A graphical representation of the mediated effect

Transcription

1 Behvior Reserh ethods, (1), 55- doi: 1.375/BR A grphil representtion of the medited effet tthew S. Fritz Virgini Polytehni Institute nd Stte University, Blksburg, Virgini nd Dvid P. Kinnon Arizon Stte University, Tempe, Arizon edition nlysis is widely used in the soil sienes. Despite the populrity of medition models, few reserhers hve used grphil methods, other thn struturl pth digrms, to represent their models. Plots of the medited effet n help reserher better understnd the results of the nlysis nd onvey these results to others. This rtile presents method for reting nd interpreting plots of the medited effet for vriety of medition models, inluding models with (1) dihotomous independent vrible, () ontinuous independent vrible, nd (3) n intertion between n independent vrible nd the mediting vrible. An empiril exmple is then presented to illustrte these plots. Smple ode for reting plots of the medited effet in R nd SAS is lso inluded, nd my be downloded from Grphil representtions of dt re often used by reserhers to explore reltions mong vribles. Plots nd grphs n be used for initil dt explortion, to exmine violtions of model ssumptions, nd to show predited sores from finl model. Despite the verstility nd usefulness of plots, reserhers hve rrely used them to illustrte the results from medition studies. Insted, reserhers present pth digrms tht show the struturl reltionship between the independent, meditor, nd dependent vribles. However, plots tht not only show the struturl properties of the medition model, but tully illustrte the mgnitude of the medited effet itself, n be very useful s visul representtion of the medited effet, whih my be more desirble thn presenting numeril results lone. These plots my be espeilly useful when investigting omplex medition models suh s inonsistent medition models in whih the diret effet nd medited effet hve opposite signs, or models in whih the meditor interts with the independent vrible, whih were first desribed by errill (199). Kinnon () used similr pproh to desribe more generl plots of the effets in medition models. The purpose of this rtile is to explin how to onstrut plots of the medited effet for study with dihotomous (e.g., rndom ssignment to tretment nd ontrol group) or ontinuous independent vrible, ontinuous meditor, nd ontinuous outome vrible. An empiril exmple is then used to show wht the finished plots look like nd how to interpret plots of the medited effet. Finlly, smple ode for reting plots of the medited effet is given for two ommon sttistil softwre pkges, SAS (SAS Institute, ) nd R (R Core Development Tem, ), to ssist reserhers in mking their own plots. The gol of this rtile is to give reserhers nother tool for presenting nd explining results from meditionl studies tht my be used in presenttions nd publitions. edition edition is the nme given to models in whih the effet of n nteedent or independent vrible (X ) on dependent vrible ( ) is trnsmitted through third intervening or mediting vrible (). In other words, X ffets, whih in turn ffets. Figure 1 shows digrm of single meditor model, where the top digrm represents the totl effet of X on nd the bottom digrm represents the medited effet of X on through. These two pth digrms n be represented using the regression equtions 5 i 1 1 X 1 e 1 (1) 5 i 1 X 1 e () 5 i 3 1 X 1 b 1 e 3, (3) where is the totl effet of X on, is the effet of X on djusted for, b is the effet of on djusted for X, is the effet of X on, i 1, i, nd i 3 re the interepts, nd e 1, e, nd e 3 re the residuls. The pth is known s the tion theory beuse X is often diretly mnipulted, wheres the b pth is known s the oneptul theory (Chen, 199; Kinnon, Tborg, & orgn-lopez, ), beuse is not usully diretly mnipulted. The medited or indiret effet is equl to b or the produt of the tion theory nd the oneptul theory oeffiients. The effet of X on tht does not pss through is lled the diret effet nd is equl to. odels where 5 re lled ompletely medited models, nd models where re lled prtilly medited. S. Fritz, mtt.fritz@vt.edu 55 Copyright Psyhonomi Soiety, In.

