Spurious correlation of ratios

Illustration of spurious correlation

Pearson states a simple example of spurious correlation:

The scatter plot above illustrates this example using 500 observations of x, y, and z. Variables x, y and z are drawn from normal distributions with means 10, 10, and 30, respectively, and standard deviations 1, 1, and 3 respectively, i.e.,

: \begin{align} x,y & \sim N(10,1) \ z & \sim N(30,3) \ \end{align}

Even though x, y, and z are statistically independent and therefore uncorrelated, in the depicted typical sample the ratios x/z and y/z have a correlation of 0.53. This is because of the common divisor (z) and can be better understood if we colour the points in the scatter plot by the z-value. Trios of (x, y, z) with relatively large z values tend to appear in the bottom left of the plot; trios with relatively small z values tend to appear in the top right.

Approximate amount of spurious correlation

Pearson derived an approximation of the correlation that would be observed between two indices (x_1/x_3 and x_2/x_4), i.e., ratios of the absolute measurements x_1, x_2, x_3, x_4:

: \rho = \frac{r_{12} v_1 v_2 - r_{14} v_1 v_4 - r_{23} v_2 v_3 + r_{34} v_3 v_4}{\sqrt{v_1^2 + v_3^2 - 2 r_{13} v_1 v_3} \sqrt{v_2^2 + v_4^2 - 2 r_{24} v_2 v_4}}

where v_i is the coefficient of variation of x_i, and r_{ij} the Pearson correlation between x_i and x_j.

This expression can be simplified for situations where there is a common divisor by setting x_3=x_4, and x_1, x_2, x_3 are uncorrelated, giving the spurious correlation:

: \rho_0 = \frac{v_3^2}{\sqrt{v_1^2 + v_3^2} \sqrt{v_2^2 + v_3^2}}.

For the special case in which all coefficients of variation are equal (as is the case in the illustrations at right), \rho_0 = 0.5

Relevance to biology and other sciences

Pearson was joined by Sir Francis Galton and Walter Frank Raphael Weldon in cautioning scientists to be wary of spurious correlation, especially in biology where it is common to scale or normalize measurements by dividing them by a particular variable or total. The danger he saw was that conclusions would be drawn from correlations that are artifacts of the analysis method, rather than actual "organic" relationships.

However, it would appear that spurious correlation (and its potential to mislead) is not yet widely understood. In 1986 John Aitchison, who pioneered the log-ratio approach to compositional data analysis wrote:

More recent publications suggest that this lack of awareness prevails, at least in molecular bioscience.

References

References

1. Pearson, Karl. (1896). "Mathematical Contributions to the Theory of Evolution – On a Form of Spurious Correlation Which May Arise When Indices Are Used in the Measurement of Organs". Proceedings of the Royal Society of London.
2. (1995). "Correlations Genuine and Spurious in Pearson and Yule". Statistical Science.
3. Aitchison, John. (1986). "The statistical analysis of compositional data". Chapman & Hall.
4. (2011). "Compositional Data Analysis: Theory and Applications". Wiley.
5. Galton, Francis. (1896). "Note to the memoir by Professor Karl Pearson, F.R.S., on spurious correlation". Proceedings of the Royal Society of London.
6. (1991). "The Spectre of 'Spurious' Correlation". Oecologia.
7. (2011). "Compositional Data Analysis: Theory and Applications". Wiley.
8. (16 March 2015). "Proportionality: A Valid Alternative to Correlation for Relative Data". PLOS Computational Biology.