154 Addendum Supplementary file – Chapter 5 Appendix: Details of statistical test procedure This appendix explains more about the significance tests in the main text. Marginal probability distributions within one person Consider, for example, the number of times an SDM vignette is classified in the low (L) tier. There are ten vignettes within a single person, of which three are SDM and seven are non-SDM. The person’s task is to assign three out of the ten vignettes to the L tier. Under the null hypothesis, the person is indifferent vignettes’ content and will randomly draw three vignettes randomly without replacement from the ten available vignettes. This can be represented as an urn with ten marbles (the ten vignettes), of which three are red (the three SDM vignettes), and seven are green, where a sample of three marbles is drawn without replacement (assigned to the L tier). The number of SDM vignettes in the sample is a random variable that we will denote by SDML C M Y CM MY CY CMY K SDM-L.pdf 1 07-03-2024 22:27 . The urn model implies that SDML C M Y CM MY CY CMY K SDM-L.pdf 1 07-03-2024 22:27 is comparable to the number or red marbles that is drawn. Therefore, the probability distribution of SDML C M Y CM MY CY CMY K SDM-L.pdf 1 07-03-2024 22:27 is a hypergeometric distribution (for which the urn model is a classic example). More specifically, P(SDML =k) = (3 k)( 7 3−k) (10 3) = 3!7!3!7! k!(3−k)!(3−k)!(4+k)!10! C M Y CM MY CY CMY K equation-39.pdf 1 07-03-2024 22:25 The probability distribution for EMOL C M Y CM MY CY CMY K EMO-L.pdf 1 07-03-2024 22:26 is the same. For TECH, the probability distribution is determined similarly, but considering that there are four rather than three TECH vignettes (i.e., four red marbles in the urn): P(TECHL =k) = (4 k)( 6 3−k) (10 3 ) = 4!6!3!7! k!(4−k)!(3−k)!(4+k)!10! C M Y CM MY CY CMY K equation-41.pdf 1 07-03-2024 22:25 These probability distributions are listed in table A1. The distributions for the high tier (SDMH, EMOH C M Y CM MY CY CMY K SDM-H, EMO-H.pdf 1 07-03-2024 22:27 and TECHH C M Y CM MY CY CMY K TECH-H.pdf 1 07-03-2024 22:27 ) are the same. Table A1 k C M Y CM MY CY CMY K k.pdf 1 07-03-2024 22:24 P(SDML =k) C M Y CM MY CY CMY K P(SDM-L-k).pdf 1 07-03-2024 22:23 P(EMOL =k) C M Y CM MY CY CMY K P(EMO-L=k).pdf 1 07-03-2024 22:23 P(TECHL =k) C M Y CM MY CY CMY K P(TECH-L=k).pdf 1 07-03-2024 22: 0 0.2917 0.2917 0.1667 1 0.5250 0.5250 0.5000 2 0.1750 0.1750 0.3000 3 0.0083 0.0083 0.0333
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