A Generalized Correlation Coefficient for the General Two-Treatment Crossover Design

This article extends the correlation methodology developed by Chinchilli et al. (2005 Chinchilli , V. M. , Phillips , B. R. , Mauger , D. T. , Szefler , S. J. ( 2005 ). A general class of correlation coefficients for the 2 × 2 crossover design . Biometr. J. 47 : 1 – 10 . [Google Scholar]) for the 2 × 2 crossover design to more complex crossover designs for clinical trials. We describe how the methodology can be adapted to a general type of two-treatment crossover design which includes either at least two sequences or at least two treatment periods or both. We then derive the asymptotic theory for the corresponding correlation statistics, investigate the statistical accuracy of the estimators via bootstrap analyses, and demonstrate their use with two real data examples.     

A Note on the Derivation of Fisher's Transformation of the Correlation Coefficient

Fisher's transformation of the bivariate-normal correlation coefficient is usually derived as a variance-stabilizing transformation and its normalizing property is then demonstrated by the reduced skewness of the distribution resulting from the transformation. In this note the transformation is derived as a normalizing transformation that incorporates variance stabilization. Some additional remarks are made on the transformation and its uses.     

半参数变系数回归模型的空间相关性检验

对半参数变系数回归模型,构造了新的空间相关性检验统计量,利用三阶矩 逼近方法导出了其检验 值的近似计算公式,蒙特卡罗模拟结果表明该统计量在检测空间相关性方面具有较高的准确性和可靠性。同时考察了误差项服从不同分布时的检验功效,体现出该检验方法的稳健性。进一步,我们还给出了检验统计量的Bootstrap方法以及检验水平的模拟效果。     

Sample Sizes in the Interval Estimation of the Correlation Coefficient

An expression is derived for the maximum length of the interval estimator of the correlation coefficient, p, under bivariate normal assumptions. The prespecification of this minimum attainable precision and the confidence level results in an expression for the sample size required. An approximate expression for the sample size is proposed and is numerically shown to be as good as or better than that based on the Fisher's Z transformation.     

The Fisher Transform of the Pearson Product Moment Correlation Coefficient and Its Square: Cumulants,Moments, and Applications

This paper extends results on the distribution of the Fisher transform of the correlation coefficient (Fisher, 1921 Fisher , R. A. ( 1921 ). On the “probable error” of a coefficient of correlation deduced from a small sample . Metron 1 : 1 – 32 . [Google Scholar]). Approaches to obtain exact moments of the Fisher transform for both null and non-null correlations are presented. We extend the classic series expansion formulae of Hotelling (1953 Hotelling , H. ( 1953 ). New light on the correlation coefficient and its transforms . Journal of the Royal Statistical Society, Ser. B 15 : 192 – 232 . [Google Scholar]) for the moments of the Fisher transform. These results are considered in the context of quadratic functions of the Fisher transform. Some applications of these results are discussed in the context of correlational hypothesis tests and confidence intervals, and a Monte Carlo experiment is used to demonstrate how application of these results impact the small sample performance of select tests on correlations.     

相关系数统计量的功能及其应用探讨——以SPSS为分析工具

相关系数分为简单相关系数、复相关系数和偏相关系数,他们的含义、功能、应用和计算机的实现操作各不相同。尽管相关系数统计量的计算可由多种软件完成,但从操作的简单性、结果的解释性方面看,SPSS、马克威等傻瓜软件应作为首选。选择因变量的影响因素时,我们主张使用偏相关系数,反对使用简单相关系数。     

Inferences Based on a Skipped Correlation Coefficient

The most popular method for trying to detect an association between two random variables is to test H ₀ ?:?ρ=0, the hypothesis that Pearson's correlation is equal to zero. It is well known, however, that Pearson's correlation is not robust, roughly meaning that small changes in any distribution, including any bivariate normal distribution as a special case, can alter its value. Moreover, the usual estimate of ρ, r, is sensitive to only a few outliers which can mask a true association. A simple alternative to testing H ₀ ?:?ρ =0 is to switch to a measure of association that guards against outliers among the marginal distributions such as Kendall's tau, Spearman's rho, a Winsorized correlation, or a so-called percentage bend correlation. But it is known that these methods fail to take into account the overall structure of the data. Many measures of association that do take into account the overall structure of the data have been proposed, but it seems that nothing is known about how they might be used to detect dependence. One such measure of association is selected, which is designed so that under bivariate normality, its estimator gives a reasonably accurate estimate of ρ. Then methods for testing the hypothesis of a zero correlation are studied.     

The Sample Coefficient of Determination in Simple Linear Regression

The behavior of the sample coefficient of determination is examined for some arrangements of independent variable values in a simple linear regression with normally distributed error terms. Numerical values of means and standard deviations are presented that provide some insight into the influence of range and arrangement of independent variable values and sample size on the size of the sample coefficient of determination. Some asymptotic results are given.     

Use of the Correlation Coefficient with Normal Probability Plots

The use of the correlation coefficient is suggested as a technique for summarizing and objectively evaluating the information contained in probability plots. Goodness-of-fit tests are constructed using this technique for several commonly used plotting positions for the normal distribution. Empirical sampling methods are used to construct the null distribution for these tests, which are then compared on the basis of power against certain nonnormal alternatives. Commonly used regression tests of fit are also included in the comparisons. The results indicate that use of the plotting position p_i = (i - .375)/(n + .25) yields a competitive regression test of fit for normality.     

