ON THE STRUCTURE OF PEARSON'S BISERIAL CORRELATION

Pearson (1909) introduced a special method for estimating the correlation coefficient between a two-level variate X and a c-level variate Y , where it is assumed that the regression of Y on X is linear. Pearson's estimator, called biserial correlation, is thus obtained from a sample of size n from a 2 × c table. On the other hand, Lancaster and Hamdan (1964) introduced the polychoric estimator of the correlation between two variates X and Y from a sample available in the form of an r×c contingency table. Later, Hamdan (1968) proved that Pearson's (1900) tetrachoric estimator is a special case of Lancaster and Hamdan polychoric estimator. The present paper applies the orthonor-mal technique which underlies Lancaster's partition of chi-squared (1949) to show that Pearson's biserial correlation is a special form of Lancaster and Hamdan's polychoric estimator.     

Bootstrap power of the generalized correlation coefficient

We present a bootstrap Monte Carlo algorithm for computing the power function of the generalized correlation coefficient. The proposed method makes no assumptions about the form of the underlying probability distribution and may be used with observed data to approximate the power function and pilot data for sample size determination. In particular, the bootstrap power functions of the Pearson product moment correlation and the Spearman rank correlation are examined. Monte Carlo experiments indicate that the proposed algorithm is reliable and compares well with the asymptotic values. An example which demonstrates how this method can be used for sample size determination and power calculations is provided.     

Correlation curve estimation for multiplicative distortion measurement errors data

A correlation curve measures the strength of the association between two variables locally at different values of covariate. This paper studies how to estimate the correlation curve under the multiplicative distortion measurement errors setting. The unobservable variables are both distorted in a multiplicative fashion by an observed confounding variable. We obtain asymptotic normality results for the estimated correlation curve. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimator. The estimated correlation curve is applied to analyze a real dataset for an illustration.     

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.     

Robust regression estimators compared via monte carlo

This paper presents results of a Monte Carlo simulation of eight families of robust regression estimators in various situations. The effects studied include long-tailed error terms, measurement error in the independent variables, various spacings of the independent variables, different sample sizes and correlation between the independent variables. An estimator that combines the best features of several of the estimators is recommended for further study.     

On blest's measure of rank correlation

Blest (2000) proposed a new nonparametric measure of correlation between two random variables. His coefficient, which is dissymmetric in its arguments, emphasizes discrepancies observed among the first ranks in the orderings induced by the variables. The authors derive the limiting distribution of Blest's index and suggest symmetric variants whose merits as statistics for testing independence are explored using asymptotic relative efficiency calculations and Monte Carlo simulations.     

A distribution-free approach to inducing rank correlation among input variables

A method for inducing a desired rank correlation matrix on a multivariate input random variable for use in a simulation study is introduced in this paper. This method is simple to use, is distribution free, preserves the exact form of the marginal distributions on the input variables, and may be used with any type of sampling scheme for which correlation of input variables is a meaningful concept. A Monte Carlo study provides an estimate of the bias and variability associated with the method. Input variables used in a model for study of geologic disposal of radioactive waste provide an example of the usefulness of this procedure. A textbook example shows how the output may be affected by the method presented in this paper.     

A Test of Independence in Two-Way Contingency Tables Based on Maximal Correlation

Maximal correlation has several desirable properties as a measure of dependence, including the fact that it vanishes if and only if the variables are independent. Except for a few special cases, it is hard to evaluate maximal correlation explicitly. We focus on two-dimensional contingency tables and discuss a procedure for estimating maximal correlation, which we use for constructing a test of independence. We compare the maximal correlation test with other tests of independence by Monte Carlo simulations. When the underlying continuous variables are dependent but uncorrelated, we point out some cases for which the new test is more powerful.     

