( or ) to be positive. Occasionally, however, a regression coefficient in this situation will be
significantlynegative. Such a variable, if significant, is called a suppressor variable.^7
Suppressor variables seem, at first glance, to be unreasonable. We know that the simple
correlation between the criterion and the variable is positive (by our definition), yet in the
resulting regression equation an increment on this variable produces a decrement in.
Moreover, it can be shown that. If is positive and is negative, the prod-
uct will be negative. Thus, by assigning a negative value, the regression solution
(which has the task of minimizing error) would appearto be reducing. This does not fit
with our preconceived ideas of what should be happening, and yet obviously there must be
some logical explanation.
Space considerations do not allow an extensive discussion of the theory of suppressor
variables, but it is important to illustrate one intuitively sensible explanation. For a more
extensive discussion of suppressor variables, see Cohen and Cohen (1983) and Darlington
(1968). (The discussion in Cohen and Cohen is particularly helpful.) Here we will take an
example from Darlington (1990). Suppose a speeded history examination (a long exam
with a short time in which to complete it) is used as a measure of some external criterion of
knowledge of history. Although knowledge of history is presumably independent of read-
ing speed, performance on the speeded test will not be. Thus, some of the variance in test
scores will reflect differences in the reading speed of the students rather than differences in
their actual knowledge. What we would really like to do is penalize students who did well
onlybecause they read quickly, and help students who did poorly onlybecause they read
slowly. This is precisely what is accomplished by having reading speed serve as a suppres-
sor variable. It is suppressing some of the error in the exam scores.
As Darlington points out, a variable will serve as a suppressor variable when it corre-
lates more highly with than with Y(where represents the residual when predicting
history knowledge from history score), and will not serve as a suppressor variable when it
correlates more highly with Ythan. Cohen, Cohen, West, and Aiken (2003) point out that
suppressor relationships are hard to find in psychology (at least statistically significant
ones), though they are easily found in biology and economics. In those fields they relate to
homeostatic mechanisms, where an increase in Xleads to an increase in Y, which in turn
causes an increase in Zwhich leads back to a decrease in Y. Although these mechanisms
are not as common in psychology, I am frequently asked about suppression effects—most
of which turn out to be statistically nonsignificant.15.10 Regression Diagnostics
In predicting state SAT performance from variables that described educational expendi-
tures and characteristics of test taking, we skipped an important step because of the need to
first lay out some of the important concepts in multiple regression. It is now time to go
back and fill that gap. Before throwing all of the observations and predictors into the model
and asking computer software to produce an answer to be written up and interpreted, we
need to look more closely at the data. We can do this by using a variety of tools supplied
by nearly all multiple regression computer programs. Once we are satisfied with the data,
we can then go on and use other available tools to help us decide which variables to include
in the model. A much more complete and readable treatment of the problem of regression
diagnostics can be Cohen et al. (2003).YrYr YrR^2
bir 0 i biR^2 =gbir 0 i r 0 i biYN
bi bi15.10 Regression Diagnostics 539(^7) Cohen and Cohen (1975) discuss two additional types of suppression, and their discussion is helpful when faced
with results that seem contrary to intuition. That discussion has been omitted in the more recent Cohen, Cohen,
West, and Aiken (2003), so you need to go back to the earlier edition.
suppressor
variable