Standardized Simple Mediation Model: A Numerical Example

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1 World Appled Scences Journal (8): 35-39, 03 ISSN IDOSI Publcatons, 03 DOI: 0.589/dos.was Standardzed Smple Medaton Model: A Numercal Example,,3, Anwar Ftranto and Habshah Md Department of Mathematcs, Faculty of Scence Unverst Putra Malaysa, Malaysa Laboratory of Appled and Computatonal Statstcs, Insttute for Mathematcal Research, Unverst Putra Malaysa 3 Department of Statstcs, Faculty of Mathematcs and Natural Scences Bogor Agrcultural Unversty, Indonesa Abstract: Medaton models fgure out how an effect occurred by hypotheszng a causal sequence. For a smple medaton model, a causal sequence s descrbed n whch an ndependent varable causes the medator whch sequentally causes the dependent varable. In ths artcle, we tred to ntroduce to use standardzed regresson coeffcent to the nvolvng the smple medaton model snce a standardzed coeffcent wll be more meanngful than an unstandardzed coeffcents. In ths artcle, we show that n smple medaton model, even though standardzed regresson coeffcents are dfferent from the unstandardzed coeffcents, but the standardzed coeffcents mantan the order of magntude of the unstandardzed regresson coeffcents for the smple medaton model. Key words: Medaton Model Standardzed Coeffcent Standard Error INTRODUCTION medators or statstcal controls, X s effect can also be ndrect va one or more medators. That s, X s effect on Medaton modelng s a powerful analytcal tool Y may be the result of X affectng the medatng that can be used to explan the nature of the relatonshp varable (s), whch n turn causally affect Y. In ths among three or more varables. In addton to that, t can approach, the relatonshp between an ndependent be used to show how a varable medates the varable and a dependent varable s decomposed nto relatonshp between levels of nterventon and outcome. drect and ndrect (medated) effects. The smplest Medaton models also explan how an effect occurred by example of medaton s a three-varable recursve model, hypotheszng a causal sequence. The basc medaton whch provdes two causal paths feedng nto a sngle model s a causal sequence n whch the ndependent dependent varable; the ndependent varable and the varable (X) causes the medator (M) whch n turn causes medatng varable drectly affect the dependent varable, the dependent varable (Y), therefore explanng how X whle the ndependent varable drectly affects the had ts effect on Y ([] and []). In general, a medatonal medator. effect s the accountng of the relatonshp between a predctor varable and an outcome. Research has Unstandardzed Smple Medaton Models: Evdence for presented basc measurement approaches for assessng medaton occurs when the relatonshp between two the effect of medaton ([3], [4] and [5]). varables can be partally or totally accounted for by There are many defntons of a medator varable, an ntervenng varable, whch s the medator. but accordng to [6], the defnton s a varable that s The sgnfcance of the medated effect can be tested causally between two varables and that accounts for wth a seres of regresson equatons. There are at least the relatonshp between those two varables. In contrast three lnear regresson equatons needed to buld a to an ndependent varable X s drect effect on dependent smple medaton model (Fg., panel I and panel ). varable Y, whch s ts effect ndependent of any Those equatons are as follows: Correspondng Author: Anwar Ftranto, Department of Mathematcs, Faculty of Scence Unverst Putra Malaysa, Malaysa. 35

