Large sample results for tests of association ii. multinomial and stratified sampling

This paper deals with the asymptotics of a class of tests for association in 2-way contingency tables based on square forms in cell frequencies, given the total number of observations (multinomial sampling) or one set of marginal totals (stratified sampling). The case when both row and column marginal totals are fixed (hypergeometric sampling) was studied in Kulinskaya (1994), The class of tests under consideration includes a number of classical measures for association, Its two subclasses are the tests based on statistics using centralized cell frequencies (asymptotically distributed as weighted sums of central chi-squares) and those using the non-centralized cell frequencies (asymptotically normal). The parameters of asymptotic distributions depend on the sampling model and on true marginal probabilities. Maximum efficiency for asymptotically normal statistics is achieved under hypergeometric sampling, If the cell frequencies or the statistic as a whole are centralized using marginal proportions as estimates for marginal probabilities, the asymptotic distribution does not differ much between models and it is equivalent to that under hypergeometric sampling. These findings give an extra justification for the use of permutation tests for association (which are based on hypergeometric sampling). As an application, several well known measures of association are analysed.     

Power-Divergence Test Statistics for Testing Linear by Linear Association

Power-divergence test statistics have been considered to test linear by linear association for two-way contingency tables. These test statistics have been compared based on designed simulation study and asymptotic results for 2 × 2, 2 × 3, and 3 × 3 tables. According to the results, there are test statistics with better properties than the well-known likelihood ratio test statistic for small and moderate samples.     

A Comparative Study of Model-based Tests of Independence for Ordinal Data Using the Bootstrap

When analyzing categorical data using loglinear models in sparse contingency tables, asymptotic results may fail. In this paper the empirical properties of three commonly used asymptotic tests of independence, based on the uniform association model for ordinal data, are investigated by means of Monte Carlo simulation. Five different bootstrapped tests of independence are presented and compared to the asymptotic tests. The comparisons are made with respect to both size and power properties of the tests. Results indicate that the asymptotic tests have poor size control. The test based on the estimated association parameter is severely conservative and the two chi-squared tests (Pearson, likelihood-ratio) are both liberal. The bootstrap tests that either use a parametric assumption or are based on non-pivotal test statistics do not perform better than the asymptotic tests in all situations. The bootstrap tests that are based on approximately pivotal statistics provide both adjustment of size and enhancement of power. These tests are therefore recommended for use in situations similar to those included in the simulation study.     

Simultaneous tests for homogeneity of two zero-inflated (beta) populations

ABSTRACT

Motivated by an example in marine science, we use Fisher’s method to combine independent likelihood ratio tests (LRTs) and asymptotic independent score tests to assess the equivalence of two zero-inflated Beta populations (mixture distributions with three parameters). For each test, test statistics for the three individual parameters are combined into a single statistic to address the overall difference between the two populations. We also develop non parametric and semiparametric permutation-based tests for simultaneously comparing two or three features of unknown populations. Simulations show that the likelihood-based tests perform well for large sample sizes and that the statistics based on combining LRT statistics outperforms the ones based on combining score test statistics. The permutation-based tests have overall better performance in terms of both power and type I error rate. Our methods are easy to implement and computationally efficient, and can be expanded to more than two populations and to other multiple parameter families. The permutation tests are entirely generic and can be useful in various applications dealing with zero (or other) inflation.     

Percentage points for directional anderson-darling goodness-of-fit tests

This paper discusses the problem of assessing the asymptotic distribution when parameters of the hypothesized distribution are estimated from a sample, pointing out a common mistake included in the paper by Sinclair, Spurr, and Ahmad (1990) which introduced two modifications of the Anderson-Darling goodness-of-fit test statistic. Their two test statistics modify the popular Anderson-Darling test statistic to be sensitive to departures of the fitted distribution from the true distribution in one or the other of the tails. This paper uses these new test statistics to develop tests of fit for the normal and exponential distributions. Easy to use formulas are given so the reader can perform these tests at any sample size without consulting exhaustive tables of percentage points. Finally a power study is given to demonstrate the test statistics’ viability against a broad range of alternatives.     

Linear rank tests under general alternatives,with application to summary statistics computed from repeated measures data

Linear rank tests are used extensively for comparing two or more groups of continuous outcomes. Tests in this class retain proper test size with minimal assumptions and can have high efficiency towards an alternative of interest. In recent years, these tests have been increasingly used in settings where an individual's observation is itself a scalar summary of several outcome measures. Here, simple distributional structures on the outcome variables can lead to complex differences between the distributions of summary statistics of the comparison groups. The local asymptotic power of linear rank tests when the groups are assumed to differ by a location or scale alternative has been studied in detail. However, not much is known about their behavior for other types of alternatives. To address this, we derive the asymptotic distribution of linear rank tests under a general contiguous alternative and then investigate the implications for location–scale families and more general settings, including an example drawn from an AIDS clinical trial where the continuous outcome is a summary statistic computed from repeated measures of a biological marker.     

