Bivariate Sample Research Articles

Estimation of the Correlation Coefficient Using Bivariate Ranked Set Sampling with Application to the Bivariate Normal Distribution

Bivariate ranked set sampling BVRSS was introduced by Al-Saleh and Zheng (2002) as a bivariate version of the ordinary ranked set sampling (RSS). The procedure can be used when we deal with two characteristics simultaneously. In this article, the BVRSS procedure is used to estimate the correlation coefficient between two variables. The proposed estimators are compared to other existing estimators based on bivariate simple random sample BVSRS. The case of bivariate normal distribution is considered in details.

Linking survey data with administrative health information: characteristics associated with consent from a neonatal intensive care unit follow-up study.

Health services and population health research often depends on the ready availability of administrative health data. However, the linkage of survey-based data to administrative data for health research purposes has raised concerns about privacy. Our aim was to compare consent rates to data linkage in two samples of caregivers and describe characteristics associated with consenters. Subjects included caregivers of children admitted at birth to neonatal intensive care units (NICU) in British Columbia and caregivers of a sample of healthy children. Caregivers were asked to sign a consent form enabling researchers to link the survey information with theirs and their child's provincially collected health records. Bivariate analysis identified sample characteristics associated with consent. These were entered into logistic regression models. The sample included 1,140 of 2,221 NICU children and 393 of 718 healthy children. The overall response rate was 55% and the response rate for located families was 67.1%. Consent to data linkage with the child data was given by 71.6% of respondents and with caregiver data by 67% of respondents. Families of healthy children were as likely to provide consent as families of NICU children. Higher rates of consent were associated with being a biological parent, not requiring survey reminders, involvement in a parent support group, not working full-time, having less healthy children, multiple births and higher income. The level of consent achieved suggests that when given a choice, most people are willing to permit researcher access to their personal health information for research purposes. There is scope for educating the public about the nature and importance of research that combines survey and administrative data to address important health questions.

Nonparametric Tests for Independence Between Lifetimes and Covariates from Censored Bivariate Normal Samples

ABSTRACT Three run-based and two rank-based nonparametric tests are proposed for testing independence between lifetimes and covariates from censored bivariate normal samples. Simulated powers for the suggested nonparametric tests and the T-test based on the sample correlation coefficient are determined and compared for samples from the standard bivariate normal distribution for various sample sizes, proportions of censoring, and correlation values. To investigate the robustness of these tests to departures from bivariate normality, simulated powers for all six statistics are found when the samples are from a bivariate exponential, bivariate t distribution, and bivariate normal distributions with scale-outliers or location-outliers. Based on these results, some comments are finally made.

On Bivariate Ranked Set Sampling for Distribution and Quantile Estimation and Quantile Interval Estimation Using Ratio Estimator

As an extension to ranked set sampling (RSS), bivariate ranked set sampling (BVRSS) was introduced by Al-Saleh and Zheng [Al-Saleh, M. F., Zheng, G. (2002). Estimation of bivariate characteristics using ranked set sampling. Aust. New Zealand J. Statist. 44:221–232] for multiple characteristics estimation. In this paper, BVRSS is investigated for estimating the joint and the marginal distribution functions using direct method of estimation. The estimation of a population quantiles of two characteristics based BVRSS, for given values of probability (p) using direct method of estimation, is investigated. Also, quantiles and quantiles intervals estimation, based on BVRSS, using ratio estimator is presented. It turns out that all the suggested estimators in this paper are more efficient than the corresponding estimators obtained by using bivariate SRS and the ordinary RSS with concomitant variable. Finally, the proposed methods are illustrated using real data set of the trees.

Robustness Properties of the Pitman–Morgan Test

The classical Pitman–Morgan test is known to be optimal for testing equality of the variances of components of a bivariate normal vector. We first show that it is also optimal for a generalized model involving the matrix spherical distribution. Then we discuss and demonstrate, both analytically and empirically, that it is nonrobust, i.e., its type I error control is inexact both asymptotically and in moderate size bivariate random samples.

Impact of censoring on sample variances in a bivariate normal model

Suppose a bivariate normal random sample of size n is subjected to Type II censoring on one of the variates so that only a set of p order statistics and their concomitants are observed. For singly and symmetrically censored samples, we obtain close approximations to the distributions of sample variances of the observed order statistics and their concomitants by gamma distributions. We use simulation, method of maximum likelihood and moment matching techniques to express the gamma parameters involved as explicit functions of n and p, and in the case of concomitants, also the correlation coefficient. We examine the goodness-of-fit of these gamma approximations and the accuracy of the associated estimates of the percentiles of the two sample variances.

