A comparison of tests of homogeneity for sparse contingency tables

Five tests of homogeneity for a 2x(k+l) contingency table are compared using Monte Carlo techniques. For these studiesit is assumed that k becomes large in such a way that thecontingency table is sparse for 2xk of the cells, but the sample size in two of the cells remains large. The test statistics studied are: the chi-square approximation to the Pearson test statistic, the chi-square approximation to the likelihood ratio statistic, the normal approximation to Zelterman's (1984)the normal approximation to Pearson's chi-square, and the normal approximation to the likelihood ratio statistic. For the range of parameters studied the chi-square approximation to Pearson's statistic performs consistently well with regard to its size and power.     

Approximations of the Distributions of Test Statistics for Homogeneity of a Product Multinomial Model

Statistics R ^a based on power divergence can be used for testing the homogeneity of a product multinomial model. All R ^a have the same chi-square limiting distribution under the null hypothesis of homogeneity. R ⁰ is the log likelihood ratio statistic and R ¹ is Pearson's X ² statistic. In this article, we consider improvement of approximation of the distribution of R ^a under the homogeneity hypothesis. The expression of the asymptotic expansion of distribution of R ^a under the homogeneity hypothesis is investigated. The expression consists of continuous and discontinuous terms. Using the continuous term of the expression, a new approximation of the distribution of R ^a is proposed. A moment-corrected type of chi-square approximation is also derived. By numerical comparison, we show that both of the approximations perform much better than that of usual chi-square approximation for the statistics R ^a when a ≤ 0, which include the log likelihood ratio statistic.     

Improved likelihood ratio tests and pearson chi-square tests for independence in two dimensional contingency tables

When an I×J contingency table has many cells having very small frequencies, the usual chi-square approximation to the upper tail of the likelihood ratio goodness-of-fit statistic, G² and Pearson chi-square statistic, X², for testing independence, are not satisfactory. In this paper we consider the problem of adjusting G² and X². Suitable adjustments are suggested on the basis of analytical investigation of asymptotic bias terms for G² and X². A Monte Carlo simulation is performed for several tables to assess the adjustments of G² and X² in order to obtain a closer approximation to the nominal level of significance.     

On the decomposition of X2 for two-way contingency tables

When samples are taken independently from I populations and the subjects classified into J categories, can the Pearson's chisquare statistic X² testing the homogeneity model on the resulting I×J two-way table be decomposed into components familiar in the analysis of variance? Will the X² testing the homogeneity model on tables derived by collapsing columns in the spirit of orthogonal comparisons in factorial experiments be asymptotically independent? The answers to both questions are generally negative. This paper gives a theoretical justification.     

Exact Tests for 2 × 2 Contingency Tables

Fisher's exact test, difference in proportions, log odds ratio, Pearson's chi-squared, and likelihood ratio are compared as test statistics for testing independence of two dichotomous factors when the associated p values are computed by using the conditional distribution given the marginals. The statistics listed above that can be used for a one-sided alternative give identical p values. For a two-sided alternative, many of the above statistics lead to different p values. The p values are shown to differ only by which tables in the opposite tail from the observed table are considered more extreme than the observed table.     

The Tale of Cochran's Rule: My Contingency Table has so Many Expected Values Smaller than 5, What Am I to Do?

In an informal way, some dilemmas in connection with hypothesis testing in contingency tables are discussed. The body of the article concerns the numerical evaluation of Cochran's Rule about the minimum expected value in r × c contingency tables with fixed margins when testing independence with Pearson's X² statistic using the χ² distribution.     

Small-sample comparison of the exact and asymptotic upper tail probabilities of chi-squared goodness-of-fit statistics: the binomila and the mixture binomial

The exact and asymptotic upper tail probabilities (α = .10, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and powerdivergence statisticD ²(λ), with λ= 2/3 are compared by complete enumeration for the binomial and the mixture binomial. For the two-component mixture binomial, three cases have been distinguished. 1. Both success probabilities and the mixing weights are unknwon. 2. One of the two success probabilities is known. And 3., the mixing weights are known. The binomial was investigated for the number of cellsk, being between 3 and 6 with sample sizes between 5 and 100, for k = 7 with sample sizes between 5 and 45, and for k = 10 with sample sizes ranging from 5 to 20. For the mixture binomial, solely k = 5 cells were considered with sample sizes from 5 to 100 and k = 8 cells with sample sizes between 4 and 20. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions for the binomial: 1. Using G2 is not recommendable. 2. At the significance levels α=.10 and α=.05X ² should be preferred over D ²; D ² is the best choice at α = .01. 3. Cochran's (1954; Biometrics, 10, 417-451) rule for the minimum expectation when using X ² seems to generalize to the binomial for G ² and D ² ; as a compromise, it gives a rather strong lower limit for the expected cell frequencies in some circumstances, but a rather liberal in others. To draw similar conclusions concerning the mixture binomial was not possible, because in that case, the accuracy of the chi-squared approximation is not only a function of the chosen test statistic and of the significance level, but also heavily depends on the numerical value of theinvolved unknown parameters and on the hypothesis to be tested. Thereto, the present study may give rise only to warnings against the application of mixture models to small samples.     

