Analysis of Longitudinal Data of Epileptic Seizure Counts – A Two‐State Hidden Markov Regression Approach

This paper discusses a two‐state hidden Markov Poisson regression (MPR) model for analyzing longitudinal data of epileptic seizure counts, which allows for the rate of the Poisson process to depend on covariates through an exponential link function and to change according to the states of a two‐state Markov chain with its transition probabilities associated with covariates through a logit link function. This paper also considers a two‐state hidden Markov negative binomial regression (MNBR) model, as an alternative, by using the negative binomial instead of Poisson distribution in the proposed MPR model when there exists extra‐Poisson variation conditional on the states of the Markov chain. The two proposed models in this paper relax the stationary requirement of the Markov chain, allow for overdispersion relative to the usual Poisson regression model and for correlation between repeated observations. The proposed methodology provides a plausible analysis for the longitudinal data of epileptic seizure counts, and the MNBR model fits the data much better than the MPR model. Maximum likelihood estimation using the EM and quasi‐Newton algorithms is discussed. A Monte Carlo study for the proposed MPR model investigates the reliability of the estimation method, the choice of probabilities for the initial states of the Markov chain, and some finite sample behaviors of the maximum likelihood estimates, suggesting that (1) the estimation method is accurate and reliable as long as the total number of observations is reasonably large, and (2) the choice of probabilities for the initial states of the Markov process has little impact on the parameter estimates.     

Comparison of Bayesian methods for flexible modeling of spatial risk surfaces in disease mapping

Bayesian hierarchical models usually model the risk surface on the same arbitrary geographical units for all data sources. Poisson/gamma random field models overcome this restriction as the underlying risk surface can be specified independently to the resolution of the data. Moreover, covariates may be considered as either excess or relative risk factors. We compare the performance of the Poisson/gamma random field model to the Markov random field (MRF)‐based ecologic regression model and the Bayesian Detection of Clusters and Discontinuities (BDCD) model, in both a simulation study and a real data example. We find the BDCD model to have advantages in situations dominated by abruptly changing risk while the Poisson/gamma random field model convinces by its flexibility in the estimation of random field structures and by its flexibility incorporating covariates. The MRF‐based ecologic regression model is inferior. WinBUGS code for Poisson/gamma random field models is provided.     

Classification method for disease risk mapping based on discrete hidden Markov random fields

Risk mapping in epidemiology enables areas with a low or high risk of disease contamination to be localized and provides a measure of risk differences between these regions. Risk mapping models for pooled data currently used by epidemiologists focus on the estimated risk for each geographical unit. They are based on a Poisson log-linear mixed model with a latent intrinsic continuous hidden Markov random field (HMRF) generally corresponding to a Gaussian autoregressive spatial smoothing. Risk classification, which is necessary to draw clearly delimited risk zones (in which protection measures may be applied), generally must be performed separately. We propose a method for direct classified risk mapping based on a Poisson log-linear mixed model with a latent discrete HMRF. The discrete hidden field (HF) corresponds to the assignment of each spatial unit to a risk class. The risk values attached to the classes are parameters and are estimated. When mapping risk using HMRFs, the conditional distribution of the observed field is modeled with a Poisson rather than a Gaussian distribution as in image segmentation. Moreover, abrupt changes in risk levels are rare in disease maps. The spatial hidden model should favor smoothed out risks, but conventional discrete Markov random fields (e.g. the Potts model) do not impose this. We therefore propose new potential functions for the HF that take into account class ordering. We use a Monte Carlo version of the expectation-maximization algorithm to estimate parameters and determine risk classes. We illustrate the method's behavior on simulated and real data sets. Our method appears particularly well adapted to localize high-risk regions and estimate the corresponding risk levels.     

On Bivariate Cumulative Damage Models With Application

Bivariate cumulative damage models are proposed where the responses given the damages are independent random variables. The bivariate damage process can be either bivariate Poisson or bivariate gamma. A bivariate continuous cumulative damage model is investigated in which the responses given the damages have gamma distributions. In this case evaluation of the joint density function and bivariate tail probability function is facilitated by expanding the gamma distributions of the conditional responses by Laguerre polynomials. This approach also leads to evaluation of associated survival models. Moments and estimating equations are discussed. In addition, a bivariate discrete cumulative damage model is investigated in which the responses given the damages have a distribution chosen from a class that includes the negative binomial, the Neyman Type‐A, the Polya‐Aeppli, and the Lagrangian Poisson. Probabilities are obtained from recursive formulas which do not involve cancellation error as all quantities are non‐negative. Moments and estimating equations are presented for these models also. The continuous and the discrete models are applied to describe the rise of systolic and diastolic blood pressure with age.     

