Identifying genomic elements underlying phenotypic adaptations is an important problem in evolutionary biology. Comparative analyses learning from convergent evolution of traits are gaining momentum in accurately detecting such elements. We previously developed a method for predicting phenotypic associations of genetic elements by contrasting patterns of sequence evolution in species showing a phenotype with those that do not. Using this method, we successfully demonstrated convergent evolutionary rate shifts in genetic elements associated with two phenotypic adaptations, namely the independent subterranean and marine transitions of terrestrial mammalian lineages. Our original method calculates gene-specific rates of evolution on branches of phylogenetic trees using linear regression. These rates represent the extent of sequence divergence on a branch after removing the expected divergence on the branch due to background factors. The rates calculated using this regression analysis exhibit an important statistical limitation, namely heteroscedasticity. We observe that the rates on branches that are longer on average show higher variance, and describe how this problem adversely affects the confidence with which we can make inferences about rate shifts. Using a combination of data transformation and weighted regression, we have developed an updated method that corrects this heteroscedasticity in the rates. We additionally illustrate the improved performance offered by the updated method at robust detection of convergent rate shifts in phylogenetic trees of protein-coding genes across mammals, as well as using simulated tree data sets. Overall, we present an important extension to our evolutionary-rates-based method that performs more robustly and consistently at detecting convergent shifts in evolutionary rates.