A multiple mediator model: Power analysis based on Monte Carlo simulation

Transcription

1 Amrican Journal of Applid Psychology 2014; 3(3): Publishd onlin Jun 20, 2014 ( doi: /jajap A multipl mdiator modl: Powr analysis basd on ont Carlo simulation Z-wi a 1, 2, Wi-nan Zng 2, * 1 Thy Hua Social Work Srvic Cntr, Luogang District, Guangzhou , China 2 Dpartmnt of Psychology, School of Humanitis and anagmnt, Guangdong dical Collg, Dongguan , China addrss: mazwimac@gmailcom (Z-wi a), winanzng@hotmailcom (Wi-nan Zng) To cit this articl: Z-wi a, Wi-nan Zng A ultipl diator odl: Powr Analysis Basd on ont Carlo Simulation Amrican Journal of Applid Psychology Vol 3, No 3, 2014, pp doi: /jajap Abstract: ost of th applid psychological rsarchrs usually conduct studis rquiring application of advancd mdiation modls, such as multipl mdiator modls Howvr, in dsigning rsarch, most of th applid rsarchrs largly ignor th statistical powr of thir studis As a rsult, powr analyss ar ignord whn rsarchrs rport thir rsults It is wll rcognizd that low powr is on possibl rason for no statistically significant rsult bing idntifid in a study orovr, studis with low statistical powr hav bn labld scintifically uslss Th currnt study dscribs how to apply ont Carlo simulation to tst th typ I rror rats and statistical powr of mdiating ffcts in a multipl mdiator modl Findings from th currnt simulation study indicatd that th ffct sizs of mdiating ffcts and sampl sizs wr two important factors influncing typ I rror rats of indirct ffcts in a multipl mdiator modl Furthrmor, th rquirmnt of sampl siz and dsird powr lvl wr strongly dpndd on th ffct siz of th indirct ffct Kywords: diation, ultipl diator odls, Statistical Powr, ont Carlo Simulation, plus 1 Introduction On cntral goal of scinc is to undrstand how procsss work rathr than simply to stablish whthr a total ffct xists [1] In othr words, it is important for applid rsarchrs to invstigat whthr th caus-ffct rlation btwn two variabls is accountd for by any intrvning variabls [1] As a rsult, th analyss of mdiating ffcts ar bing commonly and widly discussd in psychological studis and sociological rsarchs, and othr rsarch aras such as clinical mdicin and pidmiology [2, 3] Rsarchrs from diffrnt disciplins hav diffrnt undrstandings about mdiating ffcts [4] Psychological rsarchrs intrprt th W rlation as mdiation [5], and such indirct rlation is oftn trmd as indirct ffct by rsarchrs from sociology [6] Furthrmor, th intrmdiat ndpoint ffct is usually usd for dscribing th intrvning rlation by rsarchrs from pidmiology [7] τ' τ Figur 1 A singl mdiator modl A singl mdiator modl is prsntd in Figur 1 [5, 8] Th total ffct of on is shown in th top panl of Figur 1, and th path cofficint is usd for dscribing th rlation btwn and In th bottom panl of Figur 1, a nw variabl is addd into th rlation btwn and, and this nw variabl is calld mdiating variabl or mdiator Th dirct ffct of on is dscribd by path cofficint orovr, th path cofficint is usd for dscribing th rlation, and th rlation is rprsntd by th path cofficint Additionally, th indirct ffct of th initial variabl on th outcom variabl via th intrvning or mdiator variabl is dfind as, which is th product of path cofficint and path cofficint Th total ffct of on is 2 3 1

