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Estimators of the multiple correlation coefficient: local robustness and confidence intervals.

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„BAZRt‚RR stI {BthI‚t{‚ $t‚„%sjR

]Y„$RB}Y‚ ]„BZ- stI ]sY‚„$t‚ *‚YBt

=t$%‚„R$?‚ N$A„‚ I‚ „Z-‚jj‚R

VAR„s{U stk „BAZR „‚„‚RR$Bt ‚R$8sB„R s„‚ I‚ht‚I Ak 8$t$8$9$t s 8‚sRZ„‚ BP R}„‚sI BP Y‚ „‚R$IZsjR Vt s{{B8}stk$t p1T8‚sRZ„‚ B„ 8Zj$}j‚ {B„„‚js$Bt {B‚^{$‚t $R Y‚t

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8‚sRZ„‚ iBAZRt‚RR

e]Vie7 3s{Zj?‚ 7!]! stI ytR$Z I‚ 7s$R$JZ‚ =t$%‚„R$?‚ N$A„‚ I‚ „Z-‚jj‚R ]_Tdd; V% 3* iBBR‚%‚j x( Td(x( „ZRR‚jR ‚j$Z8

(2)

d yt„BIZ{$Bt

qY‚ p1TRs$R${ B„ RJZs„‚I 8Zj$}j‚ {B„„‚js$Bt {B‚^{$‚t $R s RstIs„I BBj $t s}}j$‚I

„‚„‚RR$Bt stsjkR$R VjYBZY $ $R tB sjbskR YBZYPZjjk ZR‚I $ „‚8s$tR st $tPB„8s$%‚ RZ88s„k 8‚sRZ„‚ BP Y‚ }„‚I${$%‚ }Bb‚„ BP Y‚ R‚j‚{‚I „‚„‚RR$Bt 8BI‚j ]sZ$Bts„k tB‚R sABZ p1 {st A‚ PBZtI $t €%sjR‚Y wdMxD stI G$jj‚ m 7$t‚„ wdMMD G‚ bB„, $t s

RstIs„I j$t‚s„ „‚„‚RR$Bt 8BI‚j b$Y $t‚„{‚}U

r7 )L?7d#dLN N NL?7€#€ L7 7) d,N N N , z wddD

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stI r7 Y‚ „‚R}BtR‚ %s„$sAj‚ 7Z}}BR‚ Ys Y‚ ‚„„B„R 7 s„‚ $tI‚}‚tI‚t BP Y‚ ‚-}jstsB„k

%s„$sAj‚R stI $$I s{{B„I$t B s I$R„$AZ$Bt Bw?D ) B(w?bD bY‚„‚ $R Y‚ „‚R$IZsj

R{sj‚ }s„s8‚‚„ stI B( %‚„$h‚R

w3D B( $R $R Rk88‚„${stI Zt$8BIsj s„BZtI 9‚„B b$Y R„${jk }BR$$%‚ I‚tR$k

G$Y„7w |#N7D )r7?07#|N7|N7Y‚ „‚R$IZsjR P„B8 Y‚ N‚sR 7JZs„‚R wN7D h Y‚ {jsRR${sj p1 {B‚^{$‚t $R I‚ht‚I sR p1 N7 ) d z 7)d„7w |#N7D1 z 7)dwr7rƒD1, wd1D

b$Y ƒr Y‚ Rs8}j‚ s%‚„s‚ BP Y‚ I‚}‚tI‚t %s„$sAj‚ G‚ R‚‚ Ys Y‚ tZ8‚„sB„ $t wd1D ‚JZsjR Y‚ %s„$st{‚ BP Y‚ „‚R$IZsjR $t Y‚ PZjj 8BI‚j bY$j‚ Y‚ I‚tB8$tsB„ $R Y‚ %s„$st{‚ BP Y‚ „‚R$IZsjR $t Y‚ „‚IZ{‚I 8BI‚jU

r7 )(L7 7) d,N N N , z3 wdnD

ytI‚‚I Y‚ N7 ‚R$8sB„ BP ( $t wdnD ‚JZsjR ƒr

7$t{‚ Y‚ N7 ‚R$8sB„ $R %‚„k %Zjt‚„sAj‚ $t }„‚R‚t{‚ BP BZj$‚„R $ $R tB RZ„}„$R$t Ys

p1

N7 $tY‚„$R Y$R }„BAj‚8 qY$R $R $jjZR„s‚I $t 3$Z„‚ d bY‚„‚ Y‚ %sjZ‚ BPp1N7 I‚{„‚sR‚R

