Interval Estimation for the Correlation Coefficient

ABSTRACT

The correlation coefficient (CC) is a standard measure of a possible linear association between two continuous random variables. The CC plays a significant role in many scientific disciplines. For a bivariate normal distribution, there are many types of confidence intervals for the CC, such as z-transformation and maximum likelihood-based intervals. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is not well-developed. In this paper, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval for the CC when the underlying bivariate distribution is a normal distribution, and two empirical likelihood-based intervals for the CC when the underlying bivariate distribution is unknown. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.     

Robustness and monotonicity properties of generalized correlation coefficients

A new class of generalized correlation coefficients that contains the Pearson and Kendall statistics as special cases was defined by Chinchilli et al. (2005) and applied to the estimation of correlations coefficients within the context of 2×2 cross-over designs for clinical trials. In this paper, we determine the infinitesimal robustness and local stability properties of these generalized correlation coefficients by deriving their corresponding influence functions. For cases in which the population distribution is a bivariate normal or a mixture of bivariate normal distributions we obtain explicit formulas, and establish monotonicity and sign-reverse rule properties of the generalized correlation coefficients.     

A latent variable regression model for asymmetric bivariate ordered categorical data

In many areas of medical research, especially in studies that involve paired organs, a bivariate ordered categorical response should be analyzed. Using a bivariate continuous distribution as the latent variable is an interesting strategy for analyzing these data sets. In this context, the bivariate standard normal distribution, which leads to the bivariate cumulative probit regression model, is the most common choice. In this paper, we introduce another latent variable regression model for modeling bivariate ordered categorical responses. This model may be an appropriate alternative for the bivariate cumulative probit regression model, when postulating a symmetric form for marginal or joint distribution of response data does not appear to be a valid assumption. We also develop the necessary numerical procedure to obtain the maximum likelihood estimates of the model parameters. To illustrate the proposed model, we analyze data from an epidemiologic study to identify some of the most important risk indicators of periodontal disease among students 15-19 years in Tehran, Iran.     

Regression mean residual life of a parallel system of dependent components

In this paper, we consider a system consisting of two dependent components and we are interested in the average remaining life of the component that fails last when (i) the first failure occurs at time t and (ii) the first failure occurs after time t. For both the cases, expressions are derived in the case of general bivariate normal distribution and a class of bivariate exponential distribution including bivariate exponential distribution of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Moments and properties of multiplicatively constrained bivariate lognormal distribution with applications to futures hedging

In this paper, we derive explicit expressions for marginal and product moments of a bivariate lognormal distribution when a multiplicative constraint is present. We show that the coefficients of variation always decrease regardless of the multiplicative constraint imposed. We also evaluate the effects of the constraint on the variances and covariance, and present conditions under which the correlation coefficient increases under the presence of such a multiplicative constraint. We finally apply these results to futures hedging analysis and some other financial applications.     

A Method for Multivariate Probability Distributions Construction via Parameter Dependence

The nature of stochastic dependence in the classic bivariate normal density framework is analyzed. In the case of this distribution we stress the way the conditional density of one of the random variables depends on realizations of the other. Typically, in the bivariate normal case this dependence takes the form of a parameter (here the “expected value”) of one probability density depending continuously (here linearly) on realizations of the other random variable. Our point is that such a pattern does not need to be restricted to that classical case of bivariate normal. We show that this paradigm can be generalized and viewed in ways that allows us to extend it far beyond the bivariate normal distributions class.     

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.     

The approximate null distribution of the likelihood ratio test for a mixture of two bivariate normal distributions with equal co variance

We define the mixture likelihood approach to clustering by discussing the sampling distribution of the likelihood ratio test of the null hypothesis that we have observed a sample of observations of a variable having the bivariate normal distribution versus the alternative that the variable has the bivariate normal mixture with unequal means and common within component covariance matrix. The empirical distribution of the likelihood ratio test indicates that convergence to the chi-squared distribution with 2 df is at best very slow, that the sample size should be 5000 or more for the chi-squared result to hold, and that for correlations between 0.1 and 0.9 there is little, if any, dependence of the null distribution on the correlation. Our simulation study suggests a heuristic function based on the gamma.     

