1) Narrative for E(t) for first-order scenario NC, including the operation of the

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1 Onlne Supplementary Materal Contents 1 Narratve for t for frst-order scenaro NC, ncludng the operaton of the Hypergeometrc Dstrbuton (Patl & Josh, 1968 Attrbute values and narratve formulaton of t for second-order scenaro UCN 3 Threat Expectaton t and Unpredctablty of Threatened Events Var(m 1 Narratve for t for frst-order scenaro NC, ncludng the operaton of the Hypergeometrc Dstrbuton (Patl & Josh, For Scenaro NC, t 1 p p 1 q t1 H( q 1;, ', q 1 t ' p 1 ' Expected threat t for NC comprses the sum of two expressons. The frst agan s smply the product of Pr(t 1 and (t 1 ; that s, the bn contanng t 1 s assgned wth probablty 1/p, and f assgned, t 1 s chosen. The frst term of the second expresson, (p- 1/p, represents the complement of Pr(t 1, the probablty that a t other than t 1 wll be selected. The second fracton q/(-1 represents the probablty that for any partcular t, the latter s located n a bn other than that of t 1, that s (p-1q/(-1, multpled aganst the probablty that the bn contanng t ' s assgned gven that the one contanng t 1 has not been assgned, or 1/(p-1. Each of the summed terms n the second expresson nvolves the hypergeometrc dstrbuton (e.g., Patl & Josh, 1968, used to obtan the probablty that t ' s the lowest threat value n ts bn, gven that the bn wth t 1 s not assgned and that t ' s not n t 1 s bn. The Hypergeometrc dstrbuton s used to assess the probablty of a gven t ' thus beng selected, whch s tantamount to the probablty

2 that all other elements n ts current bn have hgher threat values. Parallelng bn-model termnology, H(q-1; -, -', q-1 s the probablty that out of a random samplng of q-1 (mutually ndependent balls, wthout replacement (cf., Mlenkovc, that s, q-1 (mutually ndependent threat values t ' -- from a bn contanng a total of - balls (both t 1 and the t ' under consderaton themselves are nelgble, -' of whch are whte, -- that s, -' for whch t ' exceeds the partcular t ' under consderaton, -- all q-1 sampled balls are whte -- that s, all sampled t ' values exceed the currently entertaned t '. An analogous format of the hypergeometrc dstrbuton s used when the latter s called upon n obtanng the remanng mathematcal expectatons of threat. Note that H(q-1; -, -', q-1 s equal to 1- j q 0 H ( j;, ', q 1. Moreover, obtanng q-1 whte balls n a random samplng of sze q-1 mples obtanng 0 black balls (.e., obtanng no t ' values lower than the consdered t ', wth the equvalent probablty now expressed as H(0;, ', q 1 1 H( j;, ', q 1. Along wth the element encounters beng mutually exclusve and exhaustve, whereby consttuent probabltes nvolved n t sum to 1.0, these relatons are avalable as computatonal checks regardng t for structure NC, and by extensons prescrbed by ther host scenaro structures, all other t formulae. q 1 j1

3 Attrbute values and narratve formulaton of t for second-order scenaro UCN Scenaro UCN. Pr(t 1 = 1/Pq RSS = p; t P 1 1 q 1 p t1 H( p 1; P, P ', p 1 P q q P 1 ' t ' P 1 ( P 1 p 1 P P 1 P 1 P ' H( p 1; P, P ', p 1 t ' OSS = Pp. The value of Pr(t 1 corresponds to the nverse of the product of the set szes (P and q of the herarchy levels at whch free choce s not exercsed (U and N. The RSS corresponds to the set sze (p of the ter wth free choce. The OSS s the product of the two set szes of the ters at whch there are multple possbltes (P and p under U and C, respectvely. There are two larger expressons n the UCN t formula. The frst fracton appled to the entre frst expresson s 1/P, the probablty that t 1 s bn set s selected. Wthn the large bracket, the frst term 1/q, as combned wth 1/P, accounts for Pr(t 1, whch s scaled by t 1. The frst fracton n the second term wthn the bracket represents the complement of 1/q, the probablty that t 1 s not assgned ((q-1/q. The second fracton p/(p-1 represents the probablty that t ' s one of those assgned nto t 1 s bn set, gven that t 1 tself s not assgned. The appearance of the hypergeometrc dstrbuton wthn ths frst larger expresson conveys the probablty that t ' s the least of the t ' 's

4 assgned n t 1 s bn set, gven that t 1 tself s not assgned. The hypergeometrcdstrbuton probabltes multplyng respectve t' values are summed over values. The second larger expresson starts wth the probablty that t 1 s bn set s not assgned((p-1/p. The next fracton ((P-1p/(P-1 represents the probablty that t ' s one of the assgned t' s located n bn sets other than t 1 s bn set. The fracton 1/(P-1 then accounts for the probablty that the bn set contanng t ' s assgned, gven that t 1 's bn set has not been assgned. The second use of the hypergeometrc dstrbuton yelds the probablty that t ' s the least of the p t' s assgned n t ' s bn set gven that t 1 s bn set s not assgned. Ths second larger expresson then entals the summaton of these probabltes of t' encounters combned wth the respectve t ' values.