I„sR${sjjk bY‚t Bt‚ R$tj‚ BZj$‚„ „‚}„‚R‚t‚I Ak st $t%‚„‚I „$stj‚ $R sII‚I B Y‚ Rs8}j‚ VjYBZY Y‚ h‚I N7 j$t‚ $R tB ‚tB„8BZRjk sj‚„‚I Y‚ p1

UC I‚{„‚sR‚R P„B8 Mn0

B n(0

N‚ ZR IB tBb st ‚-}‚„$8‚t sjjBb$t ZR B RZIk Y‚ $t4Z‚t{‚ BP st BZj$‚„ $t Y‚

rTI$„‚{$Bt w{sjj‚I s %‚„${sj BZj$‚„D B„ $t Y‚ ?TI$„‚{$Bt w{sjj‚I s j‚%‚„s‚ }B$tD Bt p1 N7

V Iss R‚ bsR ‚t‚„s‚I stI $R 8Zj$}j‚ {B„„‚js$Bt {B‚^{$‚t {B8}Z‚I VP‚„bs„IR s d

(3)

x y -10 -5 0 5 10 15 20 -40 -20 0 2 0 4 0 6 0 8 0 R2=0.83 R2=0.30

3$Z„‚ dU e&‚{ BP sII$t Bt‚ R$tj‚ BZj$‚„ $tI${s‚I Ak n Bt p1 N7

}B$t b$Y {BB„I$ts‚R w(, rD $R sII‚I bY‚„‚ Y‚ %sjZ‚ BP r „st‚R $t Y‚ $t‚„%sj >x,xH sR $R R‚‚t $t Y‚ Z}}‚„ j‚P }st‚j BP 3$Z„‚ 1 qY‚ jBb‚„ j‚P }st‚j RYBbR YBb Y$R s&‚{R Y‚ %sjZ‚ BP p1

N7 V R$8$js„ ‚-‚„{$R‚ $R IBt‚ PB„ j‚%‚„s‚ }B$tR Ak j‚$t %s„k Y‚ %sjZ‚ ? BP st BZj$‚„ }js{‚I s w?,(D qY‚ %sjZ‚ BP p1

N7 I$I I‚{„‚sR‚ bY‚t sII$t Y‚R‚ }B$tR

stI Ak j‚$t ? B„ r ‚tI B $tht$k $ b$jj ‚%‚t ‚tI B 9‚„B 3$Z„‚ 1 {Bth„8R Ys p1 N7

$R Y$Yjk R‚tR$Aj‚ B R$tj‚ BZj$‚„R Ns‚„ Bt $t Y$R }s}‚„ b‚ b$jj JZst$Pk Y‚ $t4Z‚t{‚ BP sII$t BAR‚„%s$BtR Ak {B8}Z$t $t4Z‚t{‚ PZt{$BtR G‚ b$jj htI Ys Y‚„‚ {st sjRB A‚ st $t{„‚sR‚ $t p1 bY‚t sII$t BZj$‚„R

