Statistical Methods for Psychology

correlation of .30 would require 187 subjects for the same degree of power. As you can see,

a reasonable amount of power requires fairly large samples. Perhaps Darlington’s (1990)

rule of thumb is the best—“more is better.”

15.7 Geometric Representation of Multiple Regression

Any linear multiple regression problem involving ppredictors can be represented graphi-

cally in p 1 1 dimensions. Thus, with one predictor we can readily draw a two-dimensional

scatter diagram and fit a regression line through the points. With two predictors we can rep-

resent the data in three-dimensional space with a plane passing through the points. With

more than three predictors, we would have to begin to think in terms of hyperspace(mul-

tidimensional space) with the regression surface(the analog of the regression line or

plane) fitted through the points. People have enough trouble thinking in terms of three-

dimensional space, without trying to handle hyperspaces, and so we will consider here only

the two-predictor case. The generalization to the case of many predictors should be appar-

ent, even if you cannot visualize the solution.

Figure 15.3 shows a three-dimensional plot of the SAT course rating (Y) against the

predictors Expend ( ) and LogPctSAT ( ). Each member of the data set is represented as

the ball on top of a flagpole. The base of the flagpole is located at the point ( , ), and

the height of the pole is Y.

In Figure 15.3, as you move from the lower right back to the left front, the heights of the

flagpoles (and therefore the values of Y) increase. If you had the three-dimensional model

represented by this figure, you could actually pass a plane through, or near, the points so as

to give the best possible fit. Some of the flagpoles would stick up through the plane, and

X 1 X 2

X 1 X 2

534 Chapter 15 Multiple Regression

Figure 15.3 Three-dimensional representation of Yas a function of X 1 and X 2

hyperspace

regression
surface