Likelihood Ratio Test Against Stochastic Order in Three Way Contingency Tables

Trend tests in dose-response have been central problems in medicine. The likelihood ratio test is often used to test hypotheses involving a stochastic order. Stratified contingency tables are common in practice. The distribution theory of likelihood ratio test has not been full developed for stratified tables and more than two stochastically ordered distributions. Under c strata of m × r tables, for testing the conditional independence against simple stochastic order alternative, this article introduces a model-free test method and gives the asymptotic distribution of the test statistic, which is a chi-bar-squared distribution. A real data set concerning an ordered stratified table will be used to show the validity of this test method.     

Decomposition of cumulative chi-squared statistics,with some new tools for their interpretation

It is well known that the Pearson statistic \(\chi ^{2}\) can perform poorly in studying the association between ordinal categorical variables. Taguchi’s and Hirotsu’s statistics have been introduced in the literature as simple alternatives to Pearson’s chi-squared test for contingency tables with ordered categorical variables. The aim of this paper is to shed new light on these statistics, stressing their interpretations and characteristics, providing in this way new and different interpretations of these statistics. Moreover, a theoretical scheme is developed showing the links between the different proposals and classes of cumulative chi-squared statistical tests, starting from a unifying index of heterogeneity, unalikeability and variability measures. Users of statistics may find it attractive to understand well the different proposals. Some decompositions of both statistics are also highlighted. This paper presents a case study of optimizing the polysilicon deposition process in a very large-scale integrated circuit, to identify the optimal combination of factor levels. It is obtained by means of the information coming from a correspondence analysis based on Taguchi’s statistic and regression models for binary dependent variables. A new optimal combination of factor levels is obtained, different from many others proposed in the literature for this data.     

Tests of symmetry with one-sided alternatives in three-way contingency tables

For a two-dimensional contingency table of probabilities, the concept of symmetry around the main diagonal is well defined. Statistical hypothesis test of symmetry versus positive bias have also been explored. For tables of higher (three or more) dimensions, however, different concepts of symmetry are available. In this study, we consider statistical inference procedures of symmetry in partial tables versus various biases in three-dimensional tables. We find the maximum likelihood estimates of the cell probabilities and the asymptotic distribution of the likelihood ratio test statistic in each case. Simulation studies are used to investigate the sizes and powers of the tests. The methodologies developed are applied on real data sets.     

An extension of the Anderson–Darling k-sample test to arbitrary sample space partition sizes

In this paper we first show that the k-sample Anderson–Darling test is basically an average of Pearson statistics in 2?×?k contingency tables that are induced by observation-based partitions of the sample space. As an extension, we construct a family of rank test statistics, indexed by c?∈??, which is based on similarly constructed c?×?k partitions. An extensive simulation study, in which we compare the new test with others, suggests that generally very high powers are obtained with the new tests. Finally we propose a decomposition of the test statistic in interpretable components.     

Multivariate tests comparing binomial probabilities, with application to safety studies for drugs

Summary.  In magazine advertisements for new drugs, it is common to see summary tables that compare the relative frequency of several side-effects for the drug and for a placebo, based on results from placebo-controlled clinical trials. The paper summarizes ways to conduct a global test of equality of the population proportions for the drug and the vector of population proportions for the placebo. For multivariate normal responses, the Hotelling T ²-test is a well-known method for testing equality of a vector of means for two independent samples. The tests in the paper are analogues of this test for vectors of binary responses. The likelihood ratio tests can be computationally intensive or have poor asymptotic performance. Simple quadratic forms comparing the two vectors provide alternative tests. Much better performance results from using a score-type version with a null-estimated covariance matrix than from the sample covariance matrix that applies with an ordinary Wald test. For either type of statistic, asymptotic inference is often inadequate, so we also present alternative, exact permutation tests. Follow-up inferences are also discussed, and our methods are applied to safety data from a phase II clinical trial.     

NONPARAMETRIC TWO SAMPLE TESTS BASED ON AREA STATISTICS

ABSTRACT

Area statistics are sample versions of areas occurring in a probability plot of two distribution functions F and G. This paper presents a unified basis for five statistics of this type. They can be used for various testing problems in the framework of the two sample problem for independent observations, such as testing equality of distributions against inequality or testing stochastic dominance of distributions in one or either direction against nondominance. Though three of the statistics considered have already been suggested in literature, two of them are new and deserve our interest. The finite sample distributions of the statistics (under F=G) can be calculated via recursion formulae. Two tables with critical values of the new statistics are included. The asymptotic distribution of the properly normalized versions of the area statistics are functionals of the Brownian bridge. The distribution functions and quantiles thereof are obtained by Monte Carlo simulation. Finally, the power functions of the two new tests based on area statistics are compared to the power functions of the tests based on the corresponding supremum statistics, i.e., statistics of the Kolmogorov–Smirnov type.     

