A bayesian analysis of the multinomial model for a dichotomous response with nonrespondents

This paper studies the mu1tinomial model 2x2 contingency table data with some cell counts missing .Various hypotheses of interest including row-column independence are tested by using Bayes factors which represent the ratio of the posterior odds to the prior odds for the null hypothesis. The Dirichlet-Beta family of prior distributions is considered for the multinomial parameters cond itional on the complement of the null hypothesis. The Bayes factor for the incomplete data is a mixture of the Bayes factors for different possibilities for the full data.     

Objective Bayesian multiple comparisons for normal variances

This paper considers the multiple comparisons problem for normal variances. We propose a solution based on a Bayesian model selection procedure to this problem in which no subjective input is considered. We construct the intrinsic and fractional priors for which the Bayes factors and model selection probabilities are well defined. The posterior probability of each model is used as a model selection tool. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and some frequentist tests.     

INTRINSIC PRIORS FOR TWO-SAMPLE TESTS IN NORMAL POPULATIONS

ABSTRACT

There have been considerable amounts of work regarding the development of various default Bayes factors in model selection and hypothesis testing. Two commonly used criteria, the intrinsic Bayes factor and the fractional Bayes factor are compared to test two independent normal means and variances. We also derive several intrinsic priors whose Bayes factors are asymptotically equivalent to the respective Bayes factors. We demonstrate our results in simulated datasets.     

A Comparison of Bayes Factors for Separated Models: Some Simulation Results

Some alternative Bayes Factors: Intrinsic, Posterior, and Fractional have been proposed to overcome the difficulties presented when prior information is weak and improper prior are used. Additional difficulties also appear when the models are separated or non nested. This article presents both simulation results and some illustrative examples analysis comparing these alternative Bayes factors to discriminate among the Lognormal, the Weibull, the Gamma, and the Exponential distributions. Simulation results are obtained for different sample sizes generated from the distributions. Results from simulations indicates that these alternative Bayes factors are useful for comparing non nested models. The simulations also show some similar behavior and that when both models are true they choose the simplest model. Some illustrative example are also presented.     

A bayesian approach for testing periodicity of dichotomous observations

A Bayesian approach is utilized to test for periodicity in a dichotomous time series. Dichotomous data arise in a variety of circumstances when a variable takes on only two possible values. Conjugate and noninformative priors are considered as well as a hierarchical Bayes approach; the latter is considered the superior Bayes methodology. The situation of stochastic period lengths is also discussed. The generalization to the multinomial model is investigated to allow for the case that a variable takes on more than two possible values. In all cases decisions are made based on a Bayes factor. The proposed procedures are demonstrated on earthquake data in the central Virginia seismic zone     

Bayesian change-point problem using Bayes factor with hierarchical prior distribution

We consider the hierarchical Bayesian models of change-point problem in a sequence of random variables having either normal population or skew-normal population. Further, we consider the problem of detecting an influential point concerning change point using Bayes factors. Our proposed models are illustrated with the real data example, the annual flow volume data of Nile River at Aswan from 1871 to 1970. The result using our proposed models indicated the largest influential observation in the year 1888 among outliers. We have shown that it is useful to measure the influence of observations on Bayes factors. Here, we consider omitting single observation as well.     

On the Use of Bayes Factor in Frequentist Testing of a Precise Hypothesis

This article deals with Bayes factors as useful Bayesian tools in frequentist testing of a precise hypothesis. A result and several examples are included to justify the definition of Bayes factor for point null hypotheses, without merging the initial distribution with a degenerate distribution on the null hypothesis. Of special interest is the problem of testing a proportion (joint with a natural criterion to compare different tests), the possible presence of nuisance parameters, or the influence of Bayesian sufficiency on this problem. The problem of testing a precise hypothesis under a Bayesian perspective is also considered and two alternative methods to deal with are given.     

Comparison of testing procedures utilizing P-values and Bayes factors in some common situations

Bayesian alternatives to classical tests for several testing problems are considered. One-sided and two-sided sets of hypotheses are tested concerning an exponential parameter, a Binomial proportion, and a normal mean. Hierarchical Bayes and noninformative Bayes procedures are compared with the appropriate classical procedure, either the uniformly most powerful test or the likelihood ratio test, in the different situations. The hierarchical prior employed is the conjugate prior at the first stage with the mean being the test parameter and a noninformative prior at the second stage for the hyper parameter(s) of the first stage prior. Fair comparisons are attempted in which fair means the likelihood of making a type I error is approximately the same for the different testing procedures; once this condition is satisfied, the power of the different tests are compared, the larger the power, the better the test. This comparison is difficult in the two-sided case due to the unsurprising discrepancy between Bayesian and classical measures of evidence that have been discussed for years. The hierarchical Bayes tests appear to compete well with the typical classical test in the one-sided cases.     

