Asymptotic expansions for the distributions of statistics based on a correlation matrix

An asymptotic expansion is given for the distribution of the α-th largest latent root of a correlation matrix, when the observations are from a multivariate normal distribution. An asymptotic expansion for the distribution of a test statistic based on a correlation matrix, which is useful in dimensionality reduction in principal component analysis, is also given. These expansions hold when the corresponding latent root of the population correlation matrix is simple. The approach here is based on a perturbation method.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

A table of some percentage points of the distribution of a difference between independent chi-square variables

The distribution of Bell-Doksum measure of correlation is that of a difference between independent chi-square variables with equal weights. A table of percentage points computed here for the distribution may be used to test a hypothesis of no correlation between two variables. The distribution of a diffference between independent chi-square variables is also useful in studying variance component estimators and some general results corresponding to the distribution are given.     

A Measure of Dependence Between Two Compositions

We consider the problem of describing the correlation between two compositions. Using a bicompositional Dirichlet distribution, we calculate a joint correlation coefficient, based on the concept of information gain, between two compositions. Numerical values of the joint correlation coefficient are calculated for compositions of two and three components, respectively. We also present an estimator of the joint correlation coefficient for a sample from a bicompositional Dirichlet distribution. Two confidence intervals are presented and we examine their empirical confidence coefficients using a Monte Carlo study. Finally, we apply the estimator to a data set analysing the joint correlation between the 1967 and 1997, and the 1977 and 1997 compositions of the government gross domestic product for the 50 states of the USA and the District of Columbia.     

Some useful integrals and their applications in correlation analysis

The distribution of the product moment correlation coefficient based on the bivariate normal distribution is well known. Recently in many business and economic data, fat tailed distributions especially some elliptical distributions have been considered as parent populations. The normal and t-distributions are well known special cases of elliptical distribution. In this paper we derive some theorems involving double integrals and apply them to derive the probability distribution of the correlation coefficient for some elliptical populations. The general nature of the theorems indicates their potential use in probability distribution theory.     

Covariate Adjusted Correlation Analysis via Varying Coefficient Models

Abstract.  We propose covariate adjusted correlation (Cadcor) analysis to target the correlation between two hidden variables that are observed after being multiplied by an unknown function of a common observable confounding variable. The distorting effects of this confounding may alter the correlation relation between the hidden variables. Covariate adjusted correlation analysis enables consistent estimation of this correlation, by targeting the definition of correlation through the slopes of the regressions of the hidden variables on each other and by establishing a connection to varying-coefficient regression. The asymptotic distribution of the resulting adjusted correlation estimate is established. These distribution results, when combined with proposed consistent estimates of the asymptotic variance, lead to the construction of approximate confidence intervals and inference for adjusted correlations. We illustrate our approach through an application to the Boston house price data. Finite sample properties of the proposed procedures are investigated through a simulation study.     

Estimating correlation from categorized bivariate normal data

The polychoric correlation and an approximate maximum likelihood estimator of correlation are compared for count data that are assumed to be derived from an underlying bivariate normal distribution. A related chi squared test for bivariate normality is also examined.     

A simple derivation of the distribution of the multiple correlation coefficient

It is shown that the non-null distribution of the multiple correlation coefficient may be derived rather easily if the correlated normal variables are defined in a convenient vay. The invariance of the correlation distribution to linear transformations of the variables makes the results generally applicable. The distribution is derived as the well-known mixture of null distributions, and some generalizations when the variables are not normally distributed are indicated.     

Vector correlation for elliptical distributions

A measure of multivariate correlation between two sets of vectors is considered when the underlying joint distribution is a member of the class of elliptical distributions. Its asymptotic distribution is derived under different situations and these results are used to test hypotheses on vector correlation when the underlying joint distribution is non-normal.     

Tables of critical values for two group interim analyses

Tables of critical values are given, which can be used to execute interim analyses in clinical trials involving two groups when the joint distribution of the test statistics can be approximated by a multivariate normal distribution. Critical values are given for both the one and two interim analyses cases for a variety of partitions of α and correlation structures. Results of power calculations are presented, which reflect the effects of both the correlation structure and partitions of α.Several examples are given, which illustrate how to apply the tables to a variety of experiments     

Evaluation of the one-sided percentage points of the singular multivariate normal distribution

This paper presents a new theorem, as a substitute for existing results which are shown to have some errors, for evaluating the exact one-sided percentage points of the multivariate normal distribution with a singular negative product correlation structure. By extending the result from the multivariate normal distribution to the multivariate t-distribution with corresponding singular correlation structure, we tabulate the one-sided critical points for the Analysis of Means procedure.     

Hermite polynomials and truncated trivariate normal distributions

The moments of a trivariate and in general of a multivariate normal distribution, which is truncated with respect to a single variable, are obtained by using properties of Hermite polynomials. An expression for the truncated correlation coefficient is derived in terms of the true population correlation coefficient and the truncation point. The values of this truncated correlation coefficient are tabulated for given values of the true correlation coefficient and a few selected values of the truncation point. A listing of the computer program for this purpose is also given.     

