Partial Correlation, Multiple Regression and Multiple Correlation 1 partial correlation coefficients 2 the least-squares multiple regression partial slopes 3 multiple correlation coefficients 4 Pearsons r partial correlation Partial Gamma / interaction ryxYXryzYZ xz r zero-order correlations

first-orderryxz ryxz =r yx ( r yz )( rxz )2 1 r yz 2 1 rxz

----- (1)

ryxzXYZ zero-order direct relationship ryxzryxspurious common cause Spurious relationship between X and Y

Z as an intervening variable between X and Y Z X Y

ryxzryxryx 0 Z ZXY XY ryxzryxXZY

X Y Z Y Z X Y Z / Y = a + b1X1 + b2X2 --------------------------- (2) b1X1Y(the partial slope ) b2 X2 Y b1 b2 b1 = Sy S1 ry1 ry 2 r12 1 r 2 12

-------------- (3)

b2 =

Sy S2

ry 2 ry1r12 1 r 2 -------------- (4) 12

a = Y - b1- b2 SY SX SX rYX rYX

rXX b b X XY 2 X X X X XXY Y Z scores XY 0S 1 standardized partial slopesbeta-weights b*b* Y Y S b1*=b1 1 Sy

b2*=b2

S2 Sy

Zy= aZ +b1* Z1 +b2* Z2 aZ = 0 Zy= b1* Z1 +b2* Z2 ----------------- (5) 2 5 2 Y 5 Y

Y Rmultiple correlation coefficient R2 (coefficient of multiple determination) RR= ry21 + ry22 2ry1 (ry 2 )(r12 )2 1 r12

R2 = r2y1 + r2 y21 (1r2y1) (r2?)

R2r2 interval-ratio level X ? b

Y12 = r2Y1 + r2Y21

(1-r2y1)

Proportion explained by X1X2

Proportion Proportion Proportion explained by explained by unexplained by X1 X2 X1 controlling for X1

R2ijk = r2ij + r2ikj ( 1 - r2ij ) = r2ik + r2ijk ( 1- r2ik)

regression model Y= a + b1X1 + b2X2 + e ei Yi = ai + b1Xi1 + b2Xi2 + ei ei) ab Y= + 1X1 +2X2 +

Yi =

i + 1Xi1 + 2Xi2 + i

specification error 1 X Y 2 Y 3 Y Y Y X Y functional form X Y 1 2Homoskedasticity ; 3 4XX 5 Homoskedasticity Y perfect high multicollinearity (1)(3)d ab BLUE (Best Linear Unbiased Estimates) BLUE multicollinearity ()

(3)e (Partial Correlation) XXYY s XY Yi = ai + bXi2 + eiyeiy YabXY X XXXabXeX XbXX e YX 5000+1000()+e 10000 , e10000-5000-1000(2)=3000 YX XYXee1 XYX XXYXY X YX Weighted Average YX multivariate normal distribution ry12 =ry1 ( ry 2 )( r12 )2 1 ry22 1 r12

XYX r

Correction factor( ry2) (r12) YXrr YX XYX XXX X1 Y r r r rr r 0 1 r r 0 r rrr r r 0 rr 1 rr0 r A B A B A B A B AB B A spurious)

specification error spurious relation a closed system) Z X Y X Z Y

1 2 yz = xy xz yz xy xzxy xz X yz

yz.x yz (Beta) XXX X X0 Xa b X 2X

b b b b aXbXbX b bbb bb r bb

(Partial Slope) remain constant) Partial Slope direct effect)

) beta weights beta weights (Path Coefficients) P21 X P31 P53 41 X P52 P54

(Path Analysis) X2 = P21X1 X3 = P31X1 X4 = P41X1 X5 = P52X2 + P53X3 + P54X4 beta weights ij.k = bij.k S sjsi XiXj S (?)j i

ij.kl = bij.kl S S ij.k j i

ij.k =

r r r 1 rij ik 2 jk

jk

rij.k

rij.k2 = (ij.k)( ji.k)

beta weights bivariate correlation ( total correlation) Path Coeficients total correlation total correlation total correlaton total correlationrij 1 X X XX XXX 2 X X total correlation rij X X X X X totla correlation r 3 2 rij = Pik rkjk

r12 = P21r11 = P21(1) = P21 r13 = P31 r14 = P41 r23 = P31r12 = P31P21 r Path Coefficients Path Coefficients X X X r23 = P31 P21 = r13 r12 r23 - r13r12 = 0 , r23.1= 0 ( r)(rr13 r Path Coefficients dummy variable /

0 1 1 0 1 0 K 1 K 1 - - - - - / 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1

/ 1 0 0 .007 + .841 .058

0.058 intercept .007 intercept .007 .058