In this chapter, we raised our SLR modeling to a new level, considering inference techniques for relationships between two quantitative variables. The next chapter will build on these same techniques but add in additional explanatory variables for what is called multiple linear regression (MLR) modeling. For example, in the Beers vs BAC study, it would have been useful to control for the weight of the subjects since people of different sizes metabolize alcohol at different rates and body size might explain some of the variability in BAC. We still would want to study the effects of beer consumption but also would be able to control for the differences in subject’s weights. Or if they had studied both male and female students, we might need to change the slope or intercept based on gender, allowing the relationship between Beers and BAC to change between these groups. That will also be handled using MLR techniques but result in two simple linear regression equations – one for each group.
In this chapter you learned how to interpret SLR models. The next chapter will feel like it is completely new initially but it actually contains very little new material, just more complicated models that use the same concepts. There will be a couple of new issues to consider for MLR and we’ll need to learn how to work with categorical variables in a regression setting – but we actually fit linear models with categorical variables in Chapters 2, 3, and 4 so that isn’t actually completely new either.
SLR is a simple (thus its name) tool for analyzing the relationship between two quantitative variables. It contains assumptions about the estimated regression line being reasonable and about the distribution of the responses around that line to do inferences for the population regression line. Our diagnostic plots help us to carefully assess those assumptions. If we cannot trust the assumptions, then the estimated line and any inferences for the population are un-trustworthy. Transformations can fix things so that we can use SLR to fit regression models. Transformations can complicate the interpretations on the original, untransformed scale but have minimal impact on the interpretations on the transformed scale. It is important to be careful with the units of the variables, especially when dealing with transformations, as this can lead to big changes in the results depending on which scale (original or transformed) the results are being interpreted on.