Estimating Bias and RMSE of Indirect Effects using Rescaled Residual Bootstrap in Mediation Analysis

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1 Anwar Ftranto, Habsa Md Estmatng Bas and RMSE of Indrect Effects usng Rescaled Resdual Bootstrap n Medaton Analyss ANWAR FITRIANTO Laboratory of Appled and Computatonal Statstcs, Insttute for Matematcal Researc Unversty Putra Malaysa UPM Serdang, Selangor MALAYSIA anwarftranto@gmal.com HABSHAH MIDI Laboratory of Appled and Computatonal Statstcs, Insttute for Matematcal Researc Unversty Putra Malaysa UPM Serdang, Selangor MALAYSIA absamd@gmal.com Abstract: It s a common practce to estmate te parameters of medaton model by usng te Ordnary Least Squares (OLS) metod. Te constructon of T statstcs and confdence nterval estmates for makng nferences on te parameters of a medaton model, partcularly te ndrect effect, s usually are based on te assumpton tat te estmates are normally dstrbuted. Noneteless, n practce many estmates are not normal and ave a eavy taled dstrbuton wc may be te results of avng outlers n te data. An alternatve approac s to use bootstrap metod wc does not rely on te normalty assumpton. In ts paper, we proposed a new bootstrap procedure of ndrect effect n medaton model wc s resstant to outlers. Te proposed approac was based on resdual bootstrap wc ncorporated rescaled studentzed resduals, namely te Rescaled Studentzed Resdual Bootstrap usng Least Squares (ReSRB). Te Monte Carlo smulatons sowed tat te ReSRB s more effcent tan some exstng metods n te presence of outlers. Key-Words: Medaton analyss, outlers, bootstrap, studentzed resduals, ndrect effect 1 Introducton In order to mprove te understandng of te nature of relatonsp between an ndependent and dependent varables, a trd varable s often suggested to be ncluded n te model. Wen te trd varable s consdered as a medator, t s ypoteszed to be lnked n a causal can between te ndependent and dependent varables [1]. Te searc for ntermedate causal varables s called medaton analyss. Medaton analyss s common n socal scence researc, as tey elaborate upon oter relatonsps. In tese studes, partcpants are randomly assgned to receve dfferent expermental condtons, and dfferences n means n te condtons are eter consstent or nconsstent wt a medaton teory. For example, cogntve dssonance s a socal psycologcal teory wc explans tat persons make decsons at least n part to reduce nternal dscomfort or dssonance. Anoter example s related to programs to reduce drug use, wc often target medators as resstance sklls, ypoteszng tat a program tat ncreases resstance sklls wll decrease drug use [2]. Te bootstrap s now a wdely used metod ([3], [4], [5], [6], [7]). In te smplest form of bootstrappng, called te percentle bootstrap, upper and lower confdence lmts are obtaned by fndng te values of te statstcs n te 1000 samples tat correspond to te 2.5% and 97.5% percentles. Tere are several varatons of te bootstrap metod tat are useful n some stuatons. Anoter bootstrap metod, called te bas-corrected bootstrap, s mportant for medaton analyss because of ts accuracy for computng confdence ntervals for te medated effect wen te medated effect s nonzero [8]. Te metod conssts of adjustng eac bootstrap sample for potental bas n te estmate of te statstc. Te bas-corrected bootstrap metod removes bas tat arses because te true parameter value s not te medan of te dstrbuton of te bootstrap estmates. Te bas correcton s used to obtan a new upper and lower percentle used to adjust te confdence lmts n te bootstrap dstrbuton. ISSN: Issue 6, Volume 9, June 2010