2 Amrican Journal of Applid Psychology 2014; 3(3): xprssd by th sum of th indirct ffct and th dirct ffct: = + Th following rgrssion quations ar usd for dscribing th singl mdiator modl that prsntd in Figur 1: (1) (2) (3) Though th singl mdiator modl proposd by Baron and Knny is widly applid [5], most of th hypothss of actual social scinc studis or psychological studis ar complicatd That is to say th caus-ffct rlation btwn two variabls is accountd for by svral mdiators In this mannr, a singl mdiator modl should b rplacd by a multipl mdiator modl [9] j 1 2 j -1 τ τ' m 1 1 m 2 2 mj -1 j -1 mj j Figur 2 A multipl mdiator modl A multipl mdiator modl with j mdiators is shown in Figur 2 [10] Th uppr panl of Figur 2 rprsnts th total ffct of on (path cofficint ) orovr, th lowr panl of Figur 2 xprsss th dirct ffct of on (path ) Additionally, th indirct ffcts of on through th j mdiators ar shown in th lowr panl of Figur 2 Th mdiating ffct of ach mdiatd pathway in a multipl mdiator modl is dfind as ( = 1 to j), which is th product of path cofficint and path cofficint ( = 1 to j) [10] Th total mdiating ffct of a multipl mdiator modl is dfind as ( )( = 1 to j), and th total ffct of on is calculatd as ( ) Of not, multipl mdiator modls hav mor advantags than singl mdiator modls [11] For xampl, th studnt drug addiction prvntion stratgy basd on school lvl includs a plurality of intrmdiaris, such as rsistanc skills, social norms, attitud about drugs and communication skills Th multipl mdiator modl is likly to provid a mor accurat assssmnt of mdiation ffcts in many rsarch contxts [11] Howvr, most of th applid rsarchrs only focus on how to fit thir data to 2 j -1 1 j y y th modls that ar in accordanc with thir thortical assumptions, but thy largly ignor th statistical powr of thir studis whn dsigning thir rsarchs As a rsult, powr analyss ar ignord whn rsarchrs rport thir rsults [12] It is wll rcognizd that low powr is on possibl rason for no statistically significant rsult bing idntifid in a study [13] Studis with low statistical powr hav bn labld scintifically uslss [13-15] Discussions about statistical powr of singl mdiator modls ar commonly sn in mdiation litratur (g [4, 8, 11, 16, 17]) Howvr, only a handful of publishd studis focus on statistical powr of complx mdiation modls, such as multipl mdiator modls [9, 18] Only with adquat powr, th sampl siz in a study will b sufficint to find and confirm a significant mdiating ffct which is of small ffct siz [9] As a rsult, it is important to prform powr analyss for multipl mdiator modls Statistical powr, which mans th probability of accpting an altrnativ hypothsis aftr rjcting a fals null hypothsis, is an important concpt of statistics Accordingly, statistical powr is dfind as 1 - probability of a typ II rror [19, 20] [21] orovr, in psychology, a dsirabl powr is at 08 [8] On uniqu function of statistical powr is to calculat rquird sampl siz to rach 08 powr to rjct a fals null hypothsis [9] If th powr in a study is lss than 08, a rsarchr should considr incrasing th sampl siz [22] Undr crtain circumstancs, highr powr is also dsird [9, 23] In fact, powr analyss ar only maningful whn conductd prior to data collction [24] In light of this, it is important for a rsarchr to carry out powr analysis to confirm how many sampls ar rquird in th hypothsis phas of a study [23] As a rsult, it is of importanc to prform powr analysis in th dsign phas of a rsarch to avoid choosing a sampl siz that is too small to find or undrstimat th xisting small ral ffct, which possibly rsulting in a study with inadquat snsitivity, or too larg to b costly [9] Prvious rsarch has discussd on how to apply ont Carlo simulation to dtct mdiating ffcts for complx mdiator modls [9], th currnt study is purportd to invstigat statistical powr for a multipl mdiator modl with thr mdiators 2 ont Carlo Simulation In this study, powr analysis was prformd by ont Carlo simulation via plus [25] ont Carlo simulation is a flxibl and as of implmntation mthod [26] [9], which nabls rsarchrs to flxibly study on how sampl sizs, diffrnt sttings of population paramtrs, normality of distributions and missingnss of data may affct statistical powr [27] This powrful and rcommndabl mthod can b asily conductd in softwar programs that hav simulation capabilitis, such as plus, SAS, LISREL and EQS [9]

