The weighted compounding of probabilities from three independent hypothesis tests

Tail probabilities from three independent hypothesis tests can be combined to form a test statistic of the form P₁,P₂ ^θ2,P₃ ^θ3.The null distribution of the combined test statistic is presented and critical values for α=0.01 and 0.05 are provided.The power of this test is discussed for the special case ofthree independent F-tests.     

Hierarchical Bayes estimation of spatial statistics for rates

The U.S. Bureau of Labour Statistics publishes monthly unemployment rate estimates for its 50 states, the District of Columbia, and all counties, under Current Population Survey. However, the unemployment rate estimates for some states are unreliable due to low sample sizes in these states. Datta et al. (1999) proposed a hierarchical Bayes (HB) method using a time series generalization of a widely used cross-sectional model in small-area estimation. However, the geographical variation is also likely to be important. To have an efficient model, a comprehensive mixed normal model that accounts for the spatial and temporal effects is considered. A HB approach using Markov chain Monte Carlo is used for the analysis of the U.S. state-level unemployment rate estimates for January 2004-December 2007. The sensitivity of such type of analysis to prior assumptions in the Gaussian context is also studied.     

Flexible empirical Bayes estimation for wavelets

Wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression. Although Bayes estimators can provide excellent mean-squared error (MSE) properties, the selection of an effective prior is a difficult task. To address this problem, we propose empirical Bayes (EB) prior selection methods for various error distributions including the normal and the heavier-tailed Student t -distributions. Under such EB prior distributions, we obtain threshold shrinkage estimators based on model selection, and multiple-shrinkage estimators based on model averaging. These EB estimators are seen to be computationally competitive with standard classical thresholding methods, and to be robust to outliers in both the data and wavelet domains. Simulated and real examples are used to illustrate the flexibility and improved MSE performance of these methods in a wide variety of settings.     

Properties of Bayes Factors Based on Test Statistics

Abstract.  This article examines the consistency, interpretation and application of Bayes factors constructed from standard test statistics. Primary conclusions are that Bayes factors based on multinomial and normal test statistics are consistent for suitable choices of the hyperparameters used to specify alternative hypotheses, and that such constructions can be extended to obtain consistent Bayes factors based on likelihood ratio statistics. A connection between Bayes factors based on likelihood ratio statistics and the Bayesian information criterion is exposed, as is a connection between Bayes factors based on F statistics and parametric Bayes factors based on normal-inverse gamma models. Similarly, Bayes factors based on chi-squared statistics for multinomial data are shown to provide accurate approximations to Bayes factors based on multinomial/Dirichlet models. An illustration of how the simple form of these Bayes factors can be exploited to generate easily interpretable summaries of the experimental 'weight of evidence' is provided.     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

Evidence From Marginally Significant t Statistics

ABSTRACT

This article examines the evidence contained in t statistics that are marginally significant in 5% tests. The bases for evaluating evidence are likelihood ratios and integrated likelihood ratios, computed under a variety of assumptions regarding the alternative hypotheses in null hypothesis significance tests. Likelihood ratios and integrated likelihood ratios provide a useful measure of the evidence in favor of competing hypotheses because they can be interpreted as representing the ratio of the probabilities that each hypothesis assigns to observed data. When they are either very large or very small, they suggest that one hypothesis is much better than the other in predicting observed data. If they are close to 1.0, then both hypotheses provide approximately equally valid explanations for observed data. I find that p-values that are close to 0.05 (i.e., that are “marginally significant”) correspond to integrated likelihood ratios that are bounded by approximately 7 in two-sided tests, and by approximately 4 in one-sided tests.

The modest magnitude of integrated likelihood ratios corresponding to p-values close to 0.05 clearly suggests that higher standards of evidence are needed to support claims of novel discoveries and new effects.     