2 Anwar Ftranto, Habsa Md Accordng to [9], a medaton analyss nvolves studyng te followng tree regresson equatons: 1) te regresson between te ndependent varable and te medator, 2) te regresson between te predctor and te outcome, and 3) te regresson between te medator and te outcome wen controllng for te predctor. If tese tree relatons are sgnfcant, te Sobel test, [10], s used to determne weter te predctor s effect on te outcome s sgnfcantly less wen te effect of te medator s taken nto account. Detaled regresson equaton needed to establs a medaton analyss s as follows: Y + cx + (1) 1 ε 1 M + ax + (2) 2 ε 2 Y + c X + bm + (3) 3 ε 3 were Y s te dependent varable, X s te ndependent varable, M s te medatng varable or medator. Ordnary least squares regresson metods (OLS) s te common metod to estmate c, a, b, and c n te medaton analyss, snce t contans a seres of regresson analyses. In te OLS, tere are several assumptons tat ave to be fulflled for te regresson model to be vald. Wen te regresson model does not meet te fundamental assumptons, te predcton and estmaton of te model may become based. Baron and Kenny s causal steps metod of examnng te sgnfcance of te medaton effect as been crtczed [11]. Frst, some researcers ave questoned te necessty of testng te overall assocaton n step 1 [12]. Second, MacKnnon n [13] ndcates tat Baron and Kenny s causal steps metod ave Type I error rates tat are too low and ave very low power, unless te sample sze s large. To overcome te lmtatons assocated wt Baron and Kenny s, [11] argue tat wt nonsymmetrc confdence ntervals for ( a b), bootstrap metods sould be used to assess medaton. Te development of bootstrappng metods n medaton analyss as receved lttle attenton especally wen outlers are present n te data. It s wort mentonng tat te classcal bootstrap metod s easly affected by outlers [14]. Ts motvates us to develop a new robust bootstrap metod n te medaton analyss. 2 General Bootstrap Procedures Resamplng metods ave become practcal wt te general avalablty of rapd computng and new software. As computers ave become faster and easer to use, tey ave made possble more powerful and accurate grapcal and numercal metods n statstcs. Bootstrap and resamplng metods are part of ts revoluton. Tey are feasble only wt te use of software, but tey are n many ways easy to use and tey generalze to stuatons were tradtonal metods based on te teory of te Normal dstrbuton cannot be appled. Bootstrappng metod becomes more popular as powerful tools used for statstcal nference nowadays. Snce ts ntroducton by Efron [15], te bootstrap as been te object of researc n statstcs. Bootstrappng uses a sample data to estmate relevant caracterstcs of te populaton. Te samplng dstrbuton of a statstc s ten constructed emprcally by resamplng from te sample. Te resamplng procedure s desgned to parallel te process by wc sample observatons were drawn from te populaton. For example, f te data represent an ndependent random sample of sze n (or a smple random sample of sze n from a muc larger populaton), ten eac bootstrap sample selects n observatons wt replacement from te orgnal sample. Obtanng te desred statstcs, t s needed to understand te general bootstrap procedure, wc s as follows: 1. Obtan a sngle sample of sze n from te populaton under study and calculate θˆ (te estmate of θ based on te sample). 2. Generate a bootstrap sample of te same sze n by samplng wt replacement from your orgnal sample. 3. Calculate ˆ θ, te value of te estmate based on te bootstrap resample. 4. Repeat steps 2 and 3 to obtan ˆ θ, ˆ θ 2,, ˆB θ. 2.1 Bootstrap bas and standard error Two typcal measures of accuracy for θˆ s te standard error (SE) and te bas of te estmator. It can be estmated by te bootstrap samples by 1 SE boot B b 1 ˆ θ ˆ b θ B 1 2, ISSN: Issue 6, Volume 9, June 2010

3 Anwar Ftranto, Habsa Md θb b were ˆ θ 1. B B ˆ Meanwle te bas s wrtten by Bas ( ˆ θ) E( ˆ θ) θ ˆ θ ˆ θ, were ˆ θ are bootstrap copes of θˆ, as defned earler. 