6.1: Relationships between two quantitative variables

The independence test in Chapter 5 provided a technique for assessing evidence of a relationship between two categorical variables. The terms relationship and association are synonyms that, in statistics, imply that particular values on one variable tend to occur more often with some other values of the other variable or that knowing something about the level of one variable provides information about the patterns of values on the other variable. These terms are not specific to the “form” of the relationship – any pattern (strong or weak, negative or positive, easily described or complicated) satisfy the definition. There are two other aspects to using these terms in a statistical context. First, they are not directional – an association between \(x\) and \(y\) is the same as saying there is an association between \(y\) and \(x\). Second, they are not causal unless the levels of one of the variables are randomly assigned in an experimental context. We add to this terminology the idea of correlation between variables \(x\) and \(y\). Correlation, in most statistical contexts, is a measure of the specific type of relationship between the variables: the linear relationship between two quantitative variables¹⁰⁸. So as we start to review these ideas from your previous statistics course, remember that associations and relationships are more general than correlations and it is possible to have no correlation where there is a strong relationship between variables. “Correlation” is used colloquially as a synonym for relationship but we will work to reserve it for its more specialized usage here to refer specifically to the linear relationship.

Assessing and then modeling relationships between quantitative variables drives the rest of the chapters, so we should get started with some motivating examples to start to think about what relationships between quantitative variables “look like”… To motivate these methods, we will start with a study of the effects of beer consumption on blood alcohol levels (BAC, in grams of alcohol per deciliter of blood). A group of \(n = 16\) student volunteers at The Ohio State University drank a randomly assigned number of beers¹⁰⁹. Thirty minutes later, a police officer measured their BAC. Your instincts, especially as well-educated college students with some chemistry knowledge, should inform you about the direction of this relationship – that there is a positive relationship between Beers and BAC. In other words, higher values of one variable are associated with higher values of the other. Similarly, lower values of one are associated with lower values of the other. In fact there are online calculators that tell you how much your BAC increases for each extra beer consumed (for example: http://www.craftbeer.com/beer-studies/blood-alcohol-content-calculator if you plug in 1 beer). The increase in \(y\) (BAC) for a 1 unit increase in \(x\) (here, 1 more beer) is an example of a slope coefficient that is applicable if the relationship between the variables is linear and something that will be fundamental in what is called a simple linear regression model. In a simple linear regression model (simple means that there is only one explanatory variable) the slope is the expected change in the mean response for a one unit increase in the explanatory variable. You could also use the BAC calculator and the models that we are going to develop to pick a total number of beers you will consume and get a predicted BAC, which employs the entire equation we will estimate.

Before we get to the specifics of this model and how we measure correlation, we should graphically explore the relationship between Beers and BAC in a scatterplot. Figure 6.1 shows a scatterplot of the results that display the expected positive relationship. Scatterplots display the response pairs for the two quantitative variables with the explanatory variable on the \(x\)-axis and the response variable on the \(y\)-axis. The relationship between Beers and BAC appears to be relatively linear but there is possibly more variability than one might expect. For example, for students consuming 5 beers, their BACs range from 0.05 to 0.10. If you look at the online BAC calculators, you will see that other factors such as weight, sex, and beer percent alcohol can impact the results. We might also be interested in previous alcohol consumption. In Chapter 8, we will learn how to estimate the relationship between Beers and BAC after correcting or controlling for those “other variables” using multiple linear regression, where we incorporate more than one quantitative explanatory variable into the linear model (somewhat like in the 2-Way ANOVA). Some of this variability might be hard or impossible to explain regardless of the other variables available and is considered unexplained variation and goes into the residual errors in our models, just like in the ANOVA models. To make scatterplots as in Figure 6.1, you could use the base R function plot, but we will want to again access the power of ggplot2 so will use geom_point to add the points to the plot at the “x” and “y” coordinates that you provide in aes(x = ..., y = ...).

library(readr)
BB <- read_csv("http://www.math.montana.edu/courses/s217/documents/beersbac.csv")

BB %>% ggplot(mapping = aes(x = Beers, y = BAC)) +
  geom_point() +
  theme_bw()

Figure 6.1: Scatterplot of Beers consumed versus BAC.

There are a few general things to look for in scatterplots:

1. Assess the \(\underline{\textbf{direction of the relationship}}\) – is it positive or negative?
2. Consider the \(\underline{\textbf{strength of the relationship}}\). The general idea of assessing strength visually is about how hard or easy it is to see the pattern. If it is hard to see a pattern, then it is weak. If it is easy to see, then it is strong.
3. Consider the \(\underline{\textbf{linearity of the relationship}}\). Does it appear to curve or does it follow a relatively straight line? Curving relationships are called curvilinear or nonlinear and can be strong or weak just like linear relationships – it is all about how tightly the points follow the pattern you identify.
4. Check for \(\underline{\textbf{unusual observations -- outliers}}\) – by looking for points that don’t follow the overall pattern. Being large in \(x\) or \(y\) doesn’t mean that the point is an outlier. Being unusual relative to the overall pattern makes a point an outlier in this setting.
5. Check for \(\underline{\textbf{changing variability}}\) in one variable based on values of the other variable. This will tie into a constant variance assumption later in the regression models.
6. Finally, look for \(\underline{\textbf{distinct groups}}\) in the scatterplot. This might suggest that observations from two populations, say males and females, were combined but the relationship between the two quantitative variables might be different for the two groups.

Going back to Figure 6.1 it appears that there is a moderately strong linear relationship between Beers and BAC – not weak but with some variability around what appears to be a fairly clear to see straight-line relationship. There might even be a hint of a nonlinear relationship in the higher beer values. There are no clear outliers because the observation at 9 beers seems to be following the overall pattern fairly closely. There is little evidence of non-constant variance mainly because of the limited size of the data set – we’ll check this with better plots later. And there are no clearly distinct groups in this plot, possibly because the # of beers was randomly assigned. These data have one more interesting feature to be noted – that subjects managed to consume 8 or 9 beers. This seems to be a large number. I have never been able to trace this data set to the original study so it is hard to know if (1) they had this study approved by a human subjects research review board to make sure it was “safe”, (2) every subject in the study was able to consume their randomly assigned amount, and (3) whether subjects were asked to show up to the study with BACs of 0. We also don’t know the exact alcohol concentration of the beer consumed or volume. So while this is a fun example to start these methods with, a better version of this data set would be nice…

In making scatterplots, there is always a choice of a variable for the \(x\)-axis and the \(y\)-axis. It is our convention to put explanatory or independent variables (the ones used to explain or predict the responses) on the \(x\)-axis. In studies where the subjects are randomly assigned to levels of a variable, this is very clearly an explanatory variable, and we can go as far as making causal inferences with it. In observational studies, it can be less clear which variable explains which. In these cases, make the most reasonable choice based on the observed variables but remember that, when the direction of relationship is unclear, you could have switched the axes and thus the implication of which variable is explanatory.​​​​​​