Tests for Symmetry Based on the One-Sample Wilcoxon Signed Rank Statistic

ABSTRACT

The one-sample Wilcoxon signed rank test was originally designed to test for a specified median, under the assumption that the distribution is symmetric, but it can also serve as a test for symmetry if the median is known. In this article we derive the Wilcoxon statistic as the first component of Pearson's X ² statistic for independence in a particularly constructed contingency table. The second and third components are new test statistics for symmetry. In the second part of the article, the Wilcoxon test is extended so that symmetry around the median and symmetry in the tails can be examined seperately. A trimming proportion is used to split the observations in the tails from those around the median. We further extend the method so that no arbitrary choice for the trimming proportion has to be made. Finally, the new tests are compared to other tests for symmetry in a simulation study. It is concluded that our tests often have substantially greater powers than most other tests.     

Comments on “a generalization of fisher's exact test in pxq contingency tables using more concordant relations”

The purpose of this note is to criticize Nguyen (1985) for his account of the literature on the generalization of Fisher's exact test and to point out parallels with existing algorithms of the algorithm proposed by Nguyen. Subsequently we will briefly raise some questions on the methodology proposed by Nguyen.

Nguyen (1985) suggests that all literature on exact testing prior to Nguyen & Sampson (1985) is based on the “more probable” relation or Exact Probability Test (EPT) as a test statistic. This is not correct. Yates (1934 - Pearson's X²), Lewontin & Felsenstein (1965 - X²), Agresti & Wackerly (1977 - X², Kendall's tau, Kruskal & Goodman's gamma), Klotz (1966 - Wilcoxon), Klotz & Teng (1977 - Kruskall & Wallis' H), Larntz (1978 - X², loglike-lihood-ratio statistic G², Freeman & Tukey statistic), and several others have investigated exact tests with other statistics than the EPT. In fact, Bennett & Nakamura (1963) are incorrectly cited as they investigated both X² and G², rather than EPT. Also, Freeman & Halton (1951) are incorrectly cited for they generalized Fisher's exact test to pxq tables and not 2xq tables as stated. And they are even predated by Yates (1934) who extended the test to 2×3 tables.     

Small -sample comparison of the exact and symptotic upper tail probabilitiesof chi-squared goodness -of-fit statistics: pearson's x 2,likelihood ratio,and power-divergence statistic(λ = 2/3)

The exact and asymptotic upper tail probabilities ( α= .l0, .05, .01, .001) of the three chi-squared goodness-of-fit statistics Pearson's X ², likelihood ratioG ², and power-divergence statisticD ² (λ ) , with λ = 2/3, are compared numerically for simple null hypotheses not involving parameter estimation. Three types of such hypotheses were investigated (equal cell probabilities, proportional cell probabilities, some fixed small expectations together with some increasing large expectations) for the number of cells being between 3 and 15, and for sample sizes from 10 to 40, increasing by steps of one. Rating the relative accuracy of the chi-squared approximation in terms of ±10% and ±20% intervals around α led to the following conclusions: 1. Using G ² is not recommended. 2 . At the more relevant significance levels α = .10 and α = .05X ² should be preferred over D ². Solely in case of unequal cell probabilitiesD ² is the better choice at α = .O1 and α = .001. 3 . Yarnold's (1970; Journal of the Amerin Statistical Association, 65, 864-886) rule for the minimum expectation when using X ² ("If the number of cells k is 3 or more, and if r denotes the number of expectations less than 5, then the minimum expectation may be as small as 5r/k.") generalizes to D ²; it gives a good lower limit for the expected cell frequencies, however, when the number of cells is greater than 3. For k = 3 , even sample sizes over 15 may be insufficient.     

Negative exponential disparity based family of goodness-of-fit tests for multinomial models

This paper investigates a new family of goodness-of-fit tests based on the negative exponential disparities. This family includes the popular Pearson's chi-square as a member and is a subclass of the general class of disparity tests (Basu and Sarkar, 1994) which also contains the family of power divergence statistics. Pitman efficiency and finite sample power comparisons between different members of this new family are made. Three asymptotic approximations of the exact null distributions of the negative exponential disparity famiiy of tests are discussed. Some numerical results on the small sample perfomance of this family of tests are presented for the symmetric null hypothesis. It is shown that the negative exponential disparity famiiy, Like the power divergence family, produces a new goodness-of-fit test statistic that can be a very attractive alternative to the Pearson's chi-square. Some numerical results suggest that, application of this test statistic, as an alternative to Pearson's chi-square, could be preferable to the I ^2/3 statistic of Cressie and Read (1984) under the use of chi-square critical values.     

