Conditional Tests for Censored Match Pairs - 1985 Popovich and Rao

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Conditional Tests for censored matched pairsE.A. Popovich a; P.V. Rao b

a Harris Corporation, Melbourne, FL b Unversity of Florida, Gainesville, FL

To cite this Article Popovich, E.A. and Rao, P.V.'Conditional Tests for censored matched pairs', Communications inStatistics - Theory and Methods, 14: 9, 2041 — 2056To link to this Article: DOI: 10.1080/03610928508829029URL: http://dx.doi.org/10.1080/03610928508829029

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COMMUN. STATIST.-THEOR. METH., 1 4 ( 9 ) , 2 0 4 1 - 2 0 5 6 (1985)

E. A. Popov ich

t k r r i s C o r p o r a t i o n ' Melbourne, FL 32919

CONDITIONAL TESTS FOR CENSORED MATCHED PAIRS

a n d P. V . Rao

U n i v e r s i t y of F l o r i d a G a i n e s v i l l e , FL 32611

ABSTRACT

A c l a s s o f s t a t i s t i c s i s p roposed f o r t h e problem of t e s t i n g

f o r l o c a t i o n d i f f e r e n c e u s i n g randomly r i g h t c e n s o r e d matched p a i r

d a t a . Each member of t h e c l a s s p r o v i d e s a c o n d i t i o n a l l y d i s t r i b u -

t i o n - f r e e t e s t o f Ho: n o l o c a t i o n d i f f e r e n c e . S i m u l a t i o n r e s u l t s

i n d i c a t e t h a t powers o f t e s t s based o n c e r t a i n members i n t h e

c l a s s a r e a s good a s o r b e t t e r t h a n t h e power o f a t e s t proposed

by Woolson a n d Lachenbruch (1980).

INTRODUCTION

The problem of t e s t i n g f o r d i f f e r e n c e between two t r e a t m e n t s

u s i n g randomly r i g h t c e n s o r e d matched p a i r d a t a was examined by

Woolson a n d Lachenbruch (1980) among o t h e r s . Under t h e a s sumpt ion

t h a t b o t h members of a p a i r have e q u a l b u t random o b s e r v a t i o n

t i m e s , Woolson and Lachenbruch deve loped a f a m i l y of t e s t

s t a t i s t i c s f o r t e s t i n g t h e n u l l h y p o t h e s i s Ho: No d i f f e r e n c e i n

Copyright @ 1985 by Marcel Dekker, Inc.

Downloaded By: [Popovich, Edward] At: 12:40 24 November 2009

2042 POPOVICH AND RAO

t r e a t m e n t s . The Woolson-Lachenbruch (WL) s t a t i s t i c s u t i l i z e t h e

concep t of g e n e r a l i z e d r a n k v e c t o r i n t r o d u c e d by K a l b f l e i s c h a n d

P r e n t i c e (1973) . These s t a t i s t i c s were d e r i v e d by f o l l o w i n g t h e

d e r i v a t i o n of t h e l o c a l l y most power fu l (LMP) r a n k t e s t f o r

l o c a t i o n i n t h e uncensored matched p a i r s c a s e . Woolson and

I achenbruch p r e s e n t two v e r s i o n s of t h e i r s t a t i s t i c : one i s b a s e d

o n d o u b l e e x p o n e n t i a l s c o r e s a n d t h e o t h e r b a s e d o n l o g i s t i c

s c o r e s .

