Weighted Averages of Correlation Coefficients

Tags

Correlation Coefficients, self-weighting, weighted correlation coefficients

There is a great discussion on weighted averages of correlation coefficients at https://www.researchgate.net/post/average_of_Pearson_correlation_coefficient_values

My most recent comments there are given below.

“The main reason for not averaging correlation coefficients in the arithmetic sense follows.”

“Correlations coefficients cannot be averaged in the arithmetic sense as they are not additive in the arithmetic sense. This is due to the fact that a correlation coefficient is a cosine, and cosines are not additive. This can be understood by mean-centering a paired data set and computing the cosine similarity between the vectors representing the variables involved.”

“If a paired data set violates the bivariative normality assumption (often overlooked, as Seifert correctly asserted), that worsens the picture. However, even if it doesn’t violates bivariative normality the computed average is a mathematically invalid exercise. If a meta analysis study is based on these averages the results can be easily challenged on these grounds.”

“Sample-size weighting is a good start, as Seifert asserted. We can certainly do better. We may compute self-weighted averages from one, more than one, or all of the constituent terms of a correlation coefficient, to account for different types of variability information present in the paired data, which otherwise might be ignored by simply sample-size weighting or applying Fisher Transformations. Which self-weighting scheme to use depends on the source of variability information to be considered (https://www.tandfonline.com/doi/abs/10.1080/03610926.2011.654037).”

On the Non-Additivity of Correlation Coefficients Part 3: The Bias & Nature of Correlation Coefficients

Tags

bias, bivariate normality, Correlation Coefficients, normal distribution, shifted up cosines

This the third and last part of a tutorial series on the non-additivity of correlation coefficients.

Their bias & nature, transformations, and approximations to normality are discussed. The risks of blindly transforming scores to ranks or arbitrarily converting r-to-Z values/Z-to-r values (Fisher Transformations) are discussed. Shifted up cosine approximations to normality are also covered.

Not all researchers know that score-to-rank transformations can change the sampling distribution of a statistic (e.g. a correlation coefficient) and that Fisher transformations are sensitive to normality violations. Combining both types of transformations is a recipe for a statistical disaster.

Alas, some meta analysis and data analytic folks are guilty of that.

On Men and Ideas: Fisher vs. Pearson

Tags

Fisher, Fisher vs. Pearson, Pearson, Royal Statistical Society

Ronald Aylmer Fisher was considered an outsider by the statistical establishment of his time.

The links below (1-3) show his struggles & nuances with Karl Pearson, his son Egon, Bowley, their followers, and the Royal Statistical Society (RSS). His life was a story of accomplishments and noise (deceptions and nasty RSS politics). He was too ahead of his time.

That reminds me of the struggles of another maverick: Benoit Mandelbrot. Eventually and like Mandelbrot, Fisher greatness was recognized. Also like Mandelbrot, he was able to boost the signal-to-noise of his career and life.

Most statisticians consider Fisher the Father of Modern Statistics (https://en.wikipedia.org/wiki/Ronald_Fisher), even when he was not allowed to teach Statistics at the University of Cambridge (they tried to silence Fisher).

Yes, scientists too can be demeaning to other scientists, more for personal reasons than for ideas and the Scientific Method. After all, they are also mostly carbon units called “humans”.

1. Fisher in 1921 https://projecteuclid.org/download/pdfview_1/euclid.ss/1118065041

2. Fisher vs Pearson: A 1935 Exchange from Nature

3. Fisher: The Outsider
R. A. Fisher: how an outsider revolutionized statistics

hashtag#Fisher
hashtag#Pearson

Regression & Correlation Calculator: Updates and Improvements

Tags

Correlation Coefficients, Pearson Correlation, Spearman Correlation

We have updated and improved our Regression & Correlation Calculator to demonstrate, as shown in the above figure, that a Spearman’s Correlation Coefficient is just a Pearson’s Correlation Coefficient computed from ranks.

The tool uses an algorithm that converts values to ranks and averages any ties that might be present before calculating the correlations. This comes handy when we need to compute a Spearman’s Correlation Coefficient from ranks with a large number of ties.

We have explained in the “What is Computed?” section of the page’s tool that as the number of ties increases the classic textbook formula for computing Spearman’s correlations

increasingly overestimates the results, even if ties were averaged.

By contrast, computing a Spearman’s as a Pearson’s always work, even in the presence or absence of ties.

To illustrate the above, consider the following two sets:

X = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
Y = [1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

using Spearman’s classic equation r_s = 0.6364 ≈ 0.64.
By contrast, r_s = 0.5222 ≈ 0.52 when computed as a Pearson coefficient derived from ranks. This is a non trivial difference.

Accordingly, we can make a case as to why we should ditch for good Spearman’s classic formula.

We also demonstrate in the page’s tool why we should never arithmetically add or average Spearman’s correlation coefficients. The same goes for Pearson’s.

Early articles in the literature of correlation coefficients theory failed to recognize the non-additivity of Pearson’s and Spearman’s Correlation Coefficients.

Sadly to say, this is sometimes reflected in current research articles, textbooks, and online publications. The worst offenders are some marketers and teachers that, in order to protect their failing models, resist to consider up-to-date research on the topic.

PS. Updated on 09-14-2018 to include the numerical example and to rewrite some lines.

Beating a dead horse, again.

Tags

Correlation Coefficients, Fisher Transformations, SEO Myths, statistics

Happy to see that Bruce J. Ladewski’s PhD thesis, Expanding a Path Analytic Model of Quality Management to Include the Management of Safety, at

https://scholarworks.wmich.edu/cgi/viewcontent.cgi?article=4107&context=dissertations

cited the Self-Weighting Model Tutorial Part 1

and stated what we all know: that correlation coefficients are not additive.

I don’t understand why some search marketers still believe the contrary. Stay away from dumb analytic from dumb SEOs, their myths, and nonsense.

Well: what can I say? Beating a dead horse,…again.