A conditional Markov model for clustered progressive multistate processes under incomplete observation

Clustered progressive chronic disease processes arise when interest lies in modeling damage in paired organ systems (e.g., kidneys, eyes), in diseases manifest in different organ systems, or in systemic conditions for which damage may occur in several locations of the body. Multistate Markov models have considerable appeal for modeling damage in such settings, particularly when patients are only under intermittent observation. Generalizations are necessary, however, to deal with the fact that processes within subjects may not be independent. We describe a conditional Markov model in which the clustering in processes within subjects is addressed by the use of multiplicative random effects for each transition intensity. The random effects for the different transition intensities may be correlated within subjects, but are assumed to be independent for different subjects. We apply the mixed Markov model to a motivating data set of patients with psoriatic arthritis, and characterize the progressive course of damage in joints of the hand. A generalization to accommodate a subpopulation of "stayers" and extensions which facilitate regression are indicated and illustrated.     

Semiparametric analysis of correlated recurrent and terminal events

In clinical and observational studies, recurrent event data (e.g., hospitalization) with a terminal event (e.g., death) are often encountered. In many instances, the terminal event is strongly correlated with the recurrent event process. In this article, we propose a semiparametric method to jointly model the recurrent and terminal event processes. The dependence is modeled by a shared gamma frailty that is included in both the recurrent event rate and terminal event hazard function. Marginal models are used to estimate the regression effects on the terminal and recurrent event processes, and a Poisson model is used to estimate the dispersion of the frailty variable. A sandwich estimator is used to achieve additional robustness. An analysis of hospitalization data for patients in the peritoneal dialysis study is presented to illustrate the proposed method.     

Boosting distributional copula regression

Capturing complex dependence structures between outcome variables (e.g., study endpoints) is of high relevance in contemporary biomedical data problems and medical research. Distributional copula regression provides a flexible tool to model the joint distribution of multiple outcome variables by disentangling the marginal response distributions and their dependence structure. In a regression setup, each parameter of the copula model, that is, the marginal distribution parameters and the copula dependence parameters, can be related to covariates via structured additive predictors. We propose a framework to fit distributional copula regression via model-based boosting, which is a modern estimation technique that incorporates useful features like an intrinsic variable selection mechanism, parameter shrinkage and the capability to fit regression models in high-dimensional data setting, that is, situations with more covariates than observations. Thus, model-based boosting does not only complement existing Bayesian and maximum-likelihood based estimation frameworks for this model class but rather enables unique intrinsic mechanisms that can be helpful in many applied problems. The performance of our boosting algorithm for copula regression models with continuous margins is evaluated in simulation studies that cover low- and high-dimensional data settings and situations with and without dependence between the responses. Moreover, distributional copula boosting is used to jointly analyze and predict the length and the weight of newborns conditional on sonographic measurements of the fetus before delivery together with other clinical variables.     

DNA damage in non-proliferating cells subjected to ionizing irradiation at high or low dose rates

Ionizing radiation damage to the genome of a non-cycling mammalian cell is analyzed using continuous time Markov chains. Immediate damage induced by the radiation is modeled as a batch Poisson arrival process of DNA double strand breaks (DSBs). Different kinds of radiation, for example gamma rays or alpha particles, have different batch probabilities. Enzymatic modulation of the immediate damage is modeled as a Markov process similar to the processes described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete exchange model, which postulates that radiation induced DSBs can subsequently either undergo enzymatically mediated repair (restitution) or can participate pairwise in chromosome exchanges, some of which make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. A complete solution of the Markov chain is known for the case that the exchange rate constant is negligible so that no irremediable chromosome lesions are produced and DSBs are the only damage to the genome. Using PDEs for generating functions, a perturbation calculation is made assuming the exchange rate constant is small compared to the repair rate constant. Some non-perturbative results applicable to very prolonged irradiation are also obtained using matrix methods: Perron-Frobenius theory, variational methods and numerical approximations of eigenvalues. Applications to experimental results on expected values, variances and statistical distributions of DNA lesions are briefly outlined.Continuous time Markov chain models are the most systematic of those current radiation damage models which treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g. ignore diffusion limitations). They contain most other homogeneous models as special cases, limiting cases or approximations. However, applying the continuous time Markov chain models to studying spatial dependence of DSB interactions, which is generally believed to be very important in some situations, presents difficulties.     