2 5 Fritz nd Kinnon A B X X Figure 1. Pth digrms for (A) the totl effet of the independent vrible on the dependent vrible nd (B) the indiret effet of the independent vrible on the dependent vrible through the meditor vrible. models (Bron & Kenny, 19). In ordinry-lest-squres regression, 5 b 1, () lthough this reltionship does not hold for ll logisti regression or multilevel models (Kinnon, ). edition should not be onfused with n intertion, lso known s modertion, in whih the mgnitude of the effet of X on is determined by the vlue of ; in ompletely medited model, if is held onstnt, then X hs no effet on. There re mny sttistil tests of medition, nd these tests fll into three tegories: usl steps, produt of oeffiients, nd differene in oeffiients. In the usl steps tests, eh step in the usl hin is tested for signifine in sequene. For exmple, the Bron nd Kenny (19) usl steps test first tests the signifine of the effet of X b on,, then the signifine of the effet of X on,, nd then finlly the effet of on ontrolling for X, b; if ll three effets re signifint, then medition is present. The produt of oeffiients tests diretly test the indiret effet, b, for signifine, by dividing b by its stndrd error nd ompring the resulting vlue with norml distribution, or by reting onfidene intervls round the indiret effet, s in Kinnon, Fritz, Willims, nd Lokwood (7). Finlly, the differene in oeffiients tests investigte the differene between the overll effet nd the prt of the overll effet tht does not pss through the meditor,, by dividing the differene by its stndrd error nd ompring the vlue with t distribution for signifine. For more informtion on tests of medition, see - Kinnon, Lokwood, Hoffmn, West, nd Sheets () nd Kinnon, Lokwood, nd Willims (). Despite the vriety of different sttistil tests of medition, ll of the tests operte under the sme ssumptions. The first ssumption is tht the order of ustion is orret tht is, X uses, even when X is not rndom ssignment to tretment, nd uses. Also, the intertion between X nd is ssumed to be zero in Eqution 3, lthough this ssumption n be relxed, s desribed lter in this rtile. Other ssumptions re tht the dependent vribles hve norml distribution nd the residuls re independent nd normlly distributed. These ssumptions pply to ll of the plots. Plotting the edited Effet To plot medited effet for dihotomous independent vrible, Equtions 1,, nd 3 must first be estimted to find vlues for the interepts nd regression oeffiients (Kinnon, ; errill, 199). Note tht X ws dummy oded nd 1, nd for onveniene, hts were not inluded over the prmeter estimtes in the plots. Seond, 7 i3 X b (Eqution 3 for X ) i3 X b (Eqution 3 for X 1) i1 (Eqution 1 for X 1) 5 Totl Effet b edited Effet i1 (Eqution 1 for X ) i (Eqution for X ) i (Eqution for X 1) Figure. Plot of the medited effet for dihotomous X vrible.

3 Plotting the edited Effet 57 the dt re plotted with on the vertil xis nd on the horizontl xis, s shown in Figure. Next, Eqution 1 is plotted for the vlues of X (i.e., nd 1), suh tht horizontl line is pled t 5 î 1, orresponding to X 5, nd seond horizontl line is pled t 5 î 1 1 ĉ, orresponding to X 5 1. The distne between the horizontl lines represents the totl effet of X on, ĉ. Then, Eqution is plotted for both vlues of X, resulting in vertil line t 5 î nd seond vertil line t 5 î 1 â. The distne between the two vertil lines represents the tion theory nd is equl to â. Finlly, Eqution 3 is plotted t both vlues of X. The slopes of these prllel simple regression lines re equl to bˆ. Looking t the 5 î 1 â vertil line, the distne between the 5 î 1 1 ĉ horizontl line nd the point where the regression line for the X 5 group intersets the 5 î 1 â vertil line is equl to ĉ, nd the distne from this intersetion point to the 5 î 1 horizontl line is equl to the medited effet, â ˆb. Interpreting the plot is strightforwrd (Kinnon, ; Kinnon, Firhild, & Fritz, 7; errill, 199). Figure shows tht ĉ is the predited mount of hnge in for one unit hnge in X nd tht the lrger the distne between the horizontl lines, the lrger the overll effet of X on. The sme is true of â, whih is the predited mount of hnge in for one unit hnge in X; the lrger the distne between the vertil lines, the greter the effet of X on. The slope of the simple regression lines, ˆb, is then the predited hnge in for one unit hnge in, djusted for X. Regrdless of whether the effet is positive or negtive, the steeper the slope of the simple regression lines, the lrger the effet of on, djusting for X. The medited effet, â ˆb, is then the predited hnge in for n â unit inrese in, whih orresponds to one unit hnge in X. Finlly, looking t the length of â ˆb reltive to the length of ĉ indites the mount of the overll effet of X on tht is medited by ; in this exmple, the meditor ounts for bout % of the effet of X on. Regrdless of the reltive size of â ˆb, sttistil signifine of the medited effet should be investigted using one of the tests of medition desribed erlier. For ontinuous