On testing the skew normal hypothesis

In this work two goodness-of-fit tests are proposed for the skew normal distribution, based on properties of this family of distributions and the sample correlation coefficient. The critical values for the tests are obtained by using Monte Carlo simulation for several sample sizes and levels of significance. The power of the proposed tests are compared with that of the tests studied by Mateu et al. (2007) and the one studied by Meintanis (2007) for several sample sizes and considering diverse alternatives. The results show that the proposed tests have greater power than those studied by Mateu et al. (2007) and Meintanis (2007) against some alternative distributions.     

Testing for heteroscedasticity of exponential correlation mixed-effects linear models based on M-estimation

Homogeneity of variance is a basic assumption in longitudinal data analysis. However, the assumption is not necessarily appropriate. In this paper, Fisher scoring method is applied to get M-estimator in the exponential correlation mixed-effects linear model. The score tests for heteroscedasticity and correlation coefficient based on M-estimator are then studied. Monte Carlo method is applied to investigate the properties of test statistics. At last, the methods and properties are illustrated by an actual data example.     

A permutation procedure for testing the equality of pattern hypotheses across groups involving correlation or covariance matrices

This paper describes a permutation procedure to test for the equality of selected elements of a covariance or correlation matrix across groups. It involves either centring or standardising each variable within each group before randomly permuting observations between groups. Since the assumption of exchangeability of observations between groups does not strictly hold following such transformations, Monte Carlo simulations were used to compare expected and empirical rejection levels as a function of group size, the number of groups and distribution type (Normal, mixtures of Normals and Gamma with various values of the shape parameter). The Monte Carlo study showed that the estimated probability levels are close to those that would be obtained with an exact test except at very small sample sizes (5 or 10 observations per group). The test appears robust against non-normal data, different numbers of groups or variables per group and unequal sample sizes per group. Power was increased with increasing sample size, effect size and the number of elements in the matrix and power was decreased with increasingly unequal numbers of observations per group.     

Fatal shortcomings of stationary AR (1) exogeneous variables in econometric monte carlo tudies of estimators

The paper first shows that the stationary normal AR(1) process (SNAR1), the most frequently used process for generating exogenous variables in econometric Monte Carlo studies, cannot generate realistic exogenous variables, which are generally trended and similar to those generated by ARIMA (p,d,q) process withd≧1 and positive drift (trend). Then, it illustrates that in the context of AR(1) disturbances,trends in exogenous variables can frequently alter the very ranking of two competing estimators, the ordinary least squares estimator (OLS) and the Cochrane-Orcutt estimators (CO). For three common econometric models—a standard regression model, a dynamic model (i.e., a model with a lagged dependent variable), and a seemingly unrelated regression model, OLS becomes superior in many cases. This is so in spite of the fact that the CO estimator in the study utilizes the true value of the first-order autocorrelation coefficient of the disturbances. The message to be derived from these findings should be ccear. If one accepts the fact that most if not all economic time series are trended, and endorses a proposition that the fundamental if not sole purpose of Monte Carlo studies in econometrics should be to provide useful guidelines to practicing econometricians, then, he must not employ SNARl (nor anyother artificially created nontrended series) as a generator of exogenous variables in a Monte Carlo study, at least in the econometrics of autocorrelated disturbances. Alternative methods of generating stochastic exogenous variables that are trended are suggested in the paper. For almost four decades, the principle of the autoregressive transformation of a regression model with first-order autocorrelated disturbances (the Coestimation priciple) has been taken for granted as a method of correcting for the autocorrelation in the disturbances—be it in the two-stage Cochrane—Orcutt estimator, the iterative Cochrane-Orcutt estimator, or an estimator utilizing nonlinear techniques or search procedures. (Comitting the first observation due to transformation is not considered very crucial in general.) The results of the pertinent Monte Carlo studies appear to justify such a procedure only because most studies have employed SNARl exogenous variables, not trended ones. Thus, Monte Carlo experimenters must be blamed, at least partially, for this prevailining malpractice. It is hoped that they will not commit additional sins by not using realistic data in their future experiments.     