A NONPARAMETRIC TEST FOR THE EQUALITY OF DEPENDENT CORRELATION COEFFICIENTS UNDER NORMALITY

ABSTRACT

A nonparametric testing method for the equality of two correlation coefficients in trivariate normal distribution, namely, one of the variables are common, is discussed. Using a permutation test, we obtain asymptotically exact solutions. The performance of this test is compared with the likelihood ratio test and a method of using the limiting distribution of correlation coefficients.     

TESTING THE LARGEST OF A SET OF CORRELATION COEFFICIENTS

A previous paper which studied the distribution of the smallest distance between N independent random points on the surface of a sphere is generalised to higher dimensions in order to study the distribution of the largest sample correlation coefficient between a set of independent normally distributed variables. Inclusion-exclusion arguments provide accurate bounds for the tail of this distribution, and by another argument more exact bounds are also found, one of which is an improvement on the result in the previous paper. Bounds are also found for the power of the test against the alternative hypothesis that one only of the population correlation coefficients is non-zero. The test is also shown to be the likelihood ratio test against the latter alternative.     

A Simple Distribution for the Sum of Correlated,Exchangeable Binary Data

This article describes a generalization of the binomial distribution. The closed form probability function for the probability of k successes out of n correlated, exchangeable Bernoulli trials depends on the number of trials and its two parameters: the common success probability and the common correlation. The distribution is derived under the assumption that the common correlation between all pairs of Bernoulli trials remains unchanged conditional on successes in all completed trials. The distribution was developed to model bond defaults but may be suited to biostatistical applications involving clusters of binary data encountered in repeated measurements or toxicity studies of families of organisms. Maximum likelihood estimates for the parameters of the distribution are found for a set of binary data from a developmental toxicity study on litters of mice.     

Second-order properties of intraclass correlation estimators for a symmetric normal distribution

Consider the problem of estimating the intraclass correlation coefficient of a symmetric normal distribution under the squared error loss function. The general admissibility of the standard estimators of the intraclass correlation coefficient is hard to check due to their complicated sampling distributions. We follow the asymptotic decision-theoretic approach of Ghosh and Sinha (1981) and prove that the three standard intraclass correlation estimators (the maximum-likelihood estimator, the method-of-moments estimator and the first-order unbiased estimator) are second-order admissible for all p ≥ 2, p being the dimension of the distribution.     

An empirical investigation of different operating characteristics of several estimators of the intraclass correlation in the analysis of binary data

This paper is concerned with properties (bias, standard deviation, mean square error and efficiency) of twenty six estimators of the intraclass correlation in the analysis of binary data. Our main interest is to study these properties when data are generated from different distributions. For data generation we considered three over-dispersed binomial distributions, namely, the beta-binomial distribution, the probit normal binomial distribution and a mixture of two binomial distributions. The findings regarding bias, standard deviation and mean squared error of all these estimators, are that (a) in general, the distributions of biases of most of the estimators are negatively skewed. The biases are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution; (b) the standard deviations are smallest when data are generated from the beta-binomial distribution; and (c) the mean squared errors are smallest when data are generated from the beta-binomial distribution and largest when data are generated from the mixture distribution. Of the 26, nine estimators including the maximum likelihood estimator, an estimator based on the optimal quadratic estimating equations of Crowder (1987), and an analysis of variance type estimator is found to have least amount of bias, standard deviation and mean squared error. Also, the distributions of the bias, standard deviation and mean squared error for each of these estimators are, in general, more symmetric than those of the other estimators. Our findings regarding efficiency are that the estimator based on the optimal quadratic estimating equations has consistently high efficiency and least variability in the efficiency results. In the important range in which the intraclass correlation is small (≤0 5), on the average, this estimator shows best efficiency performance. The analysis of variance type estimator seems to do well for larger values of the intraclass correlation. In general, the estimator based on the optimal quadratic estimating equations seems to show best efficiency performance for data from the beta-binomial distribution and the probit normal binomial distribution, and the analysis of variance type estimator seems to do well for data from the mixture distribution.     

A test for correlation based on Kendall's tau

A consistent estimator for the variance of Kendall's tau is proposed which allows for testing the hypothesis of no correlation in a bivariate distribution. The null distribution of the test statistic is tabulated under independence, and the properties of the test are discussed.     

Correlation type gogdness-of-fit statistics with censored sampling

Some comments are made concerning the possible forms of a correlation coefficient type goodness-of-fit statistic, and their relationship with other goodness-of-fit statistics, Critical values for a correlation goodness-of-fit statistic and for the Cramer-von Mises statistic are provided for testing a completely-specified null hypothesis for both complete and censored sampling, Critical values for a correlation test statistic are provided for complete and censored sampling for testing the hypothesis of normality, two parameter exponentiality, Weibull (or, extreme value) and an exponential-power distribution, respectively. Critical values are also provided for a test of one-parameter exponentiality based on the Cramer-von Mises statistic     

THE non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix is diagonal

This article deals with the exact non-null distribution of the likelihood ratio criterion for testing the hypothesis that the covariance matrix in a multinormal distribution is diagonal. The exact non-null moments as well as the exact non-null distribution are derived. The distribution is also expressed in computable form with the help of inverse Mellin transform and calculus of residues. The results obtained in this article are useful in studying the power of testing several correlation coefficients simultaneously.