I n s p i t e of t h e f a c t t h a t t h e WL-tes ts p r o v i d e a n i n t u i t i v e l y

r e a s o n a b l e method f o r t e s t i n g Ifo, t h e i r c o m p l i c a t e d f u n c t i o n a l

form makes i t d i f f i c u l t t o i n v e s t i g a t e t h e i r t h e o r e t i c a l

p r o p e r t i e s . For example , i n v iew o f t h e n o t e by Woolson and

Lachenbruch ( 1 9 8 4 ) , i t i s n o t c l e a r t h a t t h e s e t e s t s a r e LMP i n

t h e c e n s o r e d c a s e . Tne proof of t h e i r a s y m p t o t i c n o r m a l i t y , found

i n t h e WL-paper, i s s t r i c t l y v a l i d o n l y when N l , t h e number of

uncensored p a i r s , a n d N 2 , t h e number of s i n g l y c e n s o r e d p a i r s , a r e

r e g a r d e d a s nonrandom w i t h N , + N 2 t e n d i n g t o i n f i n i t y . It i s n o t

known whe the r t h e a s y m p t o t i c n o r m a l i t y h o l d s u n c o n d i t i o n a l l y a s n ,

t h e t o t a l number o f p a i r s , t e n d s t o i n f i n i t y .

For s m a l l sample s i z e s e x a c t c r i t i c a l v a l u e s f o r t h e WL-test

c o u l d b e b a s e d on t h e i r c o n d i t i o n a l p e r m u t a t i o n d i s t r i b u t i o n ,

c o n d i t i o n a l on t h e o b s e r v e d c e n s o r i n g p a t t e r n . However, t h e

d e t e r m i n a t i o n o f t h e s e c r i t i c a l v a l u e s becomes p r o g r e s s i v e l y

i m p r a c t i c a l when max(N1 ,N2) > 8 and min(N1,N2) > 2 .

The o b j e c t i v e i n t h i s p a p e r i s t o p ropose a n a l t e r n a t i v e

c l a s s of s t a t i s t i c s , r e f e r r e d t o a s Cn i n t h e s e q u e l , f o r t e s t i n g

h0. As w i l l b e s e e n f r o m what f o l l o w s , t h e s t a t i s t i c s i n cn w i l l

b e c o m p u t a t i o n a l l y s i m p l e and w i l l n o t s u f f e r f rom t h e drawbacks

of t h e W L - s t a t i s t i c s . F o r example, a Tn E ( f n w i l l be

a s y m p t o t i c a l l y normal a s n + m under s u i t a b l e r e g u l a r i t y

c o n d i t i o n s , and c o n d i t i o n a l on t h e o b s e r v e d v a l u e s of N 1 a n d N2

w i l l b e d i s t r i b u t i o n - f r e e . Because t h e s e c o n d i t i o n a l n u l l

d i s t r i b u t i o n s a r e d i s t r i b u t i o n s o f l i n e a r c o m b i n a t i o n s o f

i n d e p e n d e n t b i n o m i a l and Wilcoxon s i g n e d r a n k s t a t i s t i c s , t h e

c r i t i c a l v a l u e s of t h e s t a t i s t i c s i n (f a r e e a s i l y t a b l u l a t e d f o r


CENSORED MATCHED P A I R S 2043

a l l r e a s o n a b l e v a l u e s o f N1 a n d N 2 . F u r t h e r m o r e , s i m u l a t i o n

r e s u l t s i n d i c a t e t h a t t h e powers of c e r t a i n members o f cn a r e a s

good a s , and i n some c a s e s b e t t e r t h a n , t h e power of t h e WL-

s t a t i s t i c s .

2. N O T A T I O N a n d PRELIMINARY RESULTS

C o n s i d e r a n expe r imen t i.n which two t r e a t m e n t s a r e a p p l i e d t o

e a c h of t h e n matched p a i r s of e x p e r i m e n t a l u n i t s r e s u l t i n g i n t h e

r e s p o n s e (x? x0 ) , i = 1,2, ..., n. L e t Ci b e t h e v a l u e of a 11' 1 2

c e n s o r i n g v a r i a b l e common f o r t h e i t h p a i r . The o b s e r v e d r e s p o n s e s

f o r t h e i t h p a i r a r e (Xil ,Xi2) and (6i1, di2) where X i j = min(XYj,

Ci) a n d 8 . . = I o r U a c c o r d i n g a s X. = X Y j o r C i . I n what 1~ l j

t h e a s sumpt ions A1 - A 5 l i s t e d b e l o *

i a t e .

f o l l o w s , v a r i o u s s u b s e t s of

w i l l be u t i l i z e d a s appropr

A l : (xY1, x:~) a r e i n d e p e r [ d e n t i d e n t i c a l l y d i s t r i b u t e d ( i . i .d.)

a s a c o n t i n u o u s b i v a r i a t e random v a r i a b l e (x:, x:).

