An empirical Bayesian method for estimating biological networks from temporal microarray data.

Gene regulatory networks refer to the interactions that occur among genes and other cellular products. The topology of these networks can be inferred from measurements of changes in gene expression over time. However, because the measurement device (i.e., microarrays) typically yields information on thousands of genes over few biological replicates, these systems are quite difficult to elucidate. An approach with proven effectiveness for inferring networks is the Dynamic Bayesian Network. We have developed an iterative empirical Bayesian procedure with a Kalman filter that estimates the posterior distributions of network parameters. We compare our method to similar existing methods on simulated data and real microarray time series data. We find that the proposed method performs comparably on both model-based and data-based simulations in considerably less computational time. The R and C code used to implement the proposed method are publicly available in the R package ebdbNet.

Moderated effect size and P-value combinations for microarray meta-analyses.

MOTIVATION: With the proliferation of microarray experiments and their availability in the public domain, the use of meta-analysis methods to combine results from different studies increases. In microarray experiments, where the sample size is often limited, meta-analysis offers the possibility to considerably increase the statistical power and give more accurate results. RESULTS: A moderated effect size combination method was proposed and compared with other meta-analysis approaches. All methods were applied to real publicly available datasets on prostate cancer, and were compared in an extensive simulation study for various amounts of inter-study variability. Although the proposed moderated effect size combination improved already existing effect size approaches, the P-value combination was found to provide a better sensitivity and a better gene ranking than the other meta-analysis methods, while effect size methods were more conservative. AVAILABILITY: An R package metaMA is available on the CRAN.

Bayesian analysis of growth curves using mixed models defined by stochastic differential equations.

Growth curve data consist of repeated measurements of a continuous growth process over time in a population of individuals. These data are classically analyzed by nonlinear mixed models. However, the standard growth functions used in this context prescribe monotone increasing growth and can fail to model unexpected changes in growth rates. We propose to model these variations using stochastic differential equations (SDEs) that are deduced from the standard deterministic growth function by adding random variations to the growth dynamics. A Bayesian inference of the parameters of these SDE mixed models is developed. In the case when the SDE has an explicit solution, we describe an easily implemented Gibbs algorithm. When the conditional distribution of the diffusion process has no explicit form, we propose to approximate it using the Euler-Maruyama scheme. Finally, we suggest validating the SDE approach via criteria based on the predictive posterior distribution. We illustrate the efficiency of our method using the Gompertz function to model data on chicken growth, the modeling being improved by the SDE approach.

A whole genome Bayesian scan for adaptive genetic divergence in West African cattle.

BACKGROUND: The recent settlement of cattle in West Africa after several waves of migration from remote centres of domestication has imposed dramatic changes in their environmental conditions, in particular through exposure to new pathogens. West African cattle populations thus represent an appealing model to unravel the genome response to adaptation to tropical conditions. The purpose of this study was to identify footprints of adaptive selection at the whole genome level in a newly collected data set comprising 36,320 SNPs genotyped in 9 West African cattle populations. RESULTS: After a detailed analysis of population structure, we performed a scan for SNP differentiation via a previously proposed Bayesian procedure including extensions to improve the detection of loci under selection. Based on these results we identified 53 genomic regions and 42 strong candidate genes. Their physiological functions were mainly related to immune response (MHC region which was found under strong balancing selection, CD79A, CXCR4, DLK1, RFX3, SEMA4A, TICAM1 and TRIM21), nervous system (NEUROD6, OLFM2, MAGI1, SEMA4A and HTR4) and skin and hair properties (EDNRB, TRSP1 and KRTAP8-1). CONCLUSION: The main possible underlying selective pressures may be related to climatic conditions but also to the host response to pathogens such as Trypanosoma(sp). Overall, these results might open the way towards the identification of important variants involved in adaptation to tropical conditions and in particular to resistance to tropical infectious diseases.

REML estimation of variance parameters in nonlinear mixed effects models using the SAEM algorithm.

