The human immunodeficiency virus (HIV) has a very high mutation rate. What are the effects of individual mutations on the fitness of the virus? Theys et al. (2018) measured the effect of each of many nonlethal mutations in the DNA polymerase (pol) gene on HIV virus fitness. Fitness was estimated from the frequency of the mutated strain in hundreds of HIV-infected patients relative to that of the ancestral HIV strain lacking all mutation. The authors compared the fitness effects of three types of mutations: “synonymous” mutations that did not change the corresponding amino acid in the protein; “minor” mutations that changed the corresponding amino acid but not its group (based on amino acid properties such as their electric charge); and “major” mutations that changed the amino acid group. Fitness values range from close to 0 (indicating a strong reduction in fitness) to nearly 1 (fitness is similar to that of the ancestral strain). Because fitness values are proportions, they were transformed to the log odds of fitness to improve their fit to assumptions of normality and equal variance. The full data set is available at the book website.

a. Computer optional. Using the data set on the book website, estimate the mean fitness of each of the three classes of mutations—synonymous, minor, and major—and their standard errors. Use the untransformed fitness measurements. Which type has the highest mean fitness in this sample?

b. Computer optional. With an appropriate method, test whether mean fitness differs between the three types of mutations (synonymous, nonsynonymous-major, nonsynonymous-minor). Use the transformed fitness values (log odds fitness).

c. Computer optional. Use an appropriate method to compare all pairs of means to determine where the differences lie. Use the log odds-transformed fitness values.

d. Below is a partial ANOVA table to test whether mean fitness differs between the three types of mutations (synonymous, nonsynonymous-major, nonsynonymous-minor) using the transformed fitness values (log odds fitness). Complete the test and state the conclusion.

e. Below is the output from a Tukey–Kramer test on the transformed data. Interpret these results.

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Source TypeType ErrorError Total Total dfdf 22 568568 570570 Sum of squares 324.8749 324.8749 855.9061 855.9061 1180.78101180.7810 Mean squares FF PP