A2: There e x i s t s a r e a l number 0 s u c h t h a t (x:, X; + 0) i s a n

exchangeab le p a i r .

A3: Ci i = 1 , 2 , . ..,n a r e i . i . d . a s a c o n t i n u o u s random v a r i a b l e

C. A l so , C a n d (x? , x;) a r e r m t u a l l y i n d e p e n d e n t .

A 4 : Ci a n d (XP1, XY2) i = 1 , 2 , . . . ,n a r e m u t u a l l y independen t .

A 5 : Let P I . } d e n o t e p r o b a b i l i t y c a l c u l a t e d w i t h t h e v a l u e o f 0 0

s a t i s f y i n g A2. Then P ~ { x : > C , i = 1 , 2 ) < 1.

L ike t h e W L - s t a t i s t i c s , t h e s t a t i s t i c s we wish t o c o n s i d e r

a r e b a s e d o n t h e o b s e r v e d d i E f e r e n c e s Di = X i l - Xi2. A

d i f f e r e n c e Di w i l l b e s a i d t o b e l o n g t o c a t e g o r y C1 i f bo th Xil

a n d Xi2 a r e uncensored , t o c a t e g o r y C2 i f e x a c t l y one o f Xil a n d

Xi2 i s uncensored , and t o c a t e g o r y C3 i f b o t h Xil and Xi2 a r e

c e n s o r e d . As n o t e d by Woolson a n d J a c h e n b r u c h (1980) a Di i n C1

w i l l e q u a l t h e t r u e d i f f e r e n c e D: = xY1 - x:*, whereas a Di i n C2

w i l l imply t h a t t h e t r u e d i f f e r e n c e DP i s l e f t o r r i g h t c e n s o r e d



a t t h e o b s e r v e d d i f f e r e n c e Di a c c o r d i n g a s 6il = 1 o r 0 . The

o r d e r r e l a t i o n s h i p be tween Di a n d D; f o r t h o s e Di i n C3 i s

unknown.

Suppose f o r k = 1 , 2 , and 3 , Nk d e n o t e s t h e o b s e r v e d number of

p a i r s i n c a t e g o r y Ck, a n d l e t (Xilk, XiZk), xi = Xilk - Xi2k, i =

1 , 2 , . . . ,Nk d e n o t e t h e s e p a i r s a n d t h e i r d i f f e r e n c e s r e s p e c t i v e l y .

C l e a r l y we h a v e ,

B e f o r e p r o c e e d i n g f u r t h e r , some comments c o n c e r n i n g

a s s u m p t i o n s A1-A5 a r e i n o r d e r . From a s s u m p t i o n A2 i t f o l l o w s

t h a t XO a n d XO + 0 h a v e i d e n t i c a l m a r g i n a l d i s t r i b u t i o n s . 1 2

Assumption A3 r e q u i r e s t h a t b o t h X: and XO have a common 1 1 1 2

c e n s o r i n g t i m e Ci which may be a n u n r e a l i s t i c a s s u m p t i o n i n some

a p p l i c a t i o n s . I f C . i s t h e c e n s o r i n g t i m e f o r xo t h e n o u r 1 j i j '

model would b e a p p r o p r i a t e i f Ci = min (Ci1,Ci2) i s r e g a r d e d a s 0 0

t h e common c e n s o r i n g t i m e f o r ( x ~ , , X ~ ~ ) . Assumpt ion A5 i m p l i e s

PO{N1 + N2 > 0) > 0 , s o t h a t w i t h p o s i t i v e p r o b a b i l i t y t h e r e w i l l

be a t l e a s t one p a i r i n c a t e g o r y C1 o r C2. Obv ious ly , i f

Pe{Nl + N 2 > 0 ) = 0 , t h e n e v e r y p a i r b e l o n g s t o c a t e g o r y C3 a n d

t e s t s b a s e d on o b s e r v e d d i f f e r e n c e s Di w i l l n o t be i n f o r m a t i v e .

