Semipartial (Part) and Partial Correlation


 Egbert Scot Logan
 4 years ago
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1 Semipatial (Pat) and Patial Coelation his discussion boows heavily fom Applied Multiple egession/coelation Analysis fo the Behavioal Sciences, by Jacob and Paticia Cohen (975 edition; thee is also an updated 003 edition now). Oveview. Patial and semipatial coelations ovide anothe means of assessing the elative impotance of independent vaiables in detemining Y. Basically, they show how much each vaiable uniquely contibutes to ove and above that which can be accounted fo by the othe IVs. We will use two apoaches fo explaining patial and semipatial coelations. he fist elies imaily on fomulas, while the second uses diagams and gaphics. o save pape shuffling, we will epeat the SPSS intout fo ou income example: egession Desciptive Statistics INCOME EDUC JOBEXP Mean Std. Deviation N Coelations Peason Coelation INCOME EDUC JOBEXP INCOME EDUC JOBEXP Model Model Summay Adjusted Std. Eo of Squae Squae the Estimate.99 a a. Pedictos: (Constant), JOBEXP, EDUC Model egession esidual otal ANOVA b Sum of Squaes df Mean Squae F Sig a a. Pedictos: (Constant), JOBEXP, EDUC b. Dependent Vaiable: INCOME Model (Constant) EDUC JOBEXP Unstandadized Coefficients a. Dependent Vaiable: INCOME Standadi zed Coefficien ts Coefficients a 95% Confidence Inteval fo B t Sig. Lowe Bound Uppe Bound Coelations Zeoode Patial Pat Collineaity Statistics VIF B Std. Eo Beta oleance Semipatial (Pat) and Patial Coelation  Page
2 Apoach : Fomulas. One of the oblems that aises in multiple egession is that of defining the contibution of each IV to the multiple coelation. One answe is ovided by the semipatial coelation s and its squae, s. (NOE: Hayes and SPSS efe to this as the pat coelation.) Patial coelations and the patial coelation squaed ( and ) ae also sometimes used. Semipatial coelations. Semipatial coelations (also called pat coelations) indicate the unique contibution of an independent vaiable. Specifically, the squaed semipatial coelation fo a vaiable tells us how much will decease if that vaiable is emoved fom the egession equation. Let H the set of all the X (independent) vaiables, G the set of all the X vaiables except X Some elevant fomulas fo the semipatial and squaed semipatial coelations ae then s b * X G b * ol s YG b *( X G ) b * ol hat is, to get X s unique contibution to, fist egess Y on all the X s. hen egess Y on all the X s except X. he diffeence between the values is the squaed semipatial coelation. O altenatively, the standadized coefficients and the oleances can be used to compute the semipatials and squaed semipatials. Note that he moe toleant a vaiable is (i.e. the less highly coelated it is with the othe IVs), the geate its unique contibution to will be. Once one vaiable is added o emoved fom an equation, all the othe semipatial coelations can change. he semipatial coelations only tell you about changes to fo one vaiable at a time. Semipatial coelations ae used in Stepwise egession Pocedues, whee the compute (athe than the analyst) decides which vaiables should go into the final equation. We will discuss Stepwise egession in moe detail shotly. Fo now, we will note that, in a fowad stepwise egession, the vaiable which would add the lagest incement to (i.e. the vaiable which would have the lagest semipatial coelation) is added next (ovided it is statistically significant). In a bacwads stepwise egession, the vaiable which would oduce the smallest decease in (i.e. the vaiable with the smallest semipatial coelation) is dopped next (ovided it is not statistically significant.) Semipatial (Pat) and Patial Coelation  Page
3 Fo computational puposes, hee ae some othe fomulas fo the two IV case only: s Y  Y  Y  Y ol b  b ol s Y Y Y Y ol b b ol Fo ou income example, s Y  Y * (.07 ).8797 b ol * , s Y  Y , s Y Y * (.07 ).3606 b ol.366* s Y  Y Compae these esults with the column SPSS labels pat co. Anothe notational fom of s used is y( ). Also, efeing bac to ou geneal fomula, it may be useful to note that YG + s, YG hat is, when Y is egessed on all the Xs, is equal to the squaed coelation of Y egessed on all the Xs except X plus the squaed semipatial coelation fo X ; and, if we would lie to now what would be if a paticula vaiable wee excluded fom the equation, just subtact s fom. Fo example, if we want to now what would be if X wee eliminated fom the equation, just compute  s Y ; and, if we want to now what would be if X wee eliminated fom the equation, compute  s Y.  s Semipatial (Pat) and Patial Coelation  Page 3