yt 7‚{$Bt 1 b‚ b$jj $t„BIZ{‚ p1T8‚sRZ„‚R bY${Y B sjBt b$Y „BAZR „‚„‚RR$Bt hR

yt4Z‚t{‚ PZt{$BtR b$jj A‚ {B8}Z‚I $t 7‚{$Bt n G‚ b$jj PB{ZR Bt „‚„‚RR$Bt ‚R$8sB„R AsR‚I Bt Y‚ 8$t$8$9s$Bt BP T‚R$8sB„R BP „‚R$IZsj R{sj‚ qY‚ BR T„BAZR 8‚sRZ„‚ BP 8Zj$}j‚ {B„„‚js$Bt bY${Y 8$t$8$9‚R Y‚ 8s-$8sj $t4Z‚t{‚ Ys st BAR‚„%s$Bt {st Ys%‚ Bt p1 $R BAs$t‚I 7‚{$Bt ; ‚-}js$tR YBb {BthI‚t{‚ $t‚„%sjR PB„ Y‚ }B}Zjs$Bt 8Zj$}j‚

{B„„‚js$Bt {B‚^{$‚t {st A‚ {BtR„Z{‚I $t Y‚ {sR‚ BP 8Zj$%s„$s‚ tB„8sj$k bY$j‚ 7‚{$Bt x { Bt{ jZI‚R

(4)

x y -4 -2 0 2 4 -4 -2 0 2 4 x y -4 -2 0 2 4 -4 -2 0 2 4 y R2 -4 -2 0 2 4 0.5 0.6 0.7 0.8 0.9 1.0

Influence of an outlier in y-direction

x R2 -4 -2 0 2 4 0.5 0.6 0.7 0.8 0.9 1.0

Influence of an outlier in x-direction

3$Z„‚ 1U e&‚{ BP sII$t Bt‚ BAR‚„%s$Bt w(, rD B s Iss {jBZI Bt Y‚ %sjZ‚ BP p1 N7 wj‚P

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N7 w„$Y }st‚jD qY‚

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N7 s Y‚ B„$$tsj Rs8}j‚

1 V „BAZR 8‚sRZ„‚ BP

p

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‚R$8sB„ I‚ht‚I Ak

w|,#|D ) s„8$t

w,#D Czw„dw, #D, 3 3 3 , „zw, #DD w1dD

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|

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p1 C ) d C1 z p „dw|,#|D, 3 3 3 , „zw|,#|D Q C1 zwrd|(, 3 3 3 , rz|(D 3 w11D n

(5)

7$t{‚ wdnD $R s RZA8BI‚j BP wddD stI Ak Y‚ I‚ht$$Bt BP Y‚ „‚„‚RR$Bt ‚R$8sB„R w1dD Bt‚ „‚sI$jk R‚‚R Ys

(p1 C d3

3B„ ‚%‚„k „‚R$IZsj R{sj‚ ‚R$8sB„ C b‚ Ys%‚ stBY‚„ „‚„‚RR$Bt ‚R$8sB„ stI stBY‚„ 8‚sT RZ„‚ BP 8Zj$}j‚ {B„„‚js$Bt zsjZ‚R BPp1

C {jBR‚ B 9‚„B $tI${s‚ Ys Y‚ ‚-}jstsB„k %s„$sAj‚R

s„‚ tB ‚-}js$t$t 8Z{Y BP Y‚ I$R}‚„R$Bt BP r bY$j‚ s %sjZ‚ BP p1 {jBR‚ B Bt‚ $tI${s‚R

Ys Y‚ I$R}‚„R$Bt BP Y‚ „‚R$IZsjR $R sj8BR 9‚„B RB st ‚-{‚jj‚t h qY$R $t‚„}„‚s$Bt $R RstIs„I AZ Y‚ 8‚st$t BP XI$R}‚„R$Bt" I‚}‚tIR Bt Y‚ R‚j‚{‚I 8‚sRZ„‚ BP R{sj‚ C

