# Chapter 5a: 1. Multiple regression uses the ordinary least squares solution (as does bi-variate regression).

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1 Chpter 5: Multiple Regression Anlysis Multiple Regression is sttisticl technique for estimting the reltionship etween dependent vrile nd two or more independent (or predictor) vriles. How does multiple regression work (consider how ivrite regression works)? Tht is, wht does it do? 1. Multiple regression uses the ordinry lest squres solution (s does i-vrite regression). Tht is, it descries line for which the (sum of squred) differences etween the predicted nd the ctul vlues of the dependent vrile re t minimum. Or, in still more technicl words, the regression model cn e thought of s representing the function tht minimizes the sum of the squred errors. pred = + 1 X 1 + B 2 X 2 + B n X n 2. Multiple regression produces model tht identifies the est weighted comintion of independent vriles to predict the dependent (or criterion) vrile. pred = + 1 X 1 + B 2 X 2 + B n X n

2 How might specifiction error come into ply here? If there re vriles left out of the eqution tht hve sustntil ffect on the dependent vrile, then the weights ( nd et) ssigned to the independent vriles tht re included will e sustntilly ffected. It is ecuse of specifiction error tht we wnt to review the literture nd exmine existing theories BEFORE creting our regression model. In sum, multiple regression llows us to: --estimte the reltive importnce of severl hypothesized predictors nd --ssess the contriution of the comined vriles to chnge the dependent vrile. Indices re often constructed so tht group of ordinl vriles cn e treted s single Intervl-Rtio ti vrile (SPSS uses the compute t to dd vriles together to crete n index). Cronch s lph is used to ssess whether the vriles to e dded re mesuring the sme concept. It mesures the internl reliility of the items nd idelly is.70 or higher. The lower the lph score the more likely tht the vriles re not mesuring the sme concept nd so should not e dded together (nlyze, scle, reliility nlysis).

3 The Regression Eqution pred = + 1 X 1 + B 2 X 2 + B n X n pred = dependent vrile or the vrile to e predicted. X = the independent or predictor vriles = rw score equtions include constnt or intercept = weights; or prtil regression coefficients. They show the reltive contriution of their independent vrile on the dependent vrile when controlling for the effects of the other predictors Using the Regression Eqution to predict success of pplicnts to grdute school If we rn regression eqution with the dependent vrile eing the GPA of ll current grdute students nd the predictor vriles eing undergrdute GPA score nd GRE score, wht would the resulting regression nlysis tell us? Suppose the resulting regression eqution (explining college GPA) ws: pred =.22 + (.5)(HS-GPA) + (.001)(GRE) Wht would this tell us? Suppose we wnt to predict the success score of one pplicnt, Erin, sed on her HS-GPA of 3.80 nd her GRE score of How would we do this? HS-GPA of 3.80 nd GRE score of Regression Eqution Predicting College GPA C-GPA pred =.22 + (.5)(HS-GPA) + (.001)(GRE) The vrite in multiple regression The vrite is the comintion of vriles on the right side of the regression eqution pred = + 1 X 1 + B 2 X 2 + B n X n Erin s Regression Eqution C-GPA pred =.22 + (.5)(3.80) + (.001)(1350) C-GPA pred = C-GPA = 3.47 (Wht does this tell us?) The vrite is the predicted score (not the ctul) The vrite is sometimes viewed s representing ltent vrile

