William M. Rogers. Michigan State University, USA ... reanalyses data presented by Bishop, Fienberg, and Holland (1975) that describe the development of ...
IN T E R N A T IO N A L J O U R N A L O F B E H A V IO R A L D E V E L O P M E N T , 1998, 22 (3), 537±557
Mo de lling Syne rgy using Manife st C ate go rical V ariable s A le xa nd er vo n E ye M ichigan S tate U n iv ersity , U SA
C h rist of Sch uste r U n iv ersity o f M ich igan, U S A
W illia m M . R o ge rs M ichigan S tate U n iv ersity , U SA
T his pa p er d iscu sses m e t ho ds t o m od e l th e con cep t of syn e rgy at t he le ve l o f m a nife st cat ego rica l va riab le s. F irst, a cla ssi® ca t io n of co n ce p ts of syn er gy is pr e se nt ed . A d ditive an d n on a dd itive co nce pt s of syne rgy a re d istin gu ishe d . M o st pr om in en t a m o ng t he no n ad d itive co nce pt s is sup e ra dd itive syne rgy. E xa m p le s a re give n fro m t he na t ur al scie nce s a nd t he so cia l scie n ce s. M o de lling focu se s on t h e re la t io nship b et wee n t he a gen t s invo lve d in a syn er ge tic p ro ce ss. T he se re la t io nships a re exp re ssed in for m o f con tr asts, e xp re ssed in e ffect co ding ve cto rs in d esign m a tr ice s fo r no n stan da r d lo gline a r m o de ls. A m e th od b y Schu ste r is u se d t o tr a nsfo rm de sign m a tr ice s su ch t ha t pa ra m e te rs re ¯ ect th e p ro po se d re la tion sh ip s. A n e xam p le re a na lyse s d a ta pr ese nt ed b y B ish op , F ien b er g, a nd H olla n d ( 1975) th a t de scr ib e t he de velo pm e n t o f th ro m b oe m b olism s in wo m e n wh o differ in t h eir pa t te rn s o f co n tr ace pt ive use a nd sm o k in g. A lte rn at ive m e t ho ds of an a lysis a re com p a re d. I m plica tion s fo r de velo pm e n ta l re se a rch ar e discu sse d .
T h e co n cep t of ``syn ergy’ ’ is im p o rt an t in a wid e va rie ty o f scie n t i® c co nt e xt s. T h e t e rm ge n era lly a pp lies t o sit u at io n s in vo lvin g t h e a ctio n o f t wo o r m o re a ge n t s in effe ctin g a n o u t co m e . T h e re la t io n sh ip b e t we en t h e a ge n t s, a n d t h eir in divid u al re lat io n sh ip s with th e o u t co m e , in p a rt a llo w t h e d e ® n itio n o f d iffe re n t t yp es o f syn e rgy. It is im p o rt an t t o m a k e a R e qu est s fo r r e p r in ts sh ou ld b e se nt to P r o fe ssor A le xan der vo n E ye , M ichiga n St ate U nive rsity, D ep ar tm en t o f P sych ology, 119 Sn yd e r H a ll, M I 48824-1117, U SA . T h e a utho rs a re ind eb ted to M ar ia W o ng a nd R o be rt A . Z u cke r fo r co m m en ting o n e ar lier ver sion s of th is p a pe r. A le xan de r vo n E ye’ s wo r k on this pa p e r wa s su ppo rte d in p ar t by N I A A A gr an t N o. Z R 01 A A 07065. c 1998 T h e I n te rn ation a l So ciety fo r th e St u d y of B eh aviou r a l D e ve lop m e nt
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d istin ctio n b e t wee n n u m erical o r stat ist ica l d e ® n itio n s o f syn ergy, a n d d e ® n itio n s b a se d o n em p irica l m e ch a n ism s. It is lik e ly t h a t a wid e va riet y o f a ge n t -o u t co m e re la t io n sh ip s can b e classi® e d in a ge n e ra l syn e rgy t a xo n o m y fo r t h e p u rp o se s o f an a lysis an d stat ist ica l con clu sio n . O n e o f t h e p u rp o se s o f th is p ap e r is t o p re se n t su ch a stat ist ica l m e t h o d, u sin g m an ife st ca t ego rical va ria b le s, wh ich wo u ld b e u sa b le in a va rie ty o f co nt e xt s. It is q u ite a n o t h er issue to exa m in e t h e e m p irical m e ch a n ism s o f syn ergy, a s th e y a re ge rm a n e t o t h e p art icu la r ® e ld in qu e stio n . F o r e xa m p le , t h e e m p irical re la t io n sh ip a m o n g p sych o t h e rap ie s in t h e ir e ffe ct s o n m e n t a l illn ess is su re ly n o t th e sam e ty p e o f em p irica l re lat io n sh ip a s e xist s a m o n g ch e m ica l in sect icid e s in t h e ir effe cts o n crop p e sts. B o t h a re n e ve rt h e less a b le t o b e a n a lyse d usin g th e sa m e m et h o d o lo gy a n d p e rh a p s t h e sa m e sta t ist ica l m o d el.
SYNERGY: DEFINITION AND USE OF TERM C lassifyin g syn e rgy rela t io n sh ip s b a se d o n n u m e rical crite ria e sse n t ia lly d e lin e at e s t h e re lat io n sh ip b e t wee n q u a n titie s a sso cia t e d with t h e age n t s a n d t h e qu a n t ity a ssocia t ed with a n o u t co m e. T h is n u m erica l re la t io n sh ip , a t le a st fro m t h e pe rsp e ct ive o f sta t ist ica l m o d ellin g, m a y fa ll in t o t wo b ro a d classi® ca tio n s: ad d itiv e a n d n on ad d itiv e. A d d itiv e sy n ergy re fe rs to a sit ua t io n in wh ich t h e to t a l e ffe ct is sim p ly t h e su m o f t h e in d ep e n d e n t e ffe ct s fo r e a ch a ge n t . In t h is sit u a tio n , t h e re n e e d n o t b e rela tio n sh ip s b e t we e n t h e a ge n t s, b u t o n ly a sim ila r re la t io n sh ip b e t wee n e a ch a ge n t a n d t h e d e sir ed e ffe ct (e .g. t e a m m e m b e rs’ con t rib u t ion s t o to t a l e ffor t in ro p e p u llin g co nt e st). A n ot h e r ca se , wh ich m igh t b e co n sid e re d a su b se t o f a d d itive syn e rgy, is co n d itio n ally ad d itiv e sy n ergy . T h is is a sit u a t io n wh e re t h e a ctio n o f t h e e nt ire se t o f a ge n t s is re q u ire d to p ro d u ce a d e sired effe ct. F or a n y a ge n t t o tra n sfe r a n effe ct, t h e a ctio n s o f o th e r age n t s a re n e ce ssa ry. A n y a ge n t a ctin g in iso lat io n can n o t p ro d u ce t h e effe ct. T h e d ist inct io n b e t we e n ad d itive an d co n d itio n a lly a dd itive co n ce pt u alisat io n s o f syn e rgy ca n b e co m e h a zy wh en t he q u a n t ita t ive n a t u re o f t h e o u t co m e is in q u e stio n . F or in stan ce , if o n e is exa m inin g a q u a n tita tive o u t co m e va ria b le with a critica l t h re sh o ld le ve l a t wh ich a q u a lit a t ive ch an ge t ak e s p lace , sim p le a d ditive e ffect s ca n a p p ea r co n d itio n ally a d d itive. In a sit u a t io n wh er e t h e t ru e re la tio n sh ip is ad d itive , t wo in d ivid ua l a ge nt s’ e ffe cts m a y fall b e lo w th re sh o ld in dividu a lly, su ggest in g n o q u alitativ e e ffe ct, wh e re as t h e ir su m e xce e d s t hre sh o ld , p ro d u cin g wh a t a p p ea rs t o b e a n e ffect co n d itio n a l o n b o t h a ge n t s’ a ctivit y. F ina lly, t h e rela t io nsh ip can b e se e n a s n o n ad d itiv e, wh ich in clu d e s t h e su b cla ssi® ca t io ns o f su p erad d itiv e, sub ad d itiv e, an d iso lated syn e rgy. Su p e rad d itive syne rgy o ccu rs wh e n th e ``wh o le is gr e at er t h an t h e su m o f p art s’ ’ , a n d su b a d d itive syn e rgy is d e n o t e d b y t h e ``wh o le be in g less t h a n
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t h e su m o f p a rt s’ ’ . N u m e rically, t h e se re la t io n sh ip s a re d e ® n e d b y b o t h a n a d d itive co m p on e n t fo r e a ch a ge n t ’ s effe ct o n t h e o u t com e , a n d a n in te ra ct ive (o r m u ltip licat ive ) co m p o n e n t fo r th e re lat io n sh ip s am o n g a ge n t s in a ffect ing t h e ou t co m e. Iso la t ed syn e rgy in vo lve s t he in te ra ct ive co m p o n e nt a m o n g a ge n t s, bu t with o ut t he in ¯ u en ce o f a n y ad d itive e ffe ct s fo r in d ivid u al age n t s. H yb rid m od e ls, wh ich co n t ain b o t h in t e ra ct ive t e rm s a n d a su b se t o f a d d itive effe cts fo r e a ch a ge nt , are a lso co n ce iva b le . T h e t yp es o f syn e rgy can b e d ist in gu ish e d b y t yp e a n d d egr e e s o f re strict io n s t he stat ist ica l m o d els p la ce on t h e rela t ion sh ip s b e t we en age n t s a n d e ffe cts. A lth o u gh a ll t yp e s im p licit ly p o sit t h a t effe cts ca u se d b y age n t s can o r do e xist, t h e y differ in h ow t h e re lat io n sh ip s b etw een a ge n ts a re co nce p t u alise d, a n d h o w t he co n cep t s o f in d ivid u al e ffe cts d iffer fro m t h e co m p o sit e e ffe ct. It sh o u ld b e n o t ed t h a t syn e rgy ha s o n ly t h u s far b e e n d e ® n ed n u m e rically, b y m a gn itu d e o f e ffe ct s o n a n o u t co m e . T h e p re se n t p a p er p ro p o se s a lso m o d ellin g t h e re la t io n ship s b e t wee n a ge n t s. A lin k age o f co n ce p t u a l a n d e m p irica l no t io n s o f syn e rgy ca n b e ga in e d b y exa m in in g scie nt i® c d o m a in s in wh ich t h e t erm ``syn e rgy’ ’ h as b e e n in vo k e d an d u sed t o e xp lain em p irica l re su lts. T h e su p e ra d d itive n o t ion o f syn ergy is co m m o n acro ss m a n y d o m a in s o f scien ti® c lit e ra tu re , a n d is o ften co n sid e re d t h e ``cla ssica l’ ’ co nce p t u al typ e o f syn e rgy. E xa m ple s ca n b e fo u n d in b o t h th e n at u ra l scie n ce s a n d so cia l scie n ces. T h e fo llo w in g p a ragr a p h s ® rst give a se le ctio n o f e xa m p le s o f t h e u se o f t h e syn e rgy co nce p t in t h e n a t u ral scie n ce s. E xa m p les fro m t h e so cial scie nce s fo llo w.
