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A nonparametric comparison of two multipleregressions by means of a weighted measure ofcorrelationIbrahim Salama a & Dana Quade aa Department of Biostatistics, University of North Carolina, Chapel Hill, NC,27514Published online: 27 Jun 2007.

To cite this article: Ibrahim Salama & Dana Quade (1982) A nonparametric comparison of two multiple regressionsby means of a weighted measure of correlation, Communications in Statistics - Theory and Methods, 11:11,1185-1195

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A NONPAKAMETKIC C0EU'AI:ISON O F TWO PtUL'IIPLE RECRESSIONS MY MANS OF A WGILII'I'LD PlEASUKE O F COKHEIATIUN

1l ra l i i111 5 a l a n ~ a and Dana Quade

DepnrLn~cnt o f B i o s t a t i s t i c s U n i v e r s i L y o f Nor th C a r o l i n a

Clrapsl Hill, NC 27514

Key Wolds and ~'hrases: n ~ u l t i p l e r e g r e s s i o n ; r ank c o r r e l a t i o n ;

S p e a r n u n ' s f o o t r u l e

Let ( R l j * - , R . ) , j = 1 , 2 , b e two r a n k i n g s o f m i t e m s ; l e t

nl J T k , k-1, ..., m, be t h e n u ~ ~ i b c r of i t e m s w i t h r a n k 5 k i n b o t h

r a n k i n g s ; and l e t T = Z'Tk/k . Tlien T is a measure of r a n k

c o r r e l a t i o n wlrich g i v e s g r e a t e r we igh t t o i t e m s o f l o w r a n k t h a n

h igh . Such a measure is p a r t i c u l a r l y u s e f u l i n c o n ~ p a r i n g t h e

o r d e r i n g of t h e r e g r e s s o r s i n two m u l t i p l e r e g r e s s i o n s . We

d i s c u s s the d i s t r l b u t l o ~ i of T , p r e s e n t i n g b o t h e x a c t t a b l e s and

i ~ r a c t i c a l a p p r o x i m a t i o n s , a d e x t e n d t h e c o n c e p t t o o t h e r

s i t u a t i u ~ i s .

1 . INI'KODIICTION -- Suppose we wish t o coinpare two p o p u l a t i o n s w i t h r e s p e c t t o

predict in^ a r e s p o n s e v a r i a b l e Y f rom r e g r e s s o r v a r i a b l e s

X, ,X2,. . . ,X I n a s k i n g whe the r t h e y a r e " s i n ~ i l a r " , however, we m'

do n o t mean w h e t h e r t h e r e g r e s s i o n coefficients are similar, s i n c e

t h e v a r i a b l e s observable i n o u r samples may o n l y b e p r o x i e s f o r

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u n o b s e r v a b l e u n d e r l y i n g v a r i a b l e s , m e a s u r e d d i f f e r e n t l y i n t h e two

p o p u l a t i o n s . R a t h e r , we a r e c o n c e r n e d w i t h t h e r a n k i n g o f t h e X ' s

by i m p o r t a n c e . T h i s m i g h t b e e s t a b l i s h e d by p e r f o r m i n g a s t e p w i s e

r e g r e s s i o n i n e a c h s a m p l e , w i t h t h e s i m i l a r i t y o f t h e two p o p u l a -

t i o n s tlie.1 m e a s u r e d by tht . r a n k correlation. As we s e e I t , I ~ o w c v e r .

t h e u s u a l r a n k c o r r e l a t i o n c o e f f i c l e r ~ t s a r e n o t i d e a l F o r t h i s

p u r p o s e , b e c a u s e t h e y g i v e e q u a l w e i g h t t o a l l i t e m s r a n k e d . F o r

e x a m p l e , s u p p o s e we h a v e t h r e e r e g r e s s o r s X1,X2,XJ, a n d i n o n e

s a m p l e t h e s t e p w i s e p r o c e d u r e p l a c e s them i n p r e c i s e l y t h a t o r d e r .