A Bivariate Distribution Family with Specified Correlation Coefficient and Given Marginals

For given continuous distribution functions F(x) and G(y) and a Pearson correlation coefficient ρ, an algorithm is provided to construct a sequence of continuous bivariate distributions with marginals equal to F(x) and G(y) and the corresponding correlation coefficient converges to ρ. The algorithm can be easily implemented using S-Plus or R. Applications are given to generate bivariate random variables with marginals including Gamma, Beta, Weibull, and uniform distributions.     

基尼系数合意值和警戒线的探讨

 近年来，很多学者对基尼系数的经验法则尤其是警戒线提出了质疑。本文首先分析了收入分配的合意类型，然后依据相对贫困、贫富差距及中等收入者的收入份额三个指标来界定合意的收入分配，最后根据合意收入分配确定基尼系数的合意值和警戒线。本文结论表明，0.3—0.4可以被认为是基尼系数的合意值,0.4作为基尼系数警戒线也是比较合理的。     

Evaluating the Significance Test When the Correlation Coefficient is Different from Zero in the Test of Hypothesis

Sample size and correlation coefficient of populations are the most important factors which influence the statistical significance of the sample correlation coefficient. It is observed that for evaluating the hypothesis when the observed value of the correlation coefficient's r is different from zero, Fisher's Z transformation may be incorrect for small samples especially when population correlation coefficient ρ has big values. In this study, a simulation program has been generated for to illustrate how the bias in the Fisher transformation of the correlation coefficient affects estimate precision when sample size is small and ρ has big value. By the simulation results, 90 and 95% confidence intervals of correlation coefficients have been created and tabled. As a result, it is suggested that especially when ρ is greater than 0.2 and sample sizes of 18 or less, Tables 1 and 2 can be used for the significance test in correlations.     

An Empirical Likelihood Estimate of the Finite Population Correlation Coefficient

In this article, the problem of the estimation of finite population correlation coefficient is considered using the empirical likelihood method. A new estimator that makes the use of both the known mean and variance of an auxiliary variable is proposed. The percent relative bias and percent relative efficiency of the proposed new estimator with respect to the usual estimator of the correlation coefficient is investigated through extensive simulation study for values of the correlation coefficient from ?0.90 to +0.90. The proposed estimator is found to perform better than the simple correlation coefficient from both the bias and relative efficiency points of views, for the population, considered in the investigation. At the end, the proposed estimator has been extended to complex survey designs. Supplementary materials for this article are available online.     

组内相关系数的理论基础及建模应用

对用于测度数据可靠性的统计指标组内相关系数进行了理论基础论证、建模步骤分析、软件实现说明和实际数据统计分析;对组内相关系数建模中涉及的方差分析模型选择、相对一致性和绝对一致性区分、分析结果解释等都进行了详细说明。     

Estimation of the Intraclass Correlation Coefficient for the Analysis of Covariance Model

For the unbalanced analysis of covariance model with one covariate, a simple formula is given for the intraclass correlation coefficient estimator that results from Henderson's Method 3 estimation of variance components. Example calculations and the corresponding interpretations are given for a study of the correlation of iron content among brothers. The example illustrates the manner in which the estimator depends on the pattern of correlation between the covariate and the variable under investigation.     

A Note on the Partial Correlation Coefficient

The question of why a geometric or coordinate-free approach to linear models has been subordinated to an algebraic approach is considered by reviewing selected papers having a geometric slant. These begin with R.A. Fisher's 1915 paper on the distribution of the correlation coefficient and continue through William Kruskal's elegant 1975 paper on the geometry of generalized inverses. The thesis is put forward that the relative unpopularity of the geometric approach is not due to an inherent inferiority but rather to a combination of inertia, poor exposition, and a resistance to abstraction.     

部分函数型线性变系数模型的序列相关检验

部分函数型线性变系数模型(PFLVCM)是近几年出现的一个比较灵活、应用广泛的新模型。在实际应用中,搜集到的经济和金融数据往往存在序列相关性。如果不考虑数据间的相关性直接对其进行建模,会影响模型中参数估计的精度和有效性。本文主要研究了PFLVCM中误差的序列相关性的检验问题,基于经验似然,把标量时间序列数据相关性检验的方法拓展到函数型数据中,提出了经验对数似然比检验统计量,并在零假设下得到了检验统计量的近似分布。通过蒙特卡洛数值模拟说明该统计量在有限样本下有良好的水平和功效。最后,把该方法用于检验美国商业用电消费数据是否有序列相关性,证明该统计量的有效性和实用性。     

An Approach of Stocks Substitution Strategy Using Fuzzy Interval Correlation Coefficient

The classical Pearson's correlation coefficient has been widely adopted in various fields of application. However, when the data are composed of fuzzy interval values, it is not feasible to use such a traditional approach to evaluate the correlation coefficient. In this study, we propose the specific calculation of fuzzy interval correlation coefficient with fuzzy interval data to measure the relationship between various stocks. As such, the study is able to offer an improving measure of investment strategy for stocks substitution via the analysis of the fuzzy interval correlation. In addition, we use empirical studies to verify the validity of our proposal on fuzzy interval correlation coefficient using data from companies in electric machinery and plastic sectors in Taiwan.     

Robustness of the Multiple Correlation Coefficient When Sampling From a Mixture of Two Multivariate Normal Populations

The density of the multiple correlation coefficient is derived by direct integration when the sample covariance matrix has a linear non-central distribution. Using the density, we deduce the null and non-null distribution of the multiple correlation coefficient when sampling from a mixture of two multivariate normal populations with the same covariance matrix. We also compute actual significance levels of the test of the hypothesis H_o : ρ_1·2…p = 0 versus H_a:ρ_1·2…p > 0, given the mixture model.