4 How do you decide which vriles to include in the vrite? One pproch is to exmine pst literture nd theory nd from this to develop theoreticl vrite. This is sometimes referred to s the stndrd (simultneous) regression method A second pproch is to exmine sttistics tht show the effects of ech vrile oth within nd out of the eqution. The sttisticl vrite is uilt sed on those vriles showing the most effect. These re sometimes clled Forwrd nd Bckwrd Stepwise Regression Lets work through n exmple of the stndrd multiple regression method. Lets suppose we hve decided tht we wnt to study jo commitment nd e le to predict whether n employee is going to e committed to her/his jo. Wht re our first steps? Lets suppose we exmine the literture, nd more specificlly the theory, explining jo commitment. We find: Commitment = empowerment + jo stisfction. + mngement support Lets further suppose we hve set of dt for nursing home employees tht uses Likert Scle (i.e., sttements tht llow for strongly gree to strongly disgree). Wht re our next steps? Commitment = empowerment + jo stisfction. + mngement support Do reliility test for ech set elow nd then, if the lph scores re cceptle, crete index vriles Commitment = 07, 37, 61r Empowerment = 27, 54, 76, 86, 90 Jo Stisfction (lredy creted) Mngement Support = 19, 43, 82 Wht s next?

5 Commitment = empowerment + jo stisfction. + mngement support Perform the regression nlysis nd interpret the dt (nlyze, regression, liner, identify dependent nd independent vriles). Brek Wht do ech of the following tell us? R 2, djusted R 2, constnt, coefficient, et, F-test, t-test It is interesting to note tht for multiple regression R 2 (the coefficient of multiple determintion) is used rther thn r (Person s correltion coefficient) to ssess the strength of this more complex reltionship (s compred to ivrite correltion) The djusted R 2 djusts for the infltion in R 2 cused y the numer of vriles in the eqution. As the smple size increses ove 20 cses per vrile, djustment is less needed (nd vice vers). Interprettion of the coefficients A multiple regression coefficient (or coefficient) mesures the mount of increse or decrese in the dependent vrile for one-unit difference in the independent vrile, controlling for the other independent vrile(s) in the eqution. Interprettion of the coefficients Idelly, the independent vriles re uncorrelted. Consequently, controlling for one of them will not ffect the reltionship etween the other independent vrile nd the dependent vrile. (X 1 nd X 2 re uncorrelted) X 1 X 2

6 X 1 c d X 2 In sum, if the two independent vriles re uncorrelted, we cn uniquely prtition the mount of vrince in due to X 1 nd X 2 nd is is voided. Smll intercorreltions etween the independent vriles will not gretly ised the coefficients. Demonstrtion of two independent vriles tht re somewht correlted with ech other nd with dependent vrile. However, lrge intercorreltions will ised the coefficients nd for this reson other mthemticl procedures re needed (we will e covering interction ffects nd multicollinerity in more depth) Interpreting the Stndrd Regression Coefficients All Bet coefficients re in z-score form nd thus cn e compred with ech other. Tht is, since the independent vriles re now in the sme metric, we cn determine their reltive ility to predict the dependent vrile. Consequently, the independent vrile with the lrgest et weight cn e viewed s hving the lrgest impct on the dependent vrile. However, one must e cutious to suggest tht one vrile hs twice the effect of nother due to vrious prolems such s shred correltion. Bets cnnot e compred or generlized cross different smples. This is ecuse ets re sensitive to correltions with other predictors. Commitment = empowerment + jo stisfction. + mngement support Solution: consider using other sttistics in ddition to the ets such s structure coefficients nd squred semiprtil correltions (these will e covered in clss) How much of the vrince in jo commitment is ccounted for?

7 Clculting stndrd regression eqution: Brek While we my put ll three vriles into the regression eqution t the sme time, the regression nlysis considers ech seprtely. More specificlly, ech vrile is entered into the regression eqution fter the others hve lredy een entered so tht the unique (dditionl) contriution of the vrile cn e clculted. Here is pictoril of regression eqution showing how X 1 nd X 2 ccount for some of the vrince of ( nd ). The unexplined vrince is lso identified. If we include X 3 into the regression eqution, the nlysis will determine wht remining portion of s vrince (the unexplined or residul vrince) cn e explined. This would e c. X 1 unexplined vrince X 1 unexplined vrince c X 3 X 2 X 2 Thus, in evluting the contriution of X 3, the predictor currently under considertion, it is the residul vrince of tht X 3 must trget fter sttisticlly removing (i.e., controlling for, prtiling out) the effects of X 1 nd X 2. This effect of single predictor fter controlling for others is referred to s prtil correltion (for X 3 this would e c + d). However, the residul uniquely explined nd unshred y ny other predictors is referred to s the squred semiprtil correltion (for X 3 this would e c squred) X 1 unexplined vrince c X 3 X 1 unexplined vrince c d X 3 X 2 X 2