independent vrible, Equtions 1,, nd 3 re gin estimted to find vlues for the interepts nd regression oeffiients, nd the dt re plotted with on the vertil xis nd X on the horizontl xis, s shown in Figure 3. Next, Equtions 1 nd re plotted for the men of X nd one unit bove the men of X, suh tht horizontl lines re plotted t 5 î 1 1 ĉx, whih is equl to the men of, nd t 5 î 1 1 ĉ(x 1 1), nd vertil lines re plotted t 5 î 1 âx, whih equls the men of, nd 5 î 1 â (X 1 1). The distne between the horizontl lines is then equl to ĉ, nd the distne between the two vertil lines is equl to â. Finlly, Eqution 3 is plotted. Beuse X is ontinuous, speifi vlues of X must be seleted to plot. In Figure 3, the vlues of X seleted re the men of X nd one stndrd devition bove the men of X, lthough ny vlues of X my be seleted nd ny number of vlues my be plotted (Aiken & West, 1991); the slope of eh of these lines is equl to ˆb. Looking t the 5 î 1 â(x 1 1) vertil line, the distne between the 5 î 1 1 ĉ(x 1 1) horizontl line nd the point where the regression line for the men of X intersets the 5 î 1 â(x 1 1) vertil line is equl to ĉ, nd the distne from the intersetion point to the 5 î 1 1 ĉx horizontl line is equl to the medited effet, â ˆb. The interprettion of the effets in the plot re identil to the se with dihotomous X vrible. In both Figure nd Figure 3, ll of the estimtes of the effets in the model, â, ˆb, ĉ, nd ĉ, were positive, but this i3 X b(enx 1SD) i3 X b(enx) i1 (enx 1) i1 (enx) b i (enx) i (enx 1) Figure 3. Plot of the medited effet for ontinuous X vrible. Note tht SD stnds for the stndrd devition of X.

4 5 Fritz nd Kinnon Empiril Exmple To better illustrte the onstrution nd use of these plots, n empiril exmple is inluded using dt from the Athletes Trining nd Lerning to Avoid Steroids (ATLAS) progrm (Goldberg et l., 199; Goldberg et l., ). The ATLAS progrm ws designed for high shool footbll plyers, with the gol of deresing their intentions to use nboli steroids by offering improved dietry behvior nd inresed strength trining self-effiy s diret lterntives to steroid use. The diet nd strengthtrining methods produed similr results to steroid use without the negtive onsequenes ssoited with using illiit drugs. ATLAS hnged these three outome vribles (intentions to use steroids, nutrition behvior, nd strength trining self-effiy) by trgeting 1 min mewill not neessrily our when plotting rel dt. When ˆb is negtive, the simple regression lines slope downwrd. When ĉ is negtive, the distne between the horizontl lines will remin the sme, but the order of the lines will be reversed nd â ˆb nd ĉ will swith ples. When â is negtive, the distne between the vertil lines will remin the sme, but the order of the lines will reverse, lthough â ˆb nd ĉ will still our on the 5 î 1 â line. For instne, in Figure, if â ws negtive, the line 5 î 1 â would be â units to the left of the 5 î line. A word of ution is needed for inonsistent medition models. In n inonsistent medition model, â ˆb nd ĉ re signifint, but opposite in sign, whih n use ĉ to be nonsignifint. As shown in Eqution, if â ˆb nd ĉ re equl in mgnitude but opposite in sign, then ĉ would be equl to. When this ours, the slopes of the simple regression lines my be positive or negtive nd the vertil lines my or my not be swithed, depending on the signs of the â nd ˆb oeffiients. It is lso possible in inonsistent medition models for the medited effet to be lrger thn the totl effet, suh tht the simple regression slopes of Eqution 3 ross the 5 î 1 â line outside of the rnge of the horizontl lines tht represent the totl effet ĉ. X Intertions In ertin ses, the effet of the meditor is not equl ross vlues of X. In these ses, n intertion between the independent vrible nd the mediting vrible, known s n X intertion, is present. A signifint X intertion indites tht the effet of on is funtion of X, nd hnges s the vlue of X hnges. Plotting medition model with n X intertion is very similr to plotting the typil medition model (Fritz & Kinnon, ; errill, 199). First, Equtions 1 nd re estimted to find vlues for the oeffiients. Rther thn estimting Eqution 3, however, the eqution 5 i 3 1 X 1 b 1 gx 1 e 3 (5) is estimted. The only differene between Eqution 3 nd Eqution 5 is the inlusion of the X intertion in Eqution 5. Plotting of the medited effet with n X intertion then diretly follows the steps used to plot medition models without n intertion; n exmple for dihotomous independent vrible is shown in Figure. In Figure, the solid line represents the simple regression line for the group oded X 5 nd the dshed line represents the simple regression line for the group oded X 5 1. The dotted line represents the simple regression line for the group oded X 5 1 when the intertion effet ĝ is set to nd the distne between the dshed nd dotted lines t speifi vlue of is equl to the intertion effet t tht vlue of, ĝ. i3 X b gx (X ) i3 X b gx (X 1) i3 X b (X 1) g X Effet i1 i1 b i i Figure. Plot of the medited effet with n X intertion.