Small-sample comparisons for powerdivergence goodness-of-fit statistics for symmetric and skewed simple null hypotheses

Power-divergence goodness-of-fit statistics have asymptotically a chi-squared distribution. Asymptotic results may not apply in small-sample situations, and the exact significance of a goodness-of-fit statistic may potentially be over- or under-stated by the asymptotic distribution. Several correction terms have been proposed to improve the accuracy of the asymptotic distribution, but their performance has only been studied for the equiprobable case. We extend that research to skewed hypotheses. Results are presented for one-way multinomials involving k = 2 to 6 cells with sample sizes N = 20, 40, 60, 80 and 100 and nominal test sizes f = 0.1, 0.05, 0.01 and 0.001. Six power-divergence goodness-of-fit statistics were investigated, and five correction terms were included in the study. Our results show that skewness itself does not affect the accuracy of the asymptotic approximation, which depends only on the magnitude of the smallest expected frequency (whether this comes from a small sample with the equiprobable hypothesis or a large sample with a skewed hypothesis). Throughout the conditions of the study, the accuracy of the asymptotic distribution seems to be optimal for Pearson's X² statistic (the power-divergence statistic of index u = 1) when k > 3 and the smallest expected frequency is as low as between 0.1 and 1.5 (depending on the particular k, N and nominal test size), but a computationally inexpensive improvement can be obtained in these cases by using a moment-corrected h² distribution. If the smallest expected frequency is even smaller, a normal correction yields accurate tests through the log-likelihood-ratio statistic G² (the power-divergence statistic of index u = 0).     

GOODNESS OF FIT FOR THE BINOMIAL DISTRIBUTION

Goodness of fit testing for the binomial distribution can be carried out using Pearson's X²_p statistic and its components. Applications of this technique are considered and compared with recently suggested empirical distribution function tests. Diagnostic use of components is discussed.     

A Comparison of Two Tests for Equality of Two Proportions

Two procedures for testing equality of two proportions are compared in terms of asymptotic efficiency. The comparison favors use of a statistic equivalent to Goodman's Y ² over the usual X ² statistic in some cases including that of equal sample sizes. Numerical comparisons indicate that the asymptotic results have some relevance for moderate sample sizes.     

A Test of Independence Based on the Likelihood of Cut-Points

We define a test statistic C _n based on the sum of the likelihood ratio statistics for testing independence in the 2 × 2 tables defined at n sample cut-points (X _i , Y _i ). The asymptotic distribution of C _n , given the cut-points, is sum of dependent χ² variables with one degree of freedom. We use the bootstrap to obtain the distribution of C _n . We compare the performance of several tests of bivariate independence, including Pearson, Spearman, and Kendall correlations, Blum-Kiefer-Rosenblatt statistic, and C _n under several copulas and given marginal distributions.     

THE MULTIPLE CORRELATION COEFFICIENT AND FISHER'S A STATISTIC1

Fisher's A statistic, often called the adjusted R² statistic, is shown to be a close approximation to the maximum likelihood estimate of the multiple correlation coefficient, p², based on the marginal distribution of R². Expansions for the estimate are obtained. The same methods lead to maximum marginal likelihood estimators for the noncentrality parameters for noncentral X² and F.     

An Efficient Method to Generate Data and Compute Exact P-Values in Goodness-of-Fit Testing

In this article, we use a characterization of the set of sample counts that do not match with the null hypothesis of the test of goodness of fit. Two direct applications arise: first, to instantaneously generate data sets whose corresponding asymptotic P-values belong to a certain pre-defined range; and second, to compute exact P-values for this test in an efficient way. We present both issues before illustrating them by analyzing a couple of data sets. Method's efficiency is also assessed by means of simulations. We focus on Pearson's X ² statistic but the case of likelihood-ratio statistic is also discussed.     

Goodness of fit for thei ordered categories discrete uniform distribution

Goodness of fit for thei ordered categories discrete uniform distribution can be carried out using Pearson's X² _pstatistic and its components. Applications of this technique are considered and comparisons made with recently suggested empirical uniform distribution     

Maximum Likelihood Estimation in the Bivariate Binomial (0,1) Distribution:Application TO 2×2 Tables

We consider a 2×2 contingency table, with dichotomized qualitative characters (A,A) and (B,B), as a sample of size n drawn from a bivariate binomial (0,1) distribution. Maximum likelihood estimates p?₁p?₂ and p? are derived for the parameters of the two marginals p₁p₂ and the coefficient of correlation. It is found that p? is identical to Pearson's (1904)?=(χ²/n)½, where ?² is Pearson's usual chi-square for the 2×2 table. The asymptotic variance-covariance matrix of p?_lp?₂and p is also derived.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.