Research Design - - Topic 16 Multiple Regression: Applications 2010 R.C. Gardner, Ph.D. General Overview. General Overview.

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1 Reserch Design - - Topic 6 ultiple Regression: Applictions 00 R.C. Grdner, Ph.D. Generl Overview Curve Fitting edition nlsis Generl Overview As we hve seen multiple regression is ver powerful nd useful dt nltic procedure. It is the bsis of RC nlsis s pplied to stndrd nlsis of vrince designs s well s those tht include continuous fctors. Erlier, it ws mentioned tht is ws the bsis of pth nlsis. This section describes three further pplictions:. Curve Fitting (polnomil regression). edition nlsis. odertion nlsis odertion Anlsis Curve Fitting The investigtion of non-liner functions using multiple regression ws introduced b Cohen (978). It permits reserchers to determine the nture of the functionl reltionship between two vribles, Y nd X. One procedure on SPSS tht performs this function is the Regression Curve Estimtion progrm. As indicted in the window in slide 5, there re number of options. We will focus on the one using orthogonl power functions. Below, re emples of three tpes of power functions. A combintion of positive liner nd positive qudrtic function indicted b significnt positive liner trend in model nd significnt positive X ** trend in model. Y X Liner Qudrtic Cubic A combintion of negtive liner nd negtive cubic component indicted b significnt negtive liner coefficient in model nd significnt negtive coefficient for X ** in model. Functions with combintions of components re indicted b significnt components for vrious models. Two emples re shown on slide 4. 4 X Y

2 The Window for the Curve Estimtion progrm is below. To test for liner, qudrtic, nd cubic components, check the three models indicted, nd lso check displ ANOVA tble. You should not select n of the other options with these three; the other options del with other tpes of functions. Dt Set for the Curve fitting emples 5 6 Selecting the three models nd the ANOVA tble option ields informtion bout the F-rtios (not shown here) for the three models nd the tests of the components. Liner component. The coefficient for X is not significnt, indicting tht there is no liner component The Curve Estimtion progrm plots the three models, even though of course, there is no evidence of liner or cubic trend here. Note tht the qudrtic component in these dt describes n inverted U shpe, which corresponds to the negtive sign ssocited with the regression coefficient for the qudrtic component in the previous tble. Note too tht the tests of significnce for the other components (i.e., X in the qudrtic tble, nd X nd X ** in the Cubic tble) re not tests of liner or qudrtic components. The re lso tests of qudrtic nd cubic components respectivel. Qudrtic component. X ** is significnt, indicting significnt qudrtic component. ** Cubic component. X ** is not significnt indicting tht there is no cubic component. ** **

3 Xsq-Xsqpre Curve Fitting: How it Works The Curve Estimtion progrm is simple ppliction of ultiple Regression. To see how it works, we compute two more vribles, Xsq nd Xcu (X-squred nd X-cubed), nd perform hierrchicl multiple regression nlsis s described in the Snt file below. Unlike the Curve Estimtion progrm, however, we could dd higher order functions if desired (cf., Cohen, 978). REGRESSION /ISSING LISTWISE /STATISTICS COEFF OUTS R ANOVA /CRITERIA=PIN(.05) POUT(.0) /NOORIGIN /DEPENDENT /ETHOD=ENTER /ETHOD=ENTER sq /ETHOD=ENTER cu. This would ield full set of results. I present onl the tests Note these results re identicl to those from the Curve Estimtion progrm. shows tht there is no evidence of liner reltionship (b = -.99, ns). shows significnt negtive qudrtic component (b = -.577, p<.0004). shows tht there is no evidence of cubic component (b = -.05, ns). sq sq cu of the regression coefficients Dependent Vrible: Rtionle: The residul of Xsq 5.00 from the vlue of Xsc predicted from X is determined b 0.00 clculting the regression of Xsq 5.00 on X. For these dt, tht eqution is: 0.00 Xsqpre = *X. The residuls re Xsq Xsqpre The plot is s shown Thus, if the regression coefficient for Y ginst sq in model is positive, it would indicte this tpe of function. If the regression coefficient is negtive, it indictes n inverted U tpe of function becuse of the negtive semiprtil correltion of Y with Xsq. Similrl, we could investigte the function for Xcu. The regression for Xcupre = b 0 + b *X + b *Xsq, nd plot of the residuls would produce curve like tht in Slide. edition Anlsis edition Anlsis is concerned with testing the hpothesis tht the reltionship between two vribles, nd, is due to the influence of third vrible,. There re more comple forms of medition nd ou re referred to webpge t for further informtion on the topic. Bron & Kenn (986) define meditor vrible s one tht ccounts for (much of) the reltion between n independent vrible nd dependent vrible. It ssumes cusl model where:. The independent vrible () cuses the dependent vrible (). The independent vrible () cuses the meditor (). The meditor () cuses the dependent vrible () Consider the digrm C = eqution C = eqution b