5 Plotting the edited Effet 59 b Control group ATLAS group Figure 5. Plot of the medited effet of exposure to the ATLAS progrm (X ) on resistne to drug offers ( ) nd nutrition behviors ( ). diting vribles, inluding knowledge of the effets of steroid use, peer tolerne of steroid use, nd pereived severity of steroid use (see Kinnon et l., 1, for omplete list of the 1 meditors). From these 1 meditor vribles nd 3 outome vribles, two exmples re illustrted; note tht the dt in these plots hve hd smll mount of rndom error dded to them, proess lled jittering, to mke them esier to red. The first exmple is tht exposure to the ATLAS progrm (X ) inresed n thlete s bility to resist offers of steroids (), whih, in turn, inresed n thlete s helthy nutrition behvior ( ). Using Cohort 1, wve C of the ATLAS dt gives estimtes nd stndrd errors for the prmeters of Equtions 1 3 of â 5.33 (SE 5.3), ˆb 5.5 (SE 5.), ĉ 5.37 (SE 5.37), nd ĉ 5.33 (SE 5.3), whih re plotted in Figure 5. The plot shows the differene in mgnitude between the diret effet ĉ nd the indiret effet â ˆb, with the diret effet mking up greter proportion of the overll effet, illustrted by the two horizontl lines. Looking t this plot, it n be seen tht the proportion of the effet of X on medited by is very smll nd not likely to be signifint. Testing for medition using the PRODCLIN progrm (Kinnon, Fritz, et l., 7) to rete onfidene intervl for â ˆb gve 95% onfidene intervl of [.11,.35], whih shows tht the indiret effet is not sttistilly signifint. The seond exmple is tht exposure to the ATLAS progrm (X ) deresed n thlete s belief in medi dvertisements bout the positive spets of steroids (), whih then inresed the thlete s strength trining self-effiy ( ). Using Cohort 1, wve C of the ATLAS dt gives estimtes for the prmeters of Equtions 1 3 of â 5. (SE 5.75), ˆb 5.3 (SE 5.31), ĉ 5.31 (SE 5.), nd ĉ 5.1 (SE 5.19), whih re plotted in Figure. Looking t this plot, we see tht the proportion of the totl effet, ĉ, tht is ttributble to the meditor is muh lrger, round %, ompred with the plot in Figure 5. Testing the signifine of the medited effet using the PRODCLIN progrm (Kinnon, Fritz, et l., 7) to rete onfidene intervl for â ˆb gives 95% onfidene intervl of [.1133,.3], whih leds to the onlusion tht the meditor is signifint. Summry nd Exmple Computer Progrms The plots disussed here re for the single meditor model only. However, mny medition models re muh more omplex, with multiple meditors or longitudinl dt. Plotting these lrger models requires plots more omplex thn the plots presented here, whih re therefore hrder to interpret. For instne, in the multiple meditor se, the meditor vribles would hve to be stndrdized to ple them on the sme metri, nd then the plots of the individul meditors would hve to be lid over one nother. The interprettion of suh plot would then depend on the number of meditors nd the signs nd mgnitudes of the individul medited effets. To help reserhers in reting medition plots for their own dt, smple ode for two ommon sttistis pkges, R (R Core Development Tem, ) nd SAS (SAS Institute, ), re inluded in the Appendix, s re instrutions for loding nd using the ode. Eletroni versions of these progrms re vilble t ripl/medite.htm. These smple progrms re ment to t s n outline for reserhers who re plotting single meditor models. Reserhers re enourged to modify the progrms s neessry to explore their dt nd to produe plots tht fit their individul needs.