Structural inference in transition measurement error models for longitudinal data

We propose a new class of models, transition measurement error models, to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error. We show that the response variable conditional on the error-prone covariate follows a complex transition mixed effects model. The naive model obtained by ignoring the measurement error correctly specifies the transition part of the model, but misspecifies the covariate effect structure and ignores the random effects. We next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes. We show that the naive estimator of the regression coefficient of the error-prone covariate is attenuated, while the naive estimators of the regression coefficients of the past responses are generally inflated. We then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method. In view of the multidimensional integration required by full maximum likelihood estimation, an EM algorithm is developed to calculate maximum likelihood estimators, in which Monte Carlo simulations are used to evaluate the conditional expectations in the E-step. We evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease. An additional simulation study shows that the Bayesian information criterion (BIC) performs well in choosing the correct transition orders of the models.     

Superposition properties of interacting ion channels.

Quantitative analysis of patch clamp data is widely based on stochastic models of single-channel kinetics. Membrane patches often contain more than one active channel of a given type, and it is usually assumed that these behave independently in order to interpret the record and infer individual channel properties. However, recent studies suggest there are significant channel interactions in some systems. We examine a model of dependence in a system of two identical channels, each modeled by a continuous-time Markov chain in which specified transition rates are dependent on the conductance state of the other channel, changing instantaneously when the other channel opens or closes. Each channel then has, e.g., a closed time density that is conditional on the other channel being open or closed, these being identical under independence. We relate the two densities by a convolution function that embodies information about, and serves to quantify, dependence in the closed class. Distributions of observable (superposition) sojourn times are given in terms of these conditional densities. The behavior of two channel systems based on two- and three-state Markov models is examined by simulation. Optimized fitting of simulated data using reasonable parameters values and sample size indicates that both positive and negative cooperativity can be distinguished from independence.     

A New Approach for Handling Longitudinal Count Data with Zero‐Inflation and Overdispersion: Poisson Geometric Process Model

For time series of count data, correlated measurements, clustering as well as excessive zeros occur simultaneously in biomedical applications. Ignoring such effects might contribute to misleading treatment outcomes. A generalized mixture Poisson geometric process (GMPGP) model and a zero‐altered mixture Poisson geometric process (ZMPGP) model are developed from the geometric process model, which was originally developed for modelling positive continuous data and was extended to handle count data. These models are motivated by evaluating the trend development of new tumour counts for bladder cancer patients as well as by identifying useful covariates which affect the count level. The models are implemented using Bayesian method with Markov chain Monte Carlo (MCMC) algorithms and are assessed using deviance information criterion (DIC).     

Learning generative models for protein fold families

We introduce a new approach to learning statistical models from multiple sequence alignments (MSA) of proteins. Our method, called GREMLIN (Generative REgularized ModeLs of proteINs), learns an undirected probabilistic graphical model of the amino acid composition within the MSA. The resulting model encodes both the position-specific conservation statistics and the correlated mutation statistics between sequential and long-range pairs of residues. Existing techniques for learning graphical models from MSA either make strong, and often inappropriate assumptions about the conditional independencies within the MSA (e.g., Hidden Markov Models), or else use suboptimal algorithms to learn the parameters of the model. In contrast, GREMLIN makes no a priori assumptions about the conditional independencies within the MSA. We formulate and solve a convex optimization problem, thus guaranteeing that we find a globally optimal model at convergence. The resulting model is also generative, allowing for the design of new protein sequences that have the same statistical properties as those in the MSA. We perform a detailed analysis of covariation statistics on the extensively studied WW and PDZ domains and show that our method out-performs an existing algorithm for learning undirected probabilistic graphical models from MSA. We then apply our approach to 71 additional families from the PFAM database and demonstrate that the resulting models significantly out-perform Hidden Markov Models in terms of predictive accuracy.     