4 Bron & Kenn propose three step procedure.. Regress the on the b + b where b = c = 0. Regress the editor on the b + b where b = = 0 c Tests of Significnce for edition If c is significnt, one cn still test the difference c c to see if it is significntl smller thn c. It cn be shown tht c-c = *b from the previous nottion. Therefore, test of c-c is test of the null hpothesis tht *b = 0. There re three forms of Z test tht hve been suggested, nd the test cn be performed on the website shown on slide. If the Z test is significnt, one concludes tht c is significntl smller, nd tht medition hs been demonstrted. There re minor differences in the tests, nd generll the Aroin test is the one most recommended. The Sobel Test. Regress the on both the editor nd the b S + S b b = b0 + b + b where b = c' * b b = b The Aroin Test (Goodmn I) Z = b S + Sb + S Sb For edition:. b c = must be significnt. b = c must be significnt * b. b = b must be significnt The Goodmn II Test Z = nd b = c must not be significnt or t lest significntl smller thn c. b S 4 + Sb S Sb Z = * b A Numericl Emple of edition Given the dt for vribles,, nd, we could test the medition model. The correltion mtri for the three vribles is: Person Correltion Sig. (-tiled) N Correltions Performing the three multiple regression runs produces the regression informtion on slide Step. Regression of on Step. Regression of on Step. Regression of on nd c. Dependent Vrible:. Dependent Vrible: b. Dependent Vrible: c Note c is significnt, is significnt, b is significnt, nd c is not significnt. Also note tht c-c = =.45 nd tht *b = (.56)(.655) =.45, s stted erlier. 6 4

5 Appling the Tests from the Webpge shown on Slide Inputting the vlues of nd b, nd their stndrd errors from the computer output ields the following vlues. In ech cse, the Z vlue is significnt, so we cn conclude tht is cting s meditor. Tht is, it is meningful to conclude tht the reltion between the nd the is due to the mediting effects of. 7 odertion Anlsis odertion is snonmous with interction. A modertor vrible is vrible tht ffects the direction nd/or strength of the reltion between predictor vrible nd criterion. Thus, modertor is third vrible tht chnges the reltion between the criterion nd predictor. Using nlsis of vrince terminolog vrible is modertor if it intercts with the other vrible to effect vlues on the criterion (cf., Cohen, 978). Given three vribles X = the criterion X = predictor () X = predictor () We cn compute fourth vrible, XX s the product of X nd X. We cn then compute two multiple regressions: R.,, nd R., Then if R.,, R., is significnt, vrible X modertes the reltion between vribles X nd X, nd vrible X modertes the reltion between vribles X nd X. 8 This cn be shown s pth digrm r, r r, X X XX b c Criterion If c is significnt, vrible is sid to be modertor. It is not necessr for n of the other coefficients to be significnt. In fct, Cohen notes tht in the presence of product term, the regression coefficients for the non-product terms ( nd b) re rbitrr nonsense (i.e., the hve no interprettive vlue). It is desirble tht the modertor be uncorrelted with the predictor, though this simpl mkes the interprettion of the interction clerer. As r increses, the interction gets less cler. 9 A Numericl Emple of odertion Z X Y XY Input Dt Z = Criterion X = Predictor Y = predictor XY = product term To test for modertion, compute the multiple regression of Z on X,Y, nd XY

6 Following is the tble of regression coefficients for this run. R =.79 X Y XY. Dependent Vrible: Z Viewing the interction It is common prctice to view the interction b solving the regression eqution for high nd low vlues of X nd Y, nd then plotting the results. Two plots re shown. Given men = 5., s.d. =.04, nd men = 4.08, s.d. =.5, low nd high vlues could be the mens ± sd, or lterntivel, the highest nd lowest vlues: Low High Low High X Y Z =.57.09* X 4.79* Y +.6* XY These coefficients indicte tht there is significnt interction between X nd Y in the prediction of Z, becuse the regression coefficient for XY =.6 is significnt t the.0004 level. The regression coefficients for X nd Y hve no prticulr mening. In fct, Cohen (978, p. 86) sttes the re rbitrr nonsense. Obviousl, the interprettion cn be slightl different, depending on wht is chosen s high or low, nd, of course, mn such figures could be plotted. In fct, the complete plot considering X nd Y s continuous vlues is: Centering It is often recommended tht the nlsis be done fter X nd Y hve been centered b subtrcting the men from ech vlue, nd forming the product term from the centered vlues. This chnges the vlues of the regression coefficients, but the test of significnce of the interction is identicl to wht it ws with the uncentered dt, s demonstrted in the following tble of regression coefficients. As before, the multiple correltion is.79. c c cc. Dependent Vrible: z Thus, s fr s the interction is concerned, it doesn t mtter if X nd Y re centred. Generll, however, the re. 4 6