Bayesian model selection using test statistics

Summary.  Existing Bayesian model selection procedures require the specification of prior distributions on the parameters appearing in every model in the selection set. In practice, this requirement limits the application of Bayesian model selection methodology. To overcome this limitation, we propose a new approach towards Bayesian model selection that uses classical test statistics to compute Bayes factors between possible models. In several test cases, our approach produces results that are similar to previously proposed Bayesian model selection and model averaging techniques in which prior distributions were carefully chosen. In addition to eliminating the requirement to specify complicated prior distributions, this method offers important computational and algorithmic advantages over existing simulation-based methods. Because it is easy to evaluate the operating characteristics of this procedure for a given sample size and specified number of covariates, our method facilitates the selection of hyperparameter values through prior-predictive simulation.     

Bayes factor consistency for unbalanced ANOVA models

In practical situations, most experimental designs often yield unbalanced data which have different numbers of observations per unit because of cost constraints, missing data, etc. In this paper, we consider the Bayesian approach to hypothesis testing or model selection under the one-way unbalanced fixed-effects analysis-of-variance (ANOVA) model. We adopt Zellner's g-prior with the beta-prime distribution for g, which results in an explicit closed-form expression of the Bayes factor without integral representation. Furthermore, we investigate the model selection consistency of the Bayes factor under three different asymptotic scenarios: either the number of units goes to infinity, the number of observations per unit goes to infinity, or both go to infinity. The results presented extend some existing ones of the Bayes factor for the balanced ANOVA models in the literature.     

Bayes model averaging with selection of regressors

Summary. When a number of distinct models contend for use in prediction, the choice of a single model can offer rather unstable predictions. In regression, stochastic search variable selection with Bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors. Here we look at Bayes model averaging incorporating variable selection for prediction. This offers similar mean-square errors of prediction but with a vastly reduced predictor space. This can greatly aid the interpretation of the model. It also reduces the cost if measured variables have costs. The development here uses decision theory in the context of the multivariate general linear model. In passing, this reduced predictor space Bayes model averaging is contrasted with single-model approximations. A fast algorithm for updating regressions in the Markov chain Monte Carlo searches for posterior inference is developed, allowing many more variables than observations to be contemplated. We discuss the merits of absolute rather than proportionate shrinkage in regression, especially when there are more variables than observations. The methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution.     

Posterior Predictive Comparisons for the Two-sample Problem

The two-sample problem of inferring whether two random samples have equal underlying distributions is formulated within the Bayesian framework as a comparison of two posterior predictive inferences rather than as a problem of model selection. The suggested approach is argued to be particularly advantageous in problems where the objective is to evaluate evidence in support of equality, along with being robust to the priors used and being capable of handling improper priors. Our approach is contrasted with the Bayes factor in a normal setting and finally, an additional example is considered where the observed samples are realizations of Markov chains.     

Robust durbin's and portmanteau goodness-of-fit test statistics and their limit distributions

Robust analogue of Durbin's(1970)statistic is derived and its limiting distribution is obtained under both null and alternative hypotheses. Also, robust version of the portmanteau goodness-of-fit test statistic for AR(p)model is given and the asymptotic distribution is derived.     

Bayesian goodness-of-fit test for censored data

We propose a Bayesian computation and inference method for the Pearson-type chi-squared goodness-of-fit test with right-censored survival data. Our test statistic is derived from the classical Pearson chi-squared test using the differences between the observed and expected counts in the partitioned bins. In the Bayesian paradigm, we generate posterior samples of the model parameter using the Markov chain Monte Carlo procedure. By replacing the maximum likelihood estimator in the quadratic form with a random observation from the posterior distribution of the model parameter, we can easily construct a chi-squared test statistic. The degrees of freedom of the test equal the number of bins and thus is independent of the dimensionality of the underlying parameter vector. The test statistic recovers the conventional Pearson-type chi-squared structure. Moreover, the proposed algorithm circumvents the burden of evaluating the Fisher information matrix, its inverse and the rank of the variance–covariance matrix. We examine the proposed model diagnostic method using simulation studies and illustrate it with a real data set from a prostate cancer study.     