6 Fritz nd Kinnon b Control group ATLAS group Figure. Plot of the medited effet of exposure to the ATLAS progrm (X ) on belief in medi dvertisements ( ) nd strength trining self-effiy ( ). Author note This reserh ws supported by Publi Helth Servie Grnt DA. Portions of this work hve been presented t the nnul meeting of the Soiety for Prevention Reserh. Correspondene onerning this rtile should be ddressed to. S. Fritz, Deprtment of Psyhology, Virgini Polytehni Institute nd Stte University, 19 Willims Hll, Blksburg, VA 1 (e-mil: mtt.fritz@vt.edu). REFERENCES Aiken, L. S., & West, S. G. (1991). ultiple regression: Testing nd interpreting intertions. Newbury Prk, CA: Sge. Bron, R.., & Kenny, D. A. (19). The modertor meditor distintion in soil psyhologil reserh: Coneptul, strtegi, nd sttistil onsidertions. Journl of Personlity & Soil Psyhology, 51, Chen, H. T. (199). Theory-driven evlutions. Thousnd Oks, CA: Sge. Fritz,. S., & Kinnon, D. P. (, June). Plots of the X intertion in models of the medited effet. Poster presented t the nnul meeting of the Soiety for Prevention Reserh, Sn Antonio, TX. Goldberg, L., Elliot, D. [L.], Clrke, G. N., Kinnon, D. P., oe, E. [L.], Zoref, L., et l. (199). Effets of multidimensionl nboli steroid prevention intervention: The Adolesents Trining nd Lerning to Avoid Steroids (ATLAS) progrm. Journl of the Amerin edil Assoition, 7, Goldberg, L., Kinnon, D. P., Elliot, D. L., oe, E. L., Clrke, G. [N.], & Cheong, J. (). The Adolesents Trining nd Lerning to Avoid Steroids progrm: Preventing drug use nd promoting helth behviors. Arhives of Peditris & Adolesent ediine, 15, Kinnon, D. P. (). An introdution to medition nlysis. hwh, NJ: Erlbum. Kinnon, D. P., Firhild, A. J., & Fritz,. S. (7). edition nlysis. Annul Review of Psyhology, 5, Kinnon, D. P., Fritz,. S., Willims, J., & Lokwood, C.. (7). Distribution of the produt onfidene limits for the indiret effet: Progrm PRODCLIN. Behvior Reserh ethods, 39, Kinnon, D. P., Goldberg, L., Clrke, G. N., Elliot, D. L., Cheong, J., Lpin, A., et l. (1). editing mehnisms in progrm to redue intentions to use nboli steroids nd improve exerise self-effiy nd dietry behvior. Prevention Siene,, 15-. Kinnon, D. P., Lokwood, C.., Hoffmn, J.., West, S. G., & Sheets, V. (). A omprison of methods to test medition nd other intervening vrible effets. Psyhologil ethods, 7, Kinnon, D. P., Lokwood, C.., & Willims, J. (). Confidene limits for the indiret effet: Distribution of the produt nd resmpling methods. ultivrite Behviorl Reserh, 39, Kinnon, D. P., Tborg,. P., & orgn-lopez, A. A. (). edition designs for tobo prevention reserh. Drug & Alohol Dependene,, S9-S3. errill, R. (199). Tretment effet evlution in non-dditive medition models. Unpublished disserttion, Arizon Stte University, Tempe. R Core Development Tem (). R (Version..) [Computer Progrm]. Vienn, Austri: The R Foundtion for Sttistil Computing. Avilble t SAS Institute (). SAS (Version 9.1) [Computer Progrm]. Cry, NC: SAS Institute, In. Arhived terils The following mterils ssoited with this rtile my be essed through the Psyhonomi Soiety s Norms, Stimuli, nd Dt rhive, To ess these files, serh the rhive for this rtile using the journl nme (Behvior Reserh ethods), the first uthor s nme (Fritz), nd the publition yer (). File: Fritz-BR-.zip Desription: The ompressed rhive file ontins files: med_plots.r, n R progrm to plot the medited effet for singlemeditor medition model; med_plots.ss, n SAS progrm to plot the medited effet for single-meditor medition model. Author s e-mil ddress: mtt.fritz@vt.edu (nusript reeived November, ; revision epted for publition rh 1, 7.)