S y n ergy in th e N atu ral S cien ces: E x am p les o f U se. B en n e t t , M la d y, F le sh n e r, a n d R o se (1996 ) fo u n d t h a t t wo che m ica l t re at m e n t s wh ich d id n o t ca u se b e h a vio u ra l im p airm e n t wh e n a dm in iste re d sep a ra t e ly, d id ca u se m ark e d b e h a vio u ra l im p a ir m e n t wh e n co -a d m in ist ere d . A rn o ld e t a l. (1996 ) stu d ie d t h e e ffe cts o f e n viro n m en t a l o e stro ge n ic co m p o u n d s, a n d fo u n d t h a t wh e n ad m in ist e re d in co n cert , oe stro ge n ic co m p o u n d s in cre a sed t h e ir in d ividu a l p o te n cie s to o ve r on e t h o u sa n d tim es th e ir o rigin a l in d ividu a l p o t en cy. D ra sn e r (1988 ), in stu d yin g t h e a n tin o cice p tive e ffe ct s o f m o rp h in e an d clo n id in e , fo u nd th a t e ith e r in iso la tio n h a d n o e ffe ct , wh er e as t h e ir co m b in a t io n sh o we d a n t in o cice p t ive e ffect s. Se a le e t a l. (1987 ) d em o n stra t ed t h a t ca ffe in e a n d ot h e r su b co n vu lsa n t co m po u n d s, wh e n co m b in e d cre a t e d co n vu lsive effe ct s in ra t s. In e p id e m io lo gy, W o rceste r (1971 ; see B ish o p e t a l., 1975 ) a sk e d wh e th e r th ro m b o e m b o lism s are m o re fre q u en t in in d ivid u a ls wh o b o t h sm o ke a n d t ak e o ra l co n t race p tive s t h an in in d ivid u a ls wh o e ith e r o n ly sm o ke o r o nly t a k e o ra l co n t ra ce p t ive s, o r n e ith e r. T h e t wo a ge n t s in t h is e xa m p le are sm o k in g a n d t h e o ral co n t race p t ive . B ish o p et al. (1975 ) a n alyse d t h is d a t a se t u sin g th e ® rst, le ss re strict ive o f t h e fo rego in g t wo
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d e ® n itio n s. R e su lts su gge ste d t h a t t h e t wo a ge nt s in d e e d ca n b e se e n a s syn ergist s. (L a te r se ctio n s o f t h is p a p e r u se t h e se d a t a aga in.) In b io ch e m ist ry, M o rse , M cK inla y, a n d Sp u rr (cite d fro m M cC u lla gh & N e ld e r, 1983) ask e d wh e t h er th e in sect icid e , ca rb o fu ran , a n d t he syn e rgist, p ip ero n yl b u t o xide , a re jo int ly m o re e f® cie nt t h an e ith e r in co n t ro llin g t h e gr a ssh o p pe r M elano p lu s san gu in ip es.
S y n ergy in th e S o cial S cien ces: E x am p les o f U se. T h e ap p lica t io ns o f t h e su p era d d itive co n ce p t are n o t lim ited to t h e n a tu ra l scien ce s. K e t t e r (1992 ) fo u n d t h a t sym p t o m s a sso cia t e d with b ip o la r d iso rd e r we re re fract o ry t o e ith e r o f t wo p sych o act ive m e dicat io n s, b u t resp o n sive t o t h e co m b in a t io n . G u n zbu rg (1995 ) ap p lie d in d ivid u al, gr o u p , a n d m a rita l t h e rap y m e t h o d s, a nd d e m o n stra t e d t h e syn e rgist ic fu n ctio n in g o f t h e t h re e in co m b in at io n . W ile n s, Sp e n ce r, B ie d erm a n , a n d W o znia k (1995 ) ra ise d th e issue o f syn ergy o f m e d ical d ru gs u sed to ge t h er fo r t re a t m e n t o f p sych iat r ic d iso rd e rs in ch ild re n a nd a d o le sce nt s. T h e au th o rs d e t ail gu id elin e s t h a t can be u se d fo r d e t e rm in in g t h e ap pro p ria te n e ss o f u sin g m u ltip le age n t s (d ru gs) in p h a rm a co t h er a py. In a co m p ara b le co n t ext , B u ck le y a n d Sch u lz (1996 ) d iscu ssed t h e p o te n t ial fo r syn e rgy b e t wee n n ove l a n t ip sych ot ic d ru gs a n d m o d e rn p sych o so cia l t h e ra p ies in t h e t re at m e n t o f sch izop h ren ia a n d affe ctive d iso rd e rs. In a p sych o th e ra p e u t ic co n t e xt , L o de o n (1986 ) at t e m p te d t o e stab lish syn ergy b et we e n t h e d e sir ed h e a lin g b e h a vio u r a n d th e sym p t o m p re sen t e d a t co nsu lta t io n . T h e re p o rt e d sym p t o m was in t wo ca ses th a t a yo u n g m ale an d a yo u n g fe m ale we re un a b le t o co n su m m a t e th e ir re sp ect ive m arr ia ge s. In a n e ffo rt to sp e cify a m a t h e m a t ica l m o d e l o f o lfa cto ry p e rce p t io n , L affo rt , E t ch e t o , P at t e , a n d M a rfa in g (1989 ) co m bin e d t he e xp erim en t a l e ffe ct ive n ess o f m ixt u re s with t h e p o we r la w e xp on e n t o f t h e co m p o n e n ts. T h e a u th o rs d e ® ne a co ef® cien t th a t ca n b e use d t o d e rive cu rve s o f o lfa ct or y iso -in te n sit y. F ro m t h e se cu rve s t h e a u t h ors co n clu d e t h at t h e co ef® cie n t ha s ``in t e gr at ive stre n gth ’ ’ t h a t a llo ws o n e t o d e pict syn e rgy o f o d o u r co n cen t ra t io n s. A lso in p sych o p h ysio lo gy, A yya an d L a wle ss (1992 ) co m p a re d e xp e rim e n t a lly th e swe e t n e ss o f sin gle swe e t en e rs a n d 50:50 m ixt u re s o f swe et e n e rs. R e su lts su gge ste d t h a t swee t e n er s act syn e rgist ically. In a d d itio n , b a sed on co m p a riso n s o f fre q u e n cie s, t h e au t h o rs in t erp re t e d t h e ir re su lts a s su p p o rt in g t h e co n cep t o f su p era d d itivit y. A t a m o re t h e o re t ica l leve l, N elso n (1996 ) d iscu ssed p h ilo so p h ica l a n d p sych o lo gia l a p p ro a ch e s t o co n scio u sne ss. T h e a u t h o r p ro p o se s t h a t t h e re se arch o n m e t a co gn itio n ca n p ro d u ce syn e rgy be t we en th e t wo
