Estimators of the Multiple Correlation Coe^cientU local ... - Core

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Tstatistic, or squared multiple correlation coefficient, is a standard tool in .... Figure 2: Effect of adding one observation (0,y) to a data cloud on the value of R. 1 .... If the denominator in the above expression equals zero, then by convention R. 1.
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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 p1 T8‚sRZ„‚ B„ 8Zj$}j‚ {B„„‚js$Bt {B‚^{$‚t $R Y‚t ‚sR$jk BAs$t‚I yt Y$R }s}‚„ jB{sj „BAZRt‚RR }„B}‚„$‚R BP Y‚R‚ „BAZR p1 T{B‚^{$‚tR s„‚ $t%‚R$s‚I y $R sjRB RYBbt 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

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p1 {B‚^{$‚t $R I‚ht‚I sR

z „7 w#|N7 D1 1 pN7 ) d  z7)d , ƒD1 7)d wr7  r

wd1D

b$Y rƒ Y‚ Rs8}j‚ s%‚„s‚ BP Y‚ I‚}‚tI‚t %s„$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 ) ( L 7

7 ) d, N N N , z3

wdnD

ytI‚‚I Y‚ N7 ‚R$8sB„ BP ( $t wdnD ‚JZsjR rƒ 7$t{‚ Y‚ N7 ‚R$8sB„ $R %‚„k %Zjt‚„sAj‚ $t }„‚R‚t{‚ BP BZj$‚„R $ $R tB RZ„}„$R$t Ys 1 $tY‚„$R Y$R }„BAj‚8 qY$R $R $jjZR„s‚I $t 3$Z„‚ d bY‚„‚ Y‚ %sjZ‚ BP p1 I‚{„‚sR‚R pN7 N7 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 I‚{„‚sR‚R P„B8 Mn0 UC

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 p1N7  V Iss R‚ bsR ‚t‚„s‚I stI $R 8Zj$}j‚ {B„„‚js$Bt {B‚^{$‚t {B8}Z‚I VP‚„bs„IR s d

80 20

R2=0.30

-40

-20

0

y

40

60

R2=0.83

-10

-5

0

5

10

15

20

x

1  3$Z„‚ dU e&‚{ BP sII$t Bt‚ R$tj‚ BZj$‚„ $tI${s‚I Ak n Bt pN7 }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 1  V R$8$js„ ‚-‚„{$R‚ $R IBt‚ PB„ j‚%‚„s‚ }B$tR Ak j‚$t %s„k Y‚ %sjZ‚ Y‚ %sjZ‚ BP pN7 1 I$I I‚{„‚sR‚ bY‚t sII$t Y‚R‚ }B$tR ? BP st BZj$‚„ }js{‚I s w?, (D qY‚ %sjZ‚ BP pN7 stI Ak j‚$t ? B„ r ‚tI B $tht$k $ b$jj ‚%‚t ‚tI B 9‚„B 3$Z„‚ 1 {Bth„8R Ys p1N7 $R Y$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 $t„BIZ{‚ p1 T8‚sRZ„‚R bY${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

1

4 y

-4

-2

0

2

4 2 y

0 -2 -4

-4

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0

2

4

-4

-2

x

0

2

4

x

0.9 0.8 R2 0.7 0.6 0.5

0.5

0.6

0.7

R2

0.8

0.9

1.0

Influence of an outlier in x-direction

1.0

Influence of an outlier in y-direction

-4

-2

0

2

4

-4

y

-2

0

2

4

x

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N7

1

V „BAZR 8‚sRZ„‚ BP p1

G‚ $t„BIZ{‚ Y‚ „BAZR p1 T8‚sRZ„‚ sR $t VtI‚„RBtT7}„‚{Y‚„ wd;D ]BtR$I‚„ s „‚„‚RR$Bt ‚R$8sB„ I‚ht‚I Ak | ) s„8$t Cz w„d w , #D, 3 3 3 , „z w , #DD w|

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|( ) s„8$t Cz wrd  ( , 3 3 3 , rz  ( D3

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k {B8}s„$t Y‚ ‚R$8s‚I „‚R$IZsj R{sj‚R $t Y‚ PZjj 8BI‚j stI $t Y‚ „‚IZ{‚I 8BI‚j wdnD b‚ I‚ht‚ Ak stsjBk b$Y Y‚ {jsRR${sj PB„8Zjs wd1D p Q | 3 3 3 , „z w| |

, #D,

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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

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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 BP p1 {jBR‚ B 9‚„B $tI${s‚ Ys Y‚ ‚-}jstsB„k %s„$sAj‚R C

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‚„‚ Cz wd , 3 3 3 , z D $R I‚ht‚I sR Y‚ RBjZ$Bt BP Y‚ PBjjBb$t ‚JZs$Bt $t XU z

d 3 7 'w D ) K3 z 7)d X 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 yP b‚ s,‚ 'w`D ) `1 wstI K ) dD Y‚t Cz $R s RZ8 BP RJZs„‚R stI b‚ ‚ Y‚ {jsRR${sj

1 PB„ w11D ytR‚sI BP s RJZs„‚I jBRR PZt{$Bt Bt‚ {BZjI s,‚ 'w`D ) /`/ bY${Y j‚sIR B pN7 Y‚ 8$t$8$9s$Bt BP Y‚ RZ8 BP sARBjZ‚ %sjZ‚R BP Y‚ „‚R$IZsjR stI k$‚jIR Y‚ Ud „‚„‚RR$Bt 1 $R $%‚t Ak ‚R$8sB„ qY‚ sRRB{$s‚I pNd X ~1 z 0# | /r  ? 

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qY‚ „‚RZj$t ‚R$8sB„ $R Y‚t st 7T‚R$8sB„ wiBZRR‚‚Zb m OBYs$ dM;D bY${Y b‚ {sjj Y‚ $b‚$Y 7T‚R$8sB„ w 7D qY‚ {BtRst  $t w1nD I‚‚„8$t‚R Y‚ A„‚s,IBbt }B$t BP Y‚ ‚R$8sB„ bY${Y $R Y‚ 8s-$8sj P„s{$Bt BP {Bts8$ts$Bt Ys st ‚R$8sB„ {st b$YRstI ;

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PB„ N7 $R JZ$‚ ‚„„s${ bY‚t Y‚ sII‚I }B$t $R {jBR‚ B Y‚ h‚I „‚„‚RR$Bt j$t‚ AZ PB„ p1 7 b‚ BAs$t s %‚„k R8BBY {Z„%‚ yt Y‚ t‚- R‚{$Bt b‚ b$jj ‚t‚„sj$9‚ stI PB„8sj$9‚ Y‚R‚ htI$tR Ak {B8}Z$t Y‚ sRRB{$s‚I $t4Z‚t{‚ PZt{$BtR

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R2

-4

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LMS: Influence of an outlier in x-direction

R2

0.70

0.80

0.80

0.90

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x

0.70

-2

0

2

4

-4

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BS: Influence of an outlier in y-direction

BS: Influence of an outlier in x-direction

0.70

0.70

R2

0.80

0.90

x

0.90

y

0.80

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0.80 0.70

0.70

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0.80

0.90

L1: Influence of an outlier in x-direction

0.90

L1: Influence of an outlier in y-direction

-4

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2

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-4

y

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