2 World Appl. Sc. J., (8): 35-39, 03 From ths analyss, both the nfluence of M on Y (path b) and the nfluence of X and Y (path c', also called the drect effect of X) can be obtaned. There are assumptons requred for dentfyng medaton, ncludng assumptons necessary for the statstcal methods used to estmate the strength of the relatonshp n the regresson models. The assumptons for the medator model are necessary to yeld accurate results. Cohen, [8], clearly sad that each medaton regresson equaton requres the usual assumptons for regresson analyss. These assumptons are the correct Fg. : Basc smple medaton model functonal form, no omtted nfluences, accurate measurement and well-behaved resduals. Unfortunately, Y = + cx +, () all those estmaton use unstandardzed lnear regresson M = + ax +, () coeffcents. The slope coeffcents, c,, a and b depends on the unts n whch the predctors and outcome are Y = 3+ c X + bm + 3, (3) measured, so that f ether or both were measured n dfferent unts, the slope coeffcents would change. where Y s the dependent varable, X s the ndependent varable, M s the medatng varable or Buldng Standardzed Smple Medaton Model: medator. Unstandardzed coeffcent c represents the The magntudes of estmated unstandardzed slope relaton between the ndependent varable to the parameters n multple regresson depend on the scales of dependent varable n the frst equaton, s the parameter the predctor and outcome varables. The unstandardzed relatng the ndependent varable to the dependent coeffcent does not tell us whch ndependent varable s varable adusted for the effects of medator, a s the strongest predctor n one sample. However, the unstandardzed coeffcent of the relatonshp between X standardzed slope s nterpreted as the estmated number and M and b s the parameter relatng the medator to the of standard devatons of change n the dependent dependent varable adusted for the effects of the varable for one standard devaton unt change n the ndependents varable to the medatng varable. ndependent varable, controllng for other ndependent Note that the, and represent error varables. Ths ndex can be compared across studes n 3 varablty and, and much the same way that standardzed mean dfference are the ntercepts. 3 effects are compared. The ntercepts are not nvolved n the estmaton of Another ssue n unstandardzed lnear regresson medated effects and could be left out of the equatons, coeffcent s that we must be cautous about nterpretng [7]. Note that, both c and are parameters relatng the the magntudes of the regresson coeffcents as ndependent varable to the dependent varable, but s ndcators of the relatve mportance of varables n the a partal effect, the adusted effects of the medator. equaton. They are partals, reflectng the varaton The varance of X s and s the error accounted for when all other predctors are n the model X and they may have dfferent measurement unts and varance, namely the varance n Y not accounted for by varances. If they are standardzed (beta weghts) they X. Panel ) shows the model n whch the ntervenng become more comparable. Therefore unstandardzed varable s ncluded. In ths model, there are three regresson coeffcents cannot be consdered to reflect relatonshps of nterest. Frst, X nfluences the medator, the relatve mportance of ndvdual varables and the M and s denoted by a, a regresson coeffcent obtaned nterpretaton of standardzed beta weghts s extremely from a smple regresson of M on X. The error varance for lmted by ther context-dependent nature ([8, 9]). ths regresson s denoted by. Second, the In lnear regresson, standardzed betas are often relatonshp between X and Y s examned agan, ths tme used to compare strength of predcton across varables. wth the medator, M, also ncluded as a predctor of Y. The predctors are placed on a common scale so that each 36

3 y x x = =,,, n World Appl. Sc. J., (8): 35-39, 03 has the same mean and standard devaton. Varables y y ( x x) ( x x). (6) havng larger absolute value of standardzed beta weghts = + + S y S y S x S y S y are consdered as stronger predctors n the equaton. Though some researchers dscourage the use of The Equaton (6) s then transformed through standardzed coeffcents, [0] or warn aganst nterpretng transformng ts varables of * y = ( y y) S, y them as ndcators of varable mportance, [], the most z = ( x x) S, x z ( ) = x x S. Meanwhle, the recent publcaton manual of the Amercan Psychologcal x Assocaton, [], they encourage ther routne applcaton respectve standard devatons are defned as: and reportng. Standardzed beta weghts are especally useful when varables are measured on an arbtrary scale. ( x x) ( y ) y and = S y = Computng Standardzed Coeffcents n Lnear Regresson: It s well-known that the standardzed ( x x). Then, the transformed response = regresson coeffcent n a bvarate regresson model s the same as the bvarate correlaton coeffcent between the ndependent and dependent varables. In multple varable, * y, s wrtten as: regresson analyss, standardzed regresson coeffcents are scale free estmates and are related to correlaton * * * * y = z + z +. (7) coeffcents, but the relatonshp s much more complex than n bvarate regresson. The relaton between the ordnary multple regresson Consder a regresson model wth two ndependent coeffcents and the standardzed regresson coeffcents varables. When two ndependent varables are ncluded then can be wrtten as: n a multple regresson model, the model can be defned as,, (4) S * x =, =,..., p - S y ( ), where are the standardzed regresson coeffcents n populaton. * where the y s the score on the dependent varable of the th subect, x and x are the values of the ndependent varables for the th subect, 0, and are populaton regresson coeffcents and s the error term assumed to be normally dstrbuted wth mean of zero and constant varance. To obtan standardzed regresson coeffcents, we can transform the varables nto standardzed form. From the multple regresson equaton wth two predctors, the mean of the dependent varable s obtaned as y = 0+ x + x. The mean-devaton form for the above regresson model of equaton (4) can be obtaned by subtractng y as follows: ( ) ( x x ) ( x x ) y y = + x + x + + x + x 0 0 = + + Then we dvde by the standard devaton of y on both sdes and multply each term on the rght sde by n the form of, to obtan the followng equaton: S x (5) Accommodatng Standardzed Coeffcents nto Smple Medaton Models: One reason to accommodate the standardzed lnear regresson coeffcent s that for varables wth no natural metrc, a scale-free standardzed coeffcent may be more meanngful than an unstandardzed coeffcent. The use of standardzed coeffcents n OLS transforms the ndependent varable nto a varable measured n standard devaton unts. Ths transformaton wll have huge mpact n medaton analyss, n general, snce a medaton model conssts of several lnear regresson equatons. The advantage to ths s that a one standard devaton change s known to be a substantal change relatve to the range of the ndependent varable. A Smple medaton model nvolves three regresson equatons as n Eq. (), Eq. () and Eq. (3) on whch t contans both smple and multple lnear regressons. When the beta coeffcents are standardzed, the nterpretaton of the medaton analyss wll be more meanngful. 37

