R Tutorial: Characterizing bivariate relationships

Scatter plots can reveal characteristics of the relationship between two variables. Any patterns and deviations from those patterns we see in these plots could give us some insight into the nature of the underlying phenomenon. Specifically, we look for four form, direction, strength, and outliers. Form is the overall shape made by the points. Since we are learning about linear regression, our primary concern is whether the form is linear or nonlinear. Direction is either positive or negative. Here the question is whether the two variables tend to move in the same direction. That is, when one goes up, the other tends to go up or in the opposite direction. We'll see examples of both in just a minute. The strength of the relationship is governed by how much scatter is present. Do the points seem to be clustered together in a way that suggests a close relationship, or are they very loosely organized? Finally, any points that don't fit the overall pattern or simply lie far away are important to investigate. These outliers may be erroneous measurements, where they can be exceptions that help clarify the general trend. Either way, outliers can be revealing, and we'll learn more about them later in the course, we're going to look at a bunch of scattered plots we found in the wild and talk about what we see so that you can start to build some experience in interpreting them. You can continue this exercise on your own by doing a Google Image search for scatterplot. As we work through these, please keep in mind that much of what we are doing at this stage involves making judgment calls. This is part of the nature of statistics, and while it can be frustrating, especially as a beginner, it is inescapable. For better or for worse, statistics is not a field. There's one right answer. There are, of course, an infinite number of indefensible claims, but many judgments are open to interpretation. Here we see a negative linear relationship between sign legibility and driver age. This makes sense to me, since people's eyesight tends to get worse as they age. I would characterize the strength of this relationship as moderately strong, since the pattern seems to be pervasive. I don't think we have any outliers here. This plot is in many ways a perfect candidate for fitting a regression model, since the form and direction are clear, but there is some variability in the observations. Conversely, in this next scatter plot, we see little evidence of any relationship at all. The direction is neither positive nor negative, nor is there any clearly identifiable form. Any perceived relationship would be exceptionally weak. From the same source, we now see a clear nonlinear negative relationship. This appears to be a very strong relationship since if you know the value of X you can do a pretty good job of predicting the value of Y. Here again, we see no outliers in this example scatter plot we see another clear nonlinear relationship. In this case the direction is negative for negative values of X but positive for positive values of X. The points are clustered fairly close together so the relationship is fairly strong. The point in the upper left hand corner might be considered an outlier since it is far from other points, but it doesn't represent a break from the overall pattern of the points. In this scatter plot we see what is known as a fan shape. The direction is certainly positive, but the spread of the points increases dramatically as we get further from the origin. If we wanted to fit a linear model to these data, we would be best served by performing a logarithmic transformation on both variables. My guess is that taking the log of both calcium and iron would result in a very nice looking scatterplot with a clear linear form. In these next exercises you'll get more practice creating scatter plots and characterizing what you see in them.