3 Bootstrappng Regresson Models In regresson analyss, tere are two basc approaces to perform bootstrappng, and te coce of wc to use depends upon te regressors [16]. One s to bootstrap te vector of te response varables and te assocated predctor varable. Suc bootstrap procedure s commonly called as par bootstrap. It was proposed by Freedman [17]. Te oter one s to frst ft te model and bootstrap te resduals wc ave been proposed for te frst tme by Wu [18]. Bootstrappng te resduals requres tat te resduals be ndependent and dentcally dstrbuted. If te regressors are fxed, as n desgn experment, te bootstrap resamplng must preserve tat structure. Eac bootstrap sample sould ave te same regressors. Ten, f ts s te case, te approprate bootstrap s te resdual bootstrap. But f te data as random regressors, bootstrap samples sould posses ts addtonal varaton from te regressors, so tat par bootstrap s approprate. Assume tat we want to ft a regresson model wt response varable Y and predctors { X,, } 1 X p. Ten we can wrte down tat we ave a sample of n cases of te form z Y, X,, X ]. [ 1 1p In par bootstrap, we select n cases randomly from te orgnal z 1, z1,, z n observatons wt replacement. Ten te regresson estmator computed for eac of te resultng bootstrap samples, z b1, z b2,, z bn, producng B sets of bootstrap regresson coeffcents, β [ β, β,, β ]. Standard errors and b 0b 1b p b confdence ntervals for te regresson coeffcents can be constructed based on B (a gven large number) bootstrap samples. But, [19] ponted out tat par bootstrap s less senstve to model assumptons suc as normalty and constant varance tan bootstrappng resduals. Drectly resamplng te observatons z mplctly treats te regressors X 1,, X p as random rater tan fxed. We may want to treat te Xs as fxed (f, e.g., te data derve from an expermental desgn). Ts s accomplsed by constructng te dstrbuton F n tat places probablty 1 n at eac e. We ten generate bootstrap resduals e for 1,2,, n, were te e are obtaned by samplng ndependently from F n. Practcally, recent book by Fox [20] mentoned te steps of dong resdual bootstrap samples n regresson as follows: 1. Estmate te regresson coeffcents ˆ β ˆ β,, ˆ 0, 1 β p for te orgnal sample, and calculate te ftted value and resdual for eac observaton: Yˆ ˆ β ˆ β X e Y Yˆ. ˆ β X p Te e s ten scaled usng studentzed resdual, e r, σ 1 were s leverage value at pont. Many researcer also used raw resdual, e, nstead of studentzed resdual, r. 2. Select bootstrap samples of te resduals, e [ r, r,, r ], and from tese, calculate b 1b 2b nb bootstrapped Y values, y [ Y, Y,, Y ], ˆ b b were Y Y + e. p b 1 2 n b b b 3. Regress te bootstrapped Y values on te fxed X values to obtan bootstrap regresson coeffcents. If, for example, estmates are calculated by leastsquares regresson, ten b b ( X X) X y b for 1 b 1,2,, B. b 0 1 p b b b 4. Te resampled b [ β, β,, β ] can be used n te usual manner to construct bootstrap standard errors and confdence ntervals for te regresson coeffcents. Bootstrappng wt fxed X draws an analogy between te ftted value Yˆ n te sample and te ISSN: Issue 6, Volume 9, June 2010

4 Anwar Ftranto, Habsa Md condtonal expectaton of Y n te populaton, and between te resdual e n te sample and te error ε n te populaton. Altoug no assumpton s made about te sape of te error dstrbuton, te bootstrappng procedure, by constructng te Y b accordng to te lnear model, mplctly assumes tat te functonal form of te model s correct. 3.1 Bootstrap Estmates of te Indrect Effect n Medaton Analyss Te prncple of bootstrappng s fundamental to Srout and Bolger s metod [11] for testng te statstcal sgnfcance of an ndrect effect. In statstcs of medaton analyss, formulas for te pont estmator of te ndrect effect (medated effect) and ts standard error can be used to test te sgnfcance of a medated effect and construct confdence ntervals for te effect n te populaton based on data from te sngle sample. One assumpton of ts approac s tat te populaton dstrbuton underlyng bot te pont and standard error estmates of te medated effect s normal. Te medated effect s a product of a and b. In fact, te dstrbuton of te product of two random varables does not ave a normal dstrbuton n all stuatons, and nformaton regardng te dstrbuton of te product of two random varables as been sown to provde more accurate confdence lmts and statstcal tests [21]. Tradtonal statstcal nference s based on assumptons about te dstrbuton of te populaton from wc te sample s drawn. In standard multple regresson analyss, te estmates of te standard errors of parameter estmates are derved based on te assumpton of normalty. Bootstrappng uses computer ntensve resamplng to make nferences rater tan makng assumptons about te populaton. In oter words, bootstrappng treats a gven sample as te populaton. Bootstrappng can be used for any statstcs, but n ts study, we focus only on te medated effect (also see [22], for a treatment of bootstrappng a medated effect). Te followng steps of a par bootstrap procedure outlned by [11] wll be presented below. Parameter estmates are based on Eq. (2) and Eq. (3). 