3 74 Z-wi a and Wi-nan Zng: A ultipl diator odl: Powr Analysis Basd on ont Carlo Simulation uthén and uthén suggstd that whn prforming ont Carlo simulation, a rsarchr should first spcify which modl to b studid Onc th modl is chosn, it is important to slct population valus and hav th population paramtrs fixd to th slctd valus Of not, th slction of population valus should bas on thory or prvious rsarch uthén and uthén suggstd that prvious studis can provid rsarchrs th bst stimats of population valus [25] Onc th modl is chosn, and aftr all paramtrs ar fixd to th slctd valus, numrous sampls ar drawn from th spcifid population Each sampl drawn from th population is fittd to th chosn modl, and thn th paramtrs of th chosn modl ar stimatd as wll as rcordd By rpating this procss thousands of tims, som of th paramtr stimats in som sampls will b significant, whras othrs will not Whn rplicating th sampling procsss adquatly, th powr stimat is th mpirical rat of statistical significanc avragd across all rplications [9, 25] uthén and uthén suggstd that at last 500 rplications should b prformd, and with incrasing rplications (such as 10,000 rplications), th rsult of th ont Carlo simulation convrgs will b mor prcis [28] 21 Simulation Study ackinnon suggstd a simulation study with a binary indpndnt variabl ld to th sam rsults as for th continuous indpndnt variabl cas [17] In light of this, th indpndnt variabl in th currnt simulation study was simulatd as a binary indpndnt variabl Th multipl mdiator modl invstigatd in th currnt simulation study had on indpndnt variabl, thr paralll mdiator variabls, and, and a singl dpndnt variabl All mdiating ffcts tstd in th currnt study had causal paths that bgin with th initial variabl, for a total of thr two-path mdiating ffcts and on dirct ffct Gnrally, th rlationship btwn two variabls can b ithr positiv or ngativ In th currnt study, all rlations wr st to b positiv As prsntd in Figur 3, all variabls and paths in th multipl mdiator modl wr namd according to th common idntification schms in th mdiation litratur [18, 29] Th paths linking to th thr mdiators,, wr dnotd as,, and rspctivly Th rsiduals of, and wr trmd as, and rspctivly Th paths linking, and to wr dnotd as, and rspctivly Th dirct ffct of on was dnotd as, and was trmd as th rsidual of Furthrmor, th mdiating ffct of on through was xprssd as orovr, rprsntd th indirct ffct of on through Additionally, th indirct ffct of on through was dfind as For clarity, th intrcpts wr omittd in th currnt simulation study [18] Figur 3 A multipl mdiator modl with thr mdiators Th thr mdiators wr rprsntd by th following rgrssion quations Th first mdiator was givn by: (4) Th scond mdiator was givn by: (5) Th third mdiator was givn by: (6) Prvious simulation study showd that, whn th mdiator was controlld, th dirct ffct of on had no influncs on th ffct siz of th mdiating ffct Accordingly, th path cofficint was fixd constant at a valu of 0 to simplify th modl in th currnt simulation study [17] Th following rgrssion quation was givn to rprsnt dpndnt variabl : 22 Population Valus τ' (7) In th currnt study, th indpndnt variabl was simulatd as a binary