Simultaneous estimation of parameters in different linear models and applications to biometric problems.

Empirical Bayes procedure is employed in simultaneous estimation of vector parameters from a number of Gauss-Markoff linear models. It is shown that with respect to quadratic loss function, empirical Bayes estimators are better than least squares estimators. While estimating the parameter for a particular linear model, a suggestion has been made for distinguishing between the loss due to decision maker and the loss due to individual. A method has been proposed but not fully studied to achieve balance between the two losses. Finally the problem of predicting future observations in a linear model has been considered.     

Invariance of least squared-error estimation under some commonly used neurophysiological data pre-processing methods

The relationship of least squared-error estimation to the commonly used data pre-processing method of stimulus locked signal averaging is discussed. First, a generalized squared-error estimate is derived. Second, two data pre-processing methods are introduced and shown analytically to be equivalent with respect to subsequent least squared-error estimation. The first method consists of fitting known functions directly to unaltered data while the second method fits to the same data after it has been time-averaged. A third method of less utility is also demonstrated to be equivalent. It consists of first fitting to sub-blocks of the unaltered data and then averaging the resulting estimates. Finally, a numerical example is presented. It substantiates the analytical contentions and points out practical considerations which might arise in the course of implementation of the estimation procedure.     

A 3-Component Mixture of Rayleigh Distributions: Properties and Estimation in Bayesian Framework

To study lifetimes of certain engineering processes, a lifetime model which can accommodate the nature of such processes is desired. The mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process as compared to simple models. This paper is about studying a 3-component mixture of the Rayleigh distributionsin Bayesian perspective. The censored sampling environment is considered due to its popularity in reliability theory and survival analysis. The expressions for the Bayes estimators and their posterior risks are derived under different scenarios. In case the case that no or little prior information is available, elicitation of hyperparameters is given. To examine, numerically, the performance of the Bayes estimators using non-informative and informative priors under different loss functions, we have simulated their statistical properties for different sample sizes and test termination times. In addition, to highlight the practical significance, an illustrative example based on a real-life engineering data is also given.     

Duncan's k-ratio Bayes rule approach to multiple comparisons: an overview

An alternative to frequentist approaches to multiple comparisons is Duncan's k-ratio Bayes rule approach. The purpose of this paper is to compile key results on k-ratio Bayes rules for a number of multiple comparison problems that heretofore, have only been available in separate papers or doctoral dissertations. Among other problems, multiple comparisons for means in one-way, two-way, and treatments-vs.-control structures will be reviewed. In the k-ratio approach, the optimal joint rule for a multiple comparisons problem is derived under the assumptions of additive losses and prior exchangeability for the component comparisons. In the component loss function for a comparison, a balance is achieved between the decision losses due to Type I and Type II errors by assuming that their ratio is k. The component loss is also linear in the magnitude of the error. Under the assumption of additive losses, the joint Bayes rule for the component comparisons applies to each comparison the Bayes test for that comparison considered alone. That is, a comparisonwise approach is optimal. However, under prior exchangeability of the comparisons, the component test critical regions adapt to omnibus patterns in the data. For example, for a balanced one-way array of normally distributed means, the Bayes critical t value for a difference between means is inversely related to the F ratio measuring heterogeneity among the means, resembling a continuous version of Fisher's F-protected least significant difference rule. For more complicated treatment structures, the Bayes critical t value for a difference depends intuitively on multiple F ratios and marginal difference(s) (if applicable), such that the critical t value warranted for the difference can range from being as conservative as that given by a familywise rule to actually being anti-conservative relative to that given by the unadjusted 5%-level Student's t test.     

Nearest Neighbor Classification Rule for Mixtures of Discrete and Continuous Random Variables

In this paper very simple nonparametric classification rule for mixtures of discrete and continuous random variables is described. It is based on the method of nearest neighbor proposed by Cover and Hart (1967). The bounds on the limit of the nearest neighbor rule risks are given. Both lower and upper bound depend on the Bayes risk and the loss function. Finally the method is compared with other existing methods on some practical data set.     

