In this chapter, methods for handling two different categorical predictors in the same model with a continuous response were developed. The methods build on techniques from Chapter ?? for the One-Way ANOVA and there are connections between the two models. This was most clearly seen in the guinea pig data set that was analyzed in both chapters. When two factors are available, it is better to start with the methods developed in this chapter because the interaction between the factors can, potentially, be separated from their main effects. The additive model is easier to interpret but should only be used when you are not convinced that there is an interaction is present. When an interaction is determined to be present, the main effects should not be interpreted and the interaction plot in combination with Tukey’s HSD provides information on the important aspects of the results.
If the interaction is retained in the model, there are two things you want to do with interpreting the interaction:
Describe the interaction, going through the changes from left to right in the interaction plot or term-plot for each level of the other variable.
Suggest optimal and worst combinations of the two variables to describe the highest and lowest possible estimated mean responses.
If there is no interaction, then the additive model provides information on each of the variables and the differences across levels of each variable are the same regardless of the levels of the other variable.
Some statisticians might have different recommendations for dealing with interactions and main effects, especially in the context of models with interactions. We have chosen to focus on tests for interactions to screen for “real” interactions and then interpret the interaction plots aided by the Tukey’s HSD for determining which combinations of levels are detectably different. Some suggest exploring the main effects tests even with interactions present. In some cases, those results are interesting but in others the results can be misleading and we wanted to avoid trying to parse out the scenarios when it might be safe to focus on the main effects in the presence of important interactions. Consider two scenarios, one where the main effects have large p-values but the interaction has a small p-value and the other where the main effects and the interaction all have small p-values. The methods discussed in this chapter allow us to effectively arrive at the interpretation of the differences in the results across the combinations of the treatments due to the interaction having a small p-value in both cases. The main effects results are secondary results at best when the interaction is important because we know that impacts of one explanatory variable is changing based on the levels of the other variable.
Chapter ?? presents a bit of a different set of statistical
methods that allow analyses of data sets similar
to those considered in the last two chapters but with a categorical response
variable. The methods are very different in application but are quite similar in overall goals
to those in Chapter ?? where differences in responses where explored
across groups. After Chapter ??, the rest of the semester will
return to fitting models using the lm function as used here, but
incorporating quantitative predictor variables and
then eventually incorporating both categorical and quantitative predictor
variables. The methods in Chapter ?? are actually quite similar
to those considered here, so the better you understand these models, the easier that material will be master.