Chi-Squared Goodness-of-Fit Tests: Cell Selection and Power

To use the Pearson chi-squared statistic to test the fit of a continuous distribution, it is necessary to partition the support of the distribution into k cells. A common practice is to partition the support into cells with equal probabilities. In that case, the power of the chi-squared test may vary substantially with the value of k. The effects of different values of k are investigated with a Monte Carlo power study of goodness-of-fit tests for distributions where location and scale parameters are estimated from the observed data. Allowing for the best choices of k, the Pearson and log-likelihood ratio chi-squared tests are shown to have similar maximum power for wide ranges of alternatives, but this can be substantially less than the power of other well-known goodness-of-fit tests.     

Multivariate extension of chi-squared univariate normality test

We propose a multivariate extension of the univariate chi-squared normality test. Using a known result for the distribution of quadratic forms in normal variables, we show that the proposed test statistic has an approximated chi-squared distribution under the null hypothesis of multivariate normality. As in the univariate case, the new test statistic is based on a comparison of observed and expected frequencies for specified events in sample space. In the univariate case, these events are the standard class intervals, but in the multivariate extension we propose these become hyper-ellipsoidal annuli in multivariate sample space. We assess the performance of the new test using Monte Carlo simulation. Keeping the type I error rate fixed, we show that the new test has power that compares favourably with other standard normality tests, though no uniformly most powerful test has been found. We recommend the new test due to its competitive advantages.     

A Note on Wald's Type Chi-squared Tests of Fit

The Wald's method for constructing chi-squared tests of fit has been formulated more accurately. It is shown that Wald's type statistics will follow the central chi-squared distribution if and only if the limit covariance matrix of standardized frequencies will not depend on unknown parameters. Several examples that illustrate this important fact are presented. In particular, it is shown that the goodness-of-fit statistic developed by Moore and Stubblebine does not follow the chi-squared limit distribution, and, hence, cannot be used for testing multivariate normality.     

A Decomposition of Pearson–Fisher and Dzhaparidze–Nikulin Statistics and Some Ideas for a More Powerful Test Construction

An explicit decomposition on asymptotically independent distributed as chi-squared with one degree of freedom components of the Pearson–Fisher and Dzhaparidze–Nikulin tests is presented. The decomposition is formally the same for both tests and is valid for any partitioning of a sample space. Vector-valued tests, components of which can be not only different scalar tests based on the same sample, but also scalar tests based on components or groups of components of the same statistic are considered. Numerical examples illustrating the idea are presented.     

A powerful and interpretable alternative to the Jarque–Bera test of normality based on 2nd-power skewness and kurtosis,using the Rao's score test on the APD family

We introduce the 2nd-power skewness and kurtosis, which are interesting alternatives to the classical Pearson's skewness and kurtosis, called 3rd-power skewness and 4th-power kurtosis in our terminology. We use the sample 2nd-power skewness and kurtosis to build a powerful test of normality. This test can also be derived as Rao's score test on the asymmetric power distribution, which combines the large range of exponential tail behavior provided by the exponential power distribution family with various levels of asymmetry. We find that our test statistic is asymptotically chi-squared distributed. We also propose a modified test statistic, for which we show numerically that the distribution can be approximated for finite sample sizes with very high precision by a chi-square. Similarly, we propose a directional test based on sample 2nd-power kurtosis only, for the situations where the true distribution is known to be symmetric. Our tests are very similar in spirit to the famous Jarque–Bera test, and as such are also locally optimal. They offer the same nice interpretation, with in addition the gold standard power of the regression and correlation tests. An extensive empirical power analysis is performed, which shows that our tests are among the most powerful normality tests. Our test is implemented in an R package called PoweR.     

An efficient algorithm for computing the noncentrality parameters of chi-squared tests

Use of Newton's method for computing the noncentrality parameter based on the specified power in sample size problems of chi-squared tests requires that we evaluate both the noncentral chi-squareddistribution function and its derivative with respect to the noncentrality parameter. A close relationship between computing formulas for them is revealed, by which their evaluations can be performed jointly. This property greatly reduces the amount of computation involved. The corresponding algorithm is provided in a step-by-step form.     

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.     

On the equivalence of some test criteria based on BAN estimators for the multivariate exponential family

For the general multivariate exponential family of distributions it is shown that Rao's test criterion based on efficient scores is algebraically identical to the general chi-squared criterion based on maximum likelihood estimates and, similarly, that the Wald statistic is algebraically identical to the general minimum modified chi-squared statistic using linearization; these results are valid also for the multisample versions. Thus, these are extensions to the general exponential family of the findings due to Silvey (1970) and Bhapkar (1966), respectively, for the special case of the multinomial family.It is also shown that the general forms of the chi-squared and modified chi-squared criteria reduce to their respective well-known forms for the multivariate symmetric power series distribution. This finding is, thus, an extension of results noted by Ferguson (1958) and Clickner (1976) for the special case of the multinomial distribution.     

Correspondence Analysis of Cumulative Frequencies Using a Decomposition of Taguchi's Statistic

Taguchi's statistic has long been known to be a more appropriate measure of association for ordinal variables than the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic for performing correspondence analysis when a two-way contingency table consists of one ordinal categorical variable. This article will explore the development of correspondence analysis using a decomposition of Taguchi's statistic.     

An Empirical Analysis of Some Nonparametric Goodness-of-Fit Tests for Censored Data

In this article, we consider some nonparametric goodness-of-fit tests for right censored samples, viz., the modified Kolmogorov, Cramer–von Mises–Smirnov, Anderson–Darling, and Nikulin–Rao–Robson χ² tests. We also consider an approach based on a transformation of the original censored sample to a complete one and the subsequent application of classical goodness-of-fit tests to the pseudo-complete sample. We then compare these tests in terms of power in the case of Type II censored data along with the power of the Neyman–Pearson test, and draw some conclusions. Finally, we present an illustrative example.     

