In regression analysis, we explain more of the variability in the dependent variable if we can discover genuine predictors that are separate and distinct. This involves orthogonal variables, independent variables that make separate and distinct contributions in the prediction of a dependent variable, as compared with the contributions of other variables. Orthogonal variables do not overlap each other. For example, the study we discussed earlier explored whether the amount of Facebook use predicted social capital. It is likely that a person’s personality also predicts social capital; for example, we would expect extroverted, or outgoing, people to be more likely to have this kind of social capital than introverted, or shy, people. It would be useful to separate the effects of the amount of Facebook use and extroversion on social capital.

The statistical technique we consider next is a way of quantifying (1) whether multiple pieces of evidence really are better than one, and (2) precisely how much better each additional piece of evidence actually is.

Just as a regression equation using one independent variable is a better predictor than the mean, a regression equation using more than one independent variable is likely to be an even better predictor. This makes sense in the same way that knowing a baseball player’s historical batting average plus knowing that the player continues to suffer from a serious injury is likely to improve our ability to predict the player’s future performance. So it is not surprising that multiple regression is far more common than simple linear regression. Multiple regression is a statistical technique that includes two or more predictor variables in a prediction equation.

Let’s examine an equation that might be used to predict a final exam grade from two variables, number of absences and score on the mathematics portion of the SAT. Table 14-8 repeats the data from Table 14-1, with the added variable of SAT score. (Note that although the scores on number of absences and final exam grade are real-life data from our statistics classes, the SAT scores are fictional.)

414

The computer gives us the printout seen in Figure 14-7. The column in which we’re interested is the one labeled “B,” under “Unstandardized Coefficients.” The first number, across from “(Constant),” is the intercept. The intercept is called constant because it does not change; it is not multiplied by any value of an independent variable. The intercept here is 33.422. The second number is the slope for the independent variable, number of absences. Number of absences is negatively correlated with final exam grade, so the slope, –3.340, is negative. The third number in this column is the slope for the independent variable of SAT score. As we might guess, SAT score and final exam grade are positively correlated; a student with a high SAT score tends to have a higher final exam grade. So the slope, 0.094, is positive. We can put these numbers into a regression equation, with X₁ representing the first variable, SAT score, and X₂ representing the second variable, final exam grade:

Ŷ = 33.422 – 3.34(X₁) + 0.094(X₂)

415

Once we develop the multiple regression equation, we can input raw scores on number of absences and mathematics SAT score to determine a student’s predicted score on Y. Imagine that our student, Allie, scored 600 on the mathematics portion of the SAT. We already know she planned to miss two classes this semester. What would we predict her final exam grade to be?

Ŷ = 33.422 – 3.34(X₁) + 0.094(X₂)

= 33.422 – 3.34(2) + 0.094(600)

= 33.422 – 6.68 + 56.4 = 83.142

Based on these two variables, we predict a final exam grade of 83.142 for Allie. How good is this multiple regression equation? From software, we calculated that the proportionate reduction in error for this equation is a whopping 0.93. By using a multiple regression equation with the independent variables of number of absences and SAT score, we have reduced 93% of the error that would result from predicting the mean of 76 for everyone.

When we calculate proportionate reduction in error for a multiple regression, the symbol changes slightly. The symbol is now R² instead of r². The capitalization of this statistic is an indication that the proportionate reduction in error is based on more than one independent variable.

With the development of increasingly more powerful computers and the availability of ever-larger amounts of electronic data, tools based on multiple regression have proliferated. Now the general public can access many of them online (Darlin, 2006). Bing Travel predicts the price of an airline ticket for specific routes, travel dates, and, most important, purchase dates. Using the same data available to travel agents, along with additional independent variables such as the weather and even which sports teams’ fans might be traveling to a championship game, Bing Travel mimics the regression equations used by the airlines. Airlines predict how much money potential travelers are willing to pay on a given date for a given flight and use these predictions to adjust their fares so they can earn the most money.

Bing Travel is an attempt at an end run, using mathematical prediction tools, to help savvy airline consumers either beat or wait out the airlines’ price hikes. In 2007, Bing Travel’s precursor, Farecast.com, claimed a 74.5% accuracy rate for its predictions. Zillow.com does for real estate what Bing Travel does for airline tickets. Using archival land records, Zillow predicts U.S. housing prices and claims to be accurate to within 10% of the actual selling price of a given home.

Amazon.com is trying to take the prediction game to a new level. In 2014 they filed a patent for something called “anticipatory shipping” (Bensinger, 2014). Amazon could take advantage of their immense database of customer information to ship products to distribution centers closer to customers, or even load products onto trucks—all before customers make a purchase! Amazon indicated that it might base its shipping behavior on customers’ searches, wish lists, and previous orders, as well as their online behavior, down to noticing the amount of time a customer spends with her cursor over a particular product. Like the future of visual displays of data, the future of the regression equation is limited only by the creativity of the rising generation of behavioral scientists and statisticians.

416