Pearson's X2 Statistic Research Articles

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

ABSTRACTIn 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.

Assessing Approximate Fit in Categorical Data Analysis

A family of Root Mean Square Error of Approximation (RMSEA) statistics is proposed for assessing the goodness of approximation in discrete multivariate analysis with applications to item response theory (IRT) models. The family includes RMSEAs to assess the approximation up to any level of association of the discrete variables. Two members of this family are RMSEA2, which uses up to bivariate moments, and the full information RMSEAn. The RMSEA2 is estimated using the M2 statistic of Maydeu-Olivares and Joe (2005, 2006), whereas for maximum likelihood estimation, RMSEAn is estimated using Pearson's X2 statistic. Using IRT models, we provide cutoff criteria of adequate, good, and excellent fit using the RMSEA2. When the data are ordinal, we find a strong linear relationship between the RMSEA2 and the Standardized Root Mean Squared Residual goodness-of-fit index. We are unable to offer cutoff criteria for the RMSEAn as its population values decrease as the number of variables and categories increase.

On Rao score and PearsonX 2 statistics in generalized linear models

The identity of the Rao score and PearsonX 2 statistics is well known in the areas where the latter was first introduced: goodness-of-fit in contingency tables and binary responses. We show in this paper that the same identity holds when the two statistics are used for testing goodness-of-fit of Generalized Linear Models. We also highlight the connections that exist between the two statistics when they are used for the comparison of nested models. Finally, we discuss some merits of these unifying results.

Modified GEE and Goodness of the Marginal Fit (GOMF) Test with Correlated Binary Responses for Contingency Tables

AbstractThis paper addresses testing the goodness of fit of models for marginal probabilities estimated by generalized estimating equations. We develop a modified version of generalized estimating equation and a goodness‐of‐fit test based on the fitted marginal means. The test statistic is easy to compute and has a simple reference distribution. Its performance is evaluated asymptotically and in small samples. It is also compared to the deviance and Pearson X2 statistics. Example applications are given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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).

A power approximation of the test of homogeneity for multinomial populations based on a normalizing transformation

Cressie and Read(1984) introduced multino~riial goodness-of-fit statisticsbased on a, class of divergence measures Ia between discrete distributions.It was also introduced that a class of statistics Ra for the test of homogeneityfor multino~riial populations based on Ia (Read and Cressie(1988)). This class includes Pearson's X2 statistic (when a = 1) and the log-likelihood ratio statistic (when a = 0). All Ra have the same chi-squared limiting null distribution. The power of the class is ordinary approximated from a noncentral chi-squared distribution that is also the same for all a. Applying the power approximation theories for the multinomial goodness-of-fit test developed by Broffitt and Randles(1977) and Drost et al.(l989), Taneichi and Sekiya(1995)proposed three approximations to the power of Ra that vary with the statisticchosen. In this paper we propose a. new approximation to the power of Ra The new approximation is a normal approximation based on a normalizing transformation of the statistics. Moreover, the proposed approximation and the other approximations are numerically compared with the power directly calculated from the product multinomial model. As a result of comparison, the new approximation is found to be effective for the statistics when a a≤0.

About divergence-based goodness-of-fit tests in the dirichlet-multinomial model

The Pearson X2 statistic, the loglikelihood ratio statistic, the Freeman-Tukey statistic, and others well known statistics are considered as particular cases of appropriately normed (φ-divergences Dφ(FNF) between empirical and hypothetical distributions. All these statistics are asymptotically x2-distributed. For the Dirichlet-multinomial F, conditions on J are found under which the first two moments of the test statistics match the moments of x2, up to terms of order N--1. By calculating the exact sizes and powers of goodness-of-fit tests for 14 selected φ-divergence statistics it is shown that these conditions are of practical importance even for small sample sizes N.

Statistical approaches for analyzing mutational spectra: some recommendations for categorical data.

