CoCoA: Conditional Correlation Models with Association Size

Many scientific questions can be formulated as hypotheses about conditional correlations. For instance, in tests of cognitive and physical performance, the trade-off between speed and accuracy motivates study of the two variables together. A natural question is whether speed-accuracy coupling depends on other variables, such as sustained attention. Classical regression techniques, which posit models in terms of covariates and outcomes, are insufficient to investigate the effect of a third variable on the symmetric relationship between speed and accuracy. In response, we propose CoCoA (Conditional Correlation Model with Association Size), a likelihood-based statistical framework to estimate the conditional correlation between speed and accuracy as a function of additional variables. We propose novel measures of the association size, which are analogous to effect sizes on the correlation scale, while adjusting for confound variables. In simulation studies, we compare likelihood-based estimators of conditional correlation to semi-parametric estimators adapted from genome association studies, and find that the former achieves lower bias and variance under both ideal settings and model assumption misspecification. Using neurocognitive data from the Philadelphia Neurodevelopmental Cohort, we demonstrate that greater sustained attention is associated with stronger speed-accuracy coupling in a complex reasoning task while controlling for age. By highlighting conditional correlations as the outcome of interest, our model provides complementary insights to traditional regression modelling and partitioned correlation analyses. ### Competing Interest Statement The authors have declared no competing interest.