Application of a mixed normal mixture model for the estimation of Mastitis-related parameters

The current methodology for estimating genetic parameters for SCC (SCS) does not account for the difference in SCS between healthy cows and cows with an intramammary infection (IMI). We propose a two-component finite mixed normal mixture model to estimate IMI prevalence, separate SCS subpopulation means, individual posterior probabilities of IMI, and SCS variance components. The theory is presented and the expectation-conditional maximization algorithm is utilized to compute maximum likelihood estimates. The methodology is illustrated on two simulated data sets based on the current knowledge of SCS parameters. Maximum likelihood estimates of IMI prevalence and SCS subpopulation means were close to simulated values, except for the estimate of IMI prevalence when both subpopulations were almost confounded. Individual posterior probabilities of IMI were always higher among infected than among healthy cows. Error and additive variance components obtained under the mixture model were closer to simulated values than restricted maximum likelihood estimates obtained assuming a homogeneous SCS distribution, especially when subpopulations were completely separated and when mixing proportion was highest. Convergence was linear and rapid when priors were chosen with caution. The advantages of the methodology are demonstrated, and its feasibility for large data sets is discussed.

Triple-goal estimates for disease mapping

Maps of regional morbidity and mortality rates play an important role in assessing environmental equity. They provide effective tools for identifying areas with potentially elevated risk, determining spatial trend, and formulating and validating aetiological hypotheses about disease. Bayes and empirical Bayes methods produce stable small-area rate estimates that retain geographic and demographic resolution. The beauty of the Bayesian approach lies in its ability to structure complicated models, inferential goals and analyses. Three inferential goals are relevant to disease mapping and risk assessment: (i) computing accurate estimates of disease rates in small geographic areas; (ii) estimating the distribution of disease rates over the region; (iii) ranking the disease rates so that environmental investigation can be prioritized. No single set of estimates can simultaneously optimize these three goals, and Shen and Louis propose a set of estimates that perform well on all three goals. These are optimal for estimating the distribution of rates and for ranking, and maintain a high accuracy in estimating area-specific rates. However, the Shen/Louis method is sensitive to choice of priors. To address this issue we introduce a robustified version of the method based on a smoothed non-parametric estimate of the prior. We evaluate the performance of this method through a simulation study, and illustrate it using a data set of county-specific lung cancer rates in Ohio.

Statistical issues in the analysis of disease mapping data

In this paper we discuss a number of issues that are pertinent to the analysis of disease mapping data. As an illustrative example we consider the mapping of larynx cancer across electoral wards in the North West Thames region of the U.K. Bayesian hierarchical models are now frequently employed to carry out such mapping. In a typical situation, a three-stage hierarchical model is specified in which the data are modelled as a function of area-specific relative risks at stage one; the collection of relative risks across the study region are modelled at stage two; and at stage three prior distributions are assigned to parameters of the stage two distribution. Such models allow area-specific disease relative risks to be 'smoothed' towards global and/or local mean levels across the study region. However, these models contain many structural and functional assumptions at different levels of the hierarchy; we aim to discuss some of these assumptions and illustrate their sensitivity. When relative risks are the endpoint of interest, it is common practice to assume that, for each of the age-sex strata of a particular area, there is a common multiplier (the relative risk) acting upon each of the stratum-specific risks in that area; we will examine this proportionality assumption. We also consider the choices of models and priors at stages two and three of the hierarchy, the effect of outlying areas, and an assessment of the level of smoothing that is being carried out. For inference, we concentrate on the description of the spatial variability in relative risks and on the association between the relative risks of larynx cancer and an area-level measure of socio-economic status.

SMEM algorithm for mixture models

We present a split-and-merge expectation-maximization (SMEM) algorithm to overcome the local maxima problem in parameter estimation of finite mixture models. In the case of mixture models, local maxima often involve having too many components of a mixture model in one part of the space and too few in another, widely separated part of the space. To escape from such configurations, we repeatedly perform simultaneous split-and-merge operations using a new criterion for efficiently selecting the split-and-merge candidates. We apply the proposed algorithm to the training of gaussian mixtures and mixtures of factor analyzers using synthetic and real data and show the effectiveness of using the split-and-merge operations to improve the likelihood of both the training data and of held-out test data. We also show the practical usefulness of the proposed algorithm by applying it to image compression and pattern recognition problems.

