studies typically vary in rigour and in relevance to the target question. We present simple methods of allowing for differences in rigour (or lack of internal bias) and relevance (or lack of external bias) in evidence synthesis. The methods are developed in the context of reanalysing a UK National Institute for Clinical Excellence technology appraisal in antenatal care, which includes eight comparative studies. Many were historically controlled, only one was a randomized trial and doses, populations and outcomes varied between studies and differed from the target UK setting. Using elicited opinion, we construct prior distributions to represent the biases in each study and perform a bias-adjusted meta-analysis. Adjustment had the effect of shifting the combined estimate away from the null by approximately 10%, and the variance of the combined estimate was almost tripled. Our generic bias modelling approach allows decisions to be based on all available evidence, with less rigorous or less relevant studies downweighted by using computationally simple methods. [source]

    Bayesian clustering and product partition models

    Fernando A. Quintana
    Summary. We present a decision theoretic formulation of product partition models (PPMs) that allows a formal treatment of different decision problems such as estimation or hypothesis testing and clustering methods simultaneously. A key observation in our construction is the fact that PPMs can be formulated in the context of model selection. The underlying partition structure in these models is closely related to that arising in connection with Dirichlet processes. This allows a straightforward adaptation of some computational strategies,originally devised for nonparametric Bayesian problems,to our framework. The resulting algorithms are more flexible than other competing alternatives that are used for problems involving PPMs. We propose an algorithm that yields Bayes estimates of the quantities of interest and the groups of experimental units. We explore the application of our methods to the detection of outliers in normal and Student t regression models, with clustering structure equivalent to that induced by a Dirichlet process prior. We also discuss the sensitivity of the results considering different prior distributions for the partitions. [source]

    A Bayesian model for longitudinal count data with non-ignorable dropout

    Niko A. Kaciroti
    Summary., Asthma is an important chronic disease of childhood. An intervention programme for managing asthma was designed on principles of self-regulation and was evaluated by a randomized longitudinal study. The study focused on several outcomes, and, typically, missing data remained a pervasive problem. We develop a pattern,mixture model to evaluate the outcome of intervention on the number of hospitalizations with non-ignorable dropouts. Pattern,mixture models are not generally identifiable as no data may be available to estimate a number of model parameters. Sensitivity analyses are performed by imposing structures on the unidentified parameters. We propose a parameterization which permits sensitivity analyses on clustered longitudinal count data that have missing values due to non-ignorable missing data mechanisms. This parameterization is expressed as ratios between event rates across missing data patterns and the observed data pattern and thus measures departures from an ignorable missing data mechanism. Sensitivity analyses are performed within a Bayesian framework by averaging over different prior distributions on the event ratios. This model has the advantage of providing an intuitive and flexible framework for incorporating the uncertainty of the missing data mechanism in the final analysis. [source]

    Modelling species diversity through species level hierarchical modelling

    Alan E. Gelfand
    Summary., Understanding spatial patterns of species diversity and the distributions of individ-ual species is a consuming problem in biogeography and conservation. The Cape floristic region of South Africa is a global hot spot of diversity and endemism, and the Protea atlas project, with about 60 000 site records across the region, provides an extraordinarily rich data set to model patterns of biodiversity. Model development is focused spatially at the scale of 1, grid cells (about 37 000 cells total for the region). We report on results for 23 species of a flowering plant family known as Proteaceae (of about 330 in the Cape floristic region) for a defined subregion. Using a Bayesian framework, we developed a two-stage, spatially explicit, hierarchical logistic regression. Stage 1 models the potential probability of presence or absence for each species at each cell, given species attributes, grid cell (site level) environmental data with species level coefficients, and a spatial random effect. The second level of the hierarchy models the probability of observing each species in each cell given that it is present. Because the atlas data are not evenly distributed across the landscape, grid cells contain variable numbers of sampling localities. Thus this model takes the sampling intensity at each site into account by assuming that the total number of times that a particular species was observed within a site follows a binomial distribution. After assigning prior distributions to all quantities in the model, samples from the posterior distribution were obtained via Markov chain Monte Carlo methods. Results are mapped as the model-estimated probability of presence for each species across the domain. This provides an alternative to customary empirical ,range-of-occupancy' displays. Summing yields the predicted richness of species over the region. Summaries of the posterior for each environmental coefficient show which variables are most important in explaining the presence of species. Our initial results describe biogeographical patterns over the modelled region remarkably well. In particular, species local population size and mode of dispersal contribute significantly to predicting patterns, along with annual precipitation, the coefficient of variation in rainfall and elevation. [source]

    Bayesian cure rate models for malignant melanoma: a case-study of Eastern Cooperative Oncology Group trial E1690

    Ming-Hui Chen
    We propose several Bayesian models for modelling time-to-event data. We consider a piecewise exponential model, a fully parametric cure rate model and a semiparametric cure rate model. For each model, we derive the likelihood function and examine some of its properties for carrying out Bayesian inference with non-informative priors. We also examine model identifiability issues and give conditions which guarantee identifiability. Also, for each model, we construct a class of informative prior distributions based on historical data, i.e. data from similar previous studies. These priors, called power priors, prove to be quite useful in this context. We examine the properties of the power priors for Bayesian inference and, in particular, we study their effect on the current analysis. Tools for model comparison and model assessment are also proposed. A detailed case-study of a recently completed melanoma clinical trial conducted by the Eastern Cooperative Oncology Group is presented and the methodology proposed is demonstrated in detail. [source]

    The Bayesian choice of crop variety and fertilizer dose

    Chris M Theobald
    Recent contributions to the theory of optimizing fertilizer doses in agricultural crop production have introduced Bayesian ideas to incorporate information on crop yield from several environments and on soil nutrients from a soil test, but they have not used a fully Bayesian formulation. We present such a formulation and demonstrate how the resulting Bayes decision procedure can be evaluated in practice by using Markov chain Monte Carlo methods. The approach incorporates expert knowledge of the crop and of regional and local soil conditions and allows a choice of crop variety as well as of fertilizer level. Alternative dose,response functions are expressed in terms of a common interpretable set of parameters to facilitate model comparisons and the specification of prior distributions. The approach is illustrated with a set of yield data from spring barley nitrogen,response trials and is found to be robust to changes in the dose,response function and the prior distribution for indigenous soil nitrogen. [source]