Objective Bayes Covariate‐Adjusted Sparse Graphical Model Selection

We present an objective Bayes method for covariance selection in Gaussian multivariate regression models having a sparse regression and covariance structure, the latter being Markov with respect to a directed acyclic graph (DAG). Our procedure can be easily complemented with a variable selection step, so that variable and graphical model selection can be performed jointly. In this way, we offer a solution to a problem of growing importance especially in the area of genetical genomics (eQTL analysis). The input of our method is a single default prior, essentially involving no subjective elicitation, while its output is a closed form marginal likelihood for every covariate‐adjusted DAG model, which is constant over each class of Markov equivalent DAGs; our procedure thus naturally encompasses covariate‐adjusted decomposable graphical models. In realistic experimental studies, our method is highly competitive, especially when the number of responses is large relative to the sample size.     

Reporting Bayes factors or probabilities to decision makers of unknown loss functions

When competing interests seek to influence a decision maker, a scientist must report a posterior probability or a Bayes factor among those consistent with the evidence. The disinterested scientist seeks to report the value that is least controversial in the sense that it is best protected from being discredited by one of the competing interests. If the loss function of the decision maker is not known but can be assumed to satisfy two invariance conditions, then the least controversial value is a weighted generalized mean of the upper and lower bounds of the interval.     

A discussion of the question: for what use are tests of hypotheses and tests of significance

The author believes that tests provide a poor model of most real problems, usually so poor that their objectivity is tangential and often too poor to be useful.     

Lower bounds on bayes factors for a linear regression model

This paper develops a Bayesian analysis of testing linear parameter restictions in a normal regression model. Instead of searching for a suitable "objective" prior for such a test, several interesting classes of informative priors are postulated, and the lower bounds on the Bayes factors are derived.     

Probability model choice in single samples from exponential families using Poisson log-linear modelling,and model comparison using Bayes and posterior Bayes factors

This paper describes a method due to Lindsey (1974a) for fitting different exponential family distributions for a single population to the same data, using Poisson log-linear modelling of the density or mass function. The method is extended to Efron's (1986) double exponential family, giving exact ML estimation of the two parameters not easily achievable directly. The problem of comparing the fit of the non-nested models is addressed by both Bayes and posterior Bayes factors (Aitkin, 1991). The latter allow direct comparisons of deviances from the fitted distributions.     

A Multiple Comparison Procedure Based on a Variant of the Schwarz Information Criterion in a Mixed Model

Repeated measurements are collected in a variety of situations and are generally characterized by a mixed model where the correlation within the subject is specified by the random effects. In such a mixed model, we propose a multiple comparison procedure based on a variant of the Schwarz information criterion (SIC; Schwarz, 1978 Schwarz , G. ( 1978 ). Estimating the dimension of a model . Ann. Statist. 6 : 461 – 464 .[Crossref], [Web of Science ®] , [Google Scholar]). The derivation of SIC indicates that SIC serves as an asymptotic approximation to a transformation of the Bayesian posterior probability of a candidate model. Therefore, an approximated posterior probability for a candidate model can be calculated based upon SIC. We suggest a variant of SIC which includes the terms which are asymptotically negligible in the derivation of SIC. The variant improves upon the performance of SIC in small and moderate sample-size applications. Based upon the proposed variant, the corresponding posterior probability can be calculated for each candidate model. A hypothesis testing for multiple comparisons involves one or more models in the candidate class, the posterior probability of the hypothesis testing is therefore evaluated as the sum of the posterior probabilities for the models associated with the testing. The approximated posterior probability based on the variant accommodates the effect of the prior on each model in the candidate class, and therefore is more effectively approximated than that based on SIC for conducting multiple comparisons. We derive the computational formula of the approximated posterior probability based on the variant in the mixed model. The applications in two real data sets demonstrate that the proposed procedure based on the SIC variant can perform effectively in multiple comparisons.     

Bayesian Nonlinear Regression for the Air Pollution Effects on Daily Clinic Visits in Small Areas of Taiwan

We consider the complete clinic visit records and environmental monitoring data at 50 townships and city districts where ambient air monitoring stations of Taiwan Air Quality Monitoring Stations are located. A Bayesian analysis is carried out using regression spline model on principal components obtained from several pollutant covariables. The appropriate model is selected using Bayesian model averaging. A brief account of our results is provided for the elderly patients group.