8 The semiprtil correltion is reported in SPSS output s the prt correltion nd must e squred to get the squred semiprtil correltion. The SSC scores do not dd up to equl R 2 ecuse they do not include the shred vrince (overlp or correltion) etween the independent vriles. It is lso interesting to note tht, in this pictoril tht shows how X 1, X 2, nd X 3 ccount for some of the vrince of,,, c nd d, dded together, represent the R 2. The explined or ccounted for vrince. X 1 unexplined vrince c d X 3 X 1 unexplined vrince c d X 3 X 2 X 2 It s lso interesting to note tht X 2 nd X 3 re correlted. When two predictors re correlted it will ffect their et weights. Therefore we will e revisiting this sitution. X 1 unexplined vrince X 2 c X 3 Repeting the process for ech predictor As previously noted, ech predictor is tken in turn. Tht is, ll other predictors re first plced in the eqution nd then the predictor of interest is entered. This llows us to determine the unique (dditionl) contriution of the predictor vrile. By repeting the procedure for ech predictor we cn clculte the unique contriution of ech. The Structure Coefficient This is the ivrite correltion etween (1) prticulr independent vrile nd (2) the predicted scores which cn lso e thought of in terms of the vrite since the vrite cretes the predicted score. This is conceptully similr to fctor nlysis where correltion is computed etween ech independent vrile nd fctor (or ltent vrile). The Structure Coefficient SPSS does not provide the structure coefficients. They cn e esily clculted y dividing the Person correltion etween the given predictor nd the ctul (mesured) dependent vrile nd R the multiple correltion. Or: Person Corr for predictor & dep. vr multiple correltion (R)

9 The Structure Coefficient The lrger the structure coefficient the etter the predictor reflects the construct underlying the vrite. Independent vriles cn e compred s to how well they reflect the underlying construct. A Comprison of Structure Coefficients nd Bet weights Bet weights tke into ccount the predictor s correltion with the other predictors in the nlysis, structure coefficients do not. Bet weights cn exceed the rnge of + 1 while structure coefficients cn not. Both cn e used to clculte the reltive contriutions of predictors. Sttisticins disgree on the vlue of structure coefficients. The F nd t tests The F-test is used s generl indictor of the proility tht ny of the predictor vriles contriute to the vrince in the dependent vrile within the popultion. The null hypothesis is tht the predictors weights re ll effectively equl to zero. Tht is, none of the predictors contriute to the vrince in the dependent vrile in the popultion. The F nd t tests t-tests re used to test the significnce of ech predictor in the eqution. The null hypothesis is tht predictor s weight is effectively equl to zero when the effects of the other predictors re tken into ccount. Tht is, it does not contriute to the vrince in the dependent vrile within the popultion. Wht do we men y significnce? Thus, one question to sk is: Does the independent vrile contriute to the R 2 when controlling for other independent vriles in the regression eqution? The sme question is sked for ech coefficient. In still other words, does the vrile s contriution to the slope (regression line) result in more error eing reduced thn when the vrile is not considered. Null Hypotheses: yx1 = 0