C o n s i d e r two p o s s i b i l i t i e s f o r t h e o r d e r i n t h e o t h e r s a m p l e :

( a ) X1,X3,X2, a n d ( b ) X2,X1,X3. E i t l ~ e r p r o d u c e s t h e same r a n k

c o r r e l a t i o n by t h e w u a l m e t h o d s . B u t ( a ) , w h e r e t h e two s a m p l e s

a g r e e o n t h e f i r s t r e g r e s s o r , s u r e l y r e p r e s e n t s more s i m i l a r i t y

t h a n ( b ) , w h e r e t h e y d i s a g r e e o n t h e f i r s t r e g r p s s o r a n d a g r e e

i n s t e a d o n t h e l a s t . T l ~ u s we s e e k a r a n k c o r r e l a t i o n c o e f f i c i e n t

w h i c h g i v e s g r e a t e r w e i g h t t o i t e m s o f low t h a n h i g h r a n k .

L e t R b e t h e r a n k of t h e i - t h v a r i a b l e i n the 1-t11 pop- 1 j

u l a t i o n , f o r i = L , ..., m a n d j = 1 , 2 ; a n d l e l I

Then Tk

is the number o f r e g r e s s o r s common t o t h e two r e g r e s s i o n s

a t t h e k - t h s t e p . w h i c h i s a n i n t u i t i v e m e a s u r e o f . s i m i l a r i t y a t

t h a t s t e p . With i d e n t i c a l p o p u l a t i o n s , t n have Tk = k f o r

k - 1 , . . . . ~ n w o u l d b e t h e mos t l i k e l y r c s u l t ; o r , f i n d i n g Tk = k

f o r k = l , .... m w o u l d i n d i c a t e s t r o n g l y t l l n t the two p o p u l a t i o n s

a r e s i m i l a r . A c o n s i d e r a b E e d e v i a t i o n f rom t h i s s l t u a t i v n w o u l d

s u g g e s t a high d e g r e e o f n o n - s i m i l a r i t y .

As a s t a t i s t i c t o m e a s u r e s i m l l a r j t y o v e r a l l s t e p s we

p r o p o s e

i n w l ~ l c h e a c h Tk is d i v i d e d by k i n o r d e r t o r e f l e c t t h e

d e c r e a s i n g i m p o r t a n c e o f t h e s t e p s . The maximum v a l u e o f T

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o c c u r s i f t h e two samples y i e l d t h e same r a n k i n g of t h e X ' S , i n

which c a s e Tk = k f o r a l l k , and T = 111. The minimuu v a l u e

o c c u r s i f ( t hough n o t o n l y l i ) ~ t r e X ' s a r e i n r e v e r s e o r d e r ; t h e n

ei.1~11 Tk = mdx(0, 2k-m), a n d

Note t h a t t (m)/m lrds t lrc same v a l u e f o r e a c h even i n t e g e r

N a s f o r t h e f o l l o w i n g odJ i n t e g e r : t ( 2 ) / 2 = t ( 3 ) / 3 = 112,

t ( 4 ) / 4 = t ( 5 ) / 5 = 5 / 1 2 , t (6116 = t ( 7 ) / 7 = 23/60, t ( 8 ) / 8 - t ( 9 ) / 9 - 307/840, e t c . I n a d d i t i o u , t(ni)/rn is n o n i n c r e a s i n g i n m, and ,

a s m + t (ni)/m -+ 1 - log,2. Thus we c o u l d r e s c a l e T t o l i e

i n Llre r a n g e (-1, 1 ) . o b t d i u i n g ( i f we u s e t h e l i ~ n i t i n g v a l u e of

L ) J. weigllLed r a n k c o r r r l o t i u n c o e f f i c i e n t C = 1 - 2(m-T)/rn(log,2),

bu t we s h a l l n o t b o t h e r t o do t h i s i n wltat f o l l o w s .

I t i s a l s o i n t e r e s t i n g t o n o t e t h a t t h e unweighted measure

S = I T k , which we would have o b t a i n e d had we n o t d i v i d e d e a c h Tk by k , i s r e l a t e d t o t h e meLrlc U Ilk - R . I whose s t a n d a r d i z e d

i l 1 2 v e r s i o n i s known a s S p e a r n ~ a n ' s f o o t r u l e ; Saldma (1981) p r o v e s t h a t

D = 111(nrtl) - 25.