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a p p ro ach es b y sp e cifyin g con str ain t s o n t h e ran ge o f a cce p t a b le t h eo rie s a n d by gu idin g d e ve lo p m e n t o f n ew t h e orie s. In bio lo gy, syne rgy is o fte n u se d t o d e scrib e t h e co o rd in a tio n o f t wo m u scles, fo r in stan ce , of ¯ e xo rs a n d e xt e n so rs. A ru in , A lm e id a , a n d L at a sch (1996) a n a lyse d th e b eh avio u r o f D o wn syn dro m e pa t ien t s wh o d isp la ye d la ck o f syn e rgy o f t h ese m u scle s in e lb o w o r wrist m o ve m en ts. T h e a u t ho rs o b se rve d t h a t t h e p a t ien t s sh o w sim u lta n e o u s b u rsts o f b o t h ¯ e xo r a n d e xe n so r, a n d d iscu ss th e o rie s e xp la in in g t h e p a t ie n t s’ a d a p t at io n t o pro b le m s t h a t ca n re su lt fro m th is b e h avio u r. H o u sto n a n d D o a n (1996 ) in ve stiga t e d t h e syn ergy of n e ga t ive a n d p o sitive p o lit ica l a d ve rt isin g ca m p aign s. T h e au t h o rs sta t e t h at b o t h t yp e s o f in fo rm at io n in cam p a ign s a ct in syn ergy an d d irect t he vo t ers’ a t t en t io n t o eith e r th e b a d o r t h e go o d fe a t u re s o f t h e co m pe t in g ca n d id a t e s. M o st p ro m in e n t in d e ve lo p m e nt a l re se arch is G o t t lie b ’ s (1992 ) co n ce p t o f syn ergy. T h e au t h o r p ro p o se s t h a t t h e ``cau se o f de ve lo p m e n t Ð wh a t m ak e s d e ve lo p m e n t h ap pe n Ð is t he re la t io n sh ip o f t h e t wo co m p o ne n ts, n o t t h e co m p o n e nt s t he m se lve s’ ’ (p p . 161±162). G o t tlie b’ s t e rm co m p o n en t is u se d in a fa sh io n pa rallel t o t h e p rese n t t erm agen t. Sa m p le co m po n e n t s/ a ge n t s a re orga n ism s, en viro n m e n t , o r t h e sen so ry syst e m . T h o u gh n o t a s p re va le n t in p u b lish ed rese a rch , su b ad d itiv e eff ects o f a ge n t co m b in a t io n s ha ve a lso b ee n d e m o n stra t ed . B ro glia a n d B ru n o ri (1994 ) in ve stiga te d t h e e ffect s o f h igh su cro se co n ce n t ra t io n an d lo w t e m p e ra t u re o n m aize p o llen via b ility. It was fo u n d t h at lo w t e m p era t u re o r h igh su cro se co n ce n tra t io n p rese n t e d in iso la t io n ca u sed m a ize p o lle n t o lo se viab ility q u ick ly. H o w eve r, b o t h p rese n t e d sim u lta n e o u sly h a d n o su ch e ffe ct. M ia sk o wsk i (1993) e xa m in e d th e e ffe ct s o f sp in al ch em ica l co nce n t ra tio n s o n n o cicep t ive t h re sh o ld s in m a le rat s. It wa s fo u nd t h a t su p ra spin a l D D P E a n d sp in al D A M G O b o t h a lte re d n o cice p t ive t h re sh o ld s in iso la t io n, b u t t h e ir co n cu rre n t a d m in ist ra t io n sh o we d n o syn e rgist ic e ffe ct s. It sh o u ld b e n o t e d t h a t t h e q u a n t ita t ive n a t u re o f o u t co m e va ria b les in t h ese e xa m p le s is su f® cien t ly va gu e as t o sugge st t h e cla ssi® ca t io n o f syn ergy in t o su b a d d itive ve rsu s su p era d d itive ca t ego rie s m ay b e a rb itra ry. W ith o u t a p ro p e r u n de rsta n d in g o f t h e m e ch a n ism s o f t h e rela t io n sh ip , t h e su b t ra ctive o r a d d itive e ffe cts o f age n t co m b in at io n s can n o t b e d et e rm in e d . It ca n , h o weve r, be sh o wn t h at t h e re la t io n sh ip is so m et h in g m o re t h an sim p le a d d itive syn e rgy. T h e e m p irica l m ech a n ism s b e h in d syn e rgist ic e ffe ct s in t h e b io lo gica l a n d b io ch e m ica l scien ce s a re va rie d, b ut t wo a re no t a b le Ð ch em ica l in te ra ct io n b e t wee n t h e a ge n t s, an d in t e ract ive re la tio n sh ip in t h e b io lo gica l syst e m . A n e xa m p le o f th e ® rst co n ce p t is th e sim p le ch em ica l re a ction . T h e p rese n ce o f m u ltip le ch e m ica l a ge n t s m a y tra n sfo rm a n y give n a ge n t in t o a sligh t ly d iffere n t ch e m ica l co m p o un d , h a vin g d iffe re n t e ffe ct s. T h e la t te r t yp e is lik e ly m o re p re d o m in a n t , t he p ro t ot yp ica l
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e xa m p le b e in g t wo che m ica l a ge n t s co m p e t ing fo r t h e sa m e rece pt o r sit e s in a n eu ro ch em ical syst em . T h e e ffe ctive n e ss o f p sych o a ct ive d ru gs is p re d ica te d o n th e fa ct t ha t t h ey b lo ck re ce p t o r sit e s fro m n a tu ra l n e u ro tr an sm itt e rs a sso cia t e d wit h b eh avio u ral d iso rd ers. T h e co n ce p t o f syn e rgy is re la t e d to t he ca u sa l W e d ge (se e So b e l, 1995; vo n E ye & B ra n dt staÈ d t e r, in p re ss). T h e W e dge co n ce p t u alise s a cau sa l re la t io n sh ip wh e re o n e o r m o re ca u se s le a d t o th e sa m e e ffe ct. T h e co n ce p t o f th e W e a k W e d ge is o p e n t o t h e in t e rp ret a t io n th a t a go a l ca n be re ach e d t h ro u gh m o re t h a n o n e ca u se . E ach o f t h e cau se s is su f® cie nt ye t n o t n e ce ssa ry fo r t h e e ffect t o b e o bse rve d . Syn e rgy can b e vie wed a s a sp e cia l case o f t h e we d ge in t h e sen se t h a t t wo o r m o re age n t s are a lways b o t h su f® cie n t a n d n e ce ssa ry fo r a n e ffe ct (se e F ig. 1) . T h e p re se n t a rt icle is co n cern e d wit h m o d e llin g syn ergy at t h e le ve l o f m an ife st ca t e go rical va riab le s. W e p ro p o se lo g-lin ea r m o d e ls fo r m o d e llin g t h e co nce p t o f syn ergy . F o cu s is o n m a n ifest ca t e go rica l va ria b les. B a sed o n a m e th o d p ro p o se d b y Schu ste r (in p re p a ra t io n ), d e sign m a t rice s a re sp e ci® e d t h a t le a d t o p a ram e te rs t h a t re ¯ ect t h e con t ra sts u nd e r stu dy. T h e m e th o d s p ro p o sed h e re : (1) o p e ra t e a t t h e le ve l o f m a n ifest va ria b les; (2) u se cat e go rica l va ria b les; a n d (3) d e ® n e re lat io n sh ip s a m o n g va riab le s a t t h e le ve l o f ca te go rie s o f va ria b les ra t h e r t h an va riab le s as su ch .
M ODELLING SYNERGY USING NONSTANDARD LOG-LINEAR M ODELS T h e p re sen t m o d e llin g a p p ro a ch h a s t wo go a ls. F irst, we spe cify m o d e ls t h a t d escrib e t he d at a in a sa tisfa cto r y fa sh io n . Se co n d, we a t te m p t
FIG. 1
Syne rge tic activity of two a gen ts.
M ODELLING SYNERGY
543
p a ram e t e r in t e rp ret at io n . A lth o u gh in m a n y in sta n ce s, re se a rche rs a re sa t is® e d with ® tt in g m o de ls, a n d p ara m et e rs are o f le sser im p o rt an ce (se e Slo an e & M o rga n , 1996 ), t h e p re se n t a pp ro a ch re qu ire s p a ra m e t e r in te rp re t a t io n . It is n o t su f® cien t t o p ro vide a m o d e l t ha t ® t s. O n e h a s t o sh o w in ad d ition th a t t h e co n t ra sts sp e ci® e d t o t est a p a rt icu la r h yp ot h e sis cap t u re sign i® ca n t p o rt io n s o f t h e va riab ilit y in a t ab le . T his sect io n ® rst d e scrib e s t h e lo n g-lin e a r m o d els u se d fo r m od e llin g. Su b seq ue n t ly, p ro b le m s o f p a ra m et e r in te rp re t a t ion will b e ad d re sse d .