If P0(N2 = N 3 = 0 ) = 1, t h e n t h e problem u n d e r c o n s i d e r a t i o n

i s t h e u s u a l uncensored matched p a i r s problem a n d t h e r e e x i s t a

number of l i n e a r s i g n e d r a n k t e s t s (Theorem I1 4.9 , Hajek ( 1 9 6 7 ) )

f o r t e s t i n g Ho:O = 0. The well-known Wilcoxon s i g n e d r a n k t e s t i s

a n i m p o r t a n t example . However, i f P0{N2 > 0) > 0, t h e n t h e

s i t u a t i o n i s d i f f e r e n t b e c a u s e t h e t r u e d i f f e r e n c e DP between a

p a i r i n C2 w i l l be r i g h t o r l e f t c e n s o r e d a t t h e o b s e r v e d

d i f f e r e n c e D i , and , c o n s e q u e n t l y , n o t a l l o b s e r v e d d i f f e r e n c e s

w i l l e q u a l t h e t r u e d i f f e r e n c e s . We s h a l l u s e t h e symbols N2= and

N2R t o d e n o t e , r e s p e c t i v e l y , t h e number o f l e f t - c e n s o r e d and t h e

number of r i g h t - c e n s o r e d d i f f e r e n c e s . Note t h a t


CENSORED MATCHED PAIRS

N2 = N Z L + N 2 R , N Z L = E6i1(l-6i2) a n d N Z R = C(1-6 ) 6 i l 1 2 '

Then Pk(B) d e n o t e s t h e p r o b a b i l i t y t h a t a n o b s e r v e d p a i r f a l l s

i n t o c a t e g o r y Ck, k = 1 ,2 , and , PZL(B) a n d P2R(8) d e n o t e ,

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

c e n s o r e d o r r i g h t - c e n s o r e d .

Lemma 2.1: Suppose a s s u m p t i o n s Al , A2, A3, a n d A4 h o l d .

( a ) If 6 = 0 a n d P l ( 0 ) > 0 , t h e n c o n d i t i o n a l o n N l = n1 > 0 ,

D l l , D12,...,D h a v e i n d e p e n d e n t i d e n t i c a l d i s t r i b u t i o n s In1

symmet r i c a b o u t 0.

( b ) I f P 2 ( 8 ) > 0, t h e n t h e c o n d i t i o n a l d i s t r i b u t i o n of NZR,

c o n d i t i o n a l o n N2 = n2 > 0 , i s b i n o m i a l w i t h p a r a m e t e r s n2

a n d PZK(8) /P2(8 ) .

( c ) C o n d i t i o n a l o n Ni = ni , i = l , 2 , 3 , t h e random v a r i a b l e s

DlI,D12,.. . ,D a n d NZR a r e r m t u a l l y i n d e p e n d e n t . In1

P roof o f Lemma ( 2 . 1 ) , wh ich i s s t r a i g h t f o r n a r d , w i l l b e l e f t o u t

f o r b r e v i t y . Lemma ( 2 . l a ) s u g g e s t s t h a t t h e obse rved d i f f e r e n c e s

i n Cl c o u l d be u t i l i z e d t o d e f i n e a c o n d i t i o n a l t e s t o f Ho: 8=O.