Bayesian Estimation of the Distribution Function of the Poisson Model

This paper presents the Bayes estimators of the Poisson distribution function based on complete and truncated data under a natural conjugate prior. Laplace transform of the incomplete gamma function and the Gauss hypergeometric function have been employed in order to overcome the intractability of the integrals. Numerical examples from biosciences are given to illustrate the results. A Monte Carlo study has been carried out to compare Bayes estimators under complete data with the corresponding maximum liklihood estimators.     

Doubly Bayesian Analysis of Confidence in Perceptual Decision-Making

Humans stand out from other animals in that they are able to explicitly report on the reliability of their internal operations. This ability, which is known as metacognition, is typically studied by asking people to report their confidence in the correctness of some decision. However, the computations underlying confidence reports remain unclear. In this paper, we present a fully Bayesian method for directly comparing models of confidence. Using a visual two-interval forced-choice task, we tested whether confidence reports reflect heuristic computations (e.g. the magnitude of sensory data) or Bayes optimal ones (i.e. how likely a decision is to be correct given the sensory data). In a standard design in which subjects were first asked to make a decision, and only then gave their confidence, subjects were mostly Bayes optimal. In contrast, in a less-commonly used design in which subjects indicated their confidence and decision simultaneously, they were roughly equally likely to use the Bayes optimal strategy or to use a heuristic but suboptimal strategy. Our results suggest that, while people’s confidence reports can reflect Bayes optimal computations, even a small unusual twist or additional element of complexity can prevent optimality.     

Bayes Estimator of Survival Probability from Randomly Censored Observations with Weibull Distribution of Time to Death

This paper deals with Bayes estimation of survival probability when the data are randomly censored. Such a situation arises in case of a clinical trial which extends for a limited period T. A fixed number of patients (n) are observed whose times to death have identical Weibull distribution with parameters β and θ. The maximum times of observation for different patients are also independent uniform variables as the patients arrive randomly throughout the trial. For the joint prior distribution of (β, θ) as suggested by Sinha and Kale (1980, page 137) Bayes estimator of survival probability at time t (0<t<T) has been obtained. Considering squared error loss function it is the mean of the survival probability with respect to the posterior distribution of (β, θ). This estimator is then compared with the maximum likelihood estimator, by simulation, for various values of β, θ and censoring percentage. The proposed estimator is found to be better under certain conditions.     

A Comparative Bayes tactic for mate assessment and choice

Models of mate choice tactics have assumed that females randomlyencounter males when collecting information and the informationis perfect. Empirical observations of four bird species showthat females selectively visit males and repeat visits to malesbefore mating. This suggests that the assumptions of previousmodels have been too restrictive. An alternative model of informationgathering and mate choice, which relaxes the assumptions ofrandom encounters and perfect information, is presented. Inthis Comparative Bayes model, the decision of when and fromwhom to collect information is made using Bayesian estimatesof each male's quality. Predictions from the model are that:(1) the occurrence of mate assessment will increase as initialuncertainty about the quality of males increases, as the costof gathering information decreases, and as the signal perceivedby the female becomes a better representation of males' actualqualities; (2) the occurrence of repeat visits to males willbe highest when signals from males are of medium reliability;and (3) the decision of which male to assess will depend onthe estimated qualities of males, prior certainty about eachmale's quality, the reliability of each male's signal, and thecosts of assessment. Simulations compare the fitness outcomesof the Comparative Bayes tactic to other mate choice tactics.The fitness from the Comparative Bayes tactic is significantlyhigher than from the fixed threshold tactic and than from thebest-of-n tactic when the cost of assessment is low. [BehavEcol 7: 451–460 (1996)]     

Minimal-risk scoring matrices for sequence analysis.