Bayesian predictive inference for time series count data

Correlated count data arise often in practice, especially in repeated measures situations or instances in which observations are collected over time. In this paper, we consider a parametric model for a time series of counts by constructing a likelihood-based version of a model similar to that of Zeger (1988, Biometrika 75, 621-629). The model has the advantage of incorporating both overdispersion and autocorrelation. We consider a Bayesian approach and propose a class of informative prior distributions for the model parameters that are useful for prediction. The prior specification is motivated from the notion of the existence of data from similar previous studies, called historical data, which is then quantified into a prior distribution for the current study. We derive the Bayesian predictive distribution and use a Bayesian criterion, called the predictive L measure, for assessing the predictions for a given time series model. The distribution of the predictive L measure is also derived, which will enable us to compare the predictive ability for each model under consideration. Our methodology is motivated by a real data set involving yearly pollen counts, which is examined in some detail.

Proportional means regression for censored medical costs

The semiparametric proportional means model specifies that the mean function for the cumulative medical cost over time conditional on a set of covariates is equal to an arbitrary baseline mean function multiplied by an exponential regression function. We demonstrate how to estimate the vector-valued regression parameter using possibly censored lifetime costs. The estimator is consistent and asymptotically normal with an easily estimable covariance matrix. Simulation studies show that the proposed methodology is appropriate for practical use. An application to AIDS is provided.

Inferring linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations

Three approaches are proposed in this study for detecting or estimating linkage disequilibrium between a polymorphic marker locus and a locus affecting quantitative genetic variation using the sample from random mating populations. It is shown that the disequilibrium over a wide range of circumstances may be detected with a power of 80% by using phenotypic records and marker genotypes of a few hundred individuals. Comparison of ANOVA and regression methods in this article to the transmission disequilibrium test (TDT) shows that, given the genetic variance explained by the trait locus, the power of TDT depends on the trait allele frequency, whereas the power of ANOVA and regression analyses is relatively independent from the allelic frequency. The TDT method is more powerful when the trait allele frequency is low, but much less powerful when it is high. The likelihood analysis provides reliable estimation of the model parameters when the QTL variance is at least 10% of the phenotypic variance and the sample size of a few hundred is used. Potential use of these estimates in mapping the trait locus is also discussed.

Standard errors for EM estimates in generalized linear models with random effects

A procedure is derived for computing standard errors of EM estimates in generalized linear models with random effects. Quadrature formulas are used to approximate the integrals in the EM algorithm, where two different approaches are pursued, i.e., Gauss-Hermite quadrature in the case of Gaussian random effects and nonparametric maximum likelihood estimation for an unspecified random effect distribution. An approximation of the expected Fisher information matrix is derived from an expansion of the EM estimating equations. This allows for inferential arguments based on EM estimates, as demonstrated by an example and simulations.

When should epidemiologic regressions use random coefficients?

Regression models with random coefficients arise naturally in both frequentist and Bayesian approaches to estimation problems. They are becoming widely available in standard computer packages under the headings of generalized linear mixed models, hierarchical models, and multilevel models. I here argue that such models offer a more scientifically defensible framework for epidemiologic analysis than the fixed-effects models now prevalent in epidemiology. The argument invokes an antiparsimony principle attributed to L. J. Savage, which is that models should be rich enough to reflect the complexity of the relations under study. It also invokes the countervailing principle that you cannot estimate anything if you try to estimate everything (often used to justify parsimony). Regression with random coefficients offers a rational compromise between these principles as well as an alternative to analyses based on standard variable-selection algorithms and their attendant distortion of uncertainty assessments. These points are illustrated with an analysis of data on diet, nutrition, and breast cancer.

A useful monotonic non-linear model with applications in medicine and epidemiology

In medicine and epidemiology monotonic curves are important as models for relations which prior knowledge or scientific reasoning dictate should increase or decrease consistently with the predictor value. An example is the monotonically increasing relation between cigarette consumption and the risk of coronary heart disease. In this paper I propose a new class of monotonic non-linear models which generalizes the well-known power and exponential transformations of a covariate. The models are cousins of the Gompertz family of growth curves and include non-sigmoid and asymmetric sigmoid curves. I explore their properties and illustrate their usefulness in three substantial medical and epidemiological data sets.

Bayesian mapping of quantitative trait loci for complex binary traits

A complex binary trait is a character that has a dichotomous expression but with a polygenic genetic background. Mapping quantitative trait loci (QTL) for such traits is difficult because of the discrete nature and the reduced variation in the phenotypic distribution. Bayesian statistics are proved to be a powerful tool for solving complicated genetic problems, such as multiple QTL with nonadditive effects, and have been successfully applied to QTL mapping for continuous traits. In this study, we show that Bayesian statistics are particularly useful for mapping QTL for complex binary traits. We model the binary trait under the classical threshold model of quantitative genetics. The Bayesian mapping statistics are developed on the basis of the idea of data augmentation. This treatment allows an easy way to generate the value of a hypothetical underlying variable (called the liability) and a threshold, which in turn allow the use of existing Bayesian statistics. The reversible jump Markov chain Monte Carlo algorithm is used to simulate the posterior samples of all unknowns, including the number of QTL, the locations and effects of identified QTL, genotypes of each individual at both the QTL and markers, and eventually the liability of each individual. The Bayesian mapping ends with an estimation of the joint posterior distribution of the number of QTL and the locations and effects of the identified QTL. Utilities of the method are demonstrated using a simulated outbred full-sib family. A computer program written in FORTRAN language is freely available on request.