Standard and Nonstandard Log-Linear M odels C o n sid e r t h e fo llo win g log-lin ea r m od e l m
= X ,
(1)
wh e re m = lo g F , F is th e a rray o f e xp e cte d ce ll fre q u en cie s, X is t h e in d icat o r (d e sign ) m at rix, an d is t he p a ram e t er ve cto r. St a n d ard , h iera rch ica l lo g-lin e ar m o d e ls of t h e form m
=
0
+ Xm
m
+ X i i,
(2)
wh e re 0 is th e p a ram e t er fo r th e con sta n t , su b scr ip t m in de xe s m a in e ffe ct s, a n d i in d e xe s in t e ra ction s, a re o fte n sp e ci® e d h ie ra rch ica lly, su ch t h a t h igh e r-o rd e r t e rm s im ply t h e exist e n ce o f lowe r o rd er t e rm s. F o r e xa m p le , t h e in t e ra ctio n , [A , B ], im p lies t h a t a ll m ain e ffe ct t e rm s fo r va ria b le A a n d a ll m a in e ffe ct t e rm s fo r va ria b le B a re p art of th e m o d e l. N o n sta nd a rd lo g-lin e ar m o d e ls (R in d sk o p f, 1990; vo n E ye , K re p p n e r, & W eû e ls, 1994 ; vo n E ye & Sp ie l, 1996 ) ca n b e give n a s m
=
0
+ Xm
m
+ Xi
i
+ Xs
s,
(3)
wh e re s in d e xe s t h e ``sp ecia l ve cto rs’ ’ (C lo gg, E lia so n , & G re go , 1990 ) t h a t can n o t b e d e rive d fro m sta n d a rd m ain effe ct an d in t e ra ctio n s. T h e se sp e cia l ve cto r s e xp re ss co n t ra sts t h at ca n b e co m p a re d to co n tra sts in a n alysis of va ria n ce p ost-ho c t e sts. W h e n m o d e llin g su ch co n ce p t s as syn e rgy we u se t h e e le m en t s in e q u at io n 3 as fo llo ws. T h e ® rst t h re e t e rm s o n th e right -h a n d sid e o f t h e e q u at io n (i.e . 0 , X m m , an d X i i ) exp re ss t h o se rela t io n sh ip s am o n g va ria b les t h at a re n o t sp e ci® c t o th e co n ce pt u n d e r stu d y. T h e se t h re e t e rm s fo rm wh a t we will ca ll t h e b ase m od el. H yp o t h esise d de viat io n s fro m t h e b ase m o d el are p a ra m e t e rise d in t h e fo u rt h t erm (i.e . X s s ). T h is te rm su b su m es t h e ve ct o rs t h at a re n e ed e d t o m o d e l t h e co n ce p t u n d e r stu dy. T h is t e rm will b e ca lled t h e sp eci® c p art o f th e m o d el.
544
VON EYE, SCHUSTER, ROGERS
Param eter Interpretation in Nonstandard LogLinear M odels P ara m e t er in t e rp re t at io n in lo g-lin e a r m o d e ls h a s b e e n a n issu e o f lo n g d e b at e s. W h e rea s so m e co n t rib u t io n s fo cu se d o n t h e m e a n in g o f p a ram e t e rs, fo r insta n ce , th e m e a n in g o f m ain effe ct p a ram e t e rs in t h e p re sen ce of in te ra ct io n p a ram e t ers (e .g. A lb a, 1988; H o lt, 1979 ; L o ng, 1984 ), o t h ers d iscu sse d t e ch n iq u e s fo r in t e rp ret at io n o f p a ra m e t e rs (e .g. E llio t t, 1988 ). T h e re aso n fo r t h e se d iscu ssio n s is o b viou s: p ara m e t ers d o n o t n e ce ssarily re¯ e ct th e h yp o t he se s sp e ci® e d in t h e d esign m a t rix, X . O n e o f th e p ro b lem s e n co un t e re d wh en in t e rp ret in g p a ra m e t e rs ca n b e d e scrib e d as fo llo ws. T h e re la t io n sh ip b et we e n t h e d esign m a t rix, X , a n d t h e p a ra m et e rs is = (X X Â)
±1
X mÂ,
(4)
wh e re bo t h , t h e b ase m o d e l a n d t he co n t ra sts o f in te re sts, are in clu d e d in X . 1 T h is rela t io nsh ip re ¯ e cts t h e co n t ra sts o f int e re st o n ly if X is o rt h o go n a l. A s so o n a s X is n o lo n ge r o rt h o go n a l, in te rpre t a t ion o f p a ram e t e rs b eco m e s co m p lica t ed if n o t im po ssib le . C o n sid er th e fo llo w in g t wo exa m p le s. E x am p le 1: P aram eters f ro m S atu rated 2 2 2 T ab le. In a 2 2 2 t a b le a sa t u ra t ed lo g-lin e ar m o d e l is sp eci® e d . T he d e sign m at rix fo r t h is m o de l a p p ea rs in T ab le 1. A p p lyin g e q u a t io n 4 o n e o b t a in s th e p a ra m e t e r in te rp re t a t io n give n in T a b le 2.
TABLE 1 Design M atrix fo r Saturated Lo g-Linear M odel in 2
2 Table
E ffect C od ing V ectors
C ell I nd ices
C onstan t
111 112 121 122 211 212 221 222
1 1 1 1 1 1 1 1
1
2
M ain E ff ects 1 1 1 1 ±1 ±1 ±1 ±1
1 1 ±1 ±1 1 1 ±1 ±1
I n teraction s 1 ±1 1 ±1 1 ±1 1 ±1
1 1 ±1 ±1 ±1 ±1 1 1
1 ±1 1 ±1 ±1 1 ±1 1
1 ±1 ±1 1 1 ±1 ±1 1
1 ±1 ±1 1 ±1 1 1 ±1
N ote tha t e q u ation 4 d e scr ib es th e re la tio n sh ip be twe en X a nd the p ar a m e ter s. E st im atio n is st ill m a xim um lik elihoo d.
545
M ODELLING SYNERGY TABLE 2 Param eters in Satu rated M odel for 2 P aram eter
2
2 Tabl e
E stim ate 1=8( m 1=8( m 1=8( m 1=8( m 1=8( m 1=8( m 1=8( m 1=8( m
0 A B C AB AC BC A BC
+ + + ± + ± ± ±
1 1 1 1 1 1 1 1
m
2
m m
2
m
2
m
2
m
2
m
m
2 2 2
m
+ + ± + ± + ± ±
3
m m
3
m
3
m
3
m
3
m
3
m
3 3
+ + ± ± ± ± + +
m
4
m m
4
m
4
m
4
m
4
m
m
4 4 4
+ ± + + ± ± + ±
m
5
m
5
m
5
m
5
m
5
m
5
m m
5 5
+ ± + ± ± + ± +
m
6
m m
6
m
6
m
6
m
6
m
m
6 6 6
+ ± ± + + ± ± +
m
7
m m
7
m
7
m
7
m
7
m
m
7 7 7
+ ± ± ± + + + ±
m
8)
m
8)
m
8)
m
8)
m
8)
m
8)
m
m
8) 8)
A co m p a riso n o f t h e estim a t e s in T a b le 2 with t h e co n t ra sts in T a ble 1 sh o w th a t t h e e stim a t e s p e rfe ctly re ¯ ect t h e sp e ci® ed e ffect s. E x am p le 2: P aram eters f ro m N o n o rth o go n al D esign M atrix . T ab le 3 d isp la ys t h e d e sign m a t rix fo r a 4 4 cro ss-t a bu la t ion . O n ly t he m a in e ffe ct s are con sid e re d fo r t h e b a se m o d e l. F o r th e sp eci® c p a rt o f t h e m o de l t wo a d d itio n a l ve ct o rs are in tr o d uce d . In co n t ra st t o th e p a ra m e t e r e stim a t es in t h e ® rst e xa m p le , t h e p re se n t e stim a te s d o n o t re ¯ e ct t h e co n t ra sts sp eci® e d in T a b le 3. T h is ca n b e
TABLE 3 Desig n M atrix for No n-standard Lo g-Linear M od el fo r 4 Cell I ndex
C o nstan t
11 12 13 14 21 22 23 24 31 32 33 34 41 42 43 44
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
M ain E ffects R ow s 1 1 1 1 0 0 0 0 0 0 0 0 ±1 ±1 ±1 ±1
0 0 0 0 1 1 1 1 0 0 0 0 ±1 ±1 ±1 ±1
0 0 0 0 0 0 0 0 1 1 1 1 ±1 ±1 ±1 ±1
4 Cross-tab u lation
M ain E ffect C o lum ns 1 0 0 ±1 1 0 0 ±1 1 0 0 ±1 1 0 0 ±1
0 1 0 ±1 0 1 0 ±1 0 1 0 ±1 0 1 0 ±1
0 0 1 ±1 0 0 1 ±1 0 0 1 ±1 0 0 1 ±1
Special Co n trasts 0 0 0 ±1 0 0 0 ±1 0 0 0 1 0 0 0 1
0 0 0 0 0 0 0 0 0 0 0 0 ±1 ±1 1 1
546
VON EYE, SCHUSTER, ROGERS
illu strat ed u sin g t h e p a ra m et e r e stim a t e fo r t h e ® rst of th e spe cia l ve cto rs. T h is e stim a t e is give n b y xl
=
m
11
16 m
31
±
8
+ 0
m
m
+ 22
+
16 m
12
16
21
+ ±
m
+
16 m
32
8
41
m
13
8 m
23
+ ±
+ 0
± 8
m
33
16 m
42
m
14
4 ± + ±
m
24
4 m
5
(5)
34
16 m
3
16
43
3m
+
44
16
.