G‚ b$jj PB{ZR Bt Y‚ {jsRR BP T‚R$8sB„R BP „‚R$IZsj R{sj‚ bY‚„‚Czwd, 3 3 3 , zD $R I‚ht‚I

sR Y‚ RBjZ$Bt BP Y‚ PBjjBb$t ‚JZs$Bt $t XU d z z 3 7)d 'wX7D ) K3 qY‚ PZt{$Bt ' $R RZ}}BR‚I B %‚„$Pk

wiD qY‚ PZt{$Bt' $R Rk88‚„${stI tBt I‚{„‚sR$t Bt >(, > 3Z„Y‚„8B„‚'w(D ) ( stI

' $R sj8BR ‚%‚„kbY‚„‚ {Bt$tZBZRjk I$&‚„‚t$sAj‚

qY‚ tZ8A‚„ K $R s R‚j‚{‚I {BtRst {YBR‚t sR oB(>'wDH B ‚tRZ„‚ {BtR$R‚t ‚R$8s$Bt BP s Y‚ 8BI‚j

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p1

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Y‚ 8$t$8$9s$Bt BP Y‚ RZ8 BP sARBjZ‚ %sjZ‚R BP Y‚ „‚R$IZsjR stI k$‚jIR Y‚ Ud „‚„‚RR$Bt

‚R$8sB„ qY‚ sRRB{$s‚I p1 Nd $R $%‚t Ak p1 Nd ) d Xz 7)d/r7?07#|Nd|Nd/ z 7)d/r78‚I$st7r7/ ~1 3

qY‚ sAB%‚ 8‚sRZ„‚ stI s %s„$st BP $ Ys%‚ A‚‚t RZI$‚I Ak {€‚st m 7$‚%‚„R wdMD V R8BBY stI ABZtI‚I ' PZt{$Bt $R Y‚ qZ,‚k $b‚$Y

'w`D ) 8$t w 1 E, `1 1 `; 11 L `E E; W 3 w1nD

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(6)

k I‚PsZj Y‚ x(0 A„‚s,IBbt %‚„R$Bt $R s,‚t AZ b‚ b$jj sjRB {BtR$I‚„ Y‚ 1x0 A„‚s,IBbt $b‚$Y 7T‚R$8sB„ w 71xD

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Czwd, 3 3 3 , zD )d//w>wdKDzHD3

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8‚I$st BP Y‚ sARBjZ‚ %sjZ‚R BP Y‚ „‚R$IZsjR stI ‚ Y‚ N‚sR ‚I$st BP 7JZs„‚R wN7D „‚„‚RR$Bt ‚R$8sB„ BP iBZRR‚‚Zb wdM;D 3B„ BY‚„ %sjZ‚R BP K b‚ b$jj R}‚s, sABZ Y‚ N‚sR +Zst$j‚ BP 7JZs„‚R ‚R$8sB„ wiBZRR‚‚Zb m N‚„Bk dM } d1;D 3B„ Y‚ N7 sR sj„‚sIk tB${‚I Ak VtI‚„RBtT7}„‚{Y‚„ wd;D w11D „‚RZjR $t

p1 N7) d X 8‚I$st7/r7 ?70#|N7|N7/ 7<!iq ~1 ,

bY‚„‚ 7<!iq RstIR PB„ YsjP BP Y‚ j‚tY BP Y‚ RYB„‚R $t‚„%sj {B%‚„$t YsjP BP Y‚ r7

BAR‚„%s$BtR

N‚ ZR „‚}‚s Y‚ ‚-}‚„$8‚t I‚R{„$A‚I $t 7‚{$Bt d PB„ Y‚ Ud ‚R$8sB„ Y‚ $b‚$Y

7T‚R$8sB„ stI Y‚ N‚sR ‚I$st BP 7JZs„‚R 3$Z„‚ n RYBbR Y‚ „‚RZjRU Y‚Ud‚R$8sB„ $R

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C A‚{B8$t {BtRst bY‚t Y‚ BZj$‚„ 8B%‚R PZ„Y‚„ sbsk P„B8 Y‚ B„$$t qY‚ {Z„%‚