Multipoint mapping of viability and segregation distorting loci using molecular markers

In line-crossing experiments, deviations from Mendelian segregation ratios are usually observed for some markers. We hypothesize that these deviations are caused by one or more segregation-distorting loci (SDL) linked to the markers. We develop both a maximum-likelihood (ML) method and a Bayesian method to map SDL using molecular markers. The ML mapping is implemented via an EM algorithm and the Bayesian method is performed via the Markov chain Monte Carlo (MCMC). The Bayesian mapping is computationally more intensive than the ML mapping but can handle more complicated models such as multiple SDL and variable number of SDL. Both methods are applied to a set of simulated data and real data from a cross of two Scots pine trees.

Introduction of the IBD information into the weighted pairwise correlation method for linkage analysis

The Weighted Pairwise Correlation (WPC) approach is a non-parametric method of linkage analysis that allows analysis of any kind of phenotypes (quantitative, binary, binary with age of onset) and to consider all pairs of relatives in a pedigree. The principle of this method is to test whether two relatives having close phenotypes also resemble at the marker locus more than expected under the null hypothesis of no linkage. So far, this marker resemblance was estimated by the proportion of alleles shared Identical By State (IBS) by the two relatives. Here, we propose a method to incorporate the Identical By Descent (IBD) information into the WPC approach. For any kind of relative pairs, the computation of the proportion of alleles shared IBD is based on the identification of the closest couple of ancestors, denoted as the reference couple. The IBD information is obtained for pairs of relatives having the same reference couple using individual genotypic vectors derived from this couple. This reconstruction of the IBD information is performed rapidly even in large pedigrees. Simulation studies conducted under various genetic models demonstrate that the use of IBD instead of IBS information leads to a large increase of power, especially in the situation of poorly informative markers.

Population-based family study designs: an interdisciplinary research framework for genetic epidemiology

Most complex traits such as cancer and coronary heart diseases are attributed either to heritable factors or to environmental factors or to both. Dissecting the genetic and environmental etiology of complex traits thus requires an interdisciplinary research strategy. Genetic studies generally involve families and investigate familial aggregations of traits, segregation of major disease genes, and locations of disease genes on the human genome, the latter of which can be identified via linkage analysis. Epidemiologic studies often use population-based case-control studies to establish the role of specific environmental factors. Integrating both objectives, genetic epidemiology is to assess the associations of environmental factors with disease status, to quantify the aggregation of cases within families, to characterize putative disease genes via segregation analysis, and to localize disease genes via linkage analysis with genetic markers. To accomplish these objectives through designed studies, we propose a class of population-based family study designs, which are formed by choosing among sampling designs at three stages. The objectives of sampling at these three stages are 1) combined aggregation and association analysis, 2) combined segregation, aggregation, and association analysis, and 3) combined linkage, segregation, aggregation, and association analysis. These designs form an interdisciplinary research framework for genetic epidemiology. Our preliminary exploration of this framework and related analytic methods indicates that population-based family study designs retain the efficiency of linkage analysis for localizing disease genes without losing the property of being population-based, and they will therefore allow an assessment of a joint contribution of genetic and environmental factors to complex traits.

Two-locus developments of the weighted pairwise correlation method for linkage analysis

Different studies of complex traits assumed to be influenced by two unliked loci found that two-locus linkage analysis is more powerful than the classical one-locus strategy. The Weighted Pairwise Correlation (WPC) approach is a nonparametric method for linkage analysis that has the advantage to analyze any kind of phenotypes and to consider extended pairs of relatives. In this report, we propose different two-locus extensions of the WPC method based on an additive or a multiplicative effect of two unlinked marker loci on the phenotype. Both methods and their corresponding statistics are easily derived from the classical WPC approach. Compared to the additive model, the multiplicative approach, which can be understood as a statistical interaction effect of the two markers, does not need to specify any additional parameter and allows one to test both the global effect of the two markers (T(AB)test) and the effect of one marker, e.g., B, taking into account the effect of the other, A (T(AB/A) test). When compared to classical one-locus tests by means of simulations, two-locus tests have comparable 0.05 type I error and are more powerful. In particular, tests based on the multiplicative approach appear to be quite interesting in addition to single locus tests to detect the combined role of two markers (T(AB)), or to investigate the role of a marker taking into account a known linked marker (T(AB/A)), especially when these markers have complex effects on the phenotype (e.g., statistical interaction).