C l e a r l y , a t e s t i n t h e f a m i l y of l i n e a r s i g n e d r a n k t e s t s (Ha jek ,

1967) c a n b e s e l e c t e d o n t h e b a s i s of a n assumed d e n s i t y f o r

DIj . However, i n t h i s p a p e r we s h a l l r e s t r i c t o u r a t t e n t i o n t o

t h e Wilcoxon s i g n e d r a n k t e s t b e c a u s e o f i t s h i g h e f f i c i e n c y f o r a

wide v a r i e t y of d i s t r i b u t i o n s a n d t h e a v a i l a b i l i t y of t a b l e s o f

c r i t i c a l v a l u e s f o r a l a r g e s e l e c t i o n of sample s i z e s .

Accord ing ly we c o n s i d e r t h e s t a t i s t i c



where , + ( u ) = 1 o r 0 a c c o r d i n g a s u > 0 o r < 0 and R; i s t h e r a n k

of 1 ~ 1 ~ 1 among I D I I I ~ I D ~ ~ I ~ ~ ~ ~ I D ~ N ~ I *

From Lemma (2. l b ) , i t f o l l o w s t h a t u n d e r Ho: 0=0, t h e

c o n d i t i o n a l d i s t r i b u t i o n o f N 2 R i s b i n o m i a l w i t h p a r a m e t e r s n2 and

L/2. T h e r e f o r e , a s e c o n d t e s t o f Ho:8 = 0, u t i l i z i n g d i f f e r e n c e s

i n C2, c o u l d b e b a s e d o n t h e s t a t i s t i c

C l e a r l y , u n d e r Ho:B=O, t h e c o n d i t i o n a l t e s t s based o n Tln a n d TZn

a r e d i s t r i b u t i o n - f r e e a n d f r o m Lemma (2 . l c ) t h e s e t e s t s a r e

i n d e p e n d e n t . h u s , a n y method o f combining two i n d e p e n d e n t t e s t s

c o u l d b e u t i l i z e d t o d e v e l o p a d i s t r i b u t i o n - f r e e c o n d i t i o n a l t e s t

o f Ho:8=0. C r i t i c a l v a l u e s f o r t h e s e t e s t s a r e e a s i l y t a b u l a t e d

by u t i l i z i n g t h e c r i t i c a l v a l u e s f o r i n d e p e n d e n t Wilcoxon-signed

r a n k a n d b i n o m i a l t e s t s . I n S e c t i o n 3 we c o n s i d e r a c l a s s o f

s t a t i s t i c s formed by c e r t a i n l i n e a r combina t ions o f Tln and T2n.

Throughout h e r e a f t e r , c o n d i t i o n a l d i s t r i b u t i o n s r e f e r t o

d i s t r i b u t i o n s c o n d i t i o n a l o n Ni = n i , i = l , 2 , 3 . It i s c l e a r t h a t

u n d e r Ho:8=0, t h e c o n d i t i o n a l mean and v a r i a n c e of Tin a r e ,

r e s p e c t i v e l y , pin(ni) a n d u? ( n . ) , i = 1 , 2 where v l n ( n l ) = i n 1

( ) n l ( n l + l ) , ~ ~ ~ ( n ~ ) = O , o f n ( n l ) = ( l / 2 4 ) n l ( n l + l ) ( 2 n l + l ) a n d

u;,(n2) = n2. L e t T;,(N~) d e n o t e t h e s t a n d a r d i z e d v e r s i o n s (unde r

Ho) o f Tin. Tha t i s ,

For t e s t i n g Ho:8 = 0 , we p ropose u s i n g

f o r m

a l i n e a r combina t ion i n t h e


CENSORED MATCHED PAIRS 2047

where {L,} i s a sequence o f random v a r i a b l e s s a t i s f y i n g t h e

c o n d i t i o n s

( i ) : 0 4 Ln < 1 ,

( i i ) : Ln = Ln(N,, N2) i s a f u n c t i o n o f N 1 a n d N 2 o n l y ,

a n d

( i i i ) : Ln = L + o ( 1 ) a s n + - f o r some c o n s t a n t L. P

We s h a l l d e n o t e by cn t h e c l a s s of s t a t i s t i c s d e f i n e d by (3 .2) .

