A Monte Carlo Investigation of the Fisher Z Transformation for Normal and Nonnormal Distributions

2000 ◽  
Vol 87 (3_suppl) ◽  
pp. 1101-1114 ◽  
Author(s):  
Kenneth J. Berry ◽  
Paul W. Mielke
Keyword(s):  
Monte Carlo ◽  
Correlation Coefficient ◽  
Bivariate Normal Distribution ◽  
Small Samples ◽  
Bivariate Distributions ◽  
Nonnormal Distributions ◽  
Monte Carlo Investigation ◽  
Sample Correlation ◽  
Heavy Tailed ◽  
Z Transformation

The Fisher transformation of the sample correlation coefficient r (1915, 1921) and two related techniques by Gayen (1951) and Jeyaratnam (1992) are examined for robustness to nonnormality. Monte Carlo analyses compare combinations of sample sizes and population parameters for seven bivariate distributions. The Fisher, Gayen, and Jeyaratnam approaches are shown to provide useful results for a bivariate normal distribution with any population correlation coefficient ρ and for nonnormal bivariate distributions when ρ = 0. In contrast, the techniques are virtually useless for nonnormal bivariate distributions when ρ#0.0. Surprisingly, small samples are found to provide better estimates than large samples for skewed and symmetric heavy-tailed bivariate distributions.

scholarly journals Bivariate Distributions Underlying Responses to Ordinal Variables

Psych ◽  
2021 ◽  
Vol 3 (4) ◽  
pp. 562-578
Author(s):  
Laura Kolbe ◽  
Frans Oort ◽  
Suzanne Jak
Keyword(s):  
Correlation Coefficient ◽  
Latent Variables ◽  
Ordinal Data ◽  
Bivariate Normal Distribution ◽  
Polychoric Correlation ◽  
Bivariate Distributions ◽  
Normal Distributions ◽  
Ordinal Variables ◽  
Bivariate Normal ◽  
Bivariate Normality

The association between two ordinal variables can be expressed with a polychoric correlation coefficient. This coefficient is conventionally based on the assumption that responses to ordinal variables are generated by two underlying continuous latent variables with a bivariate normal distribution. When the underlying bivariate normality assumption is violated, the estimated polychoric correlation coefficient may be biased. In such a case, we may consider other distributions. In this paper, we aimed to provide an illustration of fitting various bivariate distributions to empirical ordinal data and examining how estimates of the polychoric correlation may vary under different distributional assumptions. Results suggested that the bivariate normal and skew-normal distributions rarely hold in the empirical datasets. In contrast, mixtures of bivariate normal distributions were often not rejected.

Distribution of the Determinant of the Sample Correlation Matrix: Monte Carlo Type One Error Rates

1985 ◽  
Vol 10 (4) ◽  
pp. 384-388 ◽  
Author(s):  
John R. Reddon ◽  
Douglas N. Jackson ◽  
Donald Schopflocher
Keyword(s):  
Monte Carlo ◽  
Sample Size ◽  
Correlation Matrix ◽  
Error Rates ◽  
Small Samples ◽  
Multivariate Normal ◽  
Finite Sample ◽  
Sample Correlation

Computer sampling from a multivariate normal spherical population was used to evaluate the type one error rates for a test of sphericity based on the distribution of the determinant of the sample correlation matrix. The range of variates considered was 5 to 25, and the range of observations considered was 10 to 1,000. The type one error rates were estimated in each condition with 5,000 replications. The c.d.f. obtained from the asymptotic χ2 resulted in excessive type one errors in the conditions where the ratio of the sample size to the number of variates was small. The c.d.f. with finite sample corrections involving terms up to ο( N−3) performed much better in small samples but resulted in a biased test when the sample size was less than twice the number of variates.

scholarly journals Robustness of the sample correlation - the bivariate lognormal case

1999 ◽  
Vol 3 (1) ◽  
pp. 7-19 ◽  
Author(s):  
C. D. Lai ◽  
J. C. W. Rayner ◽  
T. P. Hutchinson
Keyword(s):  
Numerical Analysis ◽  
Correlation Coefficient ◽  
Simulation Analysis ◽  
Bivariate Normal Distribution ◽  
Bivariate Normal ◽  
Population Correlation ◽  
Sample Correlation ◽  
Skewness And Kurtosis ◽  
Bivariate Lognormal

The sample correlation coefficient R is almost universally used to estimate the population correlation coefficient ρ. If the pair (X,Y) has a bivariate normal distribution, this would not cause any trouble. However, if the marginals are nonnormal, particularly if they have high skewness and kurtosis, the estimated value from a sample may be quite different from the population correlation coefficient ρ.The bivariate lognormal is chosen as our case study for this robustness study. Two approaches are used: (i) by simulation and (ii) numerical computations.Our simulation analysis indicates that for the bivariate lognormal, the bias in estimating ρ can be very large if ρ≠0, and it can be substantially reduced only after a large number (three to four million) of observations. This phenomenon, though unexpected at first, was found to be consistent to our findings by our numerical analysis.

scholarly journals A Method for Simulating Nonnormal Distributions with Specified L-Skew, L-Kurtosis, and L-Correlation

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Pearson Correlation ◽  
Theoretical Development ◽  
Nonnormal Distributions ◽  
Heavy Tailed Distributions ◽  
Moment Estimates ◽  
Monte Carlo Studies ◽  
Asymmetric Distributions ◽  
Heavy Tailed ◽  
Primary Focus

This paper introduces two families of distributions referred to as the symmetric κ and asymmetric - distributions. The families are based on transformations of standard logistic pseudo-random deviates. The primary focus of the theoretical development is in the contexts of L-moments and the L-correlation. Also included is the development of a method for specifying distributions with controlled degrees of L-skew, L-kurtosis, and L-correlation. The method can be applied in a variety of settings such as Monte Carlo studies, simulation, or modeling events. It is also demonstrated that estimates of L-skew, L-kurtosis, and L-correlation are superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when moderate-to-heavy-tailed distributions are of concern.

scholarly journals Characterizing Tukey and -Distributions through -Moments and the -Correlation

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Todd C. Headrick ◽  
Mohan D. Pant
Keyword(s):  
Monte Carlo ◽  
Risk Analysis ◽  
Extreme Events ◽  
Pearson Correlation ◽  
Simulation Studies ◽  
Nonnormal Distributions ◽  
Heavy Tailed Distributions ◽  
Moment Estimates ◽  
Asymmetric Distributions ◽  
Heavy Tailed

This paper introduces the Tukey family of symmetric and asymmetric -distributions in the contexts of univariate -moments and the -correlation. Included is the development of a procedure for specifying nonnormal distributions with controlled degrees of -skew, -kurtosis, and -correlations. The procedure can be applied in a variety of settings such as modeling events (e.g., risk analysis, extreme events) and Monte Carlo or simulation studies. Further, it is demonstrated that estimates of -skew, -kurtosis, and -correlation are substantially superior to conventional product-moment estimates of skew, kurtosis, and Pearson correlation in terms of both relative bias and efficiency when heavy-tailed distributions are of concern.

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