T h is lin e a r fu n ctio n is virt u ally u n in te rp re t a b le a nd u n re la t e d t o t h e o rigin a l h yp o t h e sis, give n in t h e se co n d la st co lu m n in T ab le 3. T o o b t ain p a ra m e t e rs t h a t d o re ¯ e ct t h e co nt ra sts sp e ci® e d in th e o rigina l d e sign m at rix, X , Sch u ster (in p re p a ra tio n ) p ro p o se d t he fo llo win g t ra n sfo rm a t io n . C o n sid e r m a t rix H wh e re H Â h a s t h e sam e d im e n sio n s a s X a nd : Hm
sp e cifies t h e co nt ra sts o f in t e re st
C (X ) = C (H )Â
[C (A ) = co lu m n sp a ce of A ]
L et ±1
X = H (Â H H )Â , a n d sub stitu t e X * fo r X wh e n e stim a t in g ±1
= [(X ) X Â] ±1
(6)
. T h is le ad s t o
(X ) m ±1 ±1
±1
= [(H H )Â H H (Â H H )Â ] (H H )Â H m
(7)
= Hm . E q u at io n 7 sh o ws t h at t h e p ara m e te rs a re re la t ed t o t h e co n tra sts a s sp e ci® e d in th e o r igin al d e sign m a t rix wh e n t h e e stim a t io n u se s X * inste a d o f t h e origin a l d e sign m a t rix, X . T h u s, t h e re se a rche r o n ly h a s t o p erfo rm t h e fo llo w in g fou r ste p s t o o b ta in p ara m e t er e stim a t e s th a t ca n b e in te rp re t e d a s sp e ci® e d in th e o rigin a l m a t rix, X : (1) se t u p X ; (2) d e t erm in e X Â = H ; (3) ca lcu la t e X *; a n d (4) p e rfo rm a n a lysis usin g X * in ste a d o f X . P e rfo rm in g t h e se fo u r ste p s yie ld s re su lts with th e fo llo win g ch a ract e rist ics. F irst, t h e o ve ra ll m o d e l ® t is id e nt ica l t o m o de l ® t fro m X b e ca u se C (X ) = C (X ) = C (H )Â . Se co n d , t h e h yp ot h e se s sp e ci® e d in o rigin a l d e sign m at rix, X , are re ¯ ect e d in p a ram e t e rs. T hird , p ara m e te rs, sta n d a rd e rro rs, p a ram e t er in te rco rre la t io n s, e tc. a re p a rt of sta nd a rd o u t pu t o f re a dily a va ila b le so ftwa re (e .g. S+ , C D A S), a n d fo u rth , all stat ist ica l re su lts a re o b ta in e d sim u lta n e o u sly.
M ODELLING SYNERGY
547
W e will illu stra te t h at t h e t ra n sform a t io n d e scrib e d in e q ua t io n s 6 a n d 7 yie ld s in t e rp re ta b le pa ra m et e rs in t h e d a t a e xa m p le in th e se ctio n o n fo u r syn ergy m o d e ls. In t h e follo win g se ctio n we sh o w h o w t o m o d el syn e rgy.
M ODELLING SYNERGY A s wa s in dica t ed in t he ® rst se ctio n o f t h is pa p e r, syn e rgy re q u ires t wo o r m o re a ge n ts t o co -o p e ra t e to wa rd a co m m o n go a l. In t h e fo llo win g se ctio n s we d ist in gu ish t h e fo llo win g t hre e va ria n t s o f syn e rgy: 1. In d ep en d en t A gen ts’ S yn ergy . E a ch o f t wo o r m o re age n t s d isp la ys t h e sa m e effe ct o n t h e d e p en d e n t m e asu re. T o e xp la in t h e syn e rgy e ffe ct, a t lea st t wo a ge n t s m u st h ave sign i® ca n t e ffect s. F or a n e xa m p le co n sid er a stu d en t wh o is ta le n t ed a n d in ve sts e ffo rt in h er ho m e wo rk . U n de r t h e In d e p en d e n t A ge n t s’ Syn e rgy h yp o t h esis b o t h th e stu d en t ’ s t ale n t an d h e r e ffo rt acco u n t fo r a sign i® can t p ort io n o f su cce ss in h o m e wo rk . H o we ve r, t h e re is n o ad d itio na l effe ct wh e n b o t h ta le n t a n d e ffo rt a re t a k e n in t o a cco u n t . T h is co n cep t ca n b e vie we d p a ralle l to t h e ® rst d e ® n itio n o f syn ergy give n in t h e in t ro d u ct io n . 2. Join t A gen ts’ S yn ergy . E a ch o f t wo o r m o re a ge n t s d isp lays t h e sam e e ffe ct o n t h e d ep e n d e n t m e a su re . T wo t yp es o f e ffe ct m u st e xist t o e xp la in t h e syn e rgy e ffe ct : (1) th e e ffe ct s o f t wo o r m ore a ge n t s; a n d (2) sp e cia l in te ra ct io n s b e t we en th e a ge n t s. If th e o ry re q u ir es t he in t e ract io n (s) t o e xist , th e re is su perad d itiv ity in th e se n se give n in th e se co n d d e ® n itio n in t h e in tr o d u ction . If t h e re a re t wo age n t s, a n in t era ct ion m u st e xist b e twe e n t h e se t wo . If t h e re a re m o re t h an t wo a ge n t s, th e o ry m ay re q u ire t h a t e ith e r a ll in t e ra ctio n s m u st e xist fo r co m p le t e J o in t A ge n t s’ Syn ergy, o r o n ly t h e h igh e st-o rde r in t era ct io n. 3. Iso lated S y n ergy . H e re, t he o ry re q u ire s o n ly t h e in t e ra ctio n t o e xist . T h ere is n o n e e d fo r sin gle a ge n ts’ e ffe cts t o e xist . W h e n t h e re a re m o re t h a n t wo a ge n t s, t h e o ry m ay req u ire t h a t a ll in te ra ct io n s o r ju st t h e h igh e st-o rd e r in t e ra ctio n m u st e xist (se e e q u a t ion 2) . H yb rid fo rm s o f syn e rgy can a lso b e co n sid e red . F o r exa m p le , a h yb rid b e t we e n t h e se co n d a n d t h e t h ird va ria n t wo u ld b e a ca se wh e re , in a d d itio n t o in te ra ct io n s, o n ly so m e o f th e a ge n t s d isp la y e ffe cts o n t h e d e p en d e n t m e asu re. T o illu stra t e t h ese th re e co n ce p t s o f syn e rgy we p re sen t in th e n e xt se ct io n a d a ta e xa m p le.
Synerg y in the Developm ent of Throm boem bolism s: A Data Exam ple B ish o p e t a l. (1975 ) a n alyse d d at a p re sen t e d b y W o rce ste r (1971) o n t h e cau sa t io n o f t h rom b o e m b o lism s. T h e t h re e va ria b les, U se o f O ra l
548
VON EYE, SCHUSTER, ROGERS
C o n tra ce pt ive s (U , ye s = 1, n o = 2), Sm ok e r (S , ye s = 1, n o = 2), a n d P rese n ce o f T h ro m b o e m b o lism s (T , ye s = 1, n o = 2) we re o b se rve d in a sa m p le o f N = 174 re spo n d en t s. T h e re se ar ch q u e stion wa s wh e t h er U se o f O ral C o n t race p t ive s a n d Sm o k in g ca n b e viewe d a s syn e rgist s fo r t h e d e ve lo p m en t o f th ro m b o e m b o lism s. T a ble 4 d isp la ys th e o bse rve d fre q ue n cies o f th e cro ss-ta b u lat io n o f T , S , a n d U , a lo n g with t h e e xp e cte d cell fre q u e n cie s, e stim a t e d un d e r t h e b a se m o d e l, [S , U ][T ], an d t h e re sid u a ls.
Four Synergy M odels H ere , we will review so m e of t h e re su lts p re se n t e d b y B isho p e t al. (1975 , p p . 112±114) a n d rea n a lyse t h e se d a t a usin g t h e m e t h o d s p ro p o se d in t h is p a p er. Sp e ci® ca lly, we d iscu ss t h e fo llo win g ® ve m o d els: (1) th e b a se m o de l, [U , S ][T ], o f n o rela t io nsh ip s b e t wee n p re d ict o rs a n d crite rio n ; (2) t h e n o n h ie ra rch ica l m o d e l lo g-fre q u e n cy p ro p o sed b y B isho p e t a l. (1975 ), m
=
0
+
U 1
+
S 2
+
T 3
+
US 12
UST 123
+
,
(8)
(3) t he In d e p en d en t A ge n t s’ Syn e rgy M o d el wh e re t h e t wo a ge n ts, U a n d S , sim u lta n eo u sly h a ve e ffe cts on T , b u t t h ere is n o n ee d t o a cco m m o d at e su rp lu s va ria b ilit y wh ich wou ld in d ica t e su p era d d itivit y; (4) t h e Jo in t A ge n t s’ Syn e rgy M o d e l wh ich en rich es t h e In d e p en de n t A ge n t s’ Syn e rgy M o d e l b y inclu d ing a t e rm th a t acco u n t s fo r th e in te ra ct io n t h a t re su lts in su p e rad d itivit y; a n d (5) t h e Iso la t e d Syn ergy M od e l wh ich o n ly co n sid e rs t h e a ge n ts’ in t e ract io n s. M o d e l 2 was re p o rt e d in B ish o p e t a l. (1975 ). T h e In d ep en d ent A gen ts’ Sy n ergy M o d el. T h is on ly re q u ires a ll age n t s t o h a ve t h e d e sired e ffect . T he se e ffe ct s m a n ife st in t he sa m e ca t e go ry o r cat e go rie s. N o in te ra ct io n is re q u ire d . T h e a ge n t s wh o se effe cts a re sign i® ca n t o p e ra t e in syn ergy. T h e d e sign m at rix fo r t h is m o de l a p p ea rs in T ab le 5. TABLE 4 Cross-tabu lation o f T, S, an d U T SU
O b serv ed
E x pected
111 112 121 122 211 212 221 222
14.00 7.00 12.00 25.00 2.00 22.00 8.00 84.00
4.34 16.66 7.66 29.34 4.97 19.03 19.03 72.97
R esidu al 9.65517 ± 9.65517 4.34483 ± 4.34483 ± 2.96552 2.96552 ± 11.03448 11.03448
Std. R esid.
