The use of statistics to examine the association between fluoride in drinking water and cancer death rates

"The basis of statistical tests of significance of association between fluoride level in drinking water and cancer death rates is discussed. Reference is made to two reported studies in each of which cancer death rates of a number of [U.S.] cities were used. It is argued that between city variation should be taken into account when performing tests of significance. In one of the two studies this was done informally; in the other between city variation was ignored."     

An emplrical bayes approach in estimating the parameters of the or more geometric distributions

The estimation of the parameters of two or more geometric distribuionsis considered by usinq an empirical Bayesian approach. Robbins (1983) gave empirical Bayes estimates if the number of distributions N is large, buthere we consider the cascwhore N is small. The parameters of the prior distribution areest imated by looking at maximum like lihood and momentest imation methods.     

ALIGNING OF ESTIMATES: AN ALTERNATIVE TO EMPIRICAL BAYES METHODS

The data that are used in constructing empirical Bayes estimates can properly be regarded as arising in a two-stage sampling scheme. In this setting it is possible to modify the conventional parameter estimates so that a reduction in expected squared error is effected. In the empirical Bayes approach this is done through the use of Bayes's theorem. The alternative approach proposed in this paper specifies a class of modified estimates and then seeks to identify that member of the class which yields the minimum squared error. One advantage of this approach relative to the empirical Bayes approach is that certain problems involving multiple parameters are easily overcome. Further, it permits the use of relatively efficient methods of non-parametric estimation, such as those based on quantiles or ranks; this has not been achieved by empirical Bayes methods.     

AN ESTIMATE OF THE ASYMPTOTIC STANDARD ERROR OF THE SAMPLE MEDIAN

A method for estimating the asymptotic standard error of the sample median based on generalized least squares is outlined. The practical problems of implementing this new estimate along with those associated with two existing estimates are discussed. Finally a simulation study is presented to compare the three estimates.     

Empirical bayes estimation of the log odds ratio in 2×2 contingency tables

We consider a sequence of contingency tables whose cell probabilities may vary randomly. The distribution of cell probabilities is modelled by a Dirichlet distribution. Bayes and empirical Bayes estimates of the log odds ratio are obtained. Emphasis is placed on estimating the risks associated with the Bayes, empirical Bayes and maximum lilkelihood estimates of the log odds ratio.     

Distribution-Free Methods in Regression1

A modification to the least-squares estimating equations in simple linear regression, leading to exact distribution-free inference about slope, is outlined. The modification involves a general residual-transformation iψ and the cases of ψ— sign, rank, and the identity are discussed in some detail. It is also shown how exact inference about individual parameters in regressions with more than one design variable can be carried out by a restricted permutation argument coupled with a suitable design.     

A PERMUTATION PAIRED TEST ALLOWING FOR MISSING VALUES

The paper presents a paired test of the equality of two means, allowing for incomplete pairs. It is a permutation test depending on minimal distributional assumptions. The test may be performed on original data or after transformation of the data. Transformation to ranks is illustrated.     

A Simple Test For The Independence of Sets of Variables

A test for the mutual independence of subvectors of the p-dimensional random vector X , distributed as N( 0, S? ), is described. The test is based on the maximum likelihood estimates (MLEs) of the off-(block) diagonal elements of S?. It is shown that the resulting test statistic is much easier to compute than the likelihood ratio (LR) test statistic while retaining the same asymptotic power properties in view of the general properties of tests based on the MLEs (ML test) and the likelihood ratio (LR test).