We use a likelihood-based method for mapping mutations on a phylogeny in a way that allows for both site-specific and lineage-specific variation in selection intensity. The method accounts for many of the potential sources of bias encountered in mapping of mutations on trees while still being computationally efficient. We apply the method to a previously published influenza data set to investigate hypotheses about changes in selection intensity in influenza strains. Influenza virus is sometimes propagated in chicken cells for several generations before sequencing, a process that has been hypothesized to induce mutations adapting the virus to the lab medium. Our analysis suggests that there are approximately twice as many replacement substitutions in lineages propagated in chicken eggs as in lineages that are not. Previous studies have attempted to predict which viral strains future epidemics may arise from using inferences regarding positive selection. The assumption is that future epidemics are more likely to arise from the strains in which positive selection on the so-called "trunk lineages" of the evolutionary tree is most pervasive. However, we find no difference in the strength of selection in the trunk lineages versus other evolutionary lineages. Our results suggest that it may be more difficult to use inferences regarding the strength of selection on mutations to make predictions regarding viral epidemics than previously thought.