Multiple Regression and Compensatory Decision Processes
When selection decisions are based on predicted values from regression analyses involving two or more
predictors, it is said that the predictors are compensatory. That is, a low score on one of the predictors can
be compensated for by a high score on the other predictors.
There are other possible decision rules. For example, multiple hurdle decision processes require
applicants to pass two or more tests. Failure to pass one of the tests cannot be compensated for by passing
others. Single hurdle decision processes, on the other hand, require applicants to pass any one of a set of
tests.
One way of comparing such decision processes is with a graph of values of one of the predictors vs. the
other , with admit and deny decision areas denoted in the graph.
Consider the prediction of college GPA from HSGPA and ACT scores, for example. The output below is
from a 800+ freshman admitted to a university on which both HSGPA and (ACT) test scores were
available. The scatterplot of the two predictors, High School GPA vs.ACT score is given below.
4.5
4.0
3.5
3.0
2.5
2.0
1.5
HSGPA
1.0
.5
0.0
0
10
20
30
A CT COMPOSITE OR OR EQUIV A LENT
The result of a regression of College GPA onto the two predictors is given behow.
Model Summaryb
Model
1
R
.553a
R Square
.306
Adjusted
R Square
.304
Std. Error
of the
Es timate
.7602
a. Predictors: (Constant), SCORE ACT COMPOSITE
OR OR EQUIVALENT, HSGPA
b. Dependent Variable: OGPA1 1ST SEM GPA EXCL
FSEM
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The coefficients of HSGPA and ACT score are given below.
Coefficientsa
Model
1
(Constant)
HSGPA
SCORE ACT
COMPOSITE OR
OR EQUIVALENT
Standardi
zed
Coefficien
ts
Beta
Unstandardized
Coefficients
B
Std. Error
-.545
.170
.594
.060
6.399E-02
.008
.352
t
-3.203
9.965
Sig.
.001
.000
.271
7.677
.000
a. Dependent Variable: OGPA1 1ST SEM GPA EXCL FSEM
So, the prediction equation is
Predicted Y (1st Semester GPA) = -.545 + .594*HSGPA + .06399*ACT
This expresses Predicted Y as a function of X1 and X2. The general form is
Predicted Y = a + b1X1 + b2X2.
This equation can be rewritten, expressing X1 as a function of Predicted Y and X2, as
Predicted Y - a - b2X2 = b1X1
Predicted Y / b1 - a/b1 - b2X2/b1 = X1
Or
X1 = Predicted Y / b1 - a/b1 - (b2/b1)*X2.
Plugging in the values from the above equation . . .
HSGPA = Predicted Y / .594 - (-.545/.594) - (.06399/.594)*ACT
HSGPA = Predicted Y / .594 + .918
-
.108 * ACT.
For a given Predicted Y, this equation defines a line through the graph of HSGPA vs.
ACT which represents all those combinations of HSGPA and ACT which would yield
that predicted Y.
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A compensatory decision rule
So, for example, consider the above equation for the following predicted Y’s
Predicted Y(Col GPA)
2.0
2.5
3.0
3.5
4.0
Compensatory Equation
HSGPA = 3.367 + .918 - .108*ACT = 4.285 - .108*ACT
HSGPA = 4.209 + .918 - .108*ACT = 5.127 - .108*ACT
HSGPA = 5.050 + .918 - .108*ACT = 5.968 - .108*ACT
HSGPA = 5.892 + .918 - .108*ACT = 6.810 - .108*ACT
HSGPA = 6.734 + .918 - .108*ACT = 7.652 - .108*ACT
Here’s how the lines would look on a scatterplot of HSGPA vs. ACT. For a given desired
level of performance, the line separates those who would be admitted from those who
would be denied. Everyone above the line would be admitted. Everyone below the line
would be denied.
HSGPA/ACT
compensatory line to yield
predicted Col GPA of 4
4.5
4.0
Predicted Col GPA = 3.5
3.5
3.0
Predicted Col GPA = 3
2.5
2.0
Predicted Col GPA = 2.5
1.5
HSGPA
1.0
Predicted Col GPA = 2.0
.5
0.0
0
10
20
30
40
A CT COMPOSITE OR OR EQUIV A LENT
Obviously, extremes of compensation, e.g., a HSGPA = 2 with an ACT of 35, don’t
occur very often. And one might wonder what kind of a person did so poorly on one
predictor while doing so well on the other.
Also, expressing an admission rule using one of the formulas above would be very
confusing to incoming students. Suppose a university wanted to admit only those
students whose predicted GPA was 2.0. Imagine trying to explain to a student or his/her
parents that, “If your ACT score is 15, then your high school grade point will have to be
.4.285 - .108 times 15 or 2.66. But if your ACT is 20, then your high school grade point
will have to be only 4.285 - .108 times 20 or 2.12.” etc. etc.
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A multiple hurdle decision rule.
So, even though admissions committees use multiple regression analysis as a starting
point for admissions decisions, they often simplify the decision space to take into account
the fact that extreme cases of compensation won’t occur and that most people won’t
understand a completely compensatory system.
Often, universities use a multiple hurdle approach , with a minimum high school GPA
and a minimum test score. Such a multiple hurdle approach can also be represented by
lines in the graph of one predictor vs. another.
Suppose a university decided to require a high school gpa of 2.5 AND an ACT score of
20. Such a decision rule would be represented by the thicker lines in the graph below.
Compare them to the compensatory line for a predicted GPA of 3.0.
4.5
Admit
4.0
3.5
3.0
2.5
2.0
1.5
1.0
Predicted Col GPA = 2
H
.5
S
G 0.0
PA
0
10
20
30
ACT COMPOSITE OR OR
EQUIVALENT
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A single-hurdle decision rule
Finally, a decision rule in which EITHER one OR the other of two criteria had to be met
can also be represented by lines on the graph. Suppose a university decided that students
would be admitted if they had a high school GPA of 2.5 OR an ACT of 20. This decision
rule would be represented as follows . ..
4.5
Admit
4.0
3.5
3.0
2.5
2.0
1.5
Predicted Col GPA = 2
1.0
H
.5
S
G 0.0
PA
0
10
20
30
ACT COMPOSITE OR OR
EQUIVALENT
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