Estimation and goodness-of-fit for the Cox model with various types of censored data

The currently existing estimation methods and goodness-of-fit tests for the Cox model mainly deal with right censored data, but they do not have direct extension to other complicated types of censored data, such as doubly censored data, interval censored data, partly interval-censored data, bivariate right censored data, etc. In this article, we apply the empirical likelihood approach to the Cox model with complete sample, derive the semiparametric maximum likelihood estimators (SPMLE) for the Cox regression parameter and the baseline distribution function, and establish the asymptotic consistency of the SPMLE. Via the functional plug-in method, these results are extended in a unified approach to doubly censored data, partly interval-censored data, and bivariate data under univariate or bivariate right censoring. For these types of censored data mentioned, the estimation procedures developed here naturally lead to Kolmogorov-Smirnov goodness-of-fit tests for the Cox model. Some simulation results are presented.     

SMOOTH TESTS FOR THE BIVARIATE POISSON DISTRIBUTION

A theorem of Rayner & Best (1989) is generalised to permit the construction of smooth tests of goodness of fit without requiring a set of orthonormal functions on the hypothesised distribution. This result is used to construct smooth tests for the bivariate Poisson distribution. The test due to Crockett (1979) is similar to a smooth test that assesses the variance structure under the bivariate Poisson model; the test due to Loukas & Kemp (1986) is related to a smooth test that seeks to detect a particular linear relationship between the variances and covariance under the bivariate Poisson model. Using focused smooth tests may be more informative than using previously suggested tests. The distribution of the Loukas & Kemp (1986) statistic is not well approximated by the x²distribution for larger correlations, and a revised statistic is suggested.     

Tests for bivaroate exponentiality against bifra alternatives based on censored samples

We develop a test procedure to test the hypothesis that the distribution of the lifetime is bivariate exponential of Marshall and Olkin against that it is bivariate increasing failure rate average when the sample is of the type univariate or bivariate randomly censored.     

Small sample comparison of six bivariate survival curve estimators 1

We study the performance of six proposed bivariate survival curve estimators on simulated right censored data. The performance of the estimators is compared for data generated by three bivariate models with exponential marginal distributions. The estimators are compared in their ability to estimate correlations and survival functions probabilities. Simulated data results are presented so that the proposed estimators in this relatively new area of analysis can be explicitly compared to the known distribution of the data and the parameters of the underlying model. The results show clear differences in the performance of the estimators.     

Estimation of bivariate and marginal distributions with censored data

Summary. Consider a pair of random variables, both subject to random right censoring. New estimators for the bivariate and marginal distributions of these variables are proposed. The estimators of the marginal distributions are not the marginals of the corresponding estimator of the bivariate distribution. Both estimators require estimation of the conditional distribution when the conditioning variable is subject to censoring. Such a method of estimation is proposed. The weak convergence of the estimators proposed is obtained. A small simulation study suggests that the estimators of the marginal and bivariate distributions perform well relatively to respectively the Kaplan–Meier estimator for the marginal distribution and the estimators of Pruitt and van der Laan for the bivariate distribution. The use of the estimators in practice is illustrated by the analysis of a data set.     

Inequalities for bivariate distribution with x ≤ y and marginals given

Upper and lower bounds on the joint bivariate distribution are found when the marginals are given and under the additional condition that X ≤ Y with probability one. The upper bound is the same as for the unrestricted bivariate distribution with marginals given, For the lower bound a simple inequality is derived which is exact, that is, achievable, in many cases including normal and exponential marginals.     

An evaluation of methods for testing hypotheses relating to two endpoints in a single clinical trial

The issues and dangers involved in testing multiple hypotheses are well recognised within the pharmaceutical industry. In reporting clinical trials, strenuous efforts are taken to avoid the inflation of type I error, with procedures such as the Bonferroni adjustment and its many elaborations and refinements being widely employed. Typically, such methods are conservative. They tend to be accurate if the multiple test statistics involved are mutually independent and achieve less than the type I error rate specified if these statistics are positively correlated. An alternative approach is to estimate the correlations between the test statistics and to perform a test that is conditional on those estimates being the true correlations. In this paper, we begin by assuming that test statistics are normally distributed and that their correlations are known. Under these circumstances, we explore several approaches to multiple testing, adapt them so that type I error is preserved exactly and then compare their powers over a range of true parameter values. For simplicity, the explorations are confined to the bivariate case. Having described the relative strengths and weaknesses of the approaches under study, we use simulation to assess the accuracy of the approximate theory developed when the correlations are estimated from the study data rather than being known in advance and when data are binary so that test statistics are only approximately normally distributed.     