It i s e a s i l y s e e n t h a t u n d e r Ho, e v e r y member of cn h a s

c o n d i t i o n a l mean a n d s t a n d a r d d e v i a t i o n e q u a l t o 0 a n d 1

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

a b o u t 0. S i n c e t h e c o n d i t i o n a l n u l l d i s t r i b u t i o n of Tn(N1,N2) i s

a c o n v o l u t i o n o f independen t Wilcoxon s i g n e d r a n k and b i n o m i a l

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

t h e s t a t i s t i c s i n On a r e e a s i l y o b t a i n e d f o r s m a l l t o modera t e

v a l u e s of n i , i = 1 ,2 . 'Lheorem 3.1 s t a t e s c o n d i t i o n s unde r which

Tn E On will b e a s y m p t o t i c a l l y normal.

'Iheorem 3.1. If a s s u m p t i o n s A1 - A 5 h o l d , t h e n f o r e v e r y Tn in

On and r e a l number x,

where a ( * ) i s t h e cd f of a s t a n d a r d normal random v a r i a b l e .

A p roof o f Theorem 3 .1 may b e based on a n a d a p t a t i o n o f

'Iheorem 1 of Anscombe (1952) s t a t e d a s Lemma 3.1. Proof o f Lemma

3.1 , wh ich i s s t r a i g h t f o r w a r d , may b e found i n Popovich ( 1 9 8 3 ) .

Lemma 3.1. L e t {Tn 1 , n l = 1 , 2 , . . . , n2= 1 ,2 , . . . , b e a n a r r a y o f 1 '"2

random v a r i a b l e s s a t i s f y i n g c o n d i t i o n s ( i ) a n d ( i i ) .

CONDITION ( i ) : There e x i s t s a cd f F ( * ) s u c h t h a t a t e v e r y

c o n t i n u i t y p o i n t x of F ( * ) Downloaded By: [Popovich, Edward] At: 12:40 24 November 2009


CONDITION ( i i ) : C o r r e s p o n d i n g t o e a c h E , T- > 0, t h e r e e x i s t

c o n s t a n t s v a n d c s u c h t h a t m i n ( n l , n 2 ) > v i m p l i e s

( n f - n i ) < c n i , i = 1 ' 2 ) > 1- q .

} i s a s e q u e n c e o f i n t e g e r v a l u e d random v a r i a b l e s s u c h n t h a t p-l im n-lNin = A. A . > O , i = 1 , 2 , t h e n

1' 1

a t e v e r y c o n t i n u i t y p o i n t x o f F ( - ) .

A s k e t c h o f a p r o o f o f Theorem 3 . 1 . L e t ~ : , ( n ~ ) r e f e r t o t h e

random v a r i a b l e T ~ , ( N ~ ) c o n d i t i o n a l o n Ni = n i , i = 1 , 2 . C l e a r l y

TZn(ni) i s a s y m p t o t i c a l l y n o r m a l , a s n . + -, a n d by Lemma 2 . l ( c ) ,

~ : , ( n ~ ) a n d T i n ( n 2 ) a r e m t u a l l y i n d e p e n d e n t . T h e r e f o r e ,

i s a s y m p t o t i c a l l y n o r m a l a s m i n ( n l , n 2 ) + m. T h a t i s , {T;1,n2}

s a t i s f i e s CONIIITION ( i ) o f Lemma 3.1. Now s i n c e f o r ni + m, we

h a v e t h e r e p r e s e n t a t i o n ~ q , ( n ~ ) = ~ Z ~ ( n . 1 + o ( I ) , w h e r e LlZn(ni) P

i s a s t a n d a r d i z e d U - s t a t i s t i c b a s e d o n ni o b s e r v a t i o n s , we c a n