Nonlinear mixed effects models are now widely used in biometrical studies, especially in pharmacokinetic research or for the analysis of growth traits for agricultural and laboratory species. Most of these studies, however, are often based on ML estimation procedures, which are known to be biased downwards. A few REML extensions have been proposed, but only for approximated methods. The aim of this paper is to present a REML implementation for nonlinear mixed effects models within an exact estimation scheme, based on an integration of the fixed effects and a stochastic estimation procedure. This method was implemented via a stochastic EM, namely the SAEM algorithm. The simulation study showed that the proposed REML estimation procedure considerably reduced the bias observed with the ML estimation, as well as the residual mean squared error of the variance parameter estimations, especially in the unbalanced cases. ML and REML based estimators of fixed effects were also compared via simulation. Although the two kinds of estimates were very close in terms of bias and mean square error, predictions of individual profiles were clearly improved when using REML vs. ML. An application of this estimation procedure is presented for the modelling of growth in lines of chicken.

Modelling and estimating heterogeneous variances in threshold models for ordinal discrete data via Winbugs/Openbugs.

Analysis of discrete repeated outcomes is an important issue in biomedical studies. The aim of this paper is to propose a flexible and parsimonious model to account for heterogeneous variances for discrete outcomes. The proposed method is based on the use of a linear mixed model on the log of the standard deviation parameters. It is also shown how parameter estimation in this model can be performed with an exact procedure based on a Gibbs sampling algorithm implemented with the Winbugs/Openbugs software. A model comparison study is presented to illustrate the efficiency of this procedure using a well known example from the clinical trial literature. It was found that the proposed methodology considerably improved the predictive ability of the model while remaining very parsimonious. In particular, it was found that adding a random subject effect in the variance model significantly improved the posterior predictive p-value criterion of the model.

A Bayesian outlier criterion to detect SNPs under selection in large data sets.

BACKGROUND: The recent advent of high-throughput SNP genotyping technologies has opened new avenues of research for population genetics. In particular, a growing interest in the identification of footprints of selection, based on genome scans for adaptive differentiation, has emerged. METHODOLOGY/PRINCIPAL FINDINGS: The purpose of this study is to develop an efficient model-based approach to perform bayesian exploratory analyses for adaptive differentiation in very large SNP data sets. The basic idea is to start with a very simple model for neutral loci that is easy to implement under a bayesian framework and to identify selected loci as outliers via Posterior Predictive P-values (PPP-values). Applications of this strategy are considered using two different statistical models. The first one was initially interpreted in the context of populations evolving respectively under pure genetic drift from a common ancestral population while the second one relies on populations under migration-drift equilibrium. Robustness and power of the two resulting bayesian model-based approaches to detect SNP under selection are further evaluated through extensive simulations. An application to a cattle data set is also provided. CONCLUSIONS/SIGNIFICANCE: The procedure described turns out to be much faster than former bayesian approaches and also reasonably efficient especially to detect loci under positive selection.

The genome response to artificial selection: a case study in dairy cattle.

Dairy cattle breeds have been subjected over the last fifty years to intense artificial selection towards improvement of milk production traits. In this study, we performed a whole genome scan for differentiation using 42,486 SNPs in the three major French dairy cattle breeds (Holstein, Normande and Montbéliarde) to identify the main physiological pathways and regions which were affected by this selection. After analyzing the population structure, we estimated F(ST) within and across the three breeds for each SNP under a pure drift model. We further considered two different strategies to evaluate the effect of selection at the genome level. First, smoothing F(ST) values over each chromosome with a local variable bandwidth kernel estimator allowed identifying 13 highly significant regions subjected to strong and/or recent positive selection. Some of them contained genes within which causal variants with strong effect on milk production traits (GHR) or coloration (MC1R) have already been reported. To go further in the interpretation of the observed signatures of selection we subsequently concentrated on the annotation of differentiated genes defined according to the F(ST) value of SNPs localized close or within them. To that end we performed a comprehensive network analysis which suggested a central role of somatotropic and gonadotropic axes in the response to selection. Altogether, these observations shed light on the antagonism, at the genome level, between milk production and reproduction traits in highly producing dairy cows.

A structural mixed model for variances in differential gene expression studies.

The importance of variance modelling is now widely known for the analysis of microarray data. In particular the power and accuracy of statistical tests for differential gene expressions are highly dependent on variance modelling. The aim of this paper is to use a structural model on the variances, which includes a condition effect and a random gene effect, and to propose a simple estimation procedure for these parameters by working on the empirical variances. The proposed variance model was compared with various methods on both real and simulated data. It proved to be more powerful than the gene-by-gene analysis and more robust to the number of false positives than the homogeneous variance model. It performed well compared with recently proposed approaches such as SAM and VarMixt even for a small number of replicates, and performed similarly to Limma. The main advantage of the structural model is that, thanks to the use of a linear mixed model on the logarithm of the variances, various factors of variation can easily be incorporated in the model, which is not the case for previously proposed empirical Bayes methods. It is also very fast to compute and is adapted to the comparison of more than two conditions.