Zero-inflated Poisson and binomial regression with random effects: a case study

In a 1992 Technometrics paper, Lambert (1992, 34, 1-14) described zero-inflated Poisson (ZIP) regression, a class of models for count data with excess zeros. In a ZIP model, a count response variable is assumed to be distributed as a mixture of a Poisson(lambda) distribution and a distribution with point mass of one at zero, with mixing probability p. Both p and lambda are allowed to depend on covariates through canonical link generalized linear models. In this paper, we adapt Lambert's methodology to an upper bounded count situation, thereby obtaining a zero-inflated binomial (ZIB) model. In addition, we add to the flexibility of these fixed effects models by incorporating random effects so that, e.g., the within-subject correlation and between-subject heterogeneity typical of repeated measures data can be accommodated. We motivate, develop, and illustrate the methods described here with an example from horticulture, where both upper bounded count (binomial-type) and unbounded count (Poisson-type) data with excess zeros were collected in a repeated measures designed experiment.     

A Genetic Analysis of Mortality in Pigs

An analysis of mortality is undertaken in two breeds of pigs: Danish Landrace and Yorkshire. Zero-inflated and standard versions of hierarchical Poisson, binomial, and negative binomial Bayesian models were fitted using Markov chain Monte Carlo (MCMC). The objectives of the study were to investigate whether there is support for genetic variation for mortality and to study the quality of fit and predictive properties of the various models. In both breeds, the model that provided the best fit to the data was the standard binomial hierarchical model. The model that performed best in terms of the ability to predict the distribution of stillbirths was the hierarchical zero-inflated negative binomial model. The best fit of the binomial hierarchical model and of the zero-inflated hierarchical negative binomial model was obtained when genetic variation was included as a parameter. For the hierarchical binomial model, the estimate of the posterior mean of the additive genetic variance (posterior standard deviation in brackets) at the level of the logit of the probability of a stillbirth was 0.173(0.039) in Landrace and 0.202(0.048) in Yorkshire. The implications of these results from a breeding perspective are briefly discussed.LITTER size has been under selection in the Danish pig breeding program since the early 1990s and this resulted in considerable increase in total number born and also in the proportion of stillborn piglets (Sorensen et al. 2000; Su et al. 2007). A number of studies have reported genetic variation for mortality with heritabilities ranging from 0.03 to 0.12. These studies have either assumed normality of the sampling model for mortality (e.g., van Arendonk et al. 1996) or based inferences on a variety of threshold models (e.g., Roehe and Kalm 2000; Arango et al. 2006), and critical investigations of the quality of fit of the models used were not reported.Mortality data, regarded as a trait of the mother, show typically a large proportion of zeros (many litters do not have stillborn piglets). Formal genetic analyses of mortality in pigs accounting for this feature of the data are not available in the literature and this article attempts to fill this gap. The focus here is to study the quality of fit and predictive ability of a number of models and to investigate whether they provide statistical evidence for genetic variation for mortality. The statistical genetic analysis involves fitting various hierarchical models involving three discrete distributions: the Poisson, the binomial, and the negative binomial.The statistical analysis of counts based on discrete parametric distributions has a long and rich history (Johnson and Kotz 1969). In the case of unbounded counts, Poisson regression models are standard, whereas for bounded counts, when the response can be viewed as the number of successes out of a fixed number of trials, regression models based on the binomial distribution are often used (Hall 2000). A restriction of the Poisson model is that it imposes equality of mean and variance. Typically the distribution of counts is overdispersed. In the case of the binomial model the only free parameter is the probability of success, which results in a functional relationship between the mean and the variance. Several possible alternatives have been suggested to obtain more flexible models. For example, the negative binomial distribution has two parameters and allows the mean and variance to be fitted separately (Lawless 1987). An application of the negative binomial model in animal breeding can be found in Tempelman and Gianola (1996, 1999). In the same spirit, a robust alternative to the binomial model is the beta-binomial, which is a mixture of binomials where the unequal probabilities of success vary according to a beta-distribution. In general, hierarchical specifications are needed to explain extra variation that is not accounted for by the sampling model of the data. These involve assigning a distribution to the parameters of the sampling model, directly, as in the case of the negative binomial or beta-binomial models, or indirectly, by embedding these parameters in a linear structure that includes random effects as explanatory variables.There are situations where overdispersion is partly associated with an incidence of zero counts that is greater than expected under the sampling model, as in the present study. Hurdle models (Mullahy 1986; Winkelmann 2000) and zero-inflated models are two instances of finite mixture models commonly used to account for this characteristic of the data. In the present work the excess of zeros is studied using zero-inflated models that are described in Johnson and Kotz (1969) and extended by Lambert (1992). Ridout et al. (1998) provide a review of various zero-inflated models; recent applications of zero-inflated Poisson models in animal breeding are in Rodriguez-Motta et al. (2007) and in Naya et al. (2008). Zero-inflated models assume that the population consists of two sets of observations. In the first set, which may include zeros, observations are realizations from a discrete sampling process indexed by unknown parameters (this set is often referred to as the imperfect state); observations from the second set consist only of zeros and the parameter of interest is the proportion of these individuals. This set is often referred to as the perfect state. Either or both sets of parameters may depend on covariates.This article is organized as follows. material and methods describes the data, details of the models, and their Markov chain Monte Carlo (MCMC) implementation. This is followed by a presentation of the results of the analyses and of MCMC-driven explorative tools for model comparison. The article concludes with a discussion.     