Optimal correction for continuity in the chi-squared test in 2×2 tables

The most common asymptotic procedure for analyzing a 2 × 2 table (under the conditioning principle) is the ‰ chi-squared test with correction for continuity (c.f.c). According to the way this is applied, up to the present four methods have been obtained: one for one-tailed tests (Yates') and three for two-tailed tests (those of Mantel, Conover and Haber). In this paper two further methods are defined (one for each case), the 6 resulting methods are grouped in families, their individual behaviour studied and the optimal is selected. The conclusions are established on the assumption that the method studied is applied indiscriminately (without being subjected to validity conditions), and taking a basis of 400,000 tables (with the values of sample size n between 20 and 300 and exact P-values between 1% and 10%) and a criterion of evaluation based on the percentage of times in which the approximate P-value differs from the exact (Fisher's exact test) by an excessive amount. The optimal c.f.c. depends on n, on E (the minimum quantity expected) and on the error α to be used, but the rule of selection is not complicated and the new methods proposed are frequently selected. In the paper we also study what occurs when E ≥ 5, as well as whether the chi-squared by factor (n-1).     

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.     

Some remarks on the chi-squared test with both margins fixed

Under the hypothesis of independence, the chi-squared test statistic for independence in a two-way contingency table follows an asymptotic chi-squared distribution under both a multinomial and a product-multinomial models. Alalouf(1987) showed the same result holds for the third case where both margins are fixed. In this paper an intuitively easier way of proof using the conditional limit theorems is suggested and some points are discussed.     

A test for equality of variances with censored samples

A variance homogeneity test for type II right-censored samples is proposed. The test is based on Bartlett's statistic. The asymptotic distribution of the statistic is investigated. The limiting distribution is that of a linear combination of i.i.d. chi-square variables with 1 degree of freedom. By using simulation, the critical values of the null distribution of the modified Bartlett's statistic for testing the homogeneity of variances of two normal populations are obtained when the sample sizes and censoring levels are not equal. Also, we investigate the properties of the proposed test (size, power and robustness). Results show that the distribution of the test statistic depends on the censoring level. An example of the use of the new methodology in animal science involving reproduction in ewes is provided.     

Stringency-based ranking of normality tests

Many parametric statistical inferential procedures in finite samples depend crucially on the underlying normal distribution assumption. Dozens of normality tests are available in the literature to test the hypothesis of normality. Availability of such a large number of normality tests has generated a large number of simulation studies to find a best test but no one arrived at a definite answer as all depends critically on the alternative distributions which cannot be specified. A new framework, based on stringency concept, is devised to evaluate the performance of the existing normality tests. Mixture of t-distributions is used to generate the alternative space. The LR-tests, based on Neyman–Pearson Lemma, have been computed to construct a power envelope for calculating the stringencies of the selected normality tests. While evaluating the stringencies, Anderson–Darling (AD) statistic turns out to be the best normality test.     

A Statistical Reanalysis of the Classical Rutherford's Experiment

Modified chi-squared and some newly developed tests for the Poisson, binomial, and an approximated Feller's distribution are discussed. A reanalysis of the classical Rutherford's experimental data on alpha decay is done. Previous analyses of the data were not correct from the point of view of the theory of statistical testing. Tests used show that the data contradict to both Poisson and binomial distribution and do not contradict to a precise “binomial” approximation of Feller's distribution that takes into account a counter's dead time. This gives a plausible statistically correct confirmation of the well-established exponential law of radioactive decay.     

A randomized Baumgartner statistic for multivariate two-sample testing hypothesis

In this paper, multivariate two-sample testing problems were examined based on the Jure?ková–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.     

Cumulative Correspondence Analysis of Two-Way Ordinal Contingency Tables

A suitable measure of association for two ordered variables is the doubly cumulative chi-squared statistic (Hirotsu, 1994 Hirotsu , C. ( 1994 ). Modelling and analysing the generalized interaction . Proc. Third IEEE Conf. Control Applic. 2 : 1283 – 1288 .[Crossref] , [Google Scholar]). This statistic is obtained by considering the cumulative sum of cell frequencies across the variables. In this article, we explore the development of correspondence analysis which takes into account the presence of two ordered variables by partitioning the doubly cumulative chi-squared statistic.     

Modified Baumgartner statistics for the two-sample and multisample problems: a numerical comparison

Various non-parametric rank tests based on the Baumgartner statistic have been proposed for testing the location, scale and location–scale parameters. The modified Baumgartner statistics are not suitable for the scale shifts for a two-sample problem. Two modified Baumgartner statistics are proposed by changing the weight function. The suggested statistics are extended to the multisample problem. Some exact critical values of the suggested test statistics are evaluated. Simulations are used to investigate the power of the modified Baumgartner statistics.     

A comparison of chi-squared statistics for testing homogeneity of survey data:High School and Beyond survey

The performance of several test statistics for comparing vectors of propor tions from certain survey data was compared. The statistics were used to analyze a subsample of data from the 'High School and Beyond' survey. These tests include the Wald test statistic X2w and the modified Wald test statistic FW, the chi-squared test statistic X2rSB and its modification FRSB, a test X2dmb based on a probability model, and a method of moments approach, X2H. Data were also simulated based on two-stage cluster sampling design and the type I error level, and the power of these tests was obtained for selected combinations of parameter values. The statistics X2DMB XRSB, FRSB and X2H performed well both for a small number of clusters or a small number of units within clusters. The power performance of these tests is quite stable. Approximate intervals were constructed for design effect constants. Methods of estimating these constants based on a normality assumption worked best.