4 Patial Coelation Coefficients. Anothe ind of solution to the oblem of descibing each IV s paticipation in detemining is given by the patial coelation coefficient, and its squae,. he squaed patial answes the question How much of the Y vaiance which is not estimated by the othe IVs in the equation is estimated by this vaiable? he fomulas ae s s s s, s YG + s YG Note that, since the denominato cannot be geate than, patial coelations will be lage than semipatial coelations, except in the limiting case when othe IVs ae coelated 0 with Y in which case s. In the two IV case, may be found via s  Y s  Y + s, s Y s Y + s In the case of ou income example, s  Y , , s Y , (o confim these esults, loo at the column SPSS labels patial.) hese esults imply that 46% of the vaiation in Y (income) that was left unexplained by the simple egession of Y on X (education) has been explained by the addition hee of X (job expeience) as an explanatoy vaiable. Similaly, 83% of the vaiation in income that is left unexplained by the simple egession of Y on X is explained by the addition of X as an explanatoy vaiable. A fequently employed fom of notation to exess the patial is Y is also sometimes called the patial coefficient of detemination fo X. WANING. In a multiple egession, the metic coefficients ae sometimes efeed to as the patial egession coefficients. hese should not be confused with the patial coelation coefficients we ae discussing hee. Semipatial (Pat) and Patial Coelation  Page 4
5 Altenative fomulas fo semipatial and patial coelations: s * N K + ( N K ) Note that the only pat of the calculations that will change acoss X vaiables is the value; theefoe the X vaiable with the lagest patial and semipatial coelations will also have the lagest value (in magnitude). Examples: * s N K 9.09 * s * N K 3.77* ( N K ) ( N K ) Besides maing obvious how the patials and semipatials ae elated to, these fomulas may be useful if you want the patials and semipatials and they have not been epoted, but the othe infomation equied by the fomulas has been. Once I figued it out (which wasn t easy!) I used the fomula fo the semipatial in the pco outine I wote fo Stata. Semipatial (Pat) and Patial Coelation  Page 5
6 Apoach : Diagams and Gaphics. Hee is an altenative, moe visually oiented discussion of what semipatial and patial coelations ae and what they mean. Following ae gaphic eesentations of semipatial and patial coelations. Assume we have independent vaiables X, X, X 3, and X 4, and dependent vaiable Y. (Assume that all vaiables ae in standadized fom, i.e. have mean 0 and vaiance.) o get the semipatial coelation of X with Y, egess X on X, X 3, and X 4. he esidual fom this egession (i.e. the diffeence between the edicted value of X and the actual value) is e. he semipatial coelation, then, is the coelation between e and Y. It is called a semipatial coelation because the effects of X, X 3, and X 4 have been emoved (i.e. patialled out ) fom X but not fom Y. Semipatial (Pat) Coelation o get the patial coelation of X with Y, egess X on X, X 3, and X 4. he esidual fom this egession is again e. hen, egess Y on X, X 3, and X 4 (but NO X ). he esidual fom this egession is e y. he patial coelation is the coelation between e and e y. It is called a patial coelation because the effects of X, X 3, and X 4 have been patialled out fom both X and Y. Patial Coelation Semipatial (Pat) and Patial Coelation  Page 6