10 In sum, the t test is used to determine the significnce of the coefficient nd is clculted seprtely for ech. t = yx s = smple yx stndrd error for yx Using the T-test to determine the significnce of the intercept The question to sk is: When the independent vriles re equl to zero, does the intercept contriute to reducing the error when predicting the dependent vrile? A significnt intercept (or constnt) indictes tht ny reduction found in the smple cn e expected to exist in the popultion Testing the intercept The t test is gin used. The specific t formul is very complex nd fortuntely is clculted y sttisticl softwre pckges. Using the t-test for ivrite (zeroorder) correltion coefficients Zero-order (ivrite) correltion coefficients re often used to compre correltions etween ll possile pirs of vriles in regression eqution. For ech pir, specil form of the t-test is used to determine its significnce. The null hypothesis is tht the correltion is unrelted (correltion = 0). Bivrite correltion coefficients cn e otined through SPSS correltion commnd. Compring Nested Regression Equtions (lso sometimes referred to s hierrchil liner models) Nested regression equtions re series of two or more regression equtions where independent vriles re successively dded to n eqution to oserve chnges in the predictors reltionship to the dependent vrile (lso referred to s lock-entry regression method). Tht is, does dding unique set of independent vriles significntly increse the R 2. When compring the R 2 of n originl set of vriles to the R 2 fter dditionl vriles hve een included, the resercher is le to identify the unique vrition explined y the dditionl set of vriles. Any co-vrition etween the originl set of vriles nd the new vriles will e ttriuted to the originl vriles.

11 Exmple of Nested Regression Equtions With nurse ide jo stisfction s the dependent vrile, the resercher could run 3 regression equtions, ech dding dditionl independent vriles: Regression 1: ge, mritl sttus, t # of children in household (demogrphic vriles) Regression 2: ge, mritl sttus, # of children in household (demogrphic vriles); rting of orgniztion, time for providing cre, dequte stffing (orgniztionl vriles) Exmple of Nested Regression Equtions Regression 3: ge, mritl sttis, # of children in household (demogrphic vriles); rting of orgniztion, time for providing cre, dequte stffing (orgniztionl vriles); utonomy, competence, impct of work (empowerment) Determining the significnce of dding Nested Regression Equtions A specil formul of the F-test is used to compre the R 2 of the first eqution to the R 2 of the second to determine whether the differences etween the two cn e generlized to the popultion. (R F = 2 2 R 2 1) / ((K 2 K 1 ) (1 R 2 2) / (N K 2 1) --Suscripts ttched to R2 nd K indicte whether vlues come from the first (less-inclusive) eqution or the second. --K indictes the numer of independent vriles in the eqution. Determining the significnce of dding Nested Regression Equtions The purpose of the F test is to determine whether dditionl vrition in the dependent vrile is explined y dding the dditionl vriles. For the F test to e significnt, the difference in R2 must e lrge reltive to the numer of independent vriles dded to the second eqution. A similr ide is to conduct Significnce Test in order to Compre Two Regression Equtions The resercher my wnt to determine whether the independent vriles ffect the dependent vrile the sme for two different groups (sy men nd women) For exmple, in 1995 Ph.D. student (Dr. Leslie Stnley-Stevens) wnted to know if set of independent vriles hd the sme effect on jo stisfction for men s for women (she s now n ssocite professor t Trelton Stte University). Compring two regression equtions cn e ccomplished with the correltion difference test. This is sttisticl test to determine whether two correltion coefficients differ. In short it trnsforms oth the smple correltions to Z scores, pplies the test, nd the result is Z sttistic tht cn e checked for its significnce level.

12 Presenttion of dt using sttisticl procedures lerned in clss If the Z sttistic is significnt, then the difference found etween the two correltions is not due to chnce ut to the fct tht the two groups re ffected differently y the independent vriles. Also note mistkes in the text (s fr s I cn determine): P.161, The slnted-line res should e The drk gry res P.166, The second numericl column should e The first numericl column P.167, n increse of 2.89 points on the positive ffect mesure is, on verge, ssocited with 1- point gin in self-esteem. This should e reversed since self-esteem is the dependent vrile (one unit chnge in the independent vrile produces quntified chnge in the dependent vrile not vice vers). Suppressor Vriles Vriles tht re NOT highly correlted with the dependent vrile ut re le to heighten the numericl effect of ANOTHER predictor on the dependent vrile. In other words, when suppressor vrile is included in regression, the suppressor vrile will pper lrgely unrelted to the dependent vrile while the effect of ANOTHER predictor vrile will increse sustntilly. How Does it Work? Suppressor Vriles re difficult to conceptulize. Plese red the Meyers text for further explntion (p.182)

