Robustness of the Pearson Correlation against Violations of Assumptions

Abstract

The purpose of this study was to determine empirically effects of the violation of assumptions of normality and of measurement scales on the Pearson product-moment correlation coefficient. The effects of such violations were studied separately and in combination for samples of varying size from 5 to 60. Monte Carlo procedures were used to generate populations of scores for four basic distributions: normal, positively skewed, negatively skewed, and leptokurtic. Samples of varying sizes were then randomly selected from specific populations. Results of the study were based on distributions of rs which were calculated on 5,000 sets of samples of n = 5 or n = 15 and 3,000 sets of samples of n = 30 and n = 60. Results indicated that the Pearson r is insensitive to rather extreme violations of the basic assumptions of normality and type of measurement scale. Failure to meet the basic assumptions separately or in combinations had little effect upon the obtained distributions of rs.

Likert scales, levels of measurement and the “laws” of statistics

“…However, there have been a number of studies that are reassuring. Pearson (1931Pearson ( , 1932a, Dunlap (1931) and Havlicek and Peterson (1976) have all shown, using theoretical distributions, that the Pearson correlation is robust with respect to skewness and nonnormality. Havlicek and Peterson did the most extensive simulation study, looking at sample size from 5 to 60 (with 3,000-5,000 replications each), for normal, rectangular, and ordinal scales (the latter obtained by adding and subtracting numbers at random).…”

Consensus Conference Follow-up: Inter-rater Reliability Assessment of the Best Evidence in Emergency Medicine (BEEM) Rater Scale, a Medical Literature Rating Tool for Emergency Physicians

“…Second, several analyses of ordinal scale data using both parametric and nonparametric methods have demonstrated that both classes of tests tend to give similar results, and that parametric tests are indeed robust for use with ordinal data. 26,27 This may be in part because nonparametric tests, such as Spearman’s correlation, are mathematical extensions of their parametric counterparts (the Pearson correlation). Summary measures are reported with 95% confidence intervals (CIs) where applicable.…”

Shakespeare's Sonnets: A Case of and for Single–Case Historiometry

“…The empincal distribution of aesthetic success exhibits a skewed nght distnbution, with a few supremely popular sonnets standing out (e g , Sonnets 29,30, 73, and 116) One possible reaction to this fact is to subject the measure to a loganthmic transformation, thereby rendenng the distribution more normal, and thus more in keeping with the underlying assumptions of the inferential statistics Additionally, a log-transformed indicator would assign more weight to distinguishing sonnets in the middle range rather than differentiating the greatest from the least effective sonnets On the other hand, use of the raw index imposes no difficulties whatsoever from the standpoint of calculating descnptive statistics, and even the significance tests are highly robust under extreme departures from normality (Havlicek & Peterson, 1976) Moreover, we have strong empincal and theoretical reasons for maintaining that creativity is not normally distnbuted but rather is always charactenzed by a highly stewed distnbution with a long upper tail (Simonton, 1988c, chap 4) For instance, suppose that aesthetic success depends on the convergence of several stochastic qualities that contnbute to a work's overall impact in a multiplicative fashion If a composition scores extremely low on 2 Even when one deletes the 12 most popular sonnets, coefficient a stays high ( 87), and therefore the internal consistency of the measure cannot be ascnbed to the effect of a few supreme outliers any one of these attributes (e g , the verse is doggerel, or the images hackneyed, or the metaphors contnved), then the ultimate impact is ml Under this nonadditive model, the distribution of aesthetic success is necessanly lognormal As a consequence, the raw measure represents a true interval scale that would be destroyed only if subjected to a nonlinear transformation ^ Of course, to some extent this question is moot, given that raw and log-transformed scores usually correlate highly anyway, in the present instance 90 Even so, this amount of correspondence does not suffice to ensure identical results when the measures are correlated with other vanables, for nearly 20% of each measure's vanance is not shared Therefore, rather than impose an arbitrary solution, I continued the practice introduced earlier (Simonton, 1986b) of ensuring robustness by replicating all analyses on both raw and transformed popularity…”