Goodness-of-fit tests via ϕ-measures of divergence for censored data

Measures of divergence or discrepancy are used extensively in statistics in various fields. In this article, we are focusing on divergence measures that are based on a class of measures known as Csiszar's divergence measures. In particular, we propose a class of goodness-of-fit tests based on Csiszar's class of measures designed for censored survival or reliability data. Further, we derive the asymptotic distribution of the test statistic under simple and composite null hypotheses as well as under contiguous alternative hypotheses. Simulations are furnished and real data are analysed to show the performance of the proposed tests for different ?-divergence measures.     

Tests of symmetry derived as components of pearson's phi-squared distance measure

Rank statistics which arise as estimates of the first and third components of a frequency decomposition of Pearson's phi-squared distance measure, introduced by Eubank, LaRiccia, and Rosenstein (1987), are examined for their usefulness as tests of symmetry about a known median against various asymmetric alternatives, Pitman asymptotic relative efficiencies are used to compare the efficacies of the new statistics with classical test procedures, and empirical powers of the new tests are compared via simulation for a variety of asymmetric distributions. Statistics which arise from components based on Legendre polynomials are of particular interest. The first component is the familiar Wilcoxon signed rank statistic. The third component, which is a new statistic for this problem, exhibits a high level of sensitivity to a variety of asymmetric alternatives both asymptotically and in small sample studies.     

Contingency tables with fuzzy information

ABSTRACT

This paper extends the classical methods of analysis of a two-way contingency table to the fuzzy environment for two cases: (1) when the available sample of observations is reported as imprecise data, and (2) the case in which we prefer to categorize the variables based on linguistic terms rather than as crisp quantities. For this purpose, the α-cuts approach is used to extend the usual concepts of the test statistic and p-value to the fuzzy test statistic and fuzzy p-value. In addition, some measures of association are extended to the fuzzy version in order to evaluate the dependence in such contingency tables. Some practical examples are provided to explain the applicability of the proposed methods in real-world problems.     

Asymmetry models for square contingency tables: exact tests via algebraic statistics

Square contingency tables with the same row and column classification occur frequently in a wide range of statistical applications, e.g. whenever the members of a matched pair are classified on the same scale, which is usually ordinal. Such tables are analysed by choosing an appropriate loglinear model. We focus on the models of symmetry, triangular, diagonal and ordinal quasi symmetry. The fit of a specific model is tested by the chi-squared test or the likelihood-ratio test, where p-values are calculated from the asymptotic chi-square distribution of the test statistic or, if this seems unjustified, from the exact conditional distribution. Since the calculation of exact p-values is often not feasible, we propose alternatives based on algebraic statistics combined with MCMC methods.     

Large sample results for permutation tests of association

This paper deals with the asymptotics of permutation tests based on a certain rather general class of measures of association for R by C contingency tables, given marginal totals. This class includes the classical chi-square test, the T _b and γ indices of Goodman and Kruskall (1954) and the popular Rand (1971) index. The asymptotic distribution of this class of permutation tests for association is a weighted sum of non-central (gen-erally speaking) chi-squares. The formulae for the asymptotic moments of such tests are also given. If non-centrality holds under the null hypothe-sis of independence, the distribution in question converges to the normal distribution. The efficacies for such measures of association are obtained. Several applications are analysed in detail, including the above mentioned indices. Approximations to the permutation distribution are also discussed.     

Two classes of divergence statistics for testing uniform association

The problem of testing uniform association in cross-classifications having ordered categories is considered. Two families of test statistics, both based on divergences between certain functions of the observed data, are studied and compared. Our theoretical study is based on asymptotic properties. For each family, two consistent approximations to the null distribution of the test statistic are studied: the asymptotic null distribution and a bootstrap estimator; all the tests considered are consistent against fixed alternatives; finally, we do a local power study. Surprisingly, both families detect the same local alternatives. The finite sample performance of the tests in these two classes is numerically investigated through some simulation experiments. In the light of the obtained results, some practical recommendations are given.     

Consistent Nonparametric Tests for Lorenz Dominance

This article proposes consistent nonparametric methods for testing the null hypothesis of Lorenz dominance. The methods are based on a class of statistical functionals defined over the difference between the Lorenz curves for two samples of welfare-related variables. We present two specific test statistics belonging to the general class and derive their asymptotic properties. As the limiting distributions of the test statistics are nonstandard, we propose and justify bootstrap methods of inference. We provide methods appropriate for case where the two samples are independent as well as the case where the two samples represent different measures of welfare for one set of individuals. The small sample performance of the two tests is examined and compared in the context of a Monte Carlo study and an empirical analysis of income and consumption inequality.     

Wavelet Estimation of an Unknown Function Observed with Correlated Noise

Tests based on rank statistics are introduced to test for systematic changes in a sequence of independent observations. Proposed tests include a rank test analogous to the parametric likelihood ratio test and others analogous to parametric Bayes tests. The tests are usable with either one- or two-sided alternative hypotheses, and their asymptotic distributions are studied. The results of the general model are applied to two special cases, and their asymptotic distributions are also investigated. A Monte Carlo study verifies the applicability of asymptotic critical points in samples of moderate size, and other simulation studies compare power of the competing tests and their special-case versions. Finally, these tests are applied to a data set of traffic fatalities.