Subadditivity: Causal vs Temporal Unpacking

STUDY 2: CAUSAL VERSUS TEMPORAL UNPACKING

If implicit and explicit subadditivity arise from different mechanisms, their relative contributions should depend on how a hypothesis is partitioned. Some partitions are expected to induce primarily implicit subadditivity, whereas others are expected to produce primarily explicit subadditivity. The following study explores these effects and estimates their relative contributions.

Participants in the study were 165 Stanford students attending an introductory economics class. They answered, in a classroom setting, a few questions concerning the probability of various causes of death. The questionnaire introduced the task with the following population and mortality figures:

Each year in the United States, approximately 2 million people (or 1% of the population) die from a variety of causes. In this questionnaire, you will be asked to estimate the probability that a randomly selected death is due to one cause rather than another. Obviously, you are not expected to know the exact figures, but everyone has some idea about the prevalence of various causes of death. To give you a feel for the numbers involved, note that 1.5% of deaths each year are attributable to suicide.

This study consists of two cases. In Case 1, the focal hypothesis, homicide (H), is unpacked according to the causal agent: homicide by an acquaintance (H**a) and homicide by a stranger (H**s). In Case 2, the same focal hypothesis, H, is unpacked according to the time of occurrence: daytime homicide (H**d) and nighttime homicide (H**n). The alternative hypothesis in all judgments is accidental death (X).

In this study, the focal and alternative hypotheses are not exhaustive; the cause of death may be other than homicide or accident. Thus, participants here are asked to evaluate the conditional probability of the focal against the alternative hypothesis, assuming that one and only one of them holds. It is essential, of course, that participants understand and respect this assumption. Because the alternative hypothesis (X) in this design is held constant, probability theory requires additivity of odds, not of conditional probability. In particular, it implies R(H**a ∨ H**s, X) = R(H**a, X) + R(H**s, X), where R(A, B) denotes the probability ratio P(A, B)*/*P(B, A), provided P(B, A) ≠ 0.

We conjectured that the causal partition is more likely to bring to mind additional possibilities than the temporal partition. Homicide by an acquaintance suggests domestic violence or a partners’ quarrel, whereas homicide by a stranger suggests armed robbery or drive-by shooting. In contrast, daytime homicide and nighttime homicide are less likely to bring to mind disparate acts and hence are more readily repacked as an implicit disjunction. Consequently, we expect more implicit subadditivity in Case 1, due to enhanced availability, and more explicit subadditivity in Case 2, due to repacking of the explicit disjunction.

The study was designed as follows. In Case 1, the participants were randomly divided into three groups. One group (N = 55) evaluated the probability of the implicit disjunction that a randomly selected death is a homicide rather than an accidental death, P(H, X). A second group (N = 54) evaluated the probability of the explicit disjunction that a randomly selected death is a homicide committed by an acquaintance or a homicide committed by a stranger rather than an accidental death, P(H**a ∨ H**s, X). A third group (N = 56) evaluated the probability of the two individual components, P(H**a, X) and P(H**s, X).

Case 2 was presented to the same participants a few weeks later. The design and the procedures were the same, except for the use of the temporal partition instead of the causal partition. As previously, the participants were randomly divided into three groups. One group (N = 53) evaluated the implicit disjunction P(H, X); a second group (N = 53) evaluated the explicit disjunction P(H**d ∨ H**n, X); and a third group (N = 56) evaluated the two individual components, P(H**d, X) and P(H**n, X). The median estimates for both cases are presented in the upper part of Table 26.2.

Table 26.2. Median Probability Judgments (P) and Estimated Supports (s) and Weights (w) for the Two Partitions in Study 2

Note: H denotes homicide, X denotes accidental death, and H**a, H**s, H**d, H**n denote, respectively, homicide by an acquaintance, homicide by a stranger, homicide during daytime, and homicide during nighttime.

The table itself is not reproduced here. The lower part of Table 26.2 presents the supports and the weights derived from the median judgments, as shown later. Note that according to support theory, the odds P(A, B)/[1 − P(A, B)] = s(A)*/*s(B). Letting s(X) = 1, the support of each focal hypothesis in this study equals the odds of this hypothesis against the alternative X. For example,

s(H**a ∨ H**s) = 0.56/[1 − 0.56] = 1.27

Other support values were obtained similarly.

Support theory offers simple measures of implicit and explicit subadditivity. Let (A1*, . . .,* A**n) be a partition of the implicit hypothesis A. The ratio

w**A = s(A)/[s(A1) + ... + s(A**n)]

provides a global measure of the degree of subadditivity induced by the above partition. Note that w**A = 1 if probability judgments are additive, and w**A < 1 if they exhibit either implicit or explicit subadditivity. Thus, lower w implies greater subadditivity. To assess the separate contributions of implicit and explicit subadditivity, define

w**AI = s(A)/s(A1 ∨ ... ∨ A**n) w**AE = s(A1 ∨ ... ∨ A**n)/[s(A1) + ... + s(A**n)]

so that w**A = w**AI w**AE. Therefore, the global measure of subadditivity, w**A, is decomposed into its implicit (w**AI) and explicit (w**AE) components that can be estimated from the data.

Applying the preceding analysis to the data of Table 26.2 reveals more implicit subadditivity in Case 1 (w**H I = .76) than in Case 2 (w**H I = 1.00), and more explicit subadditivity in Case 2 (W**H E = .63) than in Case 1 (W**H E = .92). Strict implicit subadditivity was tested by comparing the supports of the implicit and explicit disjunctions in each case; that is, s(H) versus s(H**a ∨ H**s) and s(H) versus s(H**d ∨ H**n), using the Mann–Whitney statistic. Strict explicit subadditivity was tested by comparing the sum of the supports of the component hypotheses, within the data of a participant, to the support of the corresponding explicit disjunction; that is, s(H**a) + s(H**s) versus s(H**a ∨ H**s) and s(H**d) + s(H**n) versus s(H**d ∨ H**n). The analysis yielded significant strict implicit subadditivity in Case 1 (p *< .*01) but not in Case 2, and significant strict explicit subadditivity in Case 2 (p *< .*005), but not in Case 1. These findings support our conjecture that the causal partition induces more implicit subadditivity, whereas the temporal partition induces more explicit subadditivity.