Joint Estimation and Grouping of Quantile Association Curves across Multiple Responses

Large biobanks provide an opportunity to study how associations with related traits vary across age and across their distributions. We propose a quantile regression procedure that jointly groups responses and estimates their shared varying-coefficient curves on conditional percentile scales. An information criterion selects both the number of groups and their membership from grouped B-spline fits. We establish consistent group recovery, curve convergence rates, and feasible pointwise inference after group selection as the spline dimension grows. The analysis distinguishes true curve differences from improvements due to separate spline approximations of a shared curve. Errors in the estimated percentile scales partly cancel in comparisons of pooled and split fits of a shared curve, but can still contribute to the asymptotic variance of the curve estimators. In 127,040 UK Biobank women aged 40 to 70, we grouped nine metabolic traits by their associations with a type 2 diabetes polygenic risk score and found seven groups at the median, compared with three and four at the lower and upper quantiles. The method identifies shared tail associations among traits with distinct median associations, allowing their information to be pooled where the curves agree.