Thinking and Deciding

THE LENS MODEL 365
Table 15.1: Data and predictions (PRE) for regression example
Student P M F PRE Error
1 90 90 90 91.6 1.6
2 80 90 91 88.3 -2.7
3 70 90 84 85.0 1.0
4 70 70 71 70.7 -0.3
5 60 40 46 46.0 0.0
6 50 80 71 71.3 0.3
In some cases, the basic idea of the model — that everything is multiplied by a
weight representing its importance and all of the values are then added together —
might be wrong. One way in which the model could be wrong is that there might be
an interaction between two variables. This means that the importance (or weight) of
one variable depends on the value of the other. For example, in prediction of college
grades, aptitude tests might be more useful when high-school grades are low ...or
high — one could tell a story about either case.
Another way in which the model could be wrong is that some variables might
not have a simply linear effect. The importance of a variable might be different
for different parts of its range. It could even reverse direction. For example, the
effect of sugar content of a beverage on taste ratings would probably increase and
then decrease. In general, though, when reversals like this are not expected, simple,
additive models do quite well even if their assumptions are incorrect (Dawes and
Corrigan, 1974).
The lens model
Suppose we asked a professor to predict F (final exam grade) from M and P without
the benefit of the formula. We could then obtain a list of judgments that we could
place beside the true values for comparison, as shown in Table 15.2. We could ask
several questions about these judgments. For example, we could ask how close they
come to the true values, or whether the judgments themselves could be predicted
from M and P, and so on.
In this case, the formula for predicting the judgments J is J = .50·M +.49·P +
0.76. Notice that the professor weighed the two predictors about equally, although in
fact the midterm was a much better predictor than the paper (in the original formula
above). Notice also, however, that the error in predicting the professor’s judgments
was very small. She was quite consistent in her policy, even though it was off. On
the other hand, her overemphasis on the paper did not hurt much. She still did quite
well at predicting the final exam grade.