2.3. An alternative hypothesis: The world we care about consists of mostly monotone interactions

Reid Hastie and Robyn Dawes have offered a different account of the

success of linear models (2001, 58–62; see also Dawes 1988, 212–15). Their

explanation comes in three parts. Since we embrace and elaborate on the

third part of their explanation in section 3, we will focus only on the first

two parts of their explanation here. The first part of their explanation for

the success of SPRs is a principle about the relationship between proper

linear models and the world: Proper linear models can accurately represent

monotone (or ‘‘ordinal’’) interactions. We have already introduced linear

models—they are models in which the judgment made is a function of the

sum of a certain number of weighted variables. The best way to understand

what monotone interactions are is to consider a simple example.

Suppose a doctor has told you to reduce your body fat, and she recommends

a special diet D and an exercise regime E. Now, let’s suppose that D

alone, without the exercise regime, is effective at reducing body fat. This

would be the diet’s main effect. Suppose also that the exercise regime

alone, without the diet, is also effective at reducing body fat. Again, this

would be the main effect of exercise. Now let’s suppose Sam goes on the

diet D and the exercise regime E. If Sam gets the benefits of both—the

main effect of D and the main effect of E—then the interaction of D and E

is monotone. If, however, Sam gets the main effects of both plus an extra

benefit, then the interaction is not monotone. The extra benefit is often

called an interaction effect.

If we continue this absurdly simplistic example, it will be easy to see

why proper linear models can accurately represent monotone interactions.

Suppose that for a certain population of people, D will bring a loss of 1

2

pound per week while E will bring a loss of 3

4 pound per week. The

following linear model will predict how much weight loss one can expect:

W ј 1

2

d ю 3

4

e

where W is the number of pounds lost, d is the number of weeks on the

diet, and e is the number of weeks on the exercise regimen. It should be

clear that a proper linear model will do a reasonably good job of predicting

interactions that are not monotone, but for which the interaction

effects are not strong.

The second part of the Hastie-Dawes explanation is a speculation

about the world: In practical social settings (where linear models have proven

most successful), interactions are, near enough and in the main, monotone.

Those who study complex systems, nonlinear dynamics, and catastrophe

theory will note that not all of the world we’re interested in consists of

monotone interactions. The idea is that as long as we are not looking for

SPRs to predict the performance of nonlinear systems, linear models may

perform well—better than human experts. By restricting the explanation

of the success of linear models to practical, social settings, Hastie and

Dawes can take advantage of the flat maximum principle. From the reliability

of proper linear models, they can employ the flat maximum principle

to infer the reliability of improper linear models as well.

We have doubts about the Hastie-Dawes explanation for the success

of SPRs. Consider the linear model that represents the monotone weight

loss interaction. The reason this linear model is reliable is that it accurately

portrays the main causal agents and the relative influence of those agents

in subjects’ weight loss. But the robust reliability of SPRs can’t depend on

their reasonably accurate portrayal of causal reality. The reason is quite

simply that many SPRs are not even close to accurate portrayals of reality.

Consider a linear model that predicts academic performance on the basis

of grade point average and test scores. The student’s college GPA is not a

primary cause of her graduate school performance; same with her test

score. Rather, it is much more plausible to suppose that whatever complex

of factors goes into a student’s GPA and test scores is also heavily implicated

in a student’s success in graduate school. (Recall that the flat maximum

principle is operative when the cues employed by a linear model are

redundant.) So it seems unlikely that the success of SPRs depends on their

mirroring or reflecting monotone interactions. (Thanks to Michael Strevens

for this point.)

36 Epistemology and the Psychology of Human Judgment

We need to be a bit careful here. We’re not suggesting that we oppose

or doubt the possibility of successful SPRs that identify causes. (Just the

opposite.) Nor are we suggesting that successful SPRs do not depend for

their success on causal regularities. (Again, just the opposite.) Our point is

that even when we can’t ‘‘read off’’ anything like the causal structure of the

world from an SPR, it can still be highly reliable and worthy of being used.

If that’s so, then the success of SPRs can’t depend on their representing

(even approximately) the interactions that produce the item of interest.