Modified Normal-based Approximation to the Percentiles of Linear Combination of Independent Random Variables with Applications

A modified normal-based approximation for calculating the percentiles of a linear combination of independent random variables is proposed. This approximation is applicable in situations where expectations and percentiles of the individual random variables can be readily obtained. The merits of the approximation are evaluated for the chi-square and beta distributions using Monte Carlo simulation. An approximation to the percentiles of the ratio of two independent random variables is also given. Solutions based on the approximations are given for some classical problems such as interval estimation of the normal coefficient of variation, survival probability, the difference between or the ratio of two binomial proportions, and for some other problems. Furthermore, approximation to the percentiles of a doubly noncentral F distribution is also given. For all the problems considered, the approximation provides simple satisfactory solutions. Two examples are given to show applications of the approximation.     

Clustering of correlation matrices: A step-wise procedure

An alternative to the criteria proposed by King [9] for clustering correlation matrices is explored. Several simple examples examined by total enumeration employing our criterion indicate that the criterion yields clusters with high within group correlation. A step-wise routine is suggested for problems of realistic size and is applied to the examples presented in King's article. Additional evidence in the form of several Monte Carlo experiments indicates that the routine performs satisfactorily in determining the optimal separation of variables into two groups.     

Flexible Bivariate Count Data Regression Models

The article develops a semiparametric estimation method for the bivariate count data regression model. We develop a series expansion approach in which dependence between count variables is introduced by means of stochastically related unobserved heterogeneity components, and in which, unlike existing commonly used models, positive as well as negative correlations are allowed. Extensions that accommodate excess zeros, censored data, and multivariate generalizations are also given. Monte Carlo experiments and an empirical application to tobacco use confirms that the model performs well relative to existing bivariate models, in terms of various statistical criteria and in capturing the range of correlation among dependent variables. This article has supplementary materials online.     

Ridge estimation in regression problems with autocorrelated errors: A monte carlo study

This paper presents the results of a Monte Carlo study of OLS and GLS based adaptive ridge estimators for regression problems in which the independent variables are collinear and the errors are autocorrelated. It studies the effects of degree of collinearity, magnitude of error variance, orientation of the parameter vector and serial correlation of the independent variables on the mean squared error performance of these estimators. Results suggest that such estimators produce greatly improved performance in favorable portions of the parameter space. The GLS based methods are best when the independent variables are also serially correlated.     

Accent on Teaching Materials

A linear combination test for combining several tests of the correlation coefficient in the bivariate normal distribution is proposed. The linear combination test is compared with the well-known Fisher method of combining tests. It is shown by a Monte Carlo study that the linear combination test has a larger power.     

A Goodness-of-Fit Test for Location-Scale Max-Stable Distributions

In this article, a technique based on the sample correlation coefficient to construct goodness-of-fit tests for max-stable distributions with unknown location and scale parameters and finite second moment is proposed. Specific details to test for the Gumbel distribution are given, including critical values for small sample sizes as well as approximate critical values for larger sample sizes by using normal quantiles. A comparison by Monte Carlo simulation shows that the proposed test for the Gumbel hypothesis is substantially more powerful than some other known tests against some alternative distributions with positive skewness coefficient.     

A simple generalization of the Box–Muller method for obtaining a pair of correlated standard normal variables

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.     

Testing Serial Correlation in Semiparametric Varying-Coefficient Partially Linear Models

We propose two test statistics for testing serial correlation in semiparametric varying-coefficient partially linear models. The proposed test statistics are not only for testing zero first-order serial correlation, but also for testing higher-order serial correlations. Under the null hypothesis of no serial correlation, the test statistics are shown to have asymptotic normal or chi-square distributions. By using R, some Monte Carlo experiments are conducted to examine the finite sample performances of the proposed tests. Simulation results show that the estimated size and power of the proposed tests behave well.