variabl with an vn split Furthrmor, th thr mdiator variabls and th dpndnt variabl wr gnratd as continuous variabls In ordr to simplify th modl, th mans and variancs of all th continuous variabls wr gnratd as 0 and 1 rspctivly Th rsidual variancs of th dpndnt variabls wr fixd so that th total varianc of all variabls was 1 [9] Cohn s guidlins for R 2 mtric, th amount of xplaind varianc in th outcom, wr usd to gnrat th path cofficints [19] Som covarianc algbra was usd in th currnt study to dtrmin th propr valus for th rsidual variancs and ffct sizs in plus [9] Th following path cofficints wr chosn to show th computational dtails and to xprss how th path cofficint and rsidual varianc of ach variabl wr gnratd Th R 2 of,,,,, wr st as 26% (larg ffct), 13% (mdium ffct), 2% (small ffct), 26% (larg ffct), 13% (mdium ffct), 2% (small ffct) rspctivly Additionally, th dirct ffct of on was fixd to 0 Th indpndnt variabl simulatd in th currnt study was a binary variabl with an vn split As a rsult, th

4 Amrican Journal of Applid Psychology 2014; 3(3): proportions of th two possibl valus of th binary variabl wr 50% and 50% rspctivly Th rsidual varianc of was: (8) According to quation (4), ( )was givn by: ( ) () ( ) (9) As notd prviously that ( )1, thus:!"#($ %&)!"#(') ( 026,and ( )1, thus: (10) ( (11) According to quation (10), was solvd as 102 By using th sam formulas shown abov, th ( ),, ( )and wr solvd as 087, 0721, 098 and 0283 rspctivly Th quation of th varianc of was xprssd as: (12) Th variancs of all continuous variabls wr fixd qually to a valu of 1, thus was solvd as: (13) Th ( chosn for, and wr 026, 013, 002 rspctivly, thus: (14) (15) (16) Put th rsults back to quation (13), ( )was solvd as 059 Th distribution trnd of th data was simulatd as normal distribution in th currnt simulation study Th significanc of th mdiating ffct was dtctd by th sobl tst that plus prforms by dfault Th simulation mthod usd in th currnt study can b mployd in actual applid sttings, but rsarchrs should not that th slction of stimats of paramtrs should bas on pilot studis or prvious rsarchs Whn mor information is availabl, th small, mdium, and larg catgorization should also b avoidd [9] 23 s Th valus of th mdiating ffcts wr fixd to zro ( 0,1 2 3 to tst th typ I rror rats of th indirct ffcts in all combinations whn sampl sizs wr 100, 200 and 500 rspctivly In th first combination, all th path cofficints wr fixd to 0 ( = = = = = = 0) In th scond combination, th path cofficints of th indpndnt variabl to th thr mdiator variabls wr fixd qually to 0 ( = = = 0), but th path cofficints of th thr mdiator variabls to th dpndnt variabl wr gnratd as larg ffct, mdium ffct and small ffct rspctivly ( = 051, = 036 and = 014) In th third combination,, and wr gnratd as larg ffct, mdium ffct and small ffct rspctivly ( = 102, = 0721, = 0283), but th path cofficints, and wr gnratd as 0 ( = = = 0) In this study, 10,000 rplications wr usd for ach combination to insur that stability has bn rachd [25] Th typ I rror rat of th indirct ffct in ont Carlo simulation is rgardd as th proportion of 10,000 rplications for which th 95% confidnc intrval will not contain th valu of zro Accordingly, th mpirical typ I rror rats would b xpctd to b closr or qual to th tru alpha 005, or fulfill Bradly s libral critrion that typ I rror rats should rang btwn 0025 and 0075 [30] Th rsults of th typ I rror rats of th 3 combinations ovr th 10,000 rplications wr prsntd in Tabls 1, 2 and 3 rspctivly Whn = = 0 (1 2 3, th typ I rror rats of all mdiating ffcts in th first combination wr quit blow th valu