4 World Appl. Sc. J., (8): 35-39, 03 Table : Summary statstcs for unstandardzed and standardzed smple medaton model for Harrs and Rosenthal s Data. Statstcs â ˆb ĉ â ˆb ˆ ( ) Model Model Model 3 Medated Effect Unstandardzed Estmates Standardzed Estmates p value < R Numercal Example: To llustrate the mportance of the standardzed coeffcent, whch s equal to employng the standardzed betas n smple medaton One unt score ncrease n teacher expectancy s analyss, we present a numercal example avalable n [7], assocated wth roughly a half ncrease of the basc skll that s Harrs and Rosenthal s Data. Illustratons of the score. data are also avalable n [3]. The data contans of 40 There was a statstcally sgnfcant effect of teacher subects of teacher expectances and student expectancy on socal envronment. ( aˆ = 0.840; s a ˆ = 0.580) achevement. The example s orgnally based on [4] of And accordng to ts assocated standardzed coeffcent, four medaton models for how teacher expectances t can be seen that one unt score ncrease n teacher affect student achevement. They descrbed potental expectancy s assocated wth about 85% ncrease of the medatonal processes for how expectances about a score of socal clmate. The relaton of the socal clmate person s behavor lead to actual changes n behavor. (medator) on the basc skll score of students s The dependent varable (Y) was the score on a test of statstcally sgnfcant ( b ˆ = 0.437; s b ˆ = ) when basc sklls after one semester n the classroom. The ndependent varable, X, s the teacher expectancy controllng for teacher expectancy. A unt change n based on an ntellgence test gven to the student the the score of socal clmate s assocated wth an ncrease prevous year. There are two medator varables, M and of basc skll score. Meanwhle, the adusted effect M to represent socal clmate and teacher nput of teacher expectancy does not contrbute statstcally to (the average observer ratng of nput to the student). the response varable when the medator s n the model ( cˆ For the purpose of ths study, we arbtrary use M as the = 0.358; s c ˆ = 0.4). There s a drop n the value of medator. It s hypotheszed that the teacher expectancy c ˆ = compared to c ˆ = may lead the teacher to teach more materal and more dff The unstandardzed estmate of the medated effect s cult materal that may lead to ncreased student equal to ab ˆˆ = = cˆ c ˆ = Meanwhle, the achevement. standardzed medated effect of teacher expectancy RESULTS through socal clmate s equal to Some statstcs of the smple medaton analyss of the Harrs and Rosenthal s data s shown n Table. The teacher expectancy s sgnfcantly related to the score on a test of basc sklls, ( cˆ = ; s c ˆ = 0.734) provdng evdence that there s a statstcally sgnfcant relaton between the ndependent and the dependent varable. But snce that coeffcent s unstandardzed coeffcent, we must be careful n nterpretng the coeffcent, as has been mentoned n the prevous secton. Instead of that, t s wse to nterpret t based on ab ˆ** ˆ = = cˆ * c ˆ* = score of basc skll. In another word, even though standardzed regresson coeffcents are dfferent from the unstandardzed coeffcents, but the standardzed coeffcents mantan the order of magntude of the unstandardzed regresson coeffcents for the smple medaton model. ACKNOWLEDGMENTS The authors are grateful to Unversty Grants Scheme by Unverst Putra Malaysa for awardng the research grant. 38

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