Marginal homogeneity,symmetry, and independence

A contingency table of the mc form provides a convenient summary of data when c individuals in a matched set9 each belonging to a different one of c classifications, are identified as belonging to one of m categories, A study in which matched sets (c=3) of 1 case, 1 hospital control, and 1 neighborhood control are classified into one of m=4 occupational categories would be an example, Independence in the cxm tables for each of the matched sets implies symmetry in the summary mc table with consequent marginal homogeneity. Adaptation of the Mantel-Haenszel procedure for testing independence to the case of many cxm tables so as to yield a chi square with (cl)(ml) degrees of freedom (DF) provides a test of marginal homogeneity in the summary mc table. This can be viewed as a test of symmetry directed against alternatives which would make for marginal inhomogeneity and can differ     

On gart's test for interaction of second order

Gart (1972) ottered a Statistic on testing the hypothesis of no second-order interaction in a 2×s×t contingency table. The statistic was tn be used as an asymptotic chi-square with (s-1) (t-1) degrees of freedom. We show that this statistic suiters from the drawback that unless certain side conditions hold, the statistic would reject the null hypothesis with probability approaching one asymptotically even in the null case. Hence the statistic is not strictly valid for the hypothesis for which it was intended.     

On asymptotic non-normality of null distributions of mrpp statistics

Severe departures from normality occur frequently for null distributions of statistics associated with applications of mulLi-response permutation procedures (MRPP) for either small or large finite populations. This paper describes the commonly encountered situation associated with asymptotic non-normality for null distributions of MRPP statistics which does not depend on the underlying multivariate distribution. In addition, this paper establishes the existence of a non-degenerate underlying distribution for which the null distributions of MRPP statistics are asymptotically non-normal for essentially all size structure configurations. It is known that MRPP statistics are symmetric versions of a broader class of statistics, most of which are asymmetric. Because of the non-normality associated with null distributions of MRPP statistics, this paper includes necessary results for inferences based on the exact first three moments of anv statistic in this broader class (analogous to existing results for MRPP statistics).     

Counterexamples in unbalanced two-way analysis of variance

Recommended methods for analyzing unbalanced two-way data may be classified into two major categories:the parametric interpretation approach and the model comparison approach. Each approach has its advantages and its drawbacks. The main drawback of the parametric interpretation approach is non-orthogonality.For the model comparison approach the main drawback is the dependence of the hypothesis tested on the cell sizes. In this paper we provide examples to illustrate these drawbacks.     

Use of the 21% rulf in experimental design

When the probabilities underlying a contingency table are assumed to be symmetric, it is advantageous to combine the frequencies of equiprobable cells (Haber, 1981, 1982).The present work introduces models, as well as exact and asymptotic methods, for analyzing such ‘intraclass’ contingency tables in several dimensions.     

Identifiability of distributions under competing risks and complementary risks model

Let X₁,X₂,…,X_p be p random variables with cdf's F₁(x),F₂(x),…,F_p(x)respectively. Let U = min(X₁,X₂,…,X_p) and V = max(X₁,X₂,…,X_p).In this paper we study the problem of uniquely determining and estimating the marginal distributions F₁,F₂,…,F_p given the distribution of U or of V.

First the problem of competing and complementary risks are introduced with examples and the corresponding identification problems are considered when the X₁'s are independently distributed and U(V) is identified, as well as the case when U(V) is not identified. The case when the X₁'s are dependent is considered next. Finally the problem of estimation is considered.     

The behavior of estimated measures of association in small and moderate sample sizes for 2×3 tables

We consider the distributions of Goodman and Kruskal's G, Kendall's tau-b, and correlation coefficients rho and rho-s for sample sizes 10‘10’40 from 2×3 tables. The results are compared with asymptotic theory. It is found that the convergence of G to its asymptotic normal distribution is much slower than the convergence of the other measures to theirs, and that G is more likely to be significantly biased. However, the variances and biases of all four measures come close to their asymptotic values for quite moderate sample sizes.     

On concurrence of several linear regressions and applications to problems of enzyme kinetics

In comparing several regressions E(y_ij) =α_i + β_ix_ij i = 1, 2, ..., k, j = 1,2, ..., n_i, researchers are generally interested in the following five problems: whether they have (1) equal slope, (2) equal intercept, (3) coincidence, (4) common intersection on X-axis, and (5) common intersection on (X,Y) - plane. Problems (1) - (3) can be put into the framework of the general linear hypothesis and the F-test can be used. However, problems (4) and (5) cannot be put into the general linear hypothesis because they are ratios of parameters. Hence, in this paper we consider the generalized likelihood ratio test for hypothesis testing. An application to an enzyme kinetics problem in Aniline Metabolism is demonstrated