A d j. R es.
4.63206 ± 2.36584 1.57035 ± 0.80206 ± 1.33082 0.67972 ± 2.52919 1.29179
5.546 ± 5.546 1.987 ± 1.987 ± 1.609 1.609 ± 4.136 4.136
549
M ODELLING SYNERGY
TABLE 5 Desig n M atrix for In dep en dent Ag ents’ Synerg y M od el fo r Thro m bo em b olism Data T SU
F
c
T
S
U
x1
x2
111 112 121 122 211 212 221 222
14 7 12 25 2 22 8 84
1 1 1 1 1 1 1 1
±1 ±1 ±1 ±1 1 1 1 1
±1 ±1 1 1 ±1 ±1 1 1
±1 1 ±1 1 ±1 1 ±1 1
1 1 0 0 ±1 ±1 0 0
1 0 1 0 ±1 0 ±1 0
S
U 1 ±1 ±1 1 1 ±1 ±1 1
T h e con t ra sts sp e ci® c t o t he In d e p e n d e nt A ge nt s’ Syn e rgy M o d e l a re e xp re sse d in V e ct o rs, x 1 a n d x 2 . V e cto r x 1 co n t rasts cells 111 an d 112 o n t h e o n e h a n d sid e wit h ce lls 211 a n d 212 o n t h e o th e r. T h is co n tra st re ¯ ect s t h e h yp o t he sis t h a t sm o k er s su ffe r fro m t h ro m b o e m b o lism s a t in cre a se d ra t es, a n d th a t re ga rd le ss o f wh et h e r t h ey do o r d o n o t t ak e o ra l co nt ra cep tive s. T h e se co n d sp eci® c ve ct o r, x 2 , co n t ra sts ce lls 111 a n d 121 with 211 an d 221, re ¯ ect in g t h e h yp o t h esis th a t use rs o f o ra l co nt ra ce p tive s su ffer fro m t h ro m b o e m b o lism s at in cr e ase d ra t es, a n d t h a t re ga rd le ss o f wh e th e r t h e y sm ok e o r n o t . R esu lts o f t h is a n a lysis will b e in t e rp ret e d in t e rm s o f a syn e rge t ic e ffe ct if (1) t h e m o d e l ® t s a n d (2) b o th o f t h e sp e ci® c ve cto rs, x 1 an d x 2 , a re sign i® can t .
T h e Jo in t A gen ts’ S y n ergy M o d el. T h is p ro p o se s t h a t it is n o t su f® cie n t if t wo o r m o re a ge n t s wo rk t owa rd th e sa m e go a l. R a th er, t h ere m u st b e a n e ffe ct b e yo nd wh a t t h e ind e p e n d e n t a ge n t s ca n ach ie ve . T h is ``e ffect b e yo n d ’ ’ h a s be e n ca lle d su p era d d itivit y. In t he p re sen t co n t e xt we sp e cify a n a d d itio n al ve cto r to exp re ss su p e ra dd itivit y. T h is ve cto r re ¯ e ct s th e co o p e rat io n b e tw ee n t h e t wo a ge n ts. H ere , we h yp o t h e sise t h a t in d ivid u a ls wh o b o t h sm o k e an d t a k e o ra l co n tra cep t ive s su ffer fro m in crea se d ra t e s o f t h ro m b o e m b o lism s, b e yo n d wh at ca n b e e xp la in e d b y t h e e ffect s o f sm o kin g a n d t a k in g o ra l co n t ra ce pt ives a lo n e . T h e ve cto r t h a t re ¯ ect s t h is h yp o th e sis is x 3 = (1, 0, 0, 0, ± 1, 0, 0, 0). In clu d in g t h is ve cto r in to t h e d esign m a t rix in T a b le 5 yie ld s a sa tu ra t e d m o de l. T h e refo re , in o rd e r t o h a ve a m o d e l th a t can fa il a n d th a t still a llo ws u s to t est t h e h yp o t h e sis rela t ed t o su p e ra dd itivit y we can e ith e r e lim in a t e o n e o f th e ve cto rs, x 1 o r x 2 , o r p la ce e q u a lit y co n stra int s. T yp ically, e q u a lit y con stra in t s p o sit th e co n str a in t t h a t 1 = 2 fo r Q = x 1 1 + x 2 2 . If we p o se t h is co n stra in t th e re su ltin g n e w ve ct o r is x 4 = (2, 1, 1, 0, ± 2, ± 1, ± 1, 0). T h is ve cto r , h o weve r, wo u ld im p ly so m e we igh t in g. T h e re fo re, we d e cid e u sin g so m e va ria nt o f se t t ing p a ra m e t e rs
550
VON EYE, SCHUSTER, ROGERS
e q u al wh e re t h e re is no su ch weigh t in g. W e u se t h e ve cto rs, x 4 = (1, 1, 1, 0, ± 1, ± 1, ± 1, 0) , a n d x 3 in ste a d of x 1 an d x 2 , in T a ble 5. T h e co sts a n d t h e b e n e ® t s fro m p lacin g t h is co n stra in t a re o b vio u s. W e ga in t h e p o ssib ilit y t o t e st t h e o ve ra ll m o d el a n d th e su p e rad d itivit y h yp o th e sis. H o we ve r, we fo rfeit t h e o p t io n to t e st t he In d e p e nd en t A ge n t s’ Syn ergy a n d t h e Jo in t A ge n t s’ Syne rgy m od e ls a ga in st e a ch o t h e r. T h e y a re n o lo n ge r h ie rarch ica lly re la t ed t o e ach o th e r. 2 T h e Iso lated S y n ergy M o d el. T h is p ro p o se s th a t , re ga rd le ss o f wh at t h e e ffe ct s of sin gle age n t s are , syn e rgy is re¯ e cte d b y age n t s’ in t era ct io n s. T h ere fo re , e ve n if n o n e o f t h e sin gle a ge n t s ha s a n y e ffe ct, t h e re ca n still b e syn ergy b y wa y of a ge n t s’ co lla b o ra t io n (in t era ct ion ) . T h e Iso lat e d Syn e rgy M o d e l su bstitu t e s ve ct o r x 3 for ve cto r s x 1 an d x 2 in T ab le 5.
Fit of Syn ergy M odels T ab le 6 d isp la ys t h e e stim at e d exp e ct ed cell fre q u en cie s fo r t h e fo u r syn ergy m o d e ls. T ab le 7 give s a n o ve rvie w o f th e o ve ra ll m o d el ® t a ch ieve d b y e a ch of t h e fo u r m o d e ls. T h e re su lts in T ab le s 6 a n d 7 su gge st t h e B ish o p e t al.’ s (1975 ) n o n h ie ra rch ica l syn e rgy m o d e l, t h e In d e p e n de n t A ge n t s’ an d t h e Jo in t A ge n t s’ Syne rgy m o d els p ro vid e e xce lle n t ® t a n d are sign i® ca n t ly b et t e r TABLE 6 Ob served an d Estim ated Expected Cell Freq u encies for Fo ur Syn ergy M odels E xp ected Cell Frequ encies
T SU
O b serv ed Frequ encies
B FH M odel 1 a
IA S b
JA S b
ISb
111 112 121 122 211 212 221 222
14 7 12 25 2 22 8 84
14.00 6.70 12.00 25.30 2.10 22.20 7.91 83.80
12.34 8.66 13.66 23.34 3.66 20.34 6.34 85.66
14.00 11.24 7.76 25.00 2.00 17.76 12.24 84.00
14.00 8.08 5.75 30.35 2.00 20.82 14.43 78.65
a
B F H is sh or t fo r Bisho p , F ien be r g, and H olla nd ( 1975); r esu lts ar e fr om B F H , 1975, p. 113; I A S, I nd epe nd ent A gen ts’ Syne r gy M o de l; JA S, J oin t A ge nts’ Syn er gy M ode l; I S, I so late d Syne r gy M od el. b
2
T he r e is th e o p tion of p la cin g th e eq u ality con st ra in t in the I nd ep en d e nt A gen ts’ Syn er gy M o de l a s we ll. H owe ve r, th is o ption wo uld p re vent us fr om e st im at ing p a r am e ter s fo r th e two a ge nts t hat ca n diffe r in m a gn itu de an d signi® ca n ce. T h er e fo r e, we will not a pp ly this op tion in the I nde pe nd e nt A ge n ts’ M o de l.
