An Unbiased Correlation Ratio Measure

Abstract

The properties of the correlation ratio have been very thoroughly studied and reported upon. It has long been a necessary instrument in the study of the nature of regression. The work of Fisher' in 1922 made it a very precise instrument in studying the goodness of fit of second and higher degree regression lines.It, however, lacks a certain desirable simplicity of meaning in that its value, j, obtained from a sample, differs from the population value, i77, not only in a random manner due to the fluctuation of the particular sample, but also in a systematic manner which is a function of the number of arrays in which the data are recorded. This systematic difference between v7 and v is, of course, well known to the expert statistician, and allowed for in his interpretations, as, for example, is automatically the case in the use of the following formula by Fisherwhere N is the number of cases in the sample, k the number of arrays in which the dependent variable is classed, v the ordinary correlation ratio, R the ratio of the standard deviation of the differences between the points upon the regression line used and the means of the arrays of the sample to the standard deviation of the dependent variable (for a linear regression line, R is simply r, the ordinary product-moment correlation coefficient), and x2 is the ordinary x2 distributed nearly in the Pearson type III manner and with a number of degrees of freedom equal to [k -f(R)] in which f(R) is the number of linear restrictions placed upon the frequencies in determining the regression line employed, it equaling 2 in the case in straight line regression, 3 for second degree parabolic regression, etc. Entering a table giving probabilities for values of x2 with the value given by [1] and a number of degrees of freedom equal to [k -f(R)]yields a value P which is the probability that if the true regression, or regression in the population, is of the form assumed, a divergence from it as great as that observed would arise as a matter of chance. Thus P, derived for x2, is an immediately interpretable statistic. We may note the simplicity of several of the other concepts. N = the number of cases in the sample k = the number of arrays 554 PROC. N. A. S.

Measures of Effect Size for Comparative Studies: Applications, Interpretations, and Limitations

“…The magnitude of each contrast effect can be reported in three ways: eta squared (Pearson, 1905), epsilon squared (Kelley, 1935), and omega squared (Hays, 1963). Eta squared is computed as the ratio of the sum of squares for the contrast to the total sum of squares, where…”

Is Omega Squared Less Biased? a Comparison of Three Major Effect Size Indices in One-Way Anova

“…There are three major sample effect size indices in ANOVA (Grissom & Kim, 2004;D. Matsumoto, Kim, & Grissom, 2011;Keppel, 1982;Olejnik & Algina, 2000): eta squared (η 2 ); epsilon squared (ε 2 ; Kelley, 1935); and omega squared (ω 2 ; Hays, 1963) 1) . These sample indices correspond to the same population parameter, η 2 , in Equation 2.…”

“…Bothε 2 andω 2 are constructed by substituting the variance component parameters of η 2 with its bias-corrected sample estimators. The idea of Kelley's (1935)ε 2 is simply to substitute the population parameters σ 2 t and σ 2 w in Equation 2 with their corresponding unbiased estimators. To be specific, he estimated σ 2 t with SS t /(n − 1) and σ 2 w with MS w , where n is the sample size.…”

Program slice distribution functions

“…Figure 8 demonstrates the dissimilarity of the dynamic program slices in the Dunham Model for the author's suggested parameterization and suggests a correlated relationship between the standard deviation of predictions and the dissimilarity of the dynamic program slices. Figure 9 confirms and quantifies the strength of the nonlinear correlated relationship with a correlation ratio of .79 using (Kelley 1935). Because most of the PSDF construction process can be automated, a user can gain the valuable insight we've demonstrated here with little effort.…”