Estimating the product-moment correlation in samples with censoring on both variables

Bivariate samples may be subject to censoring of both random variables. For example, for two toxins measured in batches of wheat grain, there may be specific detection limits. Alternatively, censoring may be incomplete over a certain domain, with the probability of detection depending on the toxin level. In either case, data are not missing at random, and the missing data pattern bears some information on the parameters of the underlying model (informative missingness), which can be exploited for a fully efficient analysis. Estimation (after suitable data transformation) of the correlation in such samples is the subject of the present paper. We consider several estimators. The first is based on the tetrachoric correlation. It is simple to compute, but does not exploit the full information. The other two estimators exploit all information and use full maximum likelihood, but involve heavier computations. The one assumes fixed detection limits, while the other involves a logistic model for the probability of detection. For a real data set, a logistic model for the probability of detection fitted markedly better than a model with fixed detection limits, suggesting that censoring is not complete.

Number of Records in a Bivariate Sample with Application to Missouri River Flood Data

For a sequence of observations from a bivariate absolutely continuous distribution, two types of records are considered depending on whether a univariate record is established in both or in at least one of the components. The distributional properties of the associated univariate and bivariate record indicators are examined. Correlation between the number of component records and the first two moments of the number of bivariate records in a finite random sample are obtained. These are evaluated for the Farlie-Gumbel-Morgenstern and bivariate normal distributions. Large sample properties of these moments are explored. Our results are used to predict the number of record annual floods at two sites along the Missouri river during the next 50 years.

Fisher Information in an Order Statistic and its Concomitant

Let (X, Y) have an absolutely continuous distribution with parameter θ. We suggest regularity conditions on the parent distribution that permit the definition of Fisher information (FI) about θ in an X-order statistic and its Y-concomitant that are obtained from a random sample from (X, Y). We describe some general properties of the FI in such individual pairs. For the Farlie-Gumbel-Morgenstern parent with dependence parameter θ, we investigate the properties of this FI, and obtain the asymptotic relative efficiency of the maximum likelihood estimator of θ for Type II censored bivariate samples. Assuming (X, Y) is Gumbel bivariate exponential of second type, and θ is the mean of Y, we evaluate the FI in the Y-concomitant of an X-order statistic and compare it with the FI in a single Y-order statistic.

An Algorithm for Computing the Best Convex Contrast Curve of Two Bivariate Samples

An Algorithm for Computing the Best Convex Contrast Curve of Two Bivariate Samples

An Algorithm for Computing the Best Convex Contrast Curve of Two Bivariate Samples

Let (x 1, y 1),…, (x n, y n) be a sample of points in , consisting of two subsamples (x 1, y 1),…, (x n1, y n1) and (x n1+1, y n1+1),…, (x n, y n), where n = n 1 + n 2. We consider the problem of separating the two subsamples by a convex contrast curve k…that is, a real-valued convex function x → k(x). For given curve k we consider the relative frequency p 1 of points of the first sample above the graph of k similarly p 2. The goal is to choose k such that the difference in relative frequency p 1 – p 2 becomes maximal. This maximal value can be regarded as a generalized one-sided Kolmogorov-Smirnov test statistic , where p 1(A) = (1/n 1)#{i ≤ n 1 : (xi, y i) ∈ A}, p 2(A) = (1/n 2#{n 1 + 1 ≤ i ≤ n : (x i, y i) ∈ A} and is the class of all epigraphs {(x y) : k(x) ≤ y} of convex functions k. The standard Kolmogorov-Smirnov test statistic corresponds to the class consisting of all sets —that is, epigraphs of constant functions. Test statistics of this form arise in nonparametric analysis of covariance where the regression function is assumed to be convex. In this article we give a recursive relation and an algorithm based on it to compute the value KS and an optimal convex contrast curve in O(n 3) steps. The convexity assumptions on the sets of and on the nonparametric regression function are interrelated; this is explained in the large sample situation where n 1 Λ n 2 → + ∞.

Asymptotic efficiency of independence tests based on gini's rank association coefficient, spearman's footrule and their generalizations

Gini's rank association coefficient and Spearman's footrule, as statistics for testing independence in bivariate samples, are as natural as Spearman's and Kendall's rank correlation coefficients, but their efficiency properties are not well explored. We find here the expression for the local Bahadur efficiency of Gini's test and Spearman's footrule for general alternatives. Several examples are given in which both statistics behave better than Spearman's and Kendall's coefficients. Similar results are obtained the general measure of monotone dependence considered recently by Conti et al. (1996). The coincidence between Pitman and Bahadur efficiencies is also proved.

Using Median Lines in Robust Bivariate Data Analysis

Methods for robust comparison of bivariate errors-in-variables are considered. The concept of median lines is introduced for the robust estimation of principal components. Median lines separate the bivariate sample space into two equally sized parts. Statistical properties of the model parameters are derived. Robust residual analysis assesses linear relationships as well as goodness of fit and allows for the detection of potential outliers. Special emphasis is laid on graphical methods. A bivariate box-plot is proposed for exploratory data analysis. The median lines procedure is illustrated by a real example.