An inverse-probability-weighted approach to estimation of the bivariate survival function under left-truncation and right-censoring

In this note, we consider estimating the bivariate survival function when both survival times are subject to random left truncation and one of the survival times is subject to random right censoring. Motivated by Satten and Datta [2001. The Kaplan–Meier estimator as an inverse-probability-of-censoring weighted average. Amer. Statist. 55, 207–210], we propose an inverse-probability-weighted (IPW) estimator. It involves simultaneous estimation of the bivariate survival function of the truncation variables and that of the censoring variable and the truncation variable of the uncensored components. We prove that (i) when there is no censoring, the IPW estimator reduces to NPMLE of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131] and (ii) when there is random left truncation and right censoring on only one of the components and the other component is always observed, the IPW estimator reduces to the estimator of Gijbels and Gürler [1998. Covariance function of a bivariate distribution function estimator for left truncated and right censored data. Statist. Sin. 1219–1232]. Based on Theorem 3.1 of van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627], we prove that the IPW estimator is consistent under certain conditions. Finally, we examine the finite sample performance of the IPW estimator in some simulation studies. For the special case that censoring time is independent of truncation time, a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by van der Laan [1996a. Nonparametric estimation of the bivariate survival function with truncated data. J. Multivariate Anal. 58, 107–131, 1996b. Efficient estimation of the bivariate censoring model and repairing NPMLE. Ann. Statist. 24, 596–627]. For the special case (i), a simulation study is conducted to compare the performances of the IPW estimator against that of the estimator proposed by Huang et al. (2001. Nonnparametric estimation of marginal distributions under bivariate truncation with application to testing for age-of-onset application. Statist. Sin. 11, 1047–1068).     

Sample sizes for trials involving multiple correlated must-win comparisons

In Clinical trials involving multiple comparisons of interest, the importance of controlling the trial Type I error is well-understood and well-documented. Moreover, when these comparisons are themselves correlated, methodologies exist for accounting for the correlation in the trial design, when calculating the trial significance levels. However, less well-documented is the fact that there are some circumstances where multiple comparisons affect the Type II error rather than the Type I error, and failure to account for this, can result in a reduction in the overall trial power. In this paper, we describe sample size calculations for clinical trials involving multiple correlated comparisons, where all the comparisons must be statistically significant for the trial to provide evidence of effect, and show how such calculations have to account for multiplicity in the Type II error. For the situation of two comparisons, we provide a result which assumes a bivariate Normal distribution. For the general case of two or more comparisons we provide a solution using inflation factors to increase the sample size relative to the case of a single outcome. We begin with a simple case of two comparisons assuming a bivariate Normal distribution, show how to factor in correlation between comparisons and then generalise our findings to situations with two or more comparisons. These methods are easy to apply, and we demonstrate how accounting for the multiplicity in the Type II error leads, at most, to modest increases in the sample size.     

Adaptive designs for dose-finding based on efficacy–toxicity response

We propose a new adaptive procedure for dose-finding in clinical trials when both efficacy and toxicity responses are available. We model the distribution of this bivariate binary endpoint using either Gumbel bivariate logistic regression or Cox bivariate binary model. In both cases, the analytic formulae for the Fisher information matrix are obtained, that form the basis for derivation of the locally optimal and adaptive designs.     

Properties and applications of the sarmanov family of bivariate distributions

We discuss properties of the bivariate family of distributions introduced by Sarmanov (1966). It is shown that correlation coefficients of this family of distributions have wider range than those of the Farlie-Gumbel-Morgenstern distributins. Possible applications of this family of bivariate distributions as prior distributins in Bayesian inference are discussed. The density of the bivariate Sarmanov distributions with beta marginals can be expressed as a linear combination of products of independent beta densities. This pseudoconjugate property greatly reduces the complexity of posterior computations when this bivariate beta distribution is used as a prior. Multivariate extensions are derived.