405

In this section, we explore how the logic of regression is already a part of our everyday reasoning. Then we discuss why regression doesn’t allow us to designate causation as we interpret data; for instance, MSU researchers could not say that spending more time on Facebook caused students to bridge more social capital with more online connections. This discussion of causation then leads us to a familiar warning about interpreting the meaning of regression, this time due to the process called regression to the mean. Finally, we learn how to calculate effect sizes so we can make interpretations about how well a regression equation predicts behavior.

For many different reasons, predictions are full of errors, and that margin of error is factored into the regression analysis. For example, we might predict that a student would get a certain grade based on how many classes she skipped, but we could be wrong in our prediction. Other factors, such as her intelligence, the amount of sleep she got the night before, and the number of related classes she’s taken all are likely to affect her grade as well. The number of skipped classes is highly unlikely to be a perfect predictor.

Errors in prediction lead to variability—and that is something we can measure. For t tests and ANOVAs, for example, we use standard deviation and standard error to calculate the variability around the mean. For regression, we calculate the variability around the line of best fit. Figure 14-4 illustrates that there is less variability—or error—when the data points are clustered tightly around the line of best fit. There is more variability—or error—when the data points are far away from the line of best fit.

406

The amount of error around the line of best fit can be quantified. The number that describes how far away, on average, the data points are from the line of best fit is called the standard error of the estimate, a statistic that indicates the typical distance between a regression line and the actual data points. The standard error of the estimate is essentially the standard deviation of the actual data points around the regression line. We usually get the standard error of the estimate using software, so its calculation is not covered here.

In addition to understanding the ways in which regression can help us, it is important to understand the limitations associated with using regression. It is extremely rare that the data analyzed in a regression equation are from a true experiment (one that used randomization to assign participants to conditions). Typically, we cannot randomly assign participants to conditions when the independent variable is a scale variable (rather than a nominal variable), as is usually the case with regression. So, the results are subject to the same limitations in interpretation that we discussed with respect to correlation.

In Chapter 13, we introduced the A-B-C model of understanding correlation. We noted that the correlation between number of absences and exam grade could be explained if skipping class (A) harmed one’s grade (B); if a bad grade (B) led one to skip class more often (A) because of frustration; or if a third variable (C)—such as intelligence—led both to the awareness that going to class is a good thing (A) and to good grades (B). When drawing conclusions from regression, we must consider the same set of possible confounding variables that limited our confidence in the findings following a correlation.

In fact, regression, like correlation, can be wildly inaccurate in its predictions. As with the Pearson correlation coefficient, a good statistician questions causality after the statistical analysis (to identify potential confounding variables). But one more source of error can affect fair-minded interpretations of regression analyses: regression to the mean.

407

In the study that we considered earlier in this chapter (Ruhm, 2006), economic factors predicted several indicators of health. The study also reported that “the drop in tobacco use disproportionately occurs among heavy smokers, the fall in body weight among the severely obese, and the increase in exercise among those who were completely inactive” (p. 2). What Ruhm describes captures the meaning of the word regression, as defined by its early proponents. Those who were most extreme on a given variable regressed (toward the mean). In other words, they became somewhat less extreme on that variable.

Francis Galton (Darwin’s cousin) was the first to describe the phenomenon of regression to the mean, and he did so in a number of contexts (Bernstein, 1996). For example, Galton asked nine people—including Darwin—to plant sweet pea seeds in the widely scattered locations in Britain where these people lived. Galton found that the variability among the seeds he sent out to be planted was larger than among the seeds that were produced by these plants. The largest seeds produced seeds smaller than they were. The smallest seeds produced seeds larger than they were.

Similarly, among people, Galton documented that, although tall parents tend to have taller-than-average children, their children tend to be a little shorter than they are. And although short parents tend to have shorter-than-average children, their children tend to be a little taller than they are. Galton noted that if regression to the mean did not occur, with tall people and large sweet peas producing offspring even taller or larger, and short people and small sweet peas producing offspring even shorter or smaller, “the world would consist of nothing but midgets and giants” (quoted in Bernstein, 1996, p. 167).

An understanding of regression to the mean can help us make better choices in our daily lives. For example, regression to the mean is a particularly important concept to remember when we begin to save for retirement and have to choose the specific allocations of our savings. Table 14-5 shows data from Morningstar, an investment publication. The percentages represent the increase in that investment vehicle over two 5-year periods: 1984–1989 and 1989–1994 (Bernstein, 1996). As most descriptions of mutual funds remind potential investors, previous performance is not necessarily indicative of future performance. Consider regression to the mean in your own investment decisions. It might help you ride out a decrease in a mutual fund rather than panic and sell before the likely drift back toward the mean. And it might help you avoid buying into the fund that’s been on top for several years, knowing that it stands a chance of sliding back toward the mean.

408

In the previous section, we developed a regression equation to predict a final exam score from number of absences. Now we want to know: How good is this regression equation? Is it worth having students use this equation to predict their own final exam grades from the numbers of classes they plan to skip? To answer this question, we calculate a form of effect size, the proportionate reduction in error—a statistic that quantifies how much more accurate predictions are when we use the regression line instead of the mean as a prediction tool. (Note that the proportionate reduction in error is sometimes called the coefficient of determination.) More specifically, the proportionate reduction in error is a statistic that quantifies how much more accurate predictions are when scores are predicted using a specific regression equation rather than when the mean is just predicted for everyone.

Earlier in this chapter, we noted that if we did not have a regression equation, the best we could do is predict the mean for everyone, regardless of number of absences. The average final exam grade for students in this sample is 76. With no further information, we could only tell our students that our best guess for their statistics grade is a 76. There would obviously be a great deal of error if we predicted the mean for everyone. Using the mean to estimate scores is a reasonable way to proceed if that’s all the information we have. But the regression line provides a more precise picture of the relation between variables, so using a regression equation reduces error.

Less error is the same thing as having a smaller standard error of the estimate. And a smaller standard error of the estimate means that we’d be doing much better in our predictions than if we had a larger one; visually, this means that the actual scores are closer to the regression line. And with a larger standard error of the estimate, we’d be doing much worse in our predictions than if we had a smaller one; visually, the actual scores are farther away from the regression line.

But we can do more than just quantify the standard deviation around the regression line. We can determine how much better the regression equation is compared to the mean: We calculate the proportion of error that we eliminate by using the regression equation, rather than the mean, to make a prediction. (In this next section, we learn the long way to calculate this proportion in order to understand exactly what the proportion represents. Then we learn a shortcut.)