of 005, and did not fulfill Bradly s libral critrion as wll Although th sampl siz was incrasd to 500, th typ I rror rats of all indirct ffcts in Combination 1 wr 0 Th typ I rror rats of, and in Combination 2 and Combination 3 wr 0044, 0030, 0004, 0047, 0028 and 0002 rspctivly whn th sampl siz was 100 Furthrmor, whthr in small sampls (N = 100) or larg sampls (N = 500), th typ I rror rats of in Combination 2 and Combination 3 wr quit clos to 005 Additionally, th typ I rror rats of in Combination 2 and Combination 3 in all sampl sizs fulfilld Bradly s libral critrion [30] Howvr, th typ I rror rats of in Combination 2 and Combination 3 wr far from 005 in all sampl sizs Furthrmor, whn th sampl siz was incrasd, all typ I rror rats in Combination 2 and Combination 3 wr lvatd as wll orovr, whn incrasing th sampl siz to 500, th typ I rror rats of and in Combination 2 and Combination 3 rachd or closd to an alpha of 005 and th typ I rror rat of in Combination 2 fulfilld Bradly s libral critrion Howvr, th typ I rror rat of in Combination 3 was 0014 whn sampl siz was 500, which was too consrvativ Ths findings indicatd that diffrnt ffct sizs of and and diffrnt sampl sizs wr important factors influncing th typ I rror rats of mdiating ffcts in a multipl mdiator modl

5 76 Z-wi a and Wi-nan Zng: A ultipl diator odl: Powr Analysis Basd on ont Carlo Simulation Combination 1 Tabl 1 s in Combination 1 Total indirct ffct = = = = = = = Combination 2 Tabl 2 s in Combination 2 Total indirct ffct = = 0, = = 0, = = 0, = Tabl 3 s in Combination 3 Combination 3 Total indirct ffct = = 102, = = 072, = = 0283, = Powr Analysis Thr combinations, Combination 1 ( 102 (larg ffct), 0721 (mdium ffct), 0283(small ffct), 051 (larg ffct), 036 (mdium ffct), 014 (small ffct)); Combination 2 ( 0283(small ffct), 0721(mdium ffct), Combination 1 Tabl 4 Powr to Dtct Significant diating Effcts in Combination (larg ffct), 036 (mdium ffct), 014(small ffct), 051(larg ffct)) and Combination 3 ( 102 (larg ffct), 0283 (small ffct), 0721 (mdium ffct), 014 (small ffct), 051 (larg ffct), 014 (small ffct)) wr gnratd in this simulation study Th powr of vry indirct ffct in ach combination was stimatd ovr 10,000 rplications in sampl sizs of 100, 200 and 500 rspctivly Tabls 4, 5 and 6 prsntd th rsults of th powr stimats of diffrnt indirct ffcts in Combinations 1, 2 and 3 rspctivly Evn if th sampl siz was small (N=100), th powr of th total indirct ffcts in Combination 1 and Combination 2 rachd a lvl of 1 Additionally, th powr of th total indirct ffct in Combination 3 was blow 08 whn th sampl siz was 100 Howvr, ach of th mdiatd pathways showd us diffrnt information Tak Combination 1 as an xampl Th powr of total mdiating ffct in Combination 1 was xtrmly high at 1000 in a small sampl (N = 100), but th powr to dtct th mdiating ffct wr low at 0038 and 0175 whn th sampl sizs wr 100 and 200 rspctivly Th powr of in Combination 1 rachd at 0770 whn th sampl siz was incrasd to 500, but this powr was still lowr than 08 Ths findings indicatd that whn th mdiating ffct was larg, a small sampl siz can acquir adquat powr to dtct significant mdiating ffct In Combination 1, whn 0520, th powr of rachd 1000 whn th sampl siz was 100, which mans that a total sampl siz of at last 100 was rquird to rach th powr of 1000 to dtct significant mdiation Howvr, th powr of in Combination 2 was only 0231 whn th sampl siz was 100 Whn incrasing th sampl siz to 500, th powr of in Combination 2 was incrasd to 0882, which was slightly highr than th dsird 08 powr Ths findings wr consistnt with prvious rsarch [8] Powr Powr Powr Total indirct ffct = , 051, , 036, , 014, Combination 2 Tabl 5 Powr to Dtct Significant diating Effcts in Combination 2 Powr Powr Powr Total indirct ffct = , 036, , 014, , 051,