Bootstrapping correlation coefficients using univariate and bivariate sampling.

Bootstrapping correlation coefficients using univariate and bivariate sampling.

Bootstrapping correlation coefficients using univariate and bivariate sampling.

Bootstrapping correlation coefficients using univariate and bivariate sampling.

A comparison of estimators of the proportion nonconforming in univariate and multivariate normal samples

Maximum likelihood (ML) and minimum variance unbiased (MVU) estimators of the proportion nonconforming in univariate and bivariate normal random samples are compared for the case where the moments of the distribution are assumed to be unknown and each variable has lower and upper specification limits. Both types of estimator have skewed distributions when the proportion nonconforming and sample size are small, and the MVU estimator has a substantial probability of being zero in these situations. Using Pitman's closeness criterion, the ML estimators are nearly always superior to the MVU estimator for the cases considered. Using the MSE criterion, the MVU estimator is superior to the ML estimator when the distribution of the ML estimator is quite skewed. After transforming the estimators to symmetry, the ML estimator has smaller MSE than the MVU estimator.

Tests for equality of dispersion in divariate samples- review and empirical comparison

This paper investigates the robustness and power of several tests for equality of dispersion in bivariate samples. Parametric tests are shown by Monte Carlo simulation to be very sensitive to departures from the normality assumption. Of the alternative procedures, the extended Brown-Forsythe test and Spearman’s rank correlation of sums and differences are recommended as robust tests.

Testing for heterogeneity among phenotypic correlations: a comparison of methods using Monte Carlo simulations.

Heterogeneous phenotypic correlations may be suggestive of underlying changes in genetic covariance among life-history, morphology, and behavioural traits, and their detection is therefore relevant to many biological studies. Two new statistical tests are proposed and their performances compared with existing methods. Of all tests considered, the existing approximate test of homogeneity of product-moment correlations provides the greatest power to detect heterogeneous correlations, when based on Hotelling's z*-transformation. The use of this transformation and test is recommended under conditions of bivariate normality. A new distribution-free randomisation test of homogeneity of Spearman's rank correlations is described and recommended for use when the bivariate samples are taken from populations with non-normal or unknown distributions. An alternative randomisation test of homogeneity of product-moment correlations is shown to be a useful compromise between the approximate tests and the randomisation tests on Spearman's rank correlations: it is not as sensitive to departures from normality as the approximate tests, but has greater power than the rank correlation test. An example is provided that shows how choice of test will have a considerable influence on the conclusions of a particular study.

Seeing the forest from the trees: When predicting the behavior or status of groups, correlate means.

When measures of individual differences are used to predict group performance, the reporting of correlations computed on samples of individuals invites misinterpretation and dismissal of the data. In contrast, if regression equations, in which the correlations required are computed on bivariate means, as are the distribution statistics, it is difficult to underappreciate or lightly dismiss the utility of psychological predictors. Given sufficient sample size and linearity of regression, this technique produces cross-validated regression equations that forecast criterion means with almost perfect accuracy. This level of accuracy is provided by correlations approaching unity between bivariate samples of predictor and criterion means, and this holds true regardless of the magnitude of the “simple” correlation (e.g., rxy= .20, or rxy= .80). We illustrate this technique empirically using a measure of general intelligence as the predictor and other measures of individual differences and socioeconomic status as criteria. In addition to theoretical applications pertaining to group trends, this methodology also has implications for applied problems aimed at developing policy in education, medical, and psychological clinics, business, industry, the military, and other domains of public welfare. Linkages between this approach and epidemiological research reinforce its utility as a tool for making decisions about policy.

Limit theorems for induced extreme values

For a bivariate sample (Xi, Yi) of size n, let (U(x), V(x)) denote the following pair of induced extreme values: U(x) is the maximum of those Yi-values with corresponding Xi-value less than x and V(x) is the maximum of the remaining Yi-values. In the paper, we study the asymptotic behavior of the (suitably normalized) random vector (U(x), V(x)), and we consider several cases. First, we consider nonrandom x and let x=xn so that as n→∞, xn tends to the endpoint of FX(x), or so that xn tends to x0, a point in the support of FX(x). The second important situation appears when x=Xk∶n, i.e., we select Y-values on the basis of the random variable Xk∶n, the k-th order-statistic of the X-sample. Here we also consider two cases: (i) k=n−j with fixed j, and (ii) k=[np], where 0<p<1. The paper generalizes the earlier results of David, Joshi, and Nagaraja, where it is assumed that (X, Y) is in the bivariate (max-) domain of attraction of a bivariate stable law with independent marginals.