Intrinsic Priors for Testing Two Normal Means with Intrinsic Bayes Factors

ABSTRACT

In this article we consider the problem of comparing two normal means with unknown common variance using a Bayesian approach. Conventional Bayes factors with improper non informative priors are not well defined. The intrinsic Bayes factors are used to overcome such a difficulty. We derive intrinsic priors whose Bayes factors are asymptotically equivalent to the corresponding intrinsic Bayes factors. We illustrate our results with numerical examples.     

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.     

Bayes factor in testing precise hypotheses

Substitution of a mixed prior distribution by a continuous one for the point null hypothesis testing problem is discussed. Conditions are established in order to approximate the Bayes factors for the two problems. Besides, trough this approximation an assignation of priorprobabilities is suggested.     

Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.     

Objective Bayesian hypothesis testing in regression models with first-order autoregressive residuals

This article considers the objective Bayesian testing in the normal regression models with first-order autoregressive residuals. We propose some solutions based on a Bayesian model selection procedure to this problem where no subjective input is considered. We construct the proper priors for testing the autocorrelation coefficient based on measures of divergence between competing models, which is called the divergence-based (DB) priors and then propose the objective Bayesian decision-theoretic rule, which is called the Bayesian reference criterion (BRC). Finally, we derive the intrinsic test statistic for testing the autocorrelation coefficient. The behavior of the Bayes factor-based DB priors is examined by comparing with the BRC in a simulation study and an example.     

Some Faults of the Bayes Factor in Nonparametric Model Selection

We compare different Bayesian strategies for testing a parametric model versus a nonparametric alternative on the ground of their ability to solve the inconsistency problems arising when using the Bayes factor under certain conditions. A preliminary critical discussion of such an inconsistency is provided.     

Empirical bayes estimation of binomial parameter with symmetric priors

This paper deals with the problem of estimating the binomial parameter via the nonparametric empirical Bayes approach. This estimation problem has the feature that estimators which are asymptotically optimal in the usual empirical Bayes sense do not exist (Robbins (1958, 1964)), However, as pointed out by Liang (1934) and Gupta and Liang (1988), it is possible to construct asymptotically optimal empirical Bayes estimators if the unknown prior is symmetric about the point 1/2, In this paper, assuming symmetric priors a monotone empirical Bayes estimator is constructed by using the isotonic regression method. This estimator is asymptotically optimal in the usual empirical Bayes sense. The corresponding rate of convergence is investigated and shown to be of order n^-1, where n is the number of past observations at hand.     

Influence of groups of observations on bayes factors

A diagnostic for finding groups of observations influential on Bayes factors is discussed, which extends ideas in Pettit & Young (1990). Ways of reducing the combinatorial explosion involved in detecting more than one influential observation are considered. The effect of masking is also examined. Finally new graphical displays to identify these observations will be explored.     

The use of local and nonlocal priors in Bayesian test-based monitoring for single-arm phase II clinical trials

Bayesian sequential monitoring is widely used in adaptive phase II studies where the objective is to examine whether an experimental drug is efficacious. Common approaches for Bayesian sequential monitoring are based on posterior or predictive probabilities and Bayesian hypothesis testing procedures using Bayes factors. In the first part of the paper, we briefly show the connections between test-based (TB) and posterior probability-based (PB) sequential monitoring approaches. Next, we extensively investigate the choice of local and nonlocal priors for the TB monitoring procedure. We describe the pros and cons of different priors in terms of operating characteristics. We also develop a user-friendly Shiny application to implement the TB design.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.     

Objective priors for hypothesis testing in one-way random effects models

The Bayes factor is a key tool in hypothesis testing. Nevertheless, the important issue of which priors should be used to develop objective Bayes factors remains open. The authors consider this problem in the context of the one-way random effects model. They use concepts such as orthogonality, predictive matching and invariance to justify a specific form of the priors for common parameters and derive the intrinsic and divergence based prior for the new parameter. The authors show that both intrinsic priors or divergence-based priors produce consistent Bayes factors. They illustrate the methods and compare them with other proposals.