6 Amrican Journal of Applid Psychology 2014; 3(3): Combination 3 Tabl 6 Powr to Dtct Significant diating Effcts in Combination 3 Powr Powr Powr Total indirct ffct = , 014, , 051, , 014, Th rsults of th xact sampl sizs raching 08 powr to dtct significant mdiating ffcts in ach of th combinations wr prsntd in Tabl 7 In Combination 1, th indirct ffct was at 08 powr whn sampl siz was 524, which indicatd that 524 sampls wr ndd to hav 80% probability to accpt an altrnativ hypothsis aftr rjcting a fals null hypothsis orovr, th powr of th indirct ffcts and rachd a lvl of 1000 Ths findings confirmd that a smallr sampl siz is rquird to rach 08 powr to dtct significant mdiating ffct if th ffct siz of th mdiating ffct is larg Howvr, if th indirct ffct was small, a largr sampl siz is rquird to dtct significant mdiating ffct uthén and uthén suggstd that svral critria ar xamind to dtrmin sampl siz Firstly, th paramtr stimat bias and standard rror bias should not xcd 10% for any paramtr in th modl Scondly, th standard rror bias for th paramtr for which powr is bing assssd should not xcd 5% Thirdly, th covrag should rang btwn 091 and 098 If ths conditions ar mt, th sampl is chosn to kp powr clos to 08 [25] Tabl 7 Exact s Raching 08 Powr to Dtct Significant diating Effcts in th 3 Combinations Combination 1 Combination 2 Combination 3 Powr Powr (N = 524) (N = 396) (N = 392) Powr Total indirct ffct = Total indirct ffct = Total indirct ffct = , 051, , 036, , 014, , 036, , 014, , 051, , 014, , 051, , 014, uthén and uthén suggstd that, to dtrmin th paramtr bias, subtract th population valu from th paramtr stimat avrag ovr th rplications of th ont Carlo simulation (such as 10,000 rplications in th currnt study) and divid it by th population valu To dtrmin th standard rror bias, subtract th population standard rror from th standard rror stimat across th 10,000 rplications and thn divid it by th population standard rror Th covrag givs th proportion that th 95% confidnc intrval constructd by th 10,000 rplications contains th tru paramtr valu [25] Tabl 8 Paramtr and Standard Error Biass and Covrag in Combination 1 (N = 524) Combination 1 Paramtr Standard (N = 524) bias rror bias Covrag Total indirct ffct = , 051, , 036, , 014, Th paramtr and standard rror biass as wll as th covrag rats of th indirct ffcts in th 3 diffrnt combinations ovr th 10,000 rplications wr prsntd in Tabls 8, 9 and 10 rspctivly Rsults of paramtr and standard rror biass wr ngligibl, and all covrag rats rangd from 0932 to 0953 Ths rsults indicatd that th sampl sizs chosn in ach of th combinations can kp powr clos to 080 to dtct significant mdiating ffcts Tabl 9 Paramtr and Standard Error Biass and Covrag in Combination 2 (N = 396) Combination 2 (N = 396) Paramtr bias Standard rror bias Covrag Total indirct ffct = , 036, , 014, , 051, 0520 Combination 3 (N = Tabl 10 Paramtr and Standard Paramtr Standard Covrag Total indirct , 014, , 051, , 014,