Estimating incidence of dementia subtypes: assessing the impact of missed cases

In many community-based studies on the incidence of dementia, a target population is screened and a subsample is clinically evaluated at baseline and follow-up. Incidence rates are affected by missed cases at both exams and this complicates the estimation of these rates. Recent work proposes a regression-based technique for joint estimation of prevalence and incidence and suggests the use of surrogate information obtained on the entire cohort at both times to calculate the expected score equation contribution for individuals missing clinical exams at one or both times. This helps to quantify the impact of missed diagnosis upon the incidence estimates and their confidence intervals. We extend this work to the setting of subtypes of dementia for use in the Honolulu-Asia Aging Study on incidence of dementia. The technique is applied using two separate models for the effect of age on dementia incidence. Subsequently, shrinkage estimation methods are applied to provide more precise estimates of the rates. Published in 2000 by John Wiley & Sons, Ltd.

Estimating the incidence of dementia from two-phase sampling with non-ignorable missing data

Two-phase sampling designs have been used in the field of psychiatry to estimate prevalence and incidence of a rare disease such as dementia and Alzheimer's disease. In a longitudinal study on dementia, since the repeated two-phase sampling is conducted several years after the baseline wave, some subjects may die before the follow-up wave, thus their disease status prior to death is missing. There are reasons to suggest that the missing due to death is non-ignorable. Estimation of disease incidence from longitudinal dementia study has to appropriately adjust for data missing by death as well as the sampling design used at each study wave. In this paper we adopt a selection model approach to model the missing data by death and use a likelihood approach to derive incidence estimates. A modified EM algorithm is used to deal with data from sampling selection. The non-parametric jack-knife variance estimator is used to derive variance estimates for the model parameters and the incidence estimates. The proposed approaches are applied to data from the Indianapolis-Ibadan Dementia Study.

A mixture model for longitudinal data with application to assessment of noncompliance

In clinical trials of a self-administered drug, repeated measures of a laboratory marker, which is affected by study medication and collected in all treatment arms, can provide valuable information on population and individual summaries of compliance. In this paper, we introduce a general finite mixture of nonlinear hierarchical models that allows estimates of component membership probabilities and random effect distributions for longitudinal data arising from multiple subpopulations, such as from noncomplying and complying subgroups in clinical trials. We outline a sampling strategy for fitting these models, which consists of a sequence of Gibbs, Metropolis-Hastings, and reversible jump steps, where the latter is required for switching between component models of different dimensions. Our model is applied to identify noncomplying subjects in the placebo arm of a clinical trial assessing the effectiveness of zidovudine (AZT) in the treatment of patients with HIV, where noncompliance was defined as initiation of AZT during the trial without the investigators' knowledge. We fit a hierarchical nonlinear change-point model for increases in the marker MCV (mean corpuscular volume of erythrocytes) for subjects who noncomply and a constant mean random effects model for those who comply. As part of our fully Bayesian analysis, we assess the sensitivity of conclusions to prior and modeling assumptions and demonstrate how external information and covariates can be incorporated to distinguish subgroups.

Inference of population structure using multilocus genotype data

We describe a model-based clustering method for using multilocus genotype data to infer population structure and assign individuals to populations. We assume a model in which there are K populations (where K may be unknown), each of which is characterized by a set of allele frequencies at each locus. Individuals in the sample are assigned (probabilistically) to populations, or jointly to two or more populations if their genotypes indicate that they are admixed. Our model does not assume a particular mutation process, and it can be applied to most of the commonly used genetic markers, provided that they are not closely linked. Applications of our method include demonstrating the presence of population structure, assigning individuals to populations, studying hybrid zones, and identifying migrants and admixed individuals. We show that the method can produce highly accurate assignments using modest numbers of loci-e.g. , seven microsatellite loci in an example using genotype data from an endangered bird species. The software used for this article is available from http://www.stats.ox.ac.uk/ approximately pritch/home. html.

Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians

Since the early 1980s, a bewildering array of methods for constructing bootstrap confidence intervals have been proposed. In this article, we address the following questions. First, when should bootstrap confidence intervals be used. Secondly, which method should be chosen, and thirdly, how should it be implemented. In order to do this, we review the common algorithms for resampling and methods for constructing bootstrap confidence intervals, together with some less well known ones, highlighting their strengths and weaknesses. We then present a simulation study, a flow chart for choosing an appropriate method and a survival analysis example.