There are a few instances in which positive Darwinian selection has been convincingly demonstrated at the molecular level. In this study, we present a novel test for detecting excess of radical amino-acid replacements. Such excess is usually indicative of positive Darwinian selection, but may also be due to relaxed functional constraints or model misspecification. In our test, each amino-acid replacement is characterized in terms of a physicochemical distance, i.e., the degree of dissimilarity between the exchanged amino-acid residues. By using phylogenetic trees based on protein sequences, our test identifies statistically significant deviations of the mean physicochemical distance from the random expectation, either along a taxonomic lineage or across a subtree. The mean inferred distance is calculated as the average physicochemical distance over all possible ancestral sequence reconstructions weighted by their likelihood. Our method substantially improves over previous approaches by taking into account the stochastic process, tree phylogeny, among-site rate variation, and alternative ancestral reconstructions. We provide a fast linear time algorithm for applying this test to all branches and all subtrees of a given phylogenetic tree. We validate this approach by applying it to two well-studied datasets: the MHC class I glycoproteins serving as a positive control, and the house-keeping gene carbonic anhydrase I serving as a negative control.