We introduce a minimal-risk method for estimating the frequencies of amino acids at conserved positions in a protein family. Our method, called minimal-risk estimation, finds the optimal weighting between a set of observed amino acid counts and a set of pseudofrequencies, which represent prior information about the frequencies. We compute the optimal weighting by minimizing the expected distance between the estimated frequencies and the true population frequencies, measured by either a squared-error or a relative-entropy metric. Our method accounts for the source of the pseudofrequencies, which arise either from the background distribution of amino acids or from applying a substitution matrix to the observed data. Our frequency estimates therefore depend on the size and composition of the observed data as well as the source of the pseudofrequencies. We convert our frequency estimates into minimal-risk scoring matrices for sequence analysis. A large-scale cross-validation study, involving 48 variants of seven methods, shows that the best performing method is minimal-risk estimation using the squared-error metric. Our method is implemented in the package EMATRIX, which is available on the Internet at http://motif.stanford.edu/ematrix.     

An Efficient Hierarchical Generalized Linear Mixed Model for Mapping QTL of Ordinal Traits in Crop Cultivars

Many important phenotypic traits in plants are ordinal. However, relatively little is known about the methodologies for ordinal trait association studies. In this study, we proposed a hierarchical generalized linear mixed model for mapping quantitative trait locus (QTL) of ordinal traits in crop cultivars. In this model, all the main-effect QTL and QTL-by-environment interaction were treated as random, while population mean, environmental effect and population structure were fixed. In the estimation of parameters, the pseudo data normal approximation of likelihood function and empirical Bayes approach were adopted. A series of Monte Carlo simulation experiments were performed to confirm the reliability of new method. The result showed that new method works well with satisfactory statistical power and precision. The new method was also adopted to dissect the genetic basis of soybean alkaline-salt tolerance in 257 soybean cultivars obtained, by stratified random sampling, from 6 geographic ecotypes in China. As a result, 6 main-effect QTL and 3 QTL-by-environment interactions were identified.     

Higher-order spectral analysis of spontaneous speech signals in Alzheimer’s disease

An early and accurate diagnosis of Alzheimer’s disease (AD) has been progressively attracting more attention in recent years. One of the main problems of AD is the loss of language skills. This paper presents a computational framework for classifying AD patients compared to healthy control subjects using information from spontaneous speech signals. Spontaneous speech data are obtained from 30 AD patients and 30 healthy controls. Because of the nonlinear and dynamic nature of speech signals, higher order spectral features (specifically bispectrum) were used for analysis. Four classifiers (k-Nearest Neighbor, Support Vector Machine, Naïve Bayes and Decision tree) were used to classify subjects into three different levels of AD and healthy group based on their performance in terms of the HOS-based features. Ten-fold cross-validation method was used to test the reliability of the classifier results. The results showed that the proposed method had a good potential in AD diagnosis. The proposed method was also able to diagnose the earliest stage of AD with high accuracy. The method has the great advantage of being non-invasive, cost-effective, and associated with no side effects. Therefore, the proposed method can be a spontaneous speech directed test for pre-clinical evaluation of AD diagnosis.     

Bayesian Estimation Using Ranked Set Sampling

Bayesian Estimation of the parameter of a distribution is considered using Ranked set sampling (RSS). It is shown that for at least one RSS plan, the Bayes estimator has smaller Bayes risk than the Bayes estimator using simple random sampling (SRS). Furthermore, for exponential family with conjugate prior, the Bayes estimator of the mean using balanced RSS dominates, in terms of its Bayes risk, the Bayes estimator of the mean using SRS. This procedure is used to estimate the average Milk yield of four hundreds and two sheep. The empirical efficiency supports the theoretical findings.     

