On the power function of the exact test for ther×c contingency table

Summary LetN=[n _ij ] (i=1, …,r;j=1, …,c) be the matrix of observed frequencies in anr×c contingency table fromr possibly different multinomial populations with respective probabilitiesp _i =(p _i1, …,p _ic ).Freeman andHalton have proposed an exact conditional test for the hypothesisH ₀ :p _i =(p ₁, …p _c ) of the exact test is derived. Numerical values forβ(p) were previously computed for the special case:r=3,c=2 [Bennett andNakamura, 1964].     

Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way Contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared, It appears that gamma priors yield more tractable results than lognormal priors.     

The causal analysis of contingency tables: A return to first principles

Given the undoubtedly major advances in the analysis of contingency tables which have been achieved over the past ten years or so (see for example, Fienberg, 1980; Upton, 1978; Everitt, 1977; Haberman, 1978, 1979), it might seem rather unnecessary to want to return to first princples again. However, the need arises precisely because of these advances; for progress in the specifically causal analysis of contingency tables has not matched the other advances at all. Whilst Fienberg devoted a chapter to causal analysis, he made it clear that he views “the assignment of numerical values [to the arrows in a path diagram] as problematic, and [he] would limit [the analysis] to an indication of sign for causal relationships, in a fashion similar to that described by Blalock (1964)” (Fienberg, 1980, pp. 91–92). Considering how far quantitative-variable causal analysis has developed since Blalock (1964), it becomes clear that the causal analysis of qualitative data is still at a rather primitive stage. Indeed, Haberman (1978, 1979), in his two-volume survey of the analysis of qualitative data, does not mention it at all. The problem, I believe, is that log-linear and logit methodology are not particularly suited to the logic of causality in contingency tables. In order to derive a suitable method, it is necessary to uncover the logic underlying causality when applied to qualitative variables. A few others have taken seriously the idea that a direct analysis of the form of a contingency table can lead to fruitful results (see, especially, for example, Boudon, 1967), but their work has been overshadowed by the statistically more profound advances made in log-linear methods. This article is an attempt to provide a statistically rigorous analysis based on the direct interpretation of causality embodied in a contingency table.     

A generalization of Cochran's procedure for the combining of r×c contingency tables*

Abstract Cochran [3] derives a test of association when k 2 × 2 contingency tables are combined. We show in this paper how to extend Cochran's test to the combining of k r×c contingency tables using a multiple comparison technique similar to the one presented by Dunn [4]. An example is included.     

Extensions of linear diagonal-parameter symmetry and quasi-symmetry models for cumulative probabilities in square contingency tables

For square contingency tables with ordered categories, CAUSSINUS [Annales de la Faculté des Sciences de l'Université de Toulouse (1965) Vol. 29, pp. 77–182] and AGRESTI [Statistics and Probability Letters (1983) Vol. 1, pp. 313–316] considered the quasi-symmetry and the linear diagonal-parameter symmetry models, respectively, which have multiplicative forms for cell probabilities. This paper proposes two kinds of models that have the similar multiplicative forms for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (> i ) or above. The endometrial cancer data are analyzed using these models.     

On X2- tests for contingency tables of negative multinomial types

Om te toets of die twee kenmerke waarvolgens die frekwensies in 'n gebeurlikseidstabel opgestel is onafhanklik is, word gewoonlik 'n X² toets gebruik. In die opstelling van so 'n toets word die totale frekwensie as vas beskou. In hierdie artikel word aangetoon hoe 'n X²-toets nog gebruik kan word wanneer die steekproefneming voortgesit word totdat 'n vooraf vasgestelde frekwensie in een selbereik word. Die totale frekwensie is dan 'n stogastiese veranderlike. Die eienskappe van die randverdelings van 'n negatiewe multinomiaalverdeling word gebruik om die parameters wat in die X²-verdeling optree te skat.     

