Estimating Phenotypic Correlations: Correcting for Bias Due to Intraindividual Variability

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Biology (HMC)

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1 Correlations between phenotypic traits are important in a number of contexts in physiological ecology, evolutionary physiology, and behaviour. Correlations can reflect functional connections or trade-offs among performance traits (e.g. bite force, jumping distance) and can reveal causal relationships between whole-organism traits and lower-level biochemical or morphological traits.

2 However, when one or both traits exhibit intraindividual variability (i.e. repeatability < 1), conventional estimates of Pearson product-moment correlation coefficients are biased towards zero (= attenuated). The magnitude of this bias decreases with increases in the number of measurements used to calculate the mean value of the trait for each individual. The bias varies inversely with the repeatability of each trait.

3 We present an estimator for the correlation coefficient that eliminates this bias. This estimator is based on an equation originally presented in 1904 by Spearman, and applied by researchers in psychological testing and nutritional epidemiology. The estimator is a simple function of the within-and among-individual components of variance for each of the two traits.

4 Simulations show that optimal sampling effort usually involves a small number of trials per individual and a large sample of individuals (for a fixed total sample size), although correlations between traits with low repeatabilities may be more precisely estimated with a larger number of trials per individual and a smaller number of individuals.

5 In addition to reducing the accuracy of the estimate, attenuation also reduces statistical power for detecting significant correlations. However, we do not recommend using the unbiased estimator for testing whether correlations differ from zero, because this inflates Type I error rates. Instead, the uncorrected (conventional) estimator should be used for hypothesis testing.

6 The unbiased estimator is not appropriate for correlations involving maximum or minimum values for each individual (e.g. maximum sprint speed) because sampling distributions of these extreme values typically have different properties than the sampling distributions of individual mean values.

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