The effects of health histories on stochastic process models of aging and mortality

A model of human health history and aging, based on a multivariate stochastic process with both continuous diffusion and discrete jump components, is presented. Discrete changes generate non-Gaussian diffusion with time varying continuous state distributions. An approach to calculating transition rates in dynamically heterogeneous populations, which generalizes the conditional averaging of hazard rates done in fixed frailty population models, is presented to describe health processes with multiple jumps. Conditional semi-invariants are used to approximate the conditional p.d.f. of the unobserved health history components. This is useful in analyzing the age dependence of mortality and health changes at advanced age (e.g., 95 +) where homeostatic controls weaken, and physiological dynamics and survival manifest nonlinear behavior.     

A simple imputation method for longitudinal studies with non-ignorable non-responses

Missing data are a common problem in longitudinal studies in the health sciences. Motivated by data from the Muscatine Coronary Risk Factor (MCRF) study, a longitudinal study of obesity, we propose a simple imputation method for handling non-ignorable non-responses (i.e., when non-response is related to the specific values that should have been obtained) in longitudinal studies with either discrete or continuous outcomes. In the proposed approach, two regression models are specified; one for the marginal mean of the response, the other for the conditional mean of the response given non-response patterns. Statistical inference for the model parameters is based on the generalized estimating equations (GEE) approach. An appealing feature of the proposed method is that it can be readily implemented using existing, widely-available statistical software. The method is illustrated using longitudinal data on obesity from the MCRF study.     

Models for Bounded Systems with Continuous Dynamics

Summary .  Models for natural nonlinear processes, such as population dynamics, have been given much attention in applied mathematics. For example, species competition has been extensively modeled by differential equations. Often, the scientist has preferred to model the underlying dynamical processes (i.e., theoretical mechanisms) in continuous time. It is of both scientific and mathematical interest to implement such models in a statistical framework to quantify uncertainty associated with the models in the presence of observations. That is, given discrete observations arising from the underlying continuous process, the unobserved process can be formally described while accounting for multiple sources of uncertainty (e.g., measurement error, model choice, and inherent stochasticity of process parameters). In addition to continuity, natural processes are often bounded; specifically, they tend to have nonnegative support. Various techniques have been implemented to accommodate nonnegative processes, but such techniques are often limited or overly compromising. This article offers an alternative to common differential modeling practices by using a bias-corrected truncated normal distribution to model the observations and latent process, both having bounded support. Parameters of an underlying continuous process are characterized in a Bayesian hierarchical context, utilizing a fourth-order Runge–Kutta approximation.     

CGBayesNets: Conditional Gaussian Bayesian Network Learning and Inference with Mixed Discrete and Continuous Data