u t i l i z e t h e r e s u l t o f S p r o u l e ( 1 9 7 4 ) t o show t h a t ~ q ~ ( n ~ )

s a t i s f i e s c o n d i t i o n C2 o f 4nscombe ( 1 9 5 2 ) f o r i = 1 , 2 . The r m t u a l

i n d e p e n d e n c e o f ~ : ~ ( n ~ ) a n d ~ ; ~ ( n ~ ) i m p l i e s , a f t e r some a l g e b r a ,

t h a t {TG 1 s a t i s f i e s CONIIITION ( i i ) o f Lemma 3.1. Note t h a t l '"2

u n d e r Ho, P i ( 0 ) = p - l i m n - ' ~ ~ a s n + m f o r i = 1 , 2 a n d by A 5 Pi(U)

> 0 f o r a t l e a s t o n e i , i = 1 , 2 . T h e r e f o r e , f r o m Lemma 3.1, i t

f o l l o w s t h a t u n d e r HO, T i l r N 2 i s a s y m p t o t i c a l l y n o r m a l a s n + m.

Be p r o o f t h a t TN i s a s y m p t o t i c a l l y n o r m a l u n d e r HO 1 ' 2

f o l l o w s b e c a u s e a s n + -



Theorem 3.1 shows t h a t f o r l a r g e v a l u e s of n, a n uncondi-

t i o n a l t e s t b a s e d on t h e a s y m p t o t i c n o r m a l i t y o f Tn(N1,N2) c a n b e

u s e d f o r t e s t i n g HO. I n S e c t i o n 4 we c o n s i d e r t h e problem o f

s e l e c t i o n o f a s t a t i s t i c f rom En.

4 . A STATISTIC FOR TESTING Ho

S i n c e e v e r y s t a t i s t i c i n En is a l i n e a r combina t ion of two

s t a t i s t i c s which a r e c o n d i t i o n a l l y i n d e p e n d e n t a n d a s y m p t o t i c a l l y

normal , i t i s n a t u r a l t o c o n s i d e r t h e p o s s i b i l i t y of s e l e c t i n g a

s t a t i s t i c f rom K n b a s e d on c o n d i t i o n a l P i tman A s y m p t o t i c R e l a t i v e

E f f i c i e n c y (PARE) a s min (n l ,n2) + =. Theorem 10.2.3 of S e r f l i n g

(1980) c a n be used t o d e r i v e a n e x p r e s s i o n f o r L s u c h t h a t t h e

s t a t i s t i c TnL = ( l - ~ ) ' ~ T* ( n ) + Lm5 Tk ( n ) has 'Ondi- In 1 2n 2

t i o n a l PARE i n On. U n f o r t u n a t e l y t h e form of Tnl, depends on t h e

j o i n t d i s t r i b u t i o n o f (x~,x: ,c) i n a c o m p l i c a t e d manner a n d o u r

a t t e m p t s t o s i m p l i f y t h e e x p r e s s i o n a r e n o t s u c c e s s f ~ l l t h u s f a r .

A l t e r n a t e l y , o n e c a n s e l e c t s p e c i f i c l i n e a r combina t ions i n

and examine t h e i r PARE'S r e l a t i v e t o t h e W L - s t a t i s t i c s . I n

s p i t e of t h e f a c t t h a t e f f i c a c i e s o f s t a t i s t i c s i n Q n c a n he

d e r i v e d i n a s t r a i g h t - f o r w a r d manner t h e d e r i v a t i o n o f PARE'S w i t h

r e s p e c t t o t h e W L - s t a t i s t i c s d o e s n o t a p p e a r t o be a n e a s y t a s k .