13 How Does it Work? By virtue of its correltion with predictor vrile, it ccounts for (sttisticlly controls for or negtes) tht portion of the predictor vrile not relted to the dependent vrile nd thus mkes the predictor vrile etter predictor thn it would e in the sence of the suppressor vrile. Indictions of suppressor vriles include: the correltion etween it nd the dependent vrile is sustntilly smller thn its et weight its Person correltion with the dep. vrile nd its et hve different signs it my hve ner zero correltion with the dependent vrile ut yet is significnt predictor it my hve little or no correltion with the dependent vrile ut is correlted with one or more of the predictors Complete nd Intrinsic Liner Regression Liner regression cn e viewed from two perspectives: completely liner model nd n intrinsiclly liner model: 1. completely liner model exists when the level l of mesurement for ll the vriles of interest re liner Consequently, the weighted vriles cn e dded together to otin the predicted score Complete nd Intrinsic Liner Regression 2. less thn completely liner model or intrinsiclly liner model exists when there is need to lter or trnsform one or more of the vriles (e.g., squred; logged). In these regression models we cn still dd the weighted vriles together to otin the predicted score. When thinking out intrinsiclly liner models, wht re some situtions where we would wnt to lter or trnsform the dependent vrile? Wht out the independent vrile? Exmples of trnsforming the dependent vrile 1. if the dependent vrile is dichotomous, we cn do logit trnsformtion. This will cuse us to interpret the dependent vrile le in terms of proportionsons 2. If heteroscedsticity is prolem with our dt, we cn do nturl logrithm trnsformtion to solve this prolem (we will e exmining ssumptions we mke out the dt when doing regression nlyses; one of these ssumptions is tht there is no heteroscedsticity).

14 Exmples of trnsforming the independent vrile 1. if the independent vrile hs curviliner reltionship with the dependent vrile (lso referred to s polynomil regression), the vrile cn e squred, cued, or whtever is clled for. A qudrtic function hs one end in the curve (lso referred to s second-order polynomil) nd is squred while cuic function hs two ends (lso referred to s thirdorder polynomil) nd is cued. Exmples of trnsforming the independent vrile 2. Using log trnsformtion of the independent vrile is pproprite if the independent nd dependent vriles increse together up to point nd then the effects of the independent vrile no longer hve the sme incresing effect on the dependent vrile (the effect levels off t some point). Exmples of trnsforming the independent vrile 3. Dummy vriles re used if the level of mesurement of the independent vrile is nominl. The nominl vrile is trnsformed into set of dummy vriles where ech vlue of the nominl vrile ecomes seprte dummy vrile. We will e covering Dummy Vriles in more detil lter. Exmples of trnsforming the independent vrile 4. Interction terms. Regression nlysis ssumes tht the independent vriles re unrelted (independence ssumption). However, suppose tht the reltionship etween X 1 nd the dependent vrile differs for different levels of X 2. For exmple, the effect of ge on income differs depending on the level of eduction the person hs. Exmples of trnsforming the independent vrile In this exmple there is n interction etween ge nd eduction, with eduction eing referred to s modertor vrile (lso clled conditionl vrile or specifiction vrile) Exmples of trnsforming the independent vrile While the vriles re not trnsformed in the sme sense s squring or logging vrile, new interction term is introduced into the regression eqution to ccount for the inter-reltionship etween the two independent vriles. We will e exmining interction effects nd how to identify nd interpret them.

15 Nonliner Models Nonliner Models There is whole set of models tht do not tke liner form nd thus cnnot e nlyzed through procedure tht uses ordinry lest squres. These models re est hndled y using other curve-fitting techniques. An exmple where nonliner model would e used is the cse of nominl dependent vrile. In these cses, other pproches hve een developed to nlyze them such s logistic regression nd discriminnt function nlysis. We will tke look t logistic regression t the end of the semester. Brek

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