Bayesian Networks (BN) have been a popular predictive modeling formalism in bioinformatics, but their application in modern genomics has been slowed by an inability to cleanly handle domains with mixed discrete and continuous variables. Existing free BN software packages either discretize continuous variables, which can lead to information loss, or do not include inference routines, which makes prediction with the BN impossible. We present CGBayesNets, a BN package focused around prediction of a clinical phenotype from mixed discrete and continuous variables, which fills these gaps. CGBayesNets implements Bayesian likelihood and inference algorithms for the conditional Gaussian Bayesian network (CGBNs) formalism, one appropriate for predicting an outcome of interest from, e.g., multimodal genomic data. We provide four different network learning algorithms, each making a different tradeoff between computational cost and network likelihood. CGBayesNets provides a full suite of functions for model exploration and verification, including cross validation, bootstrapping, and AUC manipulation. We highlight several results obtained previously with CGBayesNets, including predictive models of wood properties from tree genomics, leukemia subtype classification from mixed genomic data, and robust prediction of intensive care unit mortality outcomes from metabolomic profiles. We also provide detailed example analysis on public metabolomic and gene expression datasets. CGBayesNets is implemented in MATLAB and available as MATLAB source code, under an Open Source license and anonymous download at http://www.cgbayesnets.com.

This is a PLOS Computational Biology Software Article

     

DL-ADR: a novel deep learning model for classifying genomic variants into adverse drug reactions

Background

Genomic variations are associated with the metabolism and the occurrence of adverse reactions of many therapeutic agents. The polymorphisms on over 2000 locations of cytochrome P450 enzymes (CYP) due to many factors such as ethnicity, mutations, and inheritance attribute to the diversity of response and side effects of various drugs. The associations of the single nucleotide polymorphisms (SNPs), the internal pharmacokinetic patterns and the vulnerability of specific adverse reactions become one of the research interests of pharmacogenomics. The conventional genomewide association studies (GWAS) mainly focuses on the relation of single or multiple SNPs to a specific risk factors which are a one-to-many relation. However, there are no robust methods to establish a many-to-many network which can combine the direct and indirect associations between multiple SNPs and a serial of events (e.g. adverse reactions, metabolic patterns, prognostic factors etc.). In this paper, we present a novel deep learning model based on generative stochastic networks and hidden Markov chain to classify the observed samples with SNPs on five loci of two genes (CYP2D6 and CYP1A2) respectively to the vulnerable population of 14 types of adverse reactions.

Methods

A supervised deep learning model is proposed in this study. The revised generative stochastic networks (GSN) model with transited by the hidden Markov chain is used. The data of the training set are collected from clinical observation. The training set is composed of 83 observations of blood samples with the genotypes respectively on CYP2D6*2, *10, *14 and CYP1A2*1C, *1 F. The samples are genotyped by the polymerase chain reaction (PCR) method. A hidden Markov chain is used as the transition operator to simulate the probabilistic distribution. The model can perform learning at lower cost compared to the conventional maximal likelihood method because the transition distribution is conditional on the previous state of the hidden Markov chain. A least square loss (LASSO) algorithm and a k-Nearest Neighbors (kNN) algorithm are used as the baselines for comparison and to evaluate the performance of our proposed deep learning model.

Results

There are 53 adverse reactions reported during the observation. They are assigned to 14 categories. In the comparison of classification accuracy, the deep learning model shows superiority over the LASSO and kNN model with a rate over 80 %. In the comparison of reliability, the deep learning model shows the best stability among the three models.

Conclusions

Machine learning provides a new method to explore the complex associations among genomic variations and multiple events in pharmacogenomics studies. The new deep learning algorithm is capable of classifying various SNPs to the corresponding adverse reactions. We expect that as more genomic variations are added as features and more observations are made, the deep learning model can improve its performance and can act as a black-box but reliable verifier for other GWAS studies.

     

On marker-assisted prediction of genetic value: beyond the ridge

Marked-assisted genetic improvement of agricultural species exploits statistical dependencies in the joint distribution of marker genotypes and quantitative traits. An issue is how molecular (e.g., dense marker maps) and phenotypic information (e.g., some measure of yield in plants) is to be used for predicting the genetic value of candidates for selection. Multiple regression, selection index techniques, best linear unbiased prediction, and ridge regression of phenotypes on marker genotypes have been suggested, as well as more elaborate methods. Here, phenotype-marker associations are modeled hierarchically via multilevel models including chromosomal effects, a spatial covariance of marked effects within chromosomes, background genetic variability, and family heterogeneity. Lorenz curves and Gini coefficients are suggested for assessing the inequality of the contribution of different marked effects to genetic variability. Classical and Bayesian methods are presented. The Bayesian approach includes a Markov chain Monte Carlo implementation. The generality and flexibility of the Bayesian method is illustrated when a Lorenz curve is to be inferred.