7 Issues As indicted erlier, the regression coefficients for the continuous vribles hve no interprettive vlue becuse the re residulized from the product term s well s ech other. If one is interested in these vlues, one must use hierrchicl procedure, nd the nture of the hierrch will influence the interprettion. There re three pproches:. Enter X followed b Y. Enter Y followed b X. Enter X nd Y in the first step. The interprettion differs with the pproch used. When, one vrible is entered first, significnt regression coefficient for tht vrible indictes tht there is direct reltionship between tht vrible nd the criterion. A significnt regression coefficient for the second vrible on step indictes tht the second vrible dds significntl to the prediction ccounted for b the first vrible on tht step. If both vribles re dded on the first step, the regression coefficients reflect the etent to which ech vrible contributes Adding X first Results obtined if both X nd Y were dded in one step Adding Y first reltive to the other Dependent Vrible: z. Dependent Vrible: z Ecept for the Constnts, the sme vlues would be obtined in steps nd if centred vlues were used. Clculting Simple Slopes A simple slope is the slope of the criterion ginst one of the predictors for given vlues of the other, for emple the slope of Z ginst X for different vlues of Y. The generl eqution is: Z = b + b X + b Y b X * Y 0 + which cn be reordered to epress Z s function of X s follows: Z = b + b Y ) + ( b b Y )* X ( 0 + Intercept Thus: Z = ( * Y ) + ( * Y ) X Slope For Low Y =.55 intercept = slope = -.78 For High Y = 5.6 intercept = slope =.70 7 Tests of Significnce for the Simple Slopes These tests mke use of estimtes of the stndrd error of the slope b clculting the stndrd devition of the ggregte (b+by in this emple) using covrinces of the regression coefficients tht cn be obtined from the multiple regression nlsis. slope Given t = S.E. S. E. = SE + SEbY ² + cov Correltions Covrinces b b, b Coefficient Correltions. Dependent Vrible: z Y

8 Computing Stndrd Errors for Low nd High Vlues of Y For Low Y =.55 SE.. = (. 005)(. 55²) + (. 07)(. 55) =. 04 =. 55 For high Y = 5.6 SE.. = (. 005)( 56²). + (. 07)( 56. ) =. 044 =. 0 Note: This emple mkes use of the non-centered dt. Similr computtions could be performed using the centered dt. The slopes nd the test of significnce would be the sme. 9 t-tests of the significnce of ech of the slopes df = N-p-=5--= For Low Y For High Y t =.78 = t = =.4p <.0.0 Thus, for low vlue of Y (i.e., stndrd devition below the men), there is slight negtive (but not significnt) slope of Z on X, while for high vlue of Y there is significnt positive slope. Of course, such tests could be performed on n vlues of Y (or X). It should be noted tht lthough Cohen, Cohen, West, & Aiken (00) believe such tests re meningful, the stte There eists no test of significnce of difference between simple slopes computed t single vlues (points) long continuum (p.80). 0 A Cutionr Note cclellnd nd Judd (99) demonstrted tht tests of interctions in field studies often hve less thn 0% of the efficienc of controlled eperiments, nd the discuss problems ssocited with tests of modertor effects s well s curve fitting. The conclude (pp ): Our nlsis of the reltive superiorit of eperimentl designs for detecting interctions implies tht unless reserchers cn select, oversmple, or control the levels of the predictor vribles, detection of relible interctions or qudrtic effects eplining n pprecible proportion of the vrition of the dependent vrible will be difficult. This does not men tht reserchers should not seek interctions in such conditions; however, the should be wre tht the odds re ginst them. References Bron, R.. & Kenn, D.A. (986). The modertor-meditor vrible distinction in socil pschologicl reserch: conceptul, strtegic nd sttisticl considertions. Journl of Personlit nd Socil Pscholog, 5, 7-8. Cohen, J. (978). Prtiled products re interctions, prtiled powers re curve components. Pschologicl Bulletin, 85, Cohen, J., Cohen, P., West, S. G. & Aiken, L.S. (00). Applied multiple regression/correltion nlsis for the behviorl sciences (third edition). hwh, NJ: Lwrence Erlbum. cclellnd, G. H. & Judd, C.. (99). Sttisticl difficulties of detecting interctions nd modertor effects. Pschologicl Bulletin, 4,

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