M ODELLING SYNERGY
551
TABLE 7 Go odn ess-of-Fi t of M od els in Tab les 4 and 6 M odel B a se [S, U ][T ] BFH IA S JA S IS
LR± x
2
35.93 0.02 2.35 6.64 12.74
df
L R± x
3 2 1 1 2
± 35.91 33.58 29.29 23.19
2
df ± 1 2 2 1
t h a n t h e b ase m o de l. T h e Iso la t ed Syn e rgy M od e l, a lth o ugh b e tt e r t h an t h e b a se m o d el, is n o t t e n a b le itse lf. T h ere fo re , it will n o t be con sid e re d fu rt he r. T o co m e t o a co nclu sio n co n ce rnin g t h e syn e rgy h yp o t he se s o n e ca n p u rsu e two a lt ern a t ive ro u t e s. T h e ® rst in vo lve s rem o ving cell 111 a s p ro p o sed b y B ish o p e t a l. (1975). T h e au t h o rs ® rst co n clu d e fro m th e ir a n alyse s t ha t ce ll 111 is a n o u t lie r (fo r d e t ails se e B ish o p et a l., 1975, p . 141; a n d W o rce ster, 1971 ). T h is re su lt ca n a lso be o b t a in ed u sin g co n® gu ra l fre q ue n cy an a lysis (L ie n ert & K ra u t h , 1975 ; vo n E ye , 1990 ). Se con d , B ish o p et al. (1975 , p . 114) co n clu d e t h a t t h e ir resu lts su p p o rt t h e h yp o th e sis th a t th e u se o f co n t ra ce p t ive s is p o sit ive ly a sso ciat e d with t h ro m b o e m b o lism , p a rt icu la rly am o n g sm o k e rs. In co n t rast, a m o n g t h o se wh o d o n o t u se o ra l con t ra ce pt ives, sm o k in g is n o t a sso cia t ed wit h t h e d isea se . T h e seco n d ro ut e in vo lve s in t e rp re t a tio n o f t h o se p a ram e t ers t h a t we re sp e ci® e d t o te st th e syn ergy m o d e l h yp o th e se s. In t h e fo llo win g p a ragr a p h s we pu rsu e t h is ro ut e .
Fittin g th e M o d el o f In d ep en d en t A gen ts’ S yn ergy. U sin g e q u at io n 4 a n d t h e tra n sfo rm at io n in e q u a tio n s 6 a n d 7, we no w sh o w th a t p a ra m e t e r in te rp re t a t io n can b e h igh ly p ro b lem a t ic wh e n de sign m a t rice s a re n o t o rt h o go n a l. C o n sid er t h e m a trix give n in T a b le 5. T h e p a ra m e te rs fo r t h is m at rix are o f t he fo rm sh own in T a b le 8. TABLE 8 Param eters for Desig n M atrix in Table 5 P aram eter 0 T S U x1 x2 S U
E x pressio n 1=8( m 1 1=8( m 1 1=8( ± m 1=8( ± m 1=4( m 1 1=4( m 1 1=8( m 1
+ m 2 + m 3 + m 4 + m 5 + m 6 + m 7 + m 8) ± m 2 ± m 3 ± 3 m 4 ± m 5 + m 6 + m 7 + 3m 8 ) 1 ± m 2 + m 3 + m 4 ± m 5 ± m 6 + m 7 + m 8) 1 + m 2 ± m 3 + m 4 ± m 5 + m 6 ± m 7 + m 8) + m 2 ± m 3 ± m 4 ± m 5 ± m 6 + m 7 + m 8) ± m 2 + m 3 ± m 4 ± m 5 + m 6 ± m 7 + m 8) ± m 2 ± m 3 + m 4 + m 5 ± m 6 ± m 7 + m 8)
552
VON EYE, SCHUSTER, ROGERS
T a b le 8 sh o ws th a t a p p licat io n o f th e d e sign m at rix give n in T a ble 5 lea d s to p a ram e t er e stim a t es th a t are , with t h e exce pt io n s o f t h e co n sta n t a n d t he m ain e ffe ct p a ram e t e rs fo r S a n d U an d t h e in t e ra ctio n b e twe e n S a n d U , n o t re¯ e ctive o f t h e h yp o t h ese s e xp re ssed in fo rm o f co n tra sts. T h is a p p lie s in p a rt icu la r t o t h e e stim a t e s o f x 1 a n d x2 . In con t ra st, a p p licat io n o f t he t ra n sfo rm a t io n give n in eq u at io n s 6 an d 7 le a d s to th e p a ra m e t e r e stim a te s give n in T a b le 10. T a b le 9 d isp lays t h e m at rix X * wh ich re su lts fro m a p p lica t io n o f e q ua t io n 6. O b vio usly, th e de sign m at rix in T ab le 9 d o e s n o t lo o k a s if it wo u ld re ¯ ect t h e h yp o th e se s sp e ci® e d in T a b le 5. H o we ve r, t h e p a ra m e t e r e stim a te s fro m X * d o re ¯ ect t h e se h yp o t h e se s, a s is illu stra t e d in T a b le 10. T h e p a ra m et e r e stim a te s fo r t h e d esign m a t rice s, X a n d X *, a p p e ar in T ab le 11. T h e o ve rall m o d e l fo r b o th va ria n t s o f X is t h e sa m e : L R ± x 2 = 2.35, d f = 1, P = .125. T h u s, we con clu d e t h a t th e m o de l d e scrib e s t h e d a t a a d eq u a t e ly. H o we ve r, a co m p ar iso n o f t h e p ara m e t er e stim a t e s su gge sts t h a t t he m o d e ls are n o t e q u iva len t . A lth o u gh t he t-va lu e s fo r so m e pa ra m et e rs a re u n ch a n ge d b et we e n X a n d X *, sp e ci® ca lly, 0, S, U , and S U , th e t-va lu e s fo r o t he rs, spe ci® ca lly T, x1, a n d x2 , d iffe r, o n e of t h e m d ram a t ica lly ( T ). A s fa r a s t h e m o d el o f TABLE 9 M atrix X* , the Result o f Applyin g Equation 6 to th e Design M atrix in Tab le 5 c
T
0.125 0.125 0.125 0.125 0.125 0.125 0.125 0.125
0.125 ± 0.125 ± 0.125 ± 0.375 ± 0.125 0.125 0.125 0.375
S
U
± 0.125 ± 0.125 0.125 0.125 ± 0.125 ± 0.125 0.125 0.125
± 0.125 0.125 ± 0.125 0.125 ± 0.125 0.125 ± 0.125 0.125
x1
x2
0.25 0.25 ± 0.25 ± 0.25 ± 0.25 ± 0.25 0.25 0.25
0.25 ± 0.25 0.25 ± 0.25 ± 0.25 0.25 ± 0.25 0.25
S
0.125 ± 0.125 ± 0.125 0.125 0.125 ± 0.125 ± 0.125 0.125
TABLE 10 Param eters fo r Design M atrix in Table 5, After Transfor m ation P aram eter m
E x pressio n
0
1
U
±m ±m ±m
x1
m
1
x2
m
1
S U
m
1
T S
+ m 2+ m 3+ m 4+ m 5+ m 6+ m 7+ m 8 1 ± m 2 ± m 3 ± m 4 + m 5 + m 6 + m 7 + m 1 ± m 2 + m 3 + m 4 ± m 5 ± m 6 + m 7 + m 1 + m 2 ± m 3 + m 4 ± m 5 + m 6 ± m 7 + m + m 2± m 5± m 6 + m 3± m 5± m 7 ± m 2± m 3+ m 4+ m 5± m 6± m 7+ m 8
U
8 8 8
553
M ODELLING SYNERGY TABLE 11 Param eter Estim ates from the Desig n M atrices X and X* P aram eter E stim ates fro m X I nte r ce pt T S U x1 x2 S U
P aram eter E stim ates fro m X *
V alue
SE
t
2.63 0.65 0.38 0.56 0.22 1.03 0.22
0.11 0.11 0.10 0.11 0.20 0.21 0.11
23.87 5.77 3.88 5.04 1.12 4.82 2.04
I nt er cep t T S U x1 x2 S U
V alue
SE
t
21.04 0.17 3.08 4.50 0.36 1.98 1.78
0.88 0.90 0.79 0.89 0.69 0.75 0.87
23.87 0.19 3.88 5.04 0.52 2.64 2.04
In d e p en d e n t A ge n t s’ Syn e rgy is co nce rn e d , we co n clu d e t h a t t h e syn e rgy h yp o th e sis m u st b e re je cte d d esp ite th e go o d o ve ra ll m o d e l ® t . O n e crite rio n fo r t h is m o d e l t o b e accep t e d was t h a t t wo o r m o re age n t s’ e ffe ct s m u st exist . T h e p re sen t resu lts, h o we ve r, su gge st t ha t o n ly o n e a ge n t ’ s e ffe ct e xist s. T h is is th e e ffe ct o f U se o f O ra l C o n t ra ce p t ive s. Sm o k in g d o e s n o t se em to h a ve a n e ffect ab o ve an d b e yo n d t ha t o f U . T h u s, with o n ly o n e effe ct p re se n t , t h e syn e rgy h yp o t h e sis fa ils t o b e co n ® rm e d. Fittin g the M o d el o f Jo in t A gen ts’ S y n ergy . T o ® t t h e m o d el o f Jo in t A ge n t s’ Syn e rgy we su b stitu t e th e ve ct o rs, x 3 a n d x 4 fo r x 1 a n d x 2 . T h e o ve ra ll m o d e l ® t is m a rgin a l: L R ±x 2 = 6.46, d f = 1, a n d P = .011. T h e m o de l ca n , th e re fo re, b e ret a ine d o n ly if o n e set s a t o 0.01. T a b le 12 p re sen t s t h e p a ram e te r e stim a t e s for t h is m o d e l fo r b o t h X a n d X *. E ve n m o re so t h a n T a ble 11, T a b le 12 su ggests th a t re su lts crea t e d with o u t th e tra n sfo rm a tio n d e scrib e d in eq ua t io n s 6 a n d 7 ca n le ad t o in accu ra t e p a ra m e t e r e stim a t es. E stim a t in g p ara m e te rs usin g X le a d s t o t h e co n clu sio n t h a t b o t h t h e co m bin e d ve cto r o f sin gle a ge n t s’ e ffe cts, x 4 , a n d t h e ve ct o r th a t d escrib es t h e su p era d d itive co m p o n en t o f t h ese a ge n t s’