1. Usng te orgnal data set of n cases as a populaton reservor, create a bootstrap sample of n cases by random samplng wt replacement, 2. Calculate a, b, and ( b) a based on ts bootstrap sample, 3. Repeat Steps 1 and 2 a total of B tmes, 4. Examne te dstrbuton of te estmates. Te probablty dstrbuton constructed n step 4 becomes te bass for makng nferences. Tus, te standard devaton of ts dstrbuton s te bootstrapped standard error of te medated effect. Bootstrap estmates of te medated effect are stragtforward to obtan. Frst, te medated effect estmate s obtaned from te sample of data. In a bootstrap analyss of a sample of B 100, for example, a new sample of 100 s taken wt replacement from te orgnal sample of 100 and te medated effect s estmated. Ten a second sample of 100 s taken from te orgnal sample, and te medated effect s estmated. Te process s repeated a large number of tmes, usually at least Te medated effect estmated n eac bootstrap sample s used to form a dstrbuton of te bootstrap medated effect estmates. 4 Te Proposed Bootstrap Approac Due to ts strengt-proven n statstcs, bootstrap resdual approac sould be taken nto account n dong bootstrap metod n medaton analyss. Te approac s needed snce medaton analyss nvolves several lnear regresson equatons. MacKnnon n [23] developed a computer program of bootstrappng medaton analyss. He used a par bootstrap regresson n obtanng statstcs a, b and ( a b). In ts orgnal form, te pars bootstrap s mplemented by randomly resamplng te data drectly wt replacement. And as as been mentoned earler, [24] also ave done te same approaces n bootstrappng medated effects. Monte Carlo evdence suggests tat nference based on ts procedure s not always accurate [9]. Ts work as motvated us to perform resduals bootstrappng algortm based on te medaton model. Here we confne our study on te bootstrap estmate of te ndrect effects. In ts paper we propose a new bootstrap algortm for te ndrect effects of te smple medaton model. Te proposed algortm s based on te studentzed resduals. Resdual bootstrap s very common tecnque n many lteratures suc as [10], [11], and [12]. Dfferent resduals formulatons ave been proposed n te lteratures, ncludng te ordnary (raw) resduals [15], standardzed resduals [13] and studentzed resduals [14]. [14] as sown tat te bootstrap from studentzed resduals s asymptotcally vald wen te number of cases ncreases and te number of varables s fxed. [16] ISSN: Issue 6, Volume 9, June 2010

5 Anwar Ftranto, Habsa Md proposed a bootstrap approac n least square regresson to construct crtcal ponts for te studentzed resduals by employng a jackknfeafter-bootstrap tecnque. Resdual vector, εˆ can be expressed n te form of at matrx H or W as follows: εˆ Y - Yˆ Y - Xβˆ Y X(X X) (I - W) Y (I - H)Y. X Y T -1 Ten te varance of te t case resdual ê can be wrtten as: Var ( eˆ ) Var{ ( 1 ) y } 2 ( 1 ) σ, were s te t leverage, 1,, n. It means tat te varance of resduals vares nversely wt leverage. Te leverage depends on x, and so f any leverages are partcularly g, resamplng te resduals wll not do a good job of estmatng te dstrbuton of e. In te resdual bootstrap resample of lnear regresson models, [14] proposed usng studentzed resdual, r, as a functon of t leverage as follows: eˆ r ˆ σ 1. (4) Equaton (4) means tat resduals can be standardzed to correct for unequal varance caused by unequal leverages by dvdng te resdual by an estmate of ts standard devaton. [17] ponted