In studies examining the patterns or spectra of mutational damage, the primary variables of interest are expressed typically as discrete counts within defined categories of damage. Various statistical methods can be applied to test for heterogeneity among the observed spectra of different classes, treatment groups and/or doses of a mutagen. These are described and compared via computer simulations to determine which are most appropriate for practical use in the evaluation of spectral data. Our results suggest that selected, simple modifications of the usual Pearson X2 statistic for contingency tables provide stable false positive error rates near the usual alpha = 0.05 level and also acceptable sensitivity to detect differences among spectra. Extensions to the problem of identifying individual differences within and among mutant spectra are noted.

Testing Colour Proportions of M&Ms

SummaryWe test the claimed colour proportions of plain M&M's candies using Pearson's x2 statistic. The null distribution of this statistic is examined through a Minitab simulation.

The asymptotic distribution of the sum of weighted squared residuals in binary choice models

It is claimed by some authors that the distribution of the sum of weighted squared residuals, used as a goodness of fit measure in binary choice models, behaves for large n as a x2n– k–1 distribution. This claim seems to be based on a false analogy with the well–known Pearson x2 statistic for frequency tables with a fixed number of cells and cell sizes tending to infinity. We derive the asymptotic (normal) distribution and show that the approximation by the x2 distribution in general will not be valid. A new x2 test is proposed based on the asymptotic normality of the measure.

Pearson's X 2 and the Loglikelihood Ratio Statistic G 2 : A Comparative Review

Summary The importance of developing useful and appropriate statistical methods for analyzing discrete multivariate data is apparent from the enormous amount of attention this subject has commanded in the literature over the last thirty years. Central to these discussions has been Pearson's X2 statistic and the loglikelihood ratio statistic G2. Our review seeks to consolidate this fragmented literature and develop a unifying theme for much of this research. The traditional X2 and G2 statistics are viewed as members of the power-divergence family of statistics, and are linked through a single real-valued parameter. The principal areas covered in this comparative survey are small-sample comparisons of X2 and G2 under both classical (fixed-cells) assumptions and sparseness assumptions, efficiency comparisons, and various modifications to the test statistics (including parameter estimation for ungrouped data, data-dependent and overlapping cell boundaries, serially dependent data, and smoothing). Finally some future areas for research are discussed.

The Moments of the Pearson Chi-Squared Statistic and the Minimum Expected Value in Two-Way Tables

Explicit expressions for the first three moments of Pearson's X2 statistic for a two-way contingency table are used to investigate the accuracy of the χ2 approximation to its distribution. The usual ’minimum expected value' criterion is shown to give a global bound on the accuracy of the moments, but to be too conservative. This is illustrated by examples. It is shown that the adequacy of the approximation depends on the significance level that is to be used as well as the accuracy of the moments. When the χ22 approximation is inadequate, corrections based on the moments can be used. Numerical studies of the usefulness of these corrections are presented.

The moments of the Pearson chi-squared statistic and the minimum expected value in two-way tables

SUMMARY Explicit expressions for the first three moments of Pearson's X2 statistic for a two-way contingency table are used to investigate the accuracy of the x2 approximation to its distribution. The usual 'minimum expected value' criterion is shown to give a global bound on the accuracy of the moments, but to be too conservative. This is illustrated by examples. It is shown that the adequacy of the approximation depends on the significance level that is to be used as well as the accuracy of the moments. When the x2 approximation is inadequate, corrections based on the moments can be used. Numerical studies of the usefulness of these corrections are presented.

Adjusted Chi-Squared Tests for Cross-Classified Tables of Survey Data

Holt, Scott & Ewings (1980) propose a modified Pearson X2 statistic for testing independence in a two-way table of survey data. It is shown that the multiplier of X2 needed for this modification is a weighted sum of cell and marginal cell design effects. A similar adjustment of the X2 and G2 statistics is possible whenever log linear models with closed form estimates are fitted to a cross-classified table of survey data.

