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Nonparametric correlation and regression: Use & misuse

(Spearman rank correlation coefficient, Kendall rank-order correlation coefficient, monotonic relationship, Sen's estimator of slope)

Statistics courses, especially for biologists, assume formulae = understanding and teach how to do  statistics, but largely ignore what those procedures assume,  and how their results mislead when those assumptions are unreasonable. The resulting misuse is, shall we say, predictable...

Use and Misuse

In biology non-parametric correlation (especially the Spearman rank correlation coefficient) is probably used as much as parametric correlation. Fortunately it tends to get rather less abused, mainly because the assumption of linearity is relaxed. All that is required is a monotonic (continuously increasing or decreasing) trend. However, this still needs to be checked for using a scatterplot. In one example where no scatterplot was given, a negative correlation was reported as positive - presumably because the author had not noticed the negative sign and had never plotted the data out. When a scatterplot is provided, we sometimes find relationships that are emphatically not monotonic, but are U-shaped or hat-shaped. We give an example of trends in fishing effort and catches of lobsters. In this case the error was made worse by calculation of the (non-parametric) Sen's estimate of slope - which assumes a linear relationship.

A related issue is the practice of attaching a least squares regression line  to the data, but giving a P-value derived from a non-parametric correlation coefficients. Both our medical examples (on suicide rates over time, and on serotonin neuron integrity in relation level of ecstasy use) commit this error. In both cases, relationships were clearly curvilinear rather than linear, and the P-values referred to a significant monotonic rather than linear relationship. In another example the same error is made by giving the value of the coefficient of determination (r2) along with the result of the non-parametric test - this is equivalent to doing a least squares regression plot because it assumes a linear relationship. We encountered a few cases where P-values were likely to have been inaccurate because sample sizes were small, and there were many ties, but overall this was not a serious problem because authors tended to use the Kendall coefficient in this situation. High levels of measurement error and inadequate sample sizes reducing the power to detect a correlation were probably bigger factors affecting the outcomes of studies.

Artefactual correlations are as big a problem with non-parametric correlation and regression as with parametric correlation and regression. In one example of a negative correlation over time between antidepressant use and the suicide rate, causality was highly questionable because a number of possible confounding factors changed over the same time period. We give two examples of a (potentially) artefactual correlation arising from there being an uneven distribution of confounding factors. In one study (on what signals birds use as cues to detect herbivore-rich trees) values from both treatment and control plots were included in the correlations investigating concentrations of individual chemicals. There was a tendency to get positive correlations because there was a higher level of all the chemicals in the control plots. A similar situation resulted from including data from both inside and outside the national park when assessing the correlation between number of responding lions and distance inside the park. In experimental studies with random allocation confounding factors should be evenly distributed over the different levels of X - but in observational studies the researcher must carefully assess the situation to avoid artefactual correlations.


What the statisticians say

Conover (1999) covers both Spearman and Kendall rank correlation coefficients, including their use to test for trend. Sprent (1998) provides a comprehensive treatment of non-parametric tests of correlation and concordance in Chapter 9. Hollander & Wolfe (1973) and Siegel (1956) both cover the Spearman and Kendall rank correlation coefficients in their texts on nonparametric statistics. Gilbert (1987) describes a number of nonparametric tests for trend including Sen's estimator of slope.

Sen (1968) introduces his non-parametric estimator of slope. Kraemer (2006) emphasizes the use of distribution-free effect sizes such as the Spearman correlation coefficient and Kendall's coefficient of concordance for ordinal versus ordinal associations Potvin & Roff (1993) provide a useful demonstration that non-parametric correlation methods are more robust to outliers. Trexler & Travis (1993) present an overview of 'non-traditional' regression analysis, including LOWESS regression, whilst Cade & Noon (2003) provide a gentle introduction to quantile regression for ecologists. Daniels (1950) proposed that Spearman's test can be used as a test of trend by pairing measurements with the time at which they were taken.

Wikipedia provides sections on the Spearman rank correlation coefficient, the Kendall rank correlation coefficient, robust regression (criticized on-line but very readable) and quantile regression. The Handbook of biological statistics covers Spearman rank correlation. Mathematics in Education and Industry provide a good basic treatment of Spearman rank correlation.