Margolin et al. (1981) provide data showing the number of revertant colonies of Salmonella in response to varying doses of quinoline. There are six different dose levels, and three independent observations per dose. The data are provided in the data frame salmonella from the aod package.
(a) Notice that the spacings among the dose levels are approximately constant on a log scale. Plot the response (y) against both dose and base-10 log-dose (log10(y+1)) and comment on any apparent trends. Note that we use base 10 here because the dose levels 10, 100, and 1000 are simply 1, 2, and 3 in base-10 log. The +1 is used in the log here because log10 0 is not defined, but log10 1 = 0. In this problem all logs should be taken this way.
(b) Estimate a Poisson regression model for y with either dose, log-dose, or a categorical version of dose (treat dose as a six-level categorical factor by using factor(dose) in the formula) as an explanatory variable. What conclusions can you draw based on tests for a dose effect?
(c) Examine the deviance/df statistics for each model. Do the model fits appear to be good? Explain.
(d) Plot the standardized residuals against fitted means for the two models that use dose in a numerical manner. Comment on what these plots tell you.
(e) Plot standardized residuals against fitted means for the model treating dose as categorical. There is no question about the linearity of the response or the appropriateness of the link function here. What does this plot suggest?