Olfactory function in Australian aboriginal children and chronic otitis media

Chronic suppurative otitis media (CSOM), a severe form of middle ear infection, affects most Australian Aboriginal children with up to 50% in some communities suffering hearing loss as a consequence. To date, there is no information on whether repeated exposure to the pathogens that characterize CSOM and that are present in the upper respiratory airway affect olfactory function. Accordingly, this study aimed to determine whether 1) there was a high prevalence of olfactory loss in Aboriginal children and 2) hearing loss is a predictor of olfactory loss. Two hundred and sixty one 9- to 12-year-old Aboriginal children from 16 rural communities reported to have high prevalences of CSOM and hearing loss were assessed for olfactory loss using a 16-odor identification test and hearing loss. One child was found to be anosmic, 4 were slightly hyposmic, and 42 had hearing loss. No relationship was found between olfactory loss and hearing loss. The test-retest reliability of the 16-odor identification test was 0.98. It was concluded that CSOM does not appear to affect olfactory function in the long term and that hearing loss in Aboriginal children is not a predictor of olfactory loss.     

Models for estimating bayes factors with applications to phylogeny and tests of monophyly

Bayes factors comparing two or more competing hypotheses are often estimated by constructing a Markov chain Monte Carlo (MCMC) sampler to explore the joint space of the hypotheses. To obtain efficient Bayes factor estimates, Carlin and Chib (1995, Journal of the Royal Statistical Society, Series B57, 473-484) suggest adjusting the prior odds of the competing hypotheses so that the posterior odds are approximately one, then estimating the Bayes factor by simple division. A byproduct is that one often produces several independent MCMC chains, only one of which is actually used for estimation. We extend this approach to incorporate output from multiple chains by proposing three statistical models. The first assumes independent sampler draws and models the hypothesis indicator function using logistic regression for various choices of the prior odds. The two more complex models relax the independence assumption by allowing for higher-lag dependence within the MCMC output. These models allow us to estimate the uncertainty in our Bayes factor calculation and to fully use several different MCMC chains even when the prior odds of the hypotheses vary from chain to chain. We apply these methods to calculate Bayes factors for tests of monophyly in two phylogenetic examples. The first example explores the relationship of an unknown pathogen to a set of known pathogens. Identification of the unknown's monophyletic relationship may affect antibiotic choice in a clinical setting. The second example focuses on HIV recombination detection. For potential clinical application, these types of analyses must be completed as efficiently as possible.     

The Empirical Bayes Classification Rules for Mixtures of Discrete and Continuous Variables

Let us consider a general population π. Each object belonging to the population π is characterized by a pair of correlated random vectors (X, Y ). Both X and Y may be mixtures of discrete and continuous random variables. It will be assumed that our population π consists of k groups π₁,….,πk, which depend on the value of the random vector Y. A certain object, which is an element of one of the k groups π₁, …, πk, has to be classified into the correct group. The knowledge of the value of the random vector Y would permit its correct classification, but the observation of this vector is difficult or dangerous and we must assign the individual on the basis of the observation of the random vector X . The classification procedure is based on randomized decision function δ* which minimizes the risk function i.e. Bayes decision function. We give also two empirical Bayes classification rules i.e. decision functions based on the sample from population π and having property that their risks converge to Bayes risk when the sample size increases.     

Hierarchical Bayes models for cDNA microarray gene expression

cDNA microarrays are used in many contexts to compare mRNA levels between samples of cells. Microarray experiments typically give us expression measurements on 1000-20 000 genes, but with few replicates for each gene. Traditional methods using means and standard deviations to detect differential expression are not satisfactory in this context. A handful of alternative statistics have been developed, including several empirical Bayes methods. In the present paper we present two full hierarchical Bayes models for detecting gene expression, of which one (D) describes our microarray data very well. We also compare the full Bayes and empirical Bayes approaches with respect to model assumptions, false discovery rates and computer running time. The proposed models are compared to existing empirical Bayes models in a simulation study and for a set of data (Yuen et al., 2002), where 27 genes have been categorized by quantitative real-time PCR. It turns out that the existing empirical Bayes methods have at least as good performance as the full Bayes ones.     

Identifying sites under positive selection with uncertain parameter estimates.

