Running Performance, \({\rm V}{\rm O}_{\rm {2max}}\), and Running Economy: The Widespread Issue of Endogenous Selection Bias

Note that although most studies express economy and efficiency directly as the oxygen cost, some studies define economy and efficiency such that a high value indicates higher efficiency [21]. For instance, in the study by Lucia et al. [20], the correlations between \({\rm V}{\rm O}_{\mathrm {2max}}\) and CE/GE are negative.

Statistical generalization from a sample to a population (of interest) depends on assumptions such as random sampling. When using convenience samples, which are commonly used in the literature, the statistical significance of the correlations may be misleading [5]. With convenience samples, not all elites or highly trained individuals are equally likely to be included in the sample, and the study participants are likely to be more alike with regard to for instance training principles (e.g., amount of high-intensity interval training) than the participants would have been had they been selected through a probability sample. I suspect that this will lead to P values that are too small and that the uncertainty in the results is underestimated. However, the literature routinely report P values without any discussion. To explain their sampling procedure, and discuss any potential bias, researchers should consider using guidelines for reporting observational studies, for instance the Strengthening the Reporting of Observational Studies in Epidemiology (STROBE) statement [53].

Following the counterfactual model of causality, causal effects are defined as contrasts between potential outcomes [33].

Linear path models allow us to calculate coefficients under the assumption of linearity and homogeneous effects (no interactions). Restricting the analysis to elite athletes is the same as adding a control for elite athletes and interactions between elite athletes and all independent variables.

The coefficient of determination (\(R^2\)) can be calculated by summing the squared semipartial correlations, which in this case is identical to the pairwise correlations (Fig. 1a): \(R^2=\pi ^2+\gamma ^2\). Since \(\pi\) and \(\gamma\) are constrained to be equal, \(\pi\) and \(\gamma\) can be calculated from the figure using \((R^2\times \frac{1}{2})^{\frac{1}{2}}\).

This study also found an inverse relationship between RE and \({\rm V}{\rm O}_{\mathrm {2max}}\) (\(r=0.42\)), which, as discussed in Sect. 3.1, is at least partially spurious.

This means that studies which estimate within-subject correlations with small sample sizes and find significant effects most likely exaggerate the correlation.

This association is also biased by endogenous selection, but the bias is likely small and the findings accordingly informative. Given the following model: anthropometric variable \(\rightarrow\) RE \(\rightarrow\) RP \(\rightarrow\) elite, then the elite variable is a descendant of the outcome variable RE, and restricting the sample to elites amounts to conditioning on the outcome variable. However, since the effects of RE on elite are less than the effects of RP on elite, the bias would most likely be small.