TABLE 12 Param eter Estim ates from the Desig n M atrices X and X* P aram eter E stim ates fro m X I nte r ce pt T S U x3 x4 S U
P aram eter E stim ates fro m X *
V alue
SE
t
2.60 0.61 0.45 0.63 1.20 0.38 0.14
0.12 0.11 0.12 0.12 0.41 0.19 0.12
21.07 5.32 3.65 5.16 2.96 2.04 1.14
I nt er cep t T S U x3 x4 S U
V alue
SE
t
20.83 0.18 3.57 5.06 1.95 1.03 1.13
0.99 0.98 0.98 0.98 0.76 0.96 0.99
21.07 0.18 3.65 5.16 2.57 1.08 1.14
554
VON EYE, SCHUSTER, ROGERS
syn ergy, x 3 , cap t u re signi® can t po rt io n s o f t he va riab ility in t h e t a b le. H o we ve r, afte r t h e p ro pe r tra n sfo rm at io n of t he d e sign m a t rix, it b e co m e s cle a r t h a t co n stra in in g t h e co n t ra st o f th e age n t s t o b e e q u a l yie ld s n o n sign i® can t pa ra m e te rs. O n ly t h e sup e ra d d itivity p ara m e t er is sign i® can t . In th e n ext sect io n we a sk whe t h e r t h e su pe ra d d itivit y con t ra st alo n e lea d s t o a sa t isfact o ry da t a d e scrip t io n . Fittin g th e M od el o f Iso lated S y n ergy . T h is re qu ire s t o o n ly u se ve ct o r x 3 in t h e sp e cia l h yp o t he se s p a rt o f t he m od e l. T a b le 13 p re se nt s t h e p a ram e t e r e stim at e s fo r t h is syn e rgy m o d e l. T a b le 7 su ggests t h a t , a lth o u gh t h e Iso la t e d Syn e rgy M o d e l is sign i® ca n t ly b e t t e r th a n t h e n u ll m o d e l, it is n o t t e n a ble b y itse lf (L R ±x 2 = 12.74, d f = 2, P = .0017 ). T h u s, p a ra m e t e rs ca n n o t b e in t e rp re te d .
Discussion of Synergy M odels O ve ra ll, t h e in ve stiga t io n o f syn e rgy h yp o t h ese s via ® ve m o d e ls su gge sts t h a t a lth o u gh som e o f t h e m o d e ls p ro vid e e xce llen t re n d e rin gs of t h e d a t a , n o n e o f t h e m p ro vid e s stro n g su pp o rt o f syn ergy h yp o th e se s. T h e B ish o p e t al. (1975 ) m o d el p ro vid e s t he be st ® t a n d su p p o rt s t h e stat e m e n t t ha t t h e u se o f o ra l co n t ra ce p t ive s ``. . . is po sit ively a sso cia t e d with t h ro m b o em b o lism , pa rt icu la rly am o n g sm o k e rs . . .’ ’ (p . 114). H o we ve r, a m on g t h o se wh o d o n ot sm o ke , sm o k in g is n ot a sso cia t e d with t h ro m b o e m b o lism . T h e M o d e l o f In d ep e n d e n t A ge n t s’ Syn e rgy su gge sts t h a t wh ile u se of o ra l co nt ra cep tive s le ad s t o t h ro m b oe m b o lism , sm o k in g d o e s n o t . T h e m o d e l o f J o in t A ge n t s’ Syn ergy pro vid e s m argin a l ® t a n d su ggests t h a t t he re m a y b e su p e rad d itivit y in t h e se n se th a t th e re is a n effe ct t h at b o th a ge n t s ha ve t o ge t h e r b eyo n d t h e sin gle a ge n t s’ e ffe cts. H o we ve r, t h e sin gle a ge n t s’ e ffe ct s h a d to b e co n strain e d t o b e e q u al fo r t h is m o d e l t o b e co m e n o n sa tu r at e d . T h e Iso la t ed Syn e rgy M o d e l wa s n o t t e n ab le . W e t h u s co nclu d e t ha t t h e su p p o rt o f t h e syn e rgy h yp o t he se s fo r t h e p re sen t d at a is we ak . TABLE 13 Param eter Estim ates from the Desig n M atrices X and X* P aram eter E stim ates fro m X V alue I nte r ce pt T S U x3 S U
2.48 0.49 0.62 0.77 1.97 ± 0.03
SE 0.13 0.11 0.11 0.11 0.45 0.11
P aram eter E stim ates fro m X * t
18.54 4.37 5.74 7.10 4.37 ± 0.27
V alue I nt er cep t T S U x3 S U
19.83 ± 0.06 4.96 6.19 2.91 3.70
SE 1.07 0.86 0.86 0.87 0.74 0.86
t 18.54 ± 0.07 5.74 7.10 3.95 4.30
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555
SUM M ARY AND OUTLOOK It was t h e go a l o f t h is p a p e r t o p ro p o se wa ys to m o d e l t he co n ce p t o f syn e rgy. T h ree in te rp re t a t io n s o f t h e co n cep t o f syne rgy were co n sid e re d . U sin g n o n stan d a rd lo g-lin e ar m o d els a n d a tra n sfo rm at io n re ce n t ly p rop o se d b y Schu ste r (in p re p a rat io n ) th e se m o d e ls we r e spe ci® e d a nd te ste d . N o n e o f t h e se m o d e ls wa s sp eci® e d in a wa y e q u iva le nt t o h iera rch ica l lo g-lin e a r m o de ls o r e q u iva le n t t o m o d e ls t h a t ca n b e crea t e d u sin g ele m e n t s fro m h iera rch ica l m od e ls (t h is was d o n e b y B ish o p et a l., 1975 ). G e n er alisat io n s are con ce iva b le in va rio u s wa ys. T h e fo llo win g m o d e ls can b e sp eci® e d usin g t he m e th o d s p re se n t ed in t h is a rt icle: (1) m o d e ls o f p a rtia l syn e rgy co u ld b e co n sid ere d whe re syn e rgy is id en t i® e d t o e xist in su b ta b le s; (2) m o d e ls o f syn ergy of d eve lo pm e n t a l a ge n t s t h a t a ct in a t im e -re la t e d wa y; fo r in sta n ce , a ge nt s ca n a ct e ith er se q u e n t ially or syn ch ro n o u sly (o r b o t h ); (3) gr o u p -sp e ci® c syn ergy; fo r e xa m p le , o n e ca n t est th e h yp o t h esis t h a t syn e rgy o ccu rs o n ly in ce rta in gr o u ps o f in d ivid u a ls, b u t n ot in o t h e rs; a n d (4) lo n git u d in a l d eve lop m e n t o f syn e rgy ; o n e ca n t e st a ssu m pt io n s co n ce rn in g t h e t im e -rela t e d effe ctive ne ss o f syn erge t ic a ge n ts. M an y m o re m o d e ls a re co n ceiva b le, e a ch o f wh ich ca n co n sid e r co va riat e s. A n ad d itio na l gr o u p o f m e t h o d s is b eyo n d th e sco p e o f t h is a rt icle , b u t n o t b e yo n d t h e sco p e of wh a t ca n b e d o n e . O n e can co nsid e r lat e n t va ria b le co n ce p t s o f syn ergy an d t e st t h e se m o d e ls u sin g t h e m et h o d s p ro p o se d , fo r exa m p le , b y V e rm u n t (1996 ). M an uscrip t r e ce ived Sep tem be r 1997 R e vise d m a n uscr ipt r ece ive d A pr il 1998
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