Dependent variable selection in phylogenetic generalized least squares regression analysis under Pagel's lambda model

Phylogenetic generalized least squares (PGLS) regression is widely used to examine evolutionary associations while accounting for phylogenetic non-independence, but it requires designating one trait as the dependent variable and the other as the independent variable. When causal relationships between traits are unclear, choosing which trait serves as the dependent variable becomes a practical concern. While studying the evolutionary relationship between bacterial growth rate and CRISPR-Cas content, we noticed that switching the roles of dependent and independent variables in PGLS analyses could lead to inconsistent conclusions in a substantial proportion of cases. To confirm this observation, we conducted 16,000 simulations of trait evolution along binary trees with 100 terminal nodes under different evolutionary models. PGLS regressions using Pagels {lambda} model were applied to each simulation, and the results consistently showed that swapping the dependent and independent variables can lead to inconsistent outcomes. We evaluated seven potential criteria for selecting the dependent variable, including log-likelihood, AIC, R2, p-value, Pagels {lambda}, Blombergs K, and the estimated {lambda} in the Pagels {lambda} model. Among these, Pagels {lambda}, Blombergs K, and the estimated {lambda} performed equally well and outperformed the others in selecting the dependent variable, providing a reliable basis for PGLS analyses when causal direction between traits is unclear.