out tat te studentzed resdual can also be used to dentfy outlers. In order to mprove te performance of te studentzed resduals, we propose to rescale te well known studentzed resduals n Eq. (4). Te rescaled studentzed resduals s as follows: ( ) r rest r eˆ ˆ σ 1 ( ). (5) We call ts metod as Rescaled Studentzed Resdual Bootstrap based on Least Squares (ReSRB). As already known, leverage value can be used for dentfyng outlers wt respect to ter x values, wc ndcates ow muc a certan pont s far apart from ts data cloud. It measures te wegt of te t observaton n terms of ts mportance n te model s ft. Te value of wll always be n between 0 and 1, and, tecncally, represents te n n at matrx, H, were, dagonal porton of a ( ) 1 ( ) H X X X X. (6) n 1 Note tat 0 1 and k, were k te number of parameters n a lnear regresson model. If te leverage value s large, te t observaton s outlyng wt respect to ts x values. Furtermore, for correctng te unequal varances caused by unequal leverages, [14] suggested usng te standardzed resduals, r, not te usual resduals. Ts works ave motvated us to proposed te new bootstrap algortm n te medaton model by ncorporatng a new mproved verson of studentzed resduals n te bootstrappng te resduals algortm. 4.1 Te proposed bootstrap algortm We wll ncorporate te proposed ReSRB n our proposed bootstrap algortm for estmatng te ndrect effects of a medaton model. As already mentoned, to te best of our knowledge, ts s te frst bootstrap algortm ntroduced n te medaton model. Te proposed bootstrap algortm s developed by resamplng te resduals from te orgnal data nstead of employng te random resamplng of te data as was done by [24], [21], and [23]. Accordng to [26], nferences based on bootstrappng by random resamplng are not always accurate. As dscussed n prevous secton, a smple medaton analyss nvolves tree regresson equatons as n Eq. (1), Eq. (2), and Eq. (3). However, n order to obtan a bootstrap medaton effects, we need to know te value of α and β wc can be obtaned usng Eq. (2) and Eq. (3), respectvely. Te proposed approac, wc s desgned for smple medaton analyss wt te presence of sngle or multple g leverage ponts, s as follows: 1. Estmate te regresson coeffcents a and b, respectvely from Eq. (2) and Eq. (3) for te screened orgnal sample usng least squares regresson, and calculate te ftted values and resduals for eac observaton: For Equaton (2), t wll be: Mˆ e ˆ2 ˆ ˆ M Mˆ. 2 + ax ISSN: Issue 6, Volume 9, June 2010

6 Anwar Ftranto, Habsa Md Ten for rescalng te resdual eˆ 2, nstead of eˆ 2 usng studentzed resdual rˆ 2,we ˆ σ 1 gve wegt to te studentzed resdual usng nverse of ts leverage value. In oter words, te studentzed resdual s magnfed by ts leverage value,. As a result, te modfed scaled resdual become eˆ 2 rˆ 2, ˆ σ 1 MX were σˆ s te estmated standard devaton of MX resdual of te M 2 + ax + e2 equaton. And for Equaton (3), t wll be: Yˆ ˆ ˆ e Y Mˆ. ˆ3 ˆ X 3 + bm + c Smlarly, ts scaled resdual for ts equaton becomes r 3 ˆ σ YMX e ˆ3 1, were σˆ YMX s te estmated standard devaton of resdual of te Y ˆ 3 + bm + cˆ X + e3 equaton. 2. Select bootstrap samples of te resduals from screened orgnal sample, 1 2 n 2 B [ r 2B1, r2 B 2,, r2 B ] n e 3 B [ r 3B, r3 B,, r3 B ], e and from tese, calculate bootstrapped M and Y values, B B 1 B 2 B n B [ Y B1, YB 2,, YB ] n m [ M, M,, M ], y, were ˆ B M 3B M + e and Y Y + e. ˆ B 2B 3. Regress te bootstrapped M and Y values on te fxed X and M, X values, respectvely to obtan bootstrap regresson coeffcents needed n medaton analyss. 4. Repeat step 2 and 3 for B tmes to obtan B bootstrap replcatons. 4.2 Smulaton desgn Te performance of te new metod of Rescaled Studentzed Resdual Bootstrap (ReSRB) s examned on smulated data to furter valdaton. A seres of smulaton was conducted. We smulated a clean data (0% outlers) szed n wt desgn structure as follows. We X ~ 0,20, M f( X) + e 1. 