7 78 Z-wi a and Wi-nan Zng: A ultipl diator odl: Powr Analysis Basd on ont Carlo Simulation 3 Discussion It is flxibl and as of implmntation to us ont Carlo simulation to prform powr analyss for multipl mdiator modls [9] Whn prforming powr analysis via ont Carlo simulation, a rsarchr should first slct which modl to b studid Th population valu which is rquird to b st for ach paramtr in th modl is ndd to b slctd bas on thory rsarch, pilot study or othr mpirical rsarchs [25] In th currnt simulation study, Cohn s guidlins dfining a small ffct, mdium ffct and larg ffct on th R 2 mtric wr usd to gnrat th cofficint of ach path Furthrmor, som covarianc algbra was usd to dtrmin propr valus for th rsidual variancs and transform th dsird R 2 ffct sizs into unstandardizd path cofficints In th actual applid sttings, rsarchrs should not furthr that a slctd path cofficint should match a rsidual varianc As an xampl, th R 2 chosn for path and path wr both 26% in Combination 1, and th unstandardizd path cofficints of and wr solvd as 102 and 051 rspctivly orovr, th mdiating ffct of on through was 0520 in Combination 1, and th rsidual varianc solvd for was 074 Whn fixing th unstandardizd path cofficints for and in th population modl, th rsidual varianc of should b fixd constant at th valu of 074 Ovrall, th typ I rror rats in this simulation study wr within th accptabl rang, which wr clos to th 005 tru valu lvl, or fll within th libral critrion proposd by Bradly that th typ I rror rats should rang btwn 0025 and 0075 W found that typ I rror rats of indirct ffcts wr affctd by th ffct sizs of and, and sampl sizs as wll Whn = 0, th typ I rror rats of all mdiating ffcts in all sampl sizs (N = 100, N = 200 and N = 500) in this simulation study wr far from th valu of 005, and did not fulfill th libral critrion proposd by Bradly Howvr, whn = 0, 0, or 0 and = 0, th typ I rror rats wr clos to 005 or flt within 0025 ~ 0075 Ths findings wr consistnt with prvious simulation study [17] Furthrmor, w noticd that th typ I rror rats of (small ffct) in both Combination 2 and Combination 3 wr too consrvativ whthr in small sampls or larg sampls Accordingly, it is confirmd that th ffct siz of mdiating ffct is a dtrminant factor influncing typ I rror rat Additionally, th typ I rror rats of th indirct ffcts in th 3 combinations wr all lvatd with th incras of sampl sizs, and wr finally clos to an alpha of 005 Although th typ I rror rats of th indirct ffcts of small ffct sizs wr consrvativ in small sampls, whn th sampl siz was incrasd, th typ I rror rats wr lvatd Findings in th currnt study wr consistnt with prvious rsarch [8] In this simulation study, w found that th rquirmnt of sampl siz and dsird powr lvl wr strongly dpndd on on dtrminant factor, th ffct siz of th indirct ffct Whn th ffct siz of th indirct ffct was small, a largr sampl siz was rquird to rach 08 powr to dtct a significant mdiating ffct Howvr, whn th ffct siz of th indirct ffct was larg, only a small sampl siz was rquird to rach 08 powr to dtct significant indirct ffcts For xampl, was st as a small mdiating ffct in Combination 1, but as a larg mdiating ffct in Combination 2 As a rsult, whn th sampl siz was 100, th powr to dtct significant mdiating ffct for in Combination 1 was 0038, whras th powr to dtct a significant mdiating ffct for in Combination 2 was 1000 Whn th sampl siz was incrasd to 500, powr of in Combination 1 to dtct a significant mdiating ffct was lowr than 08 orovr, it is worthy of noting that although th total indirct ffct in Combination 1 rachd a satisfying powr in vry sampl siz, powr of individual mdiating ffct was still (such as in Combination 1) lowr than 08 whn th sampl siz was small, which indicatd that th diffrnt stimats of powr xist in on multipl mdiator modl whn thr ar diffrnt mdiatd pathways [9] Rfrncs [1] Hays AF, Prachr KJ Quantifying and tsting indirct ffcts in simpl mdiation modls whn th constitunt paths ar nonlinar ultivariat Bhavioral Rsarch 2010;45: [2] Albrt J, Nlson S 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