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3 Testing Our Model

This model does not incorporate an error term, and is therefore deterministic. It successfully predicts the modal outcome of each experiment in Table 1.

Table 1. Overview of the 14 effects replicated by our model (Sources: [15-18] & Table 2)

Experimental Effect

Outcome

A: 200 live

B: 1/3 * 600 live or 2/3 * none live

A

C: 400 die

D: 1/3 * 600 die or 2/3 * none die

D

A: 200 live

B: 1/3 * 600 live

Indifference

C: 400 die

D: 2/3* 600 die

Indifference

A: 200 live

B: 2/3 * none live

Strong A

C: 400 die

D: 1/3 * none die

Strong D

A: 200 live

B: 1/3* all live or 2/3 * none live

Attenuated

C: 400 die

D: 1/3 * none die or 2/3 * all die

Attenuated

A: 200 live and 400 don't live

B: 1/3 * 600 live or 2/3 * none live

Indifference

C: 400 die and 200 don't die

D: 1/3 * none die or 2/3 * 600 die

Indifference

A: 400 do not live

B: 1/3 * 600 live or 2/3 * none live

B

A: 200 do not die

B: 1/3 * none die or 2/3 * 600 die

C

A: $1m with certainty

B: 0.89*$1m or 0.1*$5m or 0.01*$0

Weak A

C: 0.89 * $0 or 0.11*$1m

D: 0.90*$0 or 0.10*$5m

Strong D

3.1 Meta-analysis

We next performed a meta-analysis of 13 studies to test our prediction that changing the interpretation of an existing problem, without changing the numerical information, would change what gists are encoded. We examined studies in the literature that deviated from Tversky & Kahneman's original experimental protocol by substituting the word “all” to options B and D in the DP. The new options B & D read:

B) Some chance that all live and some chance that none live

D) Some chance that all die and some chance that none die

The effect sizes for these studies were calculated using Cramer's V – the difference between the proportions of the samples that chose each option in each frame (thus, a perfect framing effect in which every subject chose option A would have an effect size of 1.0, whereas no framing effect, in which subjects were equally likely to choose options A and B, would have an effect size of 0) – and then compared to the corresponding effect sizes for those studies in the literature that replicated Tversky & Kahneman's original protocol. Results are shown in Table 2. We analyzed only between-subjects comparisons so as to avoid the heuristic override effect [19].

Table 2. Results of our meta-analysis showing the effect of including the word "All"

Hypothetical Population

Size

Wording Reference

N

p-value (Chi Square)

Effect Size

Standard [20]

213

<0.001 (12.27)

0.24

60

Includes [21]

80

0.813 (0.07)

0.03

"All" [22]

63

0.458 (0.51)

0.09

[5]

307

<0.001 (76.75)

0.50

[23]

90

<0.001 (21.61)

0.49

600

[24]

244

<0.001 (49.41)

0.45

[25]

292

<0.001 (31.80)

0.33

[26]

148

<0.001 (29.97)

0.45

Standard [27]

105

<0.001 (20.33)

0.44

Includes [21]

100

0.005 (7.84)

0.28

"All" [22]

65

0.005 (7.96)

0.35

Standard [28]

46

0.003 (8.91)

0.44

6000

Includes [21]

88

0.055 (3.52)

0.20

"All" [22]

61

0.029 (4.78)

0.28

For hypothetical population sizes of 60 and 6000, at least one of the tests including the word “all” did not have a significant framing effect. Furthermore, when combining across all three population sizes, effect sizes for tests using the word “all” are significantly smaller than they are for tests that follow the standard DP formulation (Mann-Whitney U test; U = 44.0; p=0.004). Note that overall effect sizes for populations of size 60 are smaller than those for larger populations. We explain this by the fact that 20 is interpreted as essentially nil [29], attenuating the framing effect.

 
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