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Genomes contain conserved non-coding sequences that perform important biological functions, such as gene regulation. We present a phylogenetic method, PhyloAcc-C, that associates nucleotide substitution rates with changes in a continuous trait of interest. The method takes as input a multiple sequence alignment of conserved elements, continuous trait data observed in extant species, and a background phylogeny and substitution process. Gibbs sampling is used to assign rate categories (background, conserved, accelerated) to lineages and explore whether the assigned rate categories are associated with increases or decreases in the rate of trait evolution. We test our method using simulations and then illustrate its application using mammalian body size and lifespan data previously analyzed with respect to protein coding genes. Like other studies, we find processes such as tumor suppression, telomere maintenance, and p53 regulation to be related to changes in longevity and body size. In addition, we also find that skeletal genes, and developmental processes, such as sprouting angiogenesis, are relevant.