2 . A 1'EST FOR SIEIII.AKI1Y

R e c a l l i n g t h a t R I s t l le r a n k of t l re i - t h v a r i a b l e i n t h e i j

j - t l ~ p a p u l a t i o n , de f i n c

pi:) = P { K = r f o r j = 1 , 2 and i , r = I .... ,rn . 11

'l'l~en we call w r i t e t h e e x p e c t a t i o n of Tki a s

s o t h a t a g e n e r a l e x p r e 6 s i o n f o r t h e e x @ c t a t i o n o t t h e s t a t i s t i c

T is

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1188 SAIAMA AND QUAYJE

We s a y t h a t r e g r e s s i o n i n t h e j - t l ~ p o l ) u l n t i o n has ~ i n i j b r m

1 p =

* f o r i = I , . . ." *

i n w h i c h case

If r e g r e s s i o n i n o n e p o p u l a t i o n h a s u n i f o r m s t r u c t u r e

l o s s o f g e n e r a l i t y , l e t i t b e t h e f i r s t -- t l w n

W s p e c i a l c a s e o f u n d f o n n s t r u c t u r e is tvrliicm strscctu

me1 = -- 2 -

e , f o r wl~lc la

a l l r a n k i n g s o f t h e v a r i a b l e s a r e e q u a l l y I i k e l y , i . e . ,

P ( R , ~ = K ~ , . . . =r = I/"! f o r a l l p e r m u t a t i o n s ( r l , ..., r m ) sRmj m

of 1 . m T h c n u l l h y p o t h e s i s which we s h a l l test u s i n g

T is t h a t t h l s is t r u e f o r a t l e a s t o n e of t h e p o p u l a t i o n s .

The a l t e r n a t i v e we h a v e i n mind is t h a t t h e r e g r e s s i o r i s i n

the two p o p u l a t i o n s Rave non-uni f orm sfmilat' structurso, by whicl i

we mean t h a e ( 1 ) = ( 2 ) * pir pir

pir ( s a y ) f o r i , r , = 1, .... m, w h e r e

P i r 2 l/m f o r a t l e a s t o n e ( f , r ) . I'llcn

However, s i n c e

is f i x e d . i t f o l l o w s t h a t

w i t h t h e l o w e r bo11nJ l m l d f n g i f arid o n l y i f t h e comninn

r e g r e s s i o n s t r u c t r ~ r e fs u n i f o r m . H e n c e , ~ t n l l e r t h e a 1 t e r n n t i v e ,

E[T) > (m+1) /2 ; i . e . . o u r t e s t s t a t i s t i c h a s a g r e a t e r mean u n d e r Dow

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'IWO MULTIPLE REGRESSIONS

t h e a l t e r n a t i v e of s i m i l a r nori-uniform s t r u c t u r e s f o r t h e

r e g r e s s i o n s i n t h e two popula t iono t h a n under t h e hypothes i s of

randou~ s t r u c t u r e f o r a t l e a s t one of them.

Now c o n s i d e r some h igher moments under t h e n u l l hypothes i s .

We have

if i,ii,iB1 a l l

d i s t i n c t ,

where lsk<k'sk"Lm;thun a l i r t l e a l g e b r a y i e l d s

Thence f i n a l l y w e c a l c u l a t e

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SALAElA AND QUADE

mtl V[T] - 12

and

The conditional distribution of Tk+l, given T Iatl~ T2=t2*

. . . , Tk=tk, and assuming that the null hypothesis is' true, may be derived as follows. Beginning with the initial condition that

P(T~-o] - (m-l)/m and P{T~=~) = l/m, suppose the joint distri-

bution of TI, ..., Tk has been found. Then define four sets of

variables :

A - (1: Rillk and ~ ~ ~ s k l , B = [i: Rillk but ~ ~ ~ > k ) ,

C = (1: Ril>k but ~ ~ ~ s k ) , D = {i: Rll>k and Ri2>k).

Note that A contains tk variables, B and C contain (k-tk)

variables each, and D contains the remaining (m-2k+tk) variables.

