1.3. Random linear models

Dawes and Corrigan (1974) took five bootstrapping experiments and for

each one constructed a random linear model. Random linear models do

not pretend to assign optimum weights to variables. Instead, random weights

are assigned—with one important caveat: All the cues are defined so they

are positively correlated with the target property. They found that the random

linear models were as reliable as the proper models and more reliable

than human experts. Recall we said that there was an SPR finding that was

denied by a well-known philosopher of psychology. This is it. This philosopher

is not alone. Dawes has described one dominant reaction to the

success of random linear models: ‘‘[M]any people didn’t believe them—

until they tested out random . . . models on their own data sets’’ (Dawes

1988, 209, n. 17).

The resistance to this finding is understandable (though, as we shall

later argue, misguided). It is very natural to suppose that people who make

predictions are in some sense ‘‘calculating’’ a suboptimal formula. (Of

course, the idea isn’t that the person explicitly calculates a complex formula

in order to make a prediction; rather, the idea is that there will be an

improper formula that simulates the person’s weighing of the various lines

of evidence in making some prediction.) Since we can’t calculate in our

heads the optimum weights to attach to the relevant cues, it’s understandable

that proper models outperform humans. This picture of humans

‘‘calculating’’ suboptimal formulas, of implicitly using improper models,

The Amazing Success of Statistical Prediction Rules 29

also fits with the optimistic explanation of the bootstrapping effect. A

bootstrapping model approximates the suboptimal formula a person

uses—but the bootstrapping model doesn’t fall victim to performance

errors to which humans are prone. So far, so good. But how are we to

understand random linear models outperforming expert humans? After

all, if experts are calculating some sort of suboptimal formula, how could

they be defeated by a formula that uses weights that are both suboptimal

and random? Surely we must do better than linear models that assign just

any old weights at all. But alas, we do not. Without a plausible explanation

for this apparent anomaly, our first reaction (and perhaps even our wellconsidered

reaction) may be to refuse to believe this could be true.

Dawes and Corrigan (1974) took five bootstrapping experiments and for

each one constructed a random linear model. Random linear models do

not pretend to assign optimum weights to variables. Instead, random weights

are assigned—with one important caveat: All the cues are defined so they

are positively correlated with the target property. They found that the random

linear models were as reliable as the proper models and more reliable

than human experts. Recall we said that there was an SPR finding that was

denied by a well-known philosopher of psychology. This is it. This philosopher

is not alone. Dawes has described one dominant reaction to the

success of random linear models: ‘‘[M]any people didn’t believe them—

until they tested out random . . . models on their own data sets’’ (Dawes

1988, 209, n. 17).

The resistance to this finding is understandable (though, as we shall

later argue, misguided). It is very natural to suppose that people who make

predictions are in some sense ‘‘calculating’’ a suboptimal formula. (Of

course, the idea isn’t that the person explicitly calculates a complex formula

in order to make a prediction; rather, the idea is that there will be an

improper formula that simulates the person’s weighing of the various lines

of evidence in making some prediction.) Since we can’t calculate in our

heads the optimum weights to attach to the relevant cues, it’s understandable

that proper models outperform humans. This picture of humans

‘‘calculating’’ suboptimal formulas, of implicitly using improper models,

The Amazing Success of Statistical Prediction Rules 29

also fits with the optimistic explanation of the bootstrapping effect. A

bootstrapping model approximates the suboptimal formula a person

uses—but the bootstrapping model doesn’t fall victim to performance

errors to which humans are prone. So far, so good. But how are we to

understand random linear models outperforming expert humans? After

all, if experts are calculating some sort of suboptimal formula, how could

they be defeated by a formula that uses weights that are both suboptimal

and random? Surely we must do better than linear models that assign just

any old weights at all. But alas, we do not. Without a plausible explanation

for this apparent anomaly, our first reaction (and perhaps even our wellconsidered

reaction) may be to refuse to believe this could be true.