5 3 Threat Expectaton t and Unpredctablty of Threatened Events Implementaton of decsonal control conveys specfc values of threat expectaton t accordng to a scenaro structure s prevalng choce condtons. Decsonal-control determned t also enters nto adverse-event predctablty, as follows. Level of threat dentfed wth scenaro-element, defned as the probablty of adverse-event occurrence t, does not drectly stpulate the mpact, or magntude, of the stochastc event tself (cf., Neufeld, 1990; Paterson & Neufeld, However, mplct n the current expresson of structure-wse threat level t are two event magntudes (m, specfcally 1 unt of severty, correspondng to event occurrence, and 0 unts, correspondng to event nonoccurrence: 1 [ t (1.0 (1 t (0] 1 t t. (5 Ths dchotomous format of event magntude nevertheless s coherent wth assocatonalmemory ( categorcal memory accounts of probablty learnng (Estes, 1975; 1976; 1977, whch repeatedly have been shown to extend to predctve judgments of stressorevent occurrence (Morrson, et al, 1988; Mothersll & Neufeld, 1985; Neufeld & Herzog, 1983; Lees & Neufeld, Greater varaton n event magntude m, however, can be accommodated n the present t computatons; where the adverse event correspondng to scenaro-element s of an unque magntude m, for example, ts cross-product wth the event s probablty of occurrence t m, that s t m m, element's value of t n the present layout. Var(m, n turn becomes can be stpulated to equal the 1 m t m m [ 1 m t m m ].

6 Exercse of avalable control thus affects level of stressor-event threat, as expressed n t; however, t also mpnges on stressor-event predctablty n quantfable ways drectly ncorporatng t. Note that stress actvaton s deemed to be drven upward as predctablty of stressor characterstcs, ncludng event magntude, declnes (see, e.g., Denut & Genest, 001; Osuna, 1985; Paterson & Neufeld, 1987; Smth, 1989; Suck & Hollng, Accordngly, varance n event magntude Var(m can be shown to equal t[1-t]. Formally, m{0,1}; t [0,1]; Var( m E[ m ] [ m ] 1 [ t (1 (1 t (0 ] { 1 Pr ( t [ t (1.0 (1 t (0]} 1 ( t [ 1 t ] t [1 t]. (6 Consequently, Var(m s maxmzed where t = 0.5. Ths value turns out to be approxmated by the larger values of t obtaned n the smulaton results presented n Tables 3 and 4 (see man document; further consderatons surroundng Var(m are avalable as a.pdf document from the frst author. In ths way, those scenaro structures wth elevated threat t also are accompaned by elevated unpredctablty t[1-t]. Other ndexes of (unpredctablty, framed wthn the present quanttatve structure, convey addtonal stuaton propertes that stand to be psychologcal-stress sgnfcant. One such ndex, whch gnores event magntude, s Var(t, t [ 1 1 t ].

7 Another, whch crcumvents event magntude, s Shannon-Weaver nformaton entropy: 1 log [ Pr( t ] Pr( tm m log [ Pr( t 1 m m ]; Pr( t Pr( t m m. References not lsted n man document Denut, M., & Genest, C. (001. An extenson of Osuna's model for stress caused by watng. Journal of Mathematcal Psychology, 45, Estes, W. K. (1975. Structural aspects of assocatve models for memory. In C.N. Cofer (Ed., The structure of human memory. San Francsco: Freeman. Estes, W. K. (1976. The cogntve sde of probablty learnng. Psychologcal Revew, 83, Estes, W. K. (1977. Some functons of memory n probablty and choce behavor. In G. H. Bower (Ed., The psychology of learnng and motvaton (Vol. 10, pp New York: Academc Press. Mothersll, K.J., & Neufeld, R.W.J. (1985. Probablty learnng and copng n dysphora and obsessve-compulsve tendences. Journal of Research n Personalty, 19, Neufeld, R.W.J., & Herzog, H. (1983. Acquston of probabltes n antcpatory apprasals of stress. Personalty and Indvdual Dfferences, 4, 1-7.

8 Osuna, E. E. (1985. The psychologcal cost of watng. Journal of Mathematcal Psychology, 9, Paterson, R., & Neufeld, R.W.J. (1987. Clear danger: The percepton of threat when the parameters are known. Psychologcal Bulletn, 101, Smth, A. (1989. A revew of the effects of nose on human performance. Scandnavan Journal of Psychology, 30, Suck, R., & Hollng, H. (1997. Stress caused by watng: A theoretcal evaluaton of a mathematcal model. Journal of Mathematcal Psychology, 41,