7 Semipatial (Pat) Coelations. o bette undestand the meaning of semipatial and squaed semipatial coelations, it will be helpful to conside the following diagam (called a ballantine ). [NOE: his ballantine descibes ou cuent oblem etty well. Section 3.4 of the 975 edition of Cohen and Cohen gives seveal othe examples of how the Xs and Y can be inteelated, e.g. X and X might be uncoelated with each othe, o they might be negatively coelated with each othe but positively coelated with Y.] In this diagam, the vaiance of each vaiable is eesented by a cicle of unit aea (i.e. each vaiable is standadized to have a vaiance of ). Hence, A + B + C + D s y yy, (B + C)/ (A + B + C + D) B + C Y, (C + D)/ (A + B + C + D) C + D Y, (C + F)/ (B + C + E + F) (C + F)/ (C + D + F + G) C + F, (B + C + D) / (A + B + C + D) B + C + D Y hat is, the ovelapping of cicles eesents thei squaed coelation, e.g.. he total aea of Y coveed by the X and X aeas eesents the opotion of Y s vaiance accounted fo by the two IVs, Y. he figue shows that this aea is equal to the sum of the aeas designated B, C, and D. (NOE: Don t confuse the A and B used in the diagam with the a and b we use fo egession coefficients!) he aeas B and D eesent those potions of Y ovelapped uniquely by X and X, espectively, wheeas aea C eesents thei simultaneous ovelap with Y. he unique aeas, exessed as opotions of Y vaiance, ae squaed semipatial coelation coefficients, and each equals the incease in the squaed multiple coelation which occus when the vaiable is added to the othe IV. hus, Semipatial (Pat) and Patial Coelation  Page 7
8 s B (B +C + D)  (C + D) Y  Y, s D (B + C + D)  (B +C) he semipatial coelation s is the coelation between all of Y and X fom which X has been patialled. It is a semipatial coelation since the effects of X have been emoved fom X but not fom Y. emoving the effect is equivalent to subtacting fom X the X values estimated fom X, that is, to woing with x  x^ (whee x^ is estimated by egessing X on X ). hat is, x  x^ is the esidual obtained by egessing X on X. We will denote this as e. Hence, s ye. s is the amount that is inceased by including X in the multiple egession equation (o altenatively, it is the amount that would go down if X wee eliminated fom the equation.) In tems of ou diagam, s y A + B + C + D, (because Y is standadized) y (B + C)/ (A + B + C + D) B + C, s B / (A + B + C + D) B. hus, we emove the aea C fom X but not fom Y. Y  Y Anothe notational fom of s used is y( ), the being a shothand way of exessing X fom which X has been patialled. Patial Coelation Coefficients. Anothe ind of solution to the oblem of descibing each IV s paticipation in detemining is given by the patial coelation coefficient, and its squae,. he squaed patial coelation may be undestood best as the opotion of the vaiance of Y not associated with X which is associated with X. hat is, B (B +C + D)  (C + D) Y  A+ B (A+ B +C + D)  (C + D)  Y Y s  Y D A+ D Moe geneally, we can say that (B + C + D) (B +C) (A+ B + C + D) (B +C) Y Y Y s  s s s s, s YG + s YG Y Semipatial (Pat) and Patial Coelation  Page 8
9 he numeato fo is the squaed semipatial coelation coefficient; howeve, the base includes not all of the vaiance as in s, but only that potion of Y vaiance which is not associated with X, that is,  Y. hus, the squaed patial answes the question How much of the Y vaiance which is not estimated by the othe IVs in the equation is estimated by this vaiable? Note that, since the denominato cannot be geate than, patial coelations will be lage than semipatial coelations, except in the limiting case when othe IVs ae coelated 0 with Y in which case s. Anothe way of viewing the patial coelation is that is the coelation between X fom which X has been patialled and Y fom which X has also been patialled (i.e., the coelation between x^ and y^ ). A fequently employed fom of notation to exess the patial is Y, which conveys that X is being patialled fom both Y and X, in contast to the semipatial, which is eesented as Y( ). is also sometimes called the patial coefficient of detemination fo X. In tems of ou diagam, s y A + B + C + D, (because Y is standadized) y (B + C)/ (A + B + C + D) B + C, s B / (A + B + C + D) B B / (A + B) hus, in the squaed semipatial coelation, aeas which belong to X and which ovelap eithe X o Y (C and D) ae emoved fom X but not Y. In the squaed patial coelation, aeas which belong to X ae emoved fom both X and Y. Semipatial (Pat) and Patial Coelation  Page 9
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