Then, at step (k+l), tile variable chosen in Sample 1 dust be from

C or D, end that chosen in Sample 2 must be from B or D. We have

the following possibilities:

Varinl)le Vnrinhle Prolmbi lity Resulting in Sample 1 in Sample 2 ~ ( m - t ~ ) ~ value OF Tk+l

from: from:

c n (k-tL) 2 '

C I) (k-tk) (m-2k+tk)

D B (k-tk) (m-2k+tk)

different 'variables 1 (m-2k+tk) (ln-2k+tk-1)

' [varS%Fe I ' (m-2k+tk)

Summing up,

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TWO MULTIPLE REGRESSIONS

Thus we have derived recursive relationships by which exact tables

can be calculated. It may be seen, by the way, that the distribution

Of Tk+l actually depends only on tk. Thus {Tl,~2,. . . ,T io

a hrkov chain.

Table I presents exact critical values of T for Type I error

probability a0.10, .05, .025, .01, and .005 and number of variables

m=5(1)15. The entire exact null-hypothesis distributione of T

for m=3(1)10, and selected values from the distributions for

m=ll(l)lS, are available in Salama and Quade (1981). When outside

the range OF the table, one mlght use

as a standard normal variable. This would ignore the positive

skew~iebis in the distribution of T, however, and thus generally

produce anti-conservative P-values (too small). Instead, our

recomendation is to treat a linear function of T as a chi-

squared. Spccifically, to have (a+bT) approximately distributed 2 as x (d), equate the first three central momenta:

where for convenience the exact third aaoment of (a+bT) has been

replaced by a sligt~tly lower value. Solving theee equations

yields a - -4(mC1)/3, b = 8, and d = 8(m+1)/3, whence

This approximation fits two moments exactly and the third fairly

well.

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SAIAMA AND QUADE

Number ol Variable1

m

TABLE I

CRITICAL VAI.UES OF THE TEST srtvrrsrlc T

Type 1 Error 13ro!)ability, a

We lllustrate these approximations at 111114. For any observed

T, the normal approximation to the P-value is P(Z 2 ( ~ - 7 . 5 ) / f i I

where Z - N(0,1), and the chi-equared approximation is P{w 2 2 8 ~ - 2 0 ) where W - x (40). Then we have, obtaining the exact

P-values from Quade and Salama (1981):

T Exact Norma 1 Chi - squared P-value Approximation Aproximnt ion

9 .lo1 32 .08386 .09682

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For t l ~ i s number of v a r i a b l e s t h e nornlal a p p r o x i m a t i o n is s t r o n g l y

a n t i c o n s e r v a t i v e , b u t t h e cl11-squared a p p r o x i m a t i o n a p p e a r s

a d e q u ~ t e f o r p r a c t i c a l p u r p o s e s .

3. EXTIINS I Oh's AND DISCllSSION

A l t l ~ o u g h we have p roposed t h e s t a t i s t i c T a s a n i n d e x f o r

cornparlng t h e r a n k i ~ l g s of t h e r e g r e s s o r s i n two p o p u l a t i o n l ; , i t i c i

e q u a l l y w a l l s u i t e d f o r c o ~ n p a r i n g some p r e s p e c i f i r d r a n k i n g wi t i t

t h e r a n k i n g o h s e r v e d i n a s i n g l e p o p u l a t i o n . Wi thou t l o s s

g e n e r a l i t y , r e l a b e l t h e v a r i a b l e s s o t h a t t h e p r e t l p e c i f i e d

r a n k i n g b e c o m s I , . . . , m . Lc t Wi b e t h e o b s e r v e d r a n k o f

i - t h v a r i a b l e , and p i r = Pi l l i = r ) f o r r = 1 ,.... m. Then

a l g e b r a w i l l e s t a b l i s h t l rn t

i n

t h e

a l i t t l e

w i t l ~ E l ' f ] = (1n t1 ) /2 i f t l ~ e p o p u l a t i o n h a s u n i f o r m r e g r e s s i o n

s t r t ; c t u r c . T l ~ u s o n c s a y t r o t t i le h y p o t h e s i s o f raodom s t r u c t u r e .

unde r w l ~ i c h T o b v i o u s l y h a s t h c d i s t r i b u t i o n p r c v i o u e l y d e r i v e d ,

a g a i n s t t h e a l t e r n a t i v e t l u t E[T] > ( iat1)/2.