75X + e, generated ( ) and Y f( X M) + e 0.1X M + e errors n te model were generated from ~ N( 0,1),. Te e. Ten te clean data were contamnated wt of 5%, 10%, 15%, 20%, 25% and 30% percentage outlers n te X, M and Y drectons. To generate te outlyng observatons for eac of te tree varables, we delete te respectve observatons and replace tem wt ter orgnal values plus a fxed sft of 10 σ dstance. About 2000 bootstrap samples were drawn from a sample sze of 25, 50, 100, and 200 to represent small, medum and large sample respectvely. In ts smulaton, we consder sample sze of n 25, n 50, n 100, and n 200. Te results of te smulaton studes are sown n Table 1 tll Table 3. We expect by dong ts knd of smulated data, te performance of te proposed ReSRB metod can be legtmated. 5 Smulaton Results Table 1 Table 3 dsplay te bas and te root mean square error (RMSE) of te ndrect effect, a b, of te proposed bootstrap metod, ReSRB and te oter tree metods at varous percentage of outlers (PO) and sample sze (n). We compared te proposed metod, namely ReSRB wt dfferent metods of bootstrap. Tose metods are RRB (Raw Resdual Bootstrap), SRB (Studentzed Resdual Bootstrap), and JRB (Jackknfed Resdual Bootstrap). Our man focus s to evaluate te performances of te proposed metod, namely te ReSRB, wc s based on te bas and RMSE. We expect tat te ReSRB can reduce bas and RMSE more tan te oter exstng metods. A statstc used to estmate a populaton parameter s unbased f te samplng dstrbuton of te statstc s centered at te value of te populaton parameter; tat s, te mean or average value of te statstc equals te true value of te parameter. An unbased statstc gves a correct estmate, on average. We knew tat te mean square error (MSE) s consdered as effcency measurement. A good estmator s te one tat ave desrable propertes wc are unbasedness and effcency. From te Table 1, wc s for contamnaton n X drecton, we see tat after B 2000 bootstrap smulaton, te bas and RMSE of eac combnaton ISSN: Issue 6, Volume 9, June 2010

7 Anwar Ftranto, Habsa Md of percentage of outlers (PO) and sample sze (n) produced by te ReSRB metod are smaller tan te RRB, SRB, and JRB. Te beavor occurs on bot clean data and contamnated data. But, te results are a lttle bt dfferent on te contamnaton n M drecton. In addton, for te contamnated data, te better performance of our proposed metod wt regard of bas and RMSE does not only appen for small or medum percentage of outlers, but also at large percentage of outlers. Among te metods we ave studed, we can say tat a bad performance s sown by RRB wc as te gest bas and RMSE. Table 1 Average Bas and RMSE for te Indrect Effect, of te Smulated Data wt Outlers n X PO (%) 0% 5% 10% 15% 20% 25% 30% Bootstrap Metods n RRB SRB JRB ReSRB Bas RMSE Bas RMSE Bas RMSE Bas RMSE Table 2 Average Bas and RMSE for te Indrect Effect, of te Smulated Data wt Outlers n M PO (%) 0% Bootstrap Metods n RRB SRB JRB ReSRB Bas RMSE Bas RMSE Bas RMSE Bas RMSE ISSN: Issue 6, Volume 9, June 2010

8 Anwar Ftranto, Habsa Md 5% 10% 15% 20% 25% 30% Table 3 Average Bas and RMSE for te Indrect Effect, of te Smulated Data wt Outlers n Y PO (%) 0% 5% 10% 15% 20% Bootstrap Metods n RRBLS SRBLS JRBLS ReSRBLS Bas RMSE Bas RMSE Bas RMSE Bas RMSE ISSN: Issue 6, Volume 9, June 2010

9 Anwar Ftranto, Habsa Md 25% 30% Te advantages of te ReSRB over oter metods are even more apparent n data sets wt medum or large percentage of outlers. In summary, te results from te experments ndcate tat te ReSRB procedure performs well n most of te gven stuatons. 6 Concluson Resamplng approaces suc as bootstrap for testng medaton effects old consderable promse for medaton analyss. Tere s evdence tat te resamplng metods generally ave more accurate Type I error rates and more statstcal power tan sngle sample metods tat assume a normal dstrbuton for te medated effect [18]. Our proposed ReSRB metod as provdes an alternatve to te bootstrap for estmatng te medated effects n medaton models. It s manly based on resdual bootstrap. Emprcal result and te smulaton study ave sown tat te ReSRB s better tan te oter metods. Te rescalng term of to te studentzed resduals ntroduced n te ReSRB procedure elp