Reid Hastie and Robyn Dawes have offered a different account of the

success of linear models (2001, 58–62; see also Dawes 1988, 212–15). Their

explanation comes in three parts. Since we embrace and elaborate on the

third part of their explanation in section 3, we will focus only on the first

two parts of their explanation here. The first part of their explanation for

the success of SPRs is a principle about the relationship between proper

linear models and the world: Proper linear models can accurately represent

monotone (or ‘‘ordinal’’) interactions. We have already introduced linear

models—they are models in which the judgment made is a function of the

sum of a certain number of weighted variables. The best way to understand

what monotone interactions are is to consider a simple example.

Suppose a doctor has told you to reduce your body fat, and she recommends

a special diet D and an exercise regime E. Now, let’s suppose that D

alone, without the exercise regime, is effective at reducing body fat. This

would be the diet’s main effect. Suppose also that the exercise regime

alone, without the diet, is also effective at reducing body fat. Again, this

would be the main effect of exercise. Now let’s suppose Sam goes on the

diet D and the exercise regime E. If Sam gets the benefits of both—the

main effect of D and the main effect of E—then the interaction of D and E

is monotone. If, however, Sam gets the main effects of both plus an extra

benefit, then the interaction is not monotone. The extra benefit is often

called an interaction effect.

If we continue this absurdly simplistic example, it will be easy to see

why proper linear models can accurately represent monotone interactions.

Suppose that for a certain population of people, D will bring a loss of 1

2

pound per week while E will bring a loss of 3

4 pound per week. The

following linear model will predict how much weight loss one can expect:

W ј 1

2

d ю 3

4

e

where W is the number of pounds lost, d is the number of weeks on the

diet, and e is the number of weeks on the exercise regimen. It should be

clear that a proper linear model will do a reasonably good job of predicting

interactions that are not monotone, but for which the interaction

effects are not strong.

The second part of the Hastie-Dawes explanation is a speculation

about the world: In practical social settings (where linear models have proven

most successful), interactions are, near enough and in the main, monotone.

Those who study complex systems, nonlinear dynamics, and catastrophe

theory will note that not all of the world we’re interested in consists of

monotone interactions. The idea is that as long as we are not looking for

SPRs to predict the performance of nonlinear systems, linear models may

perform well—better than human experts. By restricting the explanation

of the success of linear models to practical, social settings, Hastie and

Dawes can take advantage of the flat maximum principle. From the reliability

of proper linear models, they can employ the flat maximum principle

to infer the reliability of improper linear models as well.

We have doubts about the Hastie-Dawes explanation for the success

of SPRs. Consider the linear model that represents the monotone weight

loss interaction. The reason this linear model is reliable is that it accurately

portrays the main causal agents and the relative influence of those agents

in subjects’ weight loss. But the robust reliability of SPRs can’t depend on

their reasonably accurate portrayal of causal reality. The reason is quite

simply that many SPRs are not even close to accurate portrayals of reality.

Consider a linear model that predicts academic performance on the basis

of grade point average and test scores. The student’s college GPA is not a

primary cause of her graduate school performance; same with her test

score. Rather, it is much more plausible to suppose that whatever complex

of factors goes into a student’s GPA and test scores is also heavily implicated

in a student’s success in graduate school. (Recall that the flat maximum

principle is operative when the cues employed by a linear model are

redundant.) So it seems unlikely that the success of SPRs depends on their

mirroring or reflecting monotone interactions. (Thanks to Michael Strevens

for this point.)

36 Epistemology and the Psychology of Human Judgment

We need to be a bit careful here. We’re not suggesting that we oppose

or doubt the possibility of successful SPRs that identify causes. (Just the

opposite.) Nor are we suggesting that successful SPRs do not depend for

their success on causal regularities. (Again, just the opposite.) Our point is

that even when we can’t ‘‘read off’’ anything like the causal structure of the

world from an SPR, it can still be highly reliable and worthy of being used.

If that’s so, then the success of SPRs can’t depend on their representing

(even approximately) the interactions that produce the item of interest.