Codon-based substitution models are routinely used to measure selective pressures acting on protein-coding genes. To this effect, the nonsynonymous to synonymous rate ratio (dN/dS = omega) is estimated. The proportion of amino-acid sites potentially under positive selection, as indicated by omega > 1, is inferred by fitting a probability distribution where some sites are permitted to have omega > 1. These sites are then inferred by means of an empirical Bayes or by a Bayes empirical Bayes approach that, respectively, ignores or accounts for sampling errors in maximum-likelihood estimates of the distribution used to infer the proportion of sites with omega > 1. Here, we extend a previous full-Bayes approach to include models with high power and low false-positive rates when inferring sites under positive selection. We propose some heuristics to alleviate the computational burden, and show that (i) full Bayes can be superior to empirical Bayes when analyzing a small data set or small simulated data, (ii) full Bayes has only a small advantage over Bayes empirical Bayes with our small test data, and (iii) Bayesian methods appear relatively insensitive to mild misspecifications of the random process generating adaptive evolution in our simulations, but in practice can prove extremely sensitive to model specification. We suggest that the codon model used to detect amino acids under selection should be carefully selected, for instance using Akaike information criterion (AIC).     

Bayes optimal informer sets for early-stage drug discovery

An important experimental design problem in early-stage drug discovery is how to prioritize available compounds for testing when very little is known about the target protein. Informer-based ranking (IBR) methods address the prioritization problem when the compounds have provided bioactivity data on other potentially relevant targets. An IBR method selects an informer set of compounds, and then prioritizes the remaining compounds on the basis of new bioactivity experiments performed with the informer set on the target. We formalize the problem as a two-stage decision problem and introduce the Bayes Optimal Informer SEt (BOISE) method for its solution. BOISE leverages a flexible model of the initial bioactivity data, a relevant loss function, and effective computational schemes to resolve the two-step design problem. We evaluate BOISE and compare it to other IBR strategies in two retrospective studies, one on protein-kinase inhibition and the other on anticancer drug sensitivity. In both empirical settings BOISE exhibits better predictive performance than available methods. It also behaves well with missing data, where methods that use matrix completion show worse predictive performance.     

Genome-enable prediction for health traits using high-density SNP panel in US Holstein cattle

The objective of this study was to compare accuracies of different Bayesian regression models in predicting molecular breeding values for health traits in Holstein cattle. The dataset was composed of 2505 records reporting the occurrence of retained fetal membranes (RFM), metritis (MET), mastitis (MAST), displaced abomasum (DA), lameness (LS), clinical endometritis (CE), respiratory disease (RD), dystocia (DYST) and subclinical ketosis (SCK) in Holstein cows, collected between 2012 and 2014 in 16 dairies located across the US. Cows were genotyped with the Illumina BovineHD (HD, 777K). The quality controls for SNP genotypes were HWE P-value of at least 1 × 10⁻¹⁰; MAF greater than 0.01 and call rate greater than 0.95. The FImpute program was used for imputation of missing SNP markers. The effect of each SNP was estimated using the Bayesian Ridge Regression (BRR), Bayes A, Bayes B and Bayes Cπ methods. The prediction quality was assessed by the area under the curve, the prediction mean square error and the correlation between genomic breeding value and the observed phenotype, using a leave-one-out cross-validation technique that avoids iterative cross-validation. The highest accuracies of predictions achieved were: RFM [Bayes B (0.34)], MET [BRR (0.36)], MAST [Bayes B (0.55), DA [Bayes Cπ (0.26)], LS [Bayes A (0.12)], CE [Bayes A (0.32)], RD [Bayes Cπ (0.23)], DYST [Bayes A (0.35)] and SCK [Bayes Cπ (0.38)] models. Except for DA, LS and RD, the predictive abilities were similar between the methods. A strong relationship between the predictive ability and the heritability of the trait was observed, where traits with higher heritability achieved higher accuracy and lower bias when compared with those with low heritability. Overall, it has been shown that a high-density SNP panel can be used successfully to predict genomic breeding values of health traits in Holstein cattle and that the model of choice will depend mostly on the genetic architecture of the trait.