Cause and Correlation in Biology: A User's Guide to Path Analysis, Structural Equations and Causal Inference with R by Bill Shipley

Author:Bill Shipley [Shipley, Bill]
Language: eng
Format: epub
Publisher: Cambridge University Press
Published: 2016-05-02T04:00:00+00:00

5.6 Horn dimensions in bighorn sheep

Marco Festa-Bianchet and his students have been following a population of bighorn sheep from the Rocky Mountains of Alberta for many years. Horn size is very important in this species, because, among other things, it affects the ability of males in combat during the rut and therefore the evolutionary fitness of these males. Every year from 1981 until 199814 the researchers measured the total length and the circumference at the base of the two horns of the captured sheep. As one might expect, these four variables are highly correlated and display the sort of allometric scaling patterns that are so ubiquitous in biology. Are these four variables simply responding to a single latent ‘size factor’? In other words, are the patterns of correlation between the four variables simply due to a single common unmeasured cause that determines increases in linear dimensions, as Chase might have supposed?

Figure 5.9 shows this hypothesis, translated into a causal graph. To fix the scale of the latent, I fixed the variance of the latent variable to 1. The data do not follow a multivariate normal distribution, even after a Ln-transformation, as shown by Mardia's normalised coefficient of kurtosis. Because of this, I use a robust estimation method for the chi-square statistic (the Satorra–Bentler chi-square); these statistics are explained in detail in Chapter 6. The data are clearly not consistent with the single common latent model in Figure 5.9, since the Satorra–Bentler chi-square statistic is 759.106 with two degrees of freedom. The probability of observing this by chance is far lower than one in a million.

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