Analysis of a simulated microarray dataset: comparison of methods for data normalisation and detection of differential expression (open access publication).

Microarrays allow researchers to measure the expression of thousands of genes in a single experiment. Before statistical comparisons can be made, the data must be assessed for quality and normalisation procedures must be applied, of which many have been proposed. Methods of comparing the normalised data are also abundant, and no clear consensus has yet been reached. The purpose of this paper was to compare those methods used by the EADGENE network on a very noisy simulated data set. With the a priori knowledge of which genes are differentially expressed, it is possible to compare the success of each approach quantitatively. Use of an intensity-dependent normalisation procedure was common, as was correction for multiple testing. Most variety in performance resulted from differing approaches to data quality and the use of different statistical tests. Very few of the methods used any kind of background correction. A number of approaches achieved a success rate of 95% or above, with relatively small numbers of false positives and negatives. Applying stringent spot selection criteria and elimination of data did not improve the false positive rate and greatly increased the false negative rate. However, most approaches performed well, and it is encouraging that widely available techniques can achieve such good results on a very noisy data set.

Analysis of the real EADGENE data set: comparison of methods and guidelines for data normalisation and selection of differentially expressed genes (open access publication).

A large variety of methods has been proposed in the literature for microarray data analysis. The aim of this paper was to present techniques used by the EADGENE (European Animal Disease Genomics Network of Excellence) WP1.4 participants for data quality control, normalisation and statistical methods for the detection of differentially expressed genes in order to provide some more general data analysis guidelines. All the workshop participants were given a real data set obtained in an EADGENE funded microarray study looking at the gene expression changes following artificial infection with two different mastitis causing bacteria: Escherichia coli and Staphylococcus aureus. It was reassuring to see that most of the teams found the same main biological results. In fact, most of the differentially expressed genes were found for infection by E. coli between uninfected and 24 h challenged udder quarters. Very little transcriptional variation was observed for the bacteria S. aureus. Lists of differentially expressed genes found by the different research teams were, however, quite dependent on the method used, especially concerning the data quality control step. These analyses also emphasised a biological problem of cross-talk between infected and uninfected quarters which will have to be dealt with for further microarray studies.

Genetic analysis of growth curves using the SAEM algorithm.

The analysis of nonlinear function-valued characters is very important in genetic studies, especially for growth traits of agricultural and laboratory species. Inference in nonlinear mixed effects models is, however, quite complex and is usually based on likelihood approximations or Bayesian methods. The aim of this paper was to present an efficient stochastic EM procedure, namely the SAEM algorithm, which is much faster to converge than the classical Monte Carlo EM algorithm and Bayesian estimation procedures, does not require specification of prior distributions and is quite robust to the choice of starting values. The key idea is to recycle the simulated values from one iteration to the next in the EM algorithm, which considerably accelerates the convergence. A simulation study is presented which confirms the advantages of this estimation procedure in the case of a genetic analysis. The SAEM algorithm was applied to real data sets on growth measurements in beef cattle and in chickens. The proposed estimation procedure, as the classical Monte Carlo EM algorithm, provides significance tests on the parameters and likelihood based model comparison criteria to compare the nonlinear models with other longitudinal methods.

Modelling the growth curve of Maine-Anjou beef cattle using heteroskedastic random coefficients models.

A heteroskedastic random coefficients model was described for analyzing weight performances between the 100th and the 650th days of age of Maine-Anjou beef cattle. This model contained both fixed effects, random linear regression and heterogeneous variance components. The objective of this study was to analyze the difference of growth curves between animals born as twin and single bull calves. The method was based on log-linear models for residual and individual variances expressed as functions of explanatory variables. An expectation-maximization (EM) algorithm was proposed for calculating restricted maximum likelihood (REML) estimates of the residual and individual components of variances and covariances. Likelihood ratio tests were used to assess hypotheses about parameters of this model. Growth of Maine-Anjou cattle was described by a third order regression on age for a mean growth curve, two correlated random effects for the individual variability and independent errors. Three sources of heterogeneity of residual variances were detected. The difference of weight performance between bulls born as single and twin bull calves was estimated to be equal to about 15 kg for the growth period considered.