Unconditional tests for association in 2 × 2 contingency tables in the total sum fixed design

The asymptotic approach and Fisher's exact approach have often been used for testing the association between two dichotomous variables. The asymptotic approach may be appropriate to use in large samples but is often criticized for being associated with unacceptable high actual type I error rates for small to medium sample sizes. Fisher's exact approach suffers from conservative type I error rates and low power. For these reasons, a number of exact unconditional approaches have been proposed, which have been seen to be generally more powerful than exact conditional counterparts. We consider the traditional unconditional approach based on maximization and compare it to our presented approach, which is based on estimation and maximization. We extend the unconditional approach based on estimation and maximization to designs with the total sum fixed. The procedures based on the Pearson chi‐square, Yates's corrected, and likelihood ratio test statistics are evaluated with regard to actual type I error rates and powers. A real example is used to illustrate the various testing procedures. The unconditional approach based on estimation and maximization performs well, having an actual level much closer to the nominal level. The Pearson chi‐square and likelihood ratio test statistics work well with this efficient unconditional approach. This approach is generally more powerful than the other p‐value calculation methods in the scenarios considered.     

The power of the Durbin-Watson test for regressions without an intercept

In the linear regression model without an intercept, the limiting power of the Durbin–Watson test (as correlation among errors increases) is shown to take only one of two values. This is either one or zero, depending on the underlying regressor matrix. Some examples and a simple rule to decide from a given regressor matrix which of these cases applies are also given.     

Increasing sample sizes do not always increase the power of UMPU-tests for 2×2 tables

We consider the uniformly most powerful unbiased (UMPU) one-sided test for the comparison of two proportions based on sample sizes m and n, i.e., the randomized version of Fisher's exact one-sided test. It will be shown that the power function of the one-sided UMPU-test based on sample sizes m and n can coincide with the power function of the UMPU-test based on sample sizes m+1 and n for certain levels on the entire parameter space. A characterization of all such cases with identical power functions is derived. Finally, this characterization is closely related to number theoretical problems concerning Fermat-like binomial equations. Some consequences for Fisher's original exact test will be discussed, too.     

Comparison of power for the D'Agostino and the Wilk-Shapiro test of normality for small and moderate samples

Summary Results are given of a sampling study of the sensitivities of two statistical procedures viz. the D'A gostino and the W ilk -S hapiro test of normality. Four alternative distributions were considered. For 5 small sample sizes the test of D'A gostino seems to be clearly dominated by the W ilk -S hapiro test for each of the 4 alternatives. For sample size n = 100 the recently developed approximate W ilk -S hapiro test seems to dominate the D'A gostino test for the U -shaped distribution; however, little can be said about the difference in power of these two tests for the other 3 alternatives. Random samples generated from the standard normal distribution showed the degree of approximation of the percentiles of each test procedure to be good.     

A linear approximation to the power function of a test

In this paper we obtain a linear approximation to the power function of a test that is very accurate for small sample sizes. This is especially useful for robust tests where not many power functions are available. The approximation is based on the von Mises expansion of the tail probability functional and on the Tail Area Influence Function (TAIF). The goals of the paper are, first to extend the definition of the TAIF to the case of non identically distributed random variables, defining the Partial Tail Area Influence Functions and the Vectorial Tail Area Influence Function; second, to obtain exact expressions for computing these new influence functions; and, finally, to find accurate approximations to the power function, that can be used in the case of non identically distributed random variables. We include some examples of the application of this linear approximation to tests that involve the Huber statistic and also saddlepoint tests, so proving that the approximations apply not only to simple problems but also to complex ones.     

Models for the probability of concordance in cross-classification tables

For cross-classification tables having an ordinal response variable, logit and probit models are formulated for the probability that a pair of subjects is concordant. For multidimensional tables, generalized models are given for the probability that the response at one setting of explanatory variables exceeds the response at another setting. Related measures of association are discussed for two-way tables.     

On the confidence region for the one-sample 156-1156-1156-1

Summary This article shows how the usual confidence region for the parameters of the multinomial distribution can be covered and closely approximated by an easily constructed polyhedron in the space of probability vectors.

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Zusammenfassung In dieser Arbeit wird gezeigt, wie der übliche Konfidenzbereich für die Parameter der Multinomialverteilung durch ein im Raum der Wahrscheinlichkeitsvektoren enthaltene leicht konstruierbare Polyeder überdeckt und gut approximiert werden kann.

     

A note on the extension of tables for the Friedman statistic

Abstract  The tables for the Friedman rank-test arc extended using a sampling method. A table is given for the traditional significance levels .01, .025, .05 and .10 for number of blocks equal to 7 and 8 and for treatments of size 6 and 7. A brief discussion of available tables is also included along with an example of an application.