to mprove te performance of te bootstrapped estmates. Our concluson s based on ts performances wt regard to bas and root of mean square error (RMSE). Te advantages of te ReSRB over oter metods are even more apparent n data sets wt medum or large percentage of outlers. In summary, te results from te smulaton experments ndcate tat te ReSRB metod performs well n most of te gven stuatons. References: [1] James, L. R., & Brett, J. M.. Medators, moderators, and tests for medaton. Journal of Appled Psycology, 69, 1984, pp [2] MacKnnon, D. P., & Dwyer, J. H. Estmatng medated effects n preventon studes. Evaluaton Revew, 17(2), 1993, [3] Efron, B. Te bootstrap and modern statstcs. Journal of te Amercan Statstcal Assocaton, 95, 2000, [4] Bebler, K.E., Mcael W., and Kerstn W.. Bootstrap evaluaton of mxed data type dscrmnant analyss. WSEAS Transactons on Systems. 3(8), 2004, pp [5] Zaarm, A. Ibram M.A.A., and Mod. S.Y. An Evaluaton of Test Statstcs for Detectng Level Cange n BL(1,1,1,1) Models. WSEAS TRANSACTIONS on Matematcs. 2(7), 2008, pp [6] Norazan, M.R, Habsa, M., and Imon, A.H.M.R. Estmatng Regresson Coeffcents usng Wegted Bootstrap wt Probablty. WSEAS Transacton on Matematcs, 7, 2009, pp [7] Hyrena, O., Margot M. P., Mark N., and Andre Y. Te Statstcal Analyss of Longtudnal Clonal Data on Olgodendrocyte Generaton. Proceedng of te Proceedngs of te 2006 WSEAS Internatonal Conference on Matematcal Bology and Ecology, Mam, Florda, USA, January 18-20, 2006, pp [8] Efron, B. Better bootstrap confdence ntervals. Journal of te Amercan Statstcal Assocaton, 82, 1987, [9] Baron, R. M., and Kenny, D. A. Te moderatormedator varable dstncton n socal psycologcal researc: Conceptual, strategc, and statstcal consderatons. Journal of Personalty and Socal Psycology, 51 (6), [10] Sobel, M. E. Asymptotc confdence ntervals for ndrect effects n structural equaton models. Socologcal Metodology, 13, 1982, [11] Srout, P. E., and Bolger, N. Medaton n expermental and nonexpermental studes: Newprocedures and recommendatons. Psycologcal Metods, 7, 2002, pp [12] MacKnnon, D. P., Krull, J. L., and Lockwood, C. M. Equvalence of te medaton, ISSN: Issue 6, Volume 9, June 2010

10 Anwar Ftranto, Habsa Md confoundng, and suppresson effect. Preventon Scence, 1, 2000, [13] MacKnnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., and Seets, V. A comparson of metods to test medaton and oter ntervenng varable effects. Psycologcal Metods, 7, 2002, [14] Imon, A. H. M. R. and Al, M. M. Bootstrappng Regresson Resduals. Journal of te Korean Data & Informaton Scence Socety, Vol. 16, No. 3, 2005, pp [15] Efron, B. Bootstrap metods: anoter look at te jacknfe". Te Annals of Statstcs 7, 1979, pp [16] Cernck, M. R. Bootstrap Metods: A Gude for Practtoners and Researcers. Second Ed. Jon Wley & Sons, Inc. Newtown, PA [17] Freedman, D. A. Bootstrappng regresson models. Annals of Statstcs, 9, 1981, [18] Wu, C. F. J. Jackknfe, Bootstrap and Oter Resamplng Metods n Regresson Analyss. Te Annals of Statstcs, Vol. 14, No. 4, 1986, pp [19] Efron, B. & Tbsran, R. An Introducton to te Bootstrap. London: Capman & Hall [20] Fox, Jon Appled Regresson Analyss and Generalzed Lnear Models. Second Ed. Sage Publcatons, Ontaro, Canada [21] MacKnnon, D. P., Lockwood C. M., & Wllams, J. Confdence lmts for te ndrect effect: Dstrbuton of te product and resamplng metods. Multvarate Beavoral Researc, 39, 2004, [22] Bollen, K. A., & Stne, R. Drect and ndrect effects: Classcal and bootstrap estmates of varablty. Socologcal Metodology, 20, 1990, [23] MacKnnon, D. P. Introducton to Statstcal Medaton Analyss. New York: Taylor & Francs [24] Mooney, C. Z. and Duval, R. D. Bootstrappng: A non-parametrc approac to statstcal nference. Newbury Park, CA: Sage [25] Horowtz, J. L. Te bootstrap". In Handbook of Econometrcs, Vol. 5. J. J. Heckman and E. E. Leamer (eds), Elsever Scence [26] Horowtz, J. L. Te bootstrap. In Handbook of Econometrcs, Vol. 5. J. J. Heckman and E. E. Leamer (eds), Elsever Scence ISSN: Issue 6, Volume 9, June 2010