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p1

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n yt4Z‚t{‚ PZt{$BtR

N‚ C A‚ Y‚ R{sj‚ PZt{$Btsj sRRB{$s‚I b$Y Cz G‚ „‚JZ$„‚ Ys C $R 3$RY‚„ {BtR$R‚t s

Y‚ ‚„„B„ 8BI‚j I$R„$AZ$Bt 8‚st$t Ys CwBD ) qY‚ R{sj‚ PZt{$Btsj „‚}„‚R‚t$t

T‚R$8sB„R BP R{sj‚ $R I‚ht‚I sR Y‚ RBjZ$Bt BP

8

'wCwBDDQBwD )K,

(7)

y R2 -4 -2 0 2 4 0.70 0.80 0.90

L1: Influence of an outlier in y-direction

x R2 -4 -2 0 2 4 0.70 0.80 0.90

L1: Influence of an outlier in x-direction

y R2 -4 -2 0 2 4 0.70 0.80 0.90

LMS: Influence of an outlier in y-direction

x R2 -4 -2 0 2 4 0.70 0.80 0.90

LMS: Influence of an outlier in x-direction

y R2 -4 -2 0 2 4 0.70 0.80 0.90

BS: Influence of an outlier in y-direction

x R2 -4 -2 0 2 4 0.70 0.80 0.90

BS: Influence of an outlier in x-direction

3$Z„‚ nU e&‚{ BP sII$t Bt‚ BAR‚„%s$Btw(, rDB s Iss {jBZI Bt Y‚ %sjZ‚ BPp1

Nd, p1 7 stI

p1

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qY‚ YB„$9Btsj IsRY‚I j$t‚R $tI${s‚ Y‚ %sjZ‚ BP p1

Nd, p1 7 stI p1N7 s Y‚ B„$$tsj Rs8}j‚

(8)

PB„ stk Zt$%s„$s‚ I$R„$AZ$BtB G‚ t‚‚I B R‚K)oB(>'wVDH B BAs$t 3$RY‚„ {BtR$R‚t{k

s Y‚ ‚„„B„ 8BI‚j I$R„$AZ$Bt 3Z„Y‚„8B„‚ b‚ b$jj I‚tB‚ Cw.D #CwBD PB„ stk „stIB8 %s„$sAj‚ . B

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p1

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p1 CwID ) d 1 C1 (wVD3 

(9)

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(10)

qY‚ $t4Z‚t{‚ PZt{$Bt BP Y‚ p1

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U-C AZ

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qY‚ „BRRT‚„„B„ R‚tR$$%$k BP Y‚p1

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BP R{sj‚

d qY‚ BAR‚„%s$Bt w?, rD PBjjBbR }‚„P‚{jk Y‚ „‚„‚RR$Bt j$t‚ r ) ?0#L yP b‚ j‚ ?

dqs,‚ # W) ( BY‚„b$R‚ {B # (

(11)

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(12)

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C WB‚ Ys tB sjj AsI j‚%‚„s‚

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C 8‚sRZ„‚R AsR‚I Bt st T‚R$8sB„ BP R{sj‚ y $R st $88‚I$s‚ {BtR‚JZ‚t{‚ BP

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(13)

RB8‚YBb RZ„}„$R$t Ys Y‚ N7 $R j‚RR jB{sjjk „BAZR Yst st N+7T‚R$8sB„ b$Y JZst$j‚ EM0 wqY$R B}$8sj JZst$j‚ BP EM0 YsR sjRB A‚‚t BAs$t‚I Ak OBYs$ m s8s„ wdD AZ $t s I$&‚„‚t {Bt‚-D G‚ b$jj {sjj Y$R 8BR TiBAZR ‚R$8sB„ Y‚ N+7n1 R$t{‚ $ YsR s A„‚s,IBbt }B$t BP n10 yt Y‚ t‚- sAj‚ b‚ $%‚ Y‚ %sjZ‚R BP Y‚ „BRRT‚„„B„ R‚tR$$%$k „‚js$%‚ B $R 8$t$8Z8 %sjZ‚

wp1

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wd1D$1$ wnxD

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1

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(14)

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(15)

Rs W(Rs) 0.2 0.4 0.6 0.8 1.0 1234 567 LMS BS L1 LS

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(16)

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(17)

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(18)

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(19)

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References

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