Power Model of Support Scaling
This model relates direct ratings of evidence strength to support inferred from probability judgments. If both ratings and their ratios preserve the same ordering, a power transformation can connect the scales and predict judged probability ratios.
THE IDEA
SCALING SUPPORT
In support theory, the support function is derived from probability judgments. Is it possible to reverse the process and predict probability judgments from direct assessments of evidence strength? Let ŝ(A) be the rating of the strength of evidence for hypothesis A. What is the relation between such ratings and the support estimated from probability judgments? Perhaps the most natural assumption is that the two scales are monotonically related; that is, ŝ(A) ≥ ŝ(B) if and only if (iff) s(A) ≥ s(B). This assumption implies, for example, that P(A, B) ≥ 1/2 iff ŝ(A) ≥ ŝ(B), but it does not determine the functional form relating ŝ and s. To specify the relation further, the model also assumes that support ratios are monotonically related, using positive values where ratios are formed. That is, ŝ(A)/ŝ(B) ≥ ŝ(C)/ŝ(D) iff s(A)/s(B) ≥ s(C)/s(D).
It can be shown that if the two monotonicity conditions are satisfied, and both scales are defined, say, on the unit interval, then there exists a constant k > 0 such that the support function derived from probability judgments and the support function assessed directly are related by a power transformation of the form s = ŝ^k. This gives rise to the power model
R(A, B) = P(A, B)/P(B, A) = [ŝ(A)/ŝ(B)]^k,
yielding
log R(A, B) = k log[ŝ(A)/ŝ(B)].
The model can be tested by comparing predicted probability judgments with judgments collected independently of the evidence-strength ratings.
When to use it
Studying how forensic reviewers’ evidence-strength ratings relate to their probability judgments
Comparing a product manager’s ratings of evidence for competing customer preferences with their probability judgments
Testing how medical researchers’ evidence-strength ratings map to their judgments about competing treatment hypotheses
FROM THEORY TO EVERYDAY LIFE
In practice
Medical Diagnosis Application
In a hypothetical judgment study, a doctor rates evidence strength for disease A (ŝ=0.9) and disease B (ŝ=0.3). Suppose prior judgment data give an estimated power parameter k of 0.5. The model predicts a judged probability ratio of R = (0.9/0.3)^0.5 = √3 ≈ 1.732, so P(A,B) = 1.732/(1+1.732) ≈ 0.634. Under these assumptions, the predicted judgment for A relative to B is about 63%.
A MOMENT TO REFLECT
Put it to work.
How does the power parameter k affect the mapping between direct evidence ratings and probability judgments? What does k > 1 imply about the sensitivity of probability estimates to evidence strength?
In what scenarios might the assumption of monotonicity between direct evidence assessments and support function be violated? Provide an example.
How could you design an experiment to estimate the power parameter k for a specific domain?
CONTINUE THE CONVERSATION
Source texts
- Heuristics and Biases The Psychology of Intuitive Judgment (Thomas Gilovich, Dale Griffin, Daniel Kahneman)