The s t a t i s t i c T c a n a l s o b e used t o t e s t f o r s i m i l a r i t y o f

r e g r e s s i o n s t r u c t u r e aurony n > 2 p o p u l a t i o n s . L e t T b e t h e

statistic c a l c u l a t e d f rom t h e o b s e r v e d r e s u l t s f o r p o p u l a t i o n s i

dud j , where i j = l n . Then a n i n t u i t i v e l y r e a s o n a b l e t e s t

s t a t i s t i c is

which c a n b e v iewed a s a U - s t a t i s t i c oC d e g r e e 2 . Suppose the n

p o p u l a t i o n s a l l have s i m i l a r r e g r e s s i o n s t r u c t u r e s . Then, i f we

we o b t a i n

Under t h e n u l l h y p o t h e s i s of random s t r u c t u r e i n a l l p o p u l a t i o n s .

w e have

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whence

m t l mt' " [ T I = - - E[T] = -1 , 611 (n- 1 ) -

S i n c e E=O, t h e a s y m p t o t l c d i s t r i b u t i o n o f T is n o t n o m a l ;

h o w e v e r , we c a n o b t a i n a c h i - s q u a r e d a p p r o x i m a t i o n a s we d i d f o r

'T. L e t

63 = E [ T ( ' ~ ) - ~ ] ~ , IAI a E [ ( T ( ~ ~ ) - u ) (T ( 2 3 ) - u ) (T(31) -p) ;

t h e n B ( T - ~ ) + D Is a p p r o x i m a t e l y d i s t r i b u t e d a s c h i - s q u a r e d w i t h

D d e g r e e s o f f r e e d o m , w h e r e -3

Some a l g e b r a e s t a t l i s h e s t h a t (1: = m(m+1)/48(m-1) e x a c t l y ; i t is

c o n v e n i e n t t o r e p l a c e t h i s b y (m+1)/48. T h e n , r e p l a c i n g t h e e x a c t

v a l u e o f 8 by ( m t 1 ) / 2 4 a s b e f o r e , we o b t a i n P, = 4n a n d

D = 4 n ( m + 1 ) / 3 ( n - I ) , s o t h a t

N o t e t h a t t h i s a g r e e s w i t h t h e p r e v i o u s r e s u l t f o r n = 2 .

The c o m p a r i s o n o f m u l t i p l e r e g r e s s i o n s is, o f c o u r s e , n o t t h e

o n l y c o n t e x t i n wli ich a w e i g h t e d c o r r e l a t i o n s t a t i s t i c m i g l ~ t b e

u s e f u l ; o n e c a n i m a g i n e o t h e r s i t u a t i o r i s i n w h i c h t h e c l o s e n e s s of

t h e r e l a t i o n s h i p amohg t h e i t e m s r a n k e d e a r l i e r is more i m p o r t a n t

t h a n among t h o s e r a n k e d l a t e r . Nor Ls 'r t h e o n l y r e a s o n a b l e

s t a t i s t i c . F o r i n s t a n c e , i n s t e a d of m o d i f y i n g S p e a r m a n ' s f o o t r u l e ,

c o n s i d e r modl Eying S p e a r m a n ' s r h o f ram

* I t is e a s i l y shown t l l a t

Rs r a n g e s f rom - 1 when t h e two r n n k i n g s

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TWO I1UI.TIPLE KE(:KESSlONS

are opposite to 1 vlien they are identical, with expectatlon 0

if eltiler population has ra~~dom structure. It is not clear. * however, lmw to choose among T, Rs, and the innumerable other

posslbilitirs.

ACKNOWI.EDCEE.IENT

'l'lie work of the first author was sponsored by grant Hll07102

f ronl tlie Nat ionill 111s~itutr of Child Health and Human Development.

Salamn, I.A., (1981). A note on Spearman's footrule (unpublished manuscript).

Salau~a. 1 . A . . 6 Quade, D., (1981). A nonparametric comparison of the structure of two n~ultiple-regression prediction situations. institute of Statistics Mimeo Series 1325, University of Nortli Carolina at Chapel Llill.

Rccelved Eloy, 1981; Rcvr red S e p t f m l r r , 1 9 8 1 .

Hecorrur~r~dcd by H. E . O r l ~ l k , llniversi t y o f Victorla, Victoria, B.C . , C d n d d d .

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