Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means.

Inequality Constrained Analysis of Variance: A Bayesian Approach.

If you have ever wondered when visiting the pharmacy how the dosage of your prescription is determined this book will answer your questions. Dosing information on drug labels is based on discussion between the pharmaceutical manufacturer and the drug regulatory agency, and the label is a summary of results obtained from many scientific experiments. The book introduces the drug development process, the design and the analysis of clinical trials. Many of the discussions are based on applications of statistical methods in the design and analysis of dose response studies. Important procedural steps from a pharmaceutical industry perspective are also examined.

Dose Finding in Drug Development

A Multivariate Analogue of the One-Sided Testについての一注意 (多次元統計解析の数理的研究)

Bayesian model selection of informative hypotheses for repeated measurements

We discuss various statistical distributions of earthquake numbers. Previously we derived several discrete distributions to describe earthquake numbers for the branching model of earthquake occurrence: these distributions are the Poisson, geometric, logarithmic, and the negative binomial (NBD). The theoretical model is the `birth and immigration' population process. The first three distributions above can be considered special cases of the NBD. In particular, a point branching process along the magnitude (or log seismic moment) axis with independent events (immigrants) explains the magnitude/moment-frequency relation and the NBD of earthquake counts in large time/space windows, as well as the dependence of the NBD parameters on the magnitude threshold (magnitude of an earthquake catalogue completeness). We discuss applying these distributions, especially the NBD, to approximate event numbers in earthquake catalogues. There are many different representations of the NBD. Most can be traced either to the Pascal distribution or to the mixture of the Poisson distribution with the gamma law. We discuss advantages and drawbacks of both representations for statistical analysis of earthquake catalogues. We also consider applying the NBD to earthquake forecasts and describe the limits of the application for the given equations. In contrast to the one-parameter Poisson distribution so widely used to describe earthquake occurrence, the NBD has two parameters. The second parameter can be used to characterize clustering or over-dispersion of a process. We determine the parameter values and their uncertainties for several local and global catalogues, and their subdivisions in various time intervals, magnitude thresholds, spatial windows, and tectonic categories. The theoretical model of how the clustering parameter depends on the corner (maximum) magnitude can be used to predict future earthquake number distribution in regions where very large earthquakes have not yet occurred.

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Statistical distributions of earthquake numbers: consequence of branching process

Suppose we have independent random samples from each of k populations specified by scalar-valued, unknown parameters θ 1 ,...,θ k satisfying the simple order restriction θ 1 ≤...≤ θ k . Our concern is to seek distinct parameters among θ 1 θ k based on the data. To find a configuration of distinct parameters among the 0's, one may consider employing Akaike's information criterion (Akaike, 1973). However, the criterion is not appropriate for the order-restricted maximum likelihood estimator of θ = (θ 1 ,..., θ k ), since the normality or the asymptotic normality of the estimator is not valid. In this paper an information criterion is proposed for detecting the configuration of the true parameters with the simple order restriction. This method may also be applied for detecting a changepoint in a sequence of parameters with a monotone trend. A Monte Carlo study indicates that our new criterion is effective, compared to Akaike's information criterion, for detecting the configuration of normal means satisfying the simple order restriction.

An information criterion for parameters under a simple order restriction

For estimating the dispersion parameter of the negative binomial distribution, the conditional maximum likelihood estimator is compared with the maximum likelihood and moment estimators. The biases, mean squared errors and the Kullback-Leibler risks of the estimators are examined by simulation studies for a single population and multiple ones with a common parameter. We observe that the conditional maximum likelihood estimator is superior to the others in frequency for a single population as a whole, and it is more definite for multiple populations.

Estimation for the negative binomial distribution based on the conditional likelihood

The possibility that the conditional maximum likelihood estimator (CMLE) is superior to the unconditional maximum likelihood estimator (UMLE) is discussed in examples where the residual likelihood is obstructive. We observe relatively smaller risks of the CMLE for a finite sample size. The models in the study include the normal, inverse Gauss, gamma, two-parameter exponential, logit, negative binomial and two-parameter geometric ones.

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Possible superiority of the conditional mle over the unconditional mle

A new method for estimating a set of odds ratios under an order restriction based on estimating equations is proposed. The method is applied to those of the conditional maximum likelihood estimators and the Mantel-Haenszel estimators. The estimators derived from the conditional likelihood estimating equations are shown to maximize the conditional likelihoods. It is also seen that the restricted estimators converge almost surely to the respective odds ratios when the respective sample sizes become large regularly. The restricted estimators are compared with the unrestricted maximum likelihood estimators by a Monte Carlo simulation. The simulation studies show that the restricted estimates improve the mean squared errors remarkably, while the Mantel-Haenszel type estimates are competitive with the conditional maximum likelihood estimates, being slightly worse.

Estimation of odds ratios under order restrictions

Testing hypotheses on the marginal probabilities of a two-way contingency table is discussed. Three statistics are considered for testing the hypothesis of specified probabilities in the margins against alternatives with certain kind of order restriction. The properties of these statistics are discussed and their asymptotic behaviors are compared in depth. An appliction which motivated the consideration of the original testing problem is illustrated with a practical data.

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Kazuo Anraku

Papers

An information criterion for parameters under a simple order restriction

Estimation for the negative binomial distribution based on the conditional likelihood

Possible superiority of the conditional mle over the unconditional mle

Estimation of odds ratios under order restrictions

Tests for the marginal probabilities in the two-way contingency table under restricted alternatives