Adjusted chi-squared tests for cross-classified tables of survey data

SUMMARY Holt, Scott & Ewings (1980) propose a modified Pearson X2 statistic for testing independence in a two-way table of survey data. It is shown that the multiplier of X2 needed for this modification is a weighted sum of cell and marginal cell design effects. A similar adjustment of the X2 and G2 statistics is possible whenever log linear models with closed form estimates are fitted to a cross-classified table of survey data.

Two-Stage Chi Square Goodness-of-Fit Test

Abstract Consider sampling in two stages for testing a simple hypothesis about a population whose elements may be classified into a finite number of classes. The joint asymptotic distribution of the two Pearson X2 statistics based on the first and combined samples is derived and used to construct a two-stage test. The advantage of this test relative to the conventional one-stage test is discussed in terms of its asymptotic power. Tables of critical values for 1 percent and 5 percent level tests for 1–10 degrees of freedom are given as well as modifications for cases involving unknown parameters.

Two-Stage Chi Square Goodness-of-Fit Test

Abstract Consider sampling in two stages for testing a simple hypothesis about a population whose elements may be classified into a finite number of classes. The joint asymptotic distribution of the two Pearson X2 statistics based on the first and combined samples is derived and used to construct a two-stage test. The advantage of this test relative to the conventional one-stage test is discussed in terms of its asymptotic power. Tables of critical values for 1 percent and 5 percent level tests for 1–10 degrees of freedom are given as well as modifications for cases involving unknown parameters.

Pearson chi-squared test of fit with random intervals

SUMMARY A modified form of the Pearson x2 test statistic is considered where estimators based on the ungrouped sample are employed in the test statistic as well as in determining the class interval end points. In the case of continuous distributions with location and scale parameters, it is shown here that the null distribution of such a modified test statistic does not depend on the parameters if these are estimated by sample mean and variance. By utilizing known results concerning the distribution of a weighted sum of independent chi-squared variates, a table of certain percentage points of the asymptotic distribution of this modified test statistic is developed in order to facilitate its use for testing normality. case of continuous distributions. In the first place, how should the class intervals be formed and how many should there be? Secondly, if there are unknown parameters, how should they be estimated and what is their effect on the test? Aside from the complexities arising in deriving estimators ofthe parameters, the resulting distribution ofX2, the Pearson x2 statistic, can be quite different from that of x2, depending upon the method of estimation, as was observed by Chernoff & Lehmann (1954). Thirdly, the estimation of class probabilities when parameters are unknown is also usually complicated in the case of continuous dis- tributions. In considering the problem of how the class intervals should be formed in using the X2 statistic, A.R. Roy in an unpublished report, and Watson (1957, 1958, 1959) relaxed the requirement of fixed class boundaries and utilized estimators of the parameters to determine the end points of the intervals. The estimators employed for this purpose as well as for substitution in the X2 statistic were based on the ungrouped sample. In the present paper we extend a result of Roy to show, in the case of unknown location and scale parameters, that the asymptotic null distribution of the test statistic does not involve the unknown parameters when they are estimated by the sample mean and sample variance respectively. As an application of the technique developed here, we provide a table of some percentage points of the asymptotic distribution of the Pearson statistic modified in the above manner, based on equal probability classes, and used to test for normality. The number of class intervals to be employed in the test should depend on the alternative distributions. This requires knowledge of the corresponding asymptotic nonnull distribution of the test statistic and will be given for various alternatives in a subsequent

On Testing for Goodness-of-Fit of the Negative Binomial Distribution when Expectations are Small

In the case of fitting the negative binomial distribution, it is shown, by means of an example, that (a) the method of Nass [1959] provides a more suitable goodness-of-fit criterion than either Pearson's X2 statistic or the log-likelihood ratio; (b) the scope and power of all three criteria are considerably enhanced by relaxing the commonly used rule that all frequency classes should have an expectation greater than 5.