The d i f f i c u l t y h e r e i s due t o t h e t h e o r e t i c a l l y c o m p l i c a t e d

f u n c t i o n a l forms o f W L - s t a t i s t i c s . Consequen t ly , we r e s o r t e d t o a

s i m u l a t i o n s t u d y t o compare f o u r s p e c i f i c s t a t i s t i c s i n cn w i t h

T-WL, t h e X L - s t a t i s t i c b a s e d on l o g i s t i c s c o r e s . The W L - s t a t i s t i c

based on d o u b l e e x p o n e n t i a l s c o r e s was n o t i n c l u d e d i n o u r s t u d y

mainly b e c a u s e i n t h e uncensored c a s e , t h e s i g n e d r a n k t e s t i s

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

d i s t r i b u t i o n s .



When s e l e c t i n g s t a t i s t i c s from %, primary c o n s i d e r a t i o n was

g iven t o t h e u s u a l methods of forming l i n e a r combinat ions of two

s t a t i s t i c s . The f i r s t s t a t i s t i c s e l e c t e d from c, corresponds t o

e q u a l weigh ts , i .e. Ln = .5:

I f Ln i s t a k e n p r o p o r t i o n a l t o N 2 , t h a t i s , Ln= N ~ ( N ~ + N ~ ) - . ~ ,

we g e t t h e s t a t i s t i c

T-SQK = (N1+ N ~ ) - . ~ [ N . ~ ~ T ; ~ ( N ~ ) + N .52 T ; ~ ( N ~ ) ] ,

w h i l e Ln p r o p o r t i o n a l t o N; y i e l d s t h e s t a t i s t i c

T-SS = (N: + N $ ) - ' ~ [ N ~ T ~ , ( N ~ ) + N ~ T ; ~ ( N ~ ) I .

F i n a l l y , s e l e c t i n g Ln p r o p o r t i o n a l t o o2 (N ) , t h a t i s wi th 2n 1

- 1 Ln = a ; n ( ~ l ) [o tn(Nl ) + O ; ~ ( N ~ ) ] , we g e t t h e f o u r t h s t a t i s t i c

Fol lowing Woolson and Lachenbruch (l98O), we assumed t h e l o g

l i n e a r model:

where I$ > O i s a n unknown parameter , V i l and V i 2 a r e i . i . d .

nonnegat ive random v a r i a b l e s , and W . i s a n independent nonnegat ive

random v a r i a b l e f o r a l l i. It i s c l e a r t h a t i f 8 = l o g I$ t h e n ,

0 0 Xil - Xi2 = 9 + ( l o g Vil - l o g Vi2), i = 1 ,2 , .. . ,n;

0 0 consequent ly , t h e d i s t r i b u t i o n a l form of X i l - Xi2 depends on t h e

d i s t r i b u t i o n a l form of l o g Vil - l o g Vi2.



0 0 Four d i s t r i b u t i o n a l forms f o r XiL - Xi2 were s i m u l a t e d i n

t h i s s t u d y . n e l o g i s t i c d i s t r i b u t i o n was s i m u l a t e d by g e n e r a t i n g

V i l a n d V i 2 w i t h i n d e p e n d e n t e x p o n e n t i a l d i s t r i b u t i o n s . Ano the r

l i g h t - t a i l e d d i s t r i b u t i o n , t h e no rma l d i s t r i b u t i o n , was s i m u l a t e d

by g e n e r a t i n g l o g V i l a n d l o g V i 2 a s i n d e p e n d e n t s t a n d a r d normal

v a r i a b l e s . Two h e a v y - t a i l e d d i s t r i b u t i o n s , t h e d o u b l e e x p o n e n t i a l

d i s t r i b u t i o n a n d t h e RambergSchmeiser-Tukey (RST) Lambda D i s -

t r i b u t i o n (Rand les and Wolfe, 1979, p. 4 1 6 ) were a l s o s t u d i e d .

The d o u b l e e x p o n e n t i a l d i s t r i b u t i o n was a r r i v e d a t by g e n e r a t i n g

l o g V i l a n d l o g Vi2 w i t h i n d e p e n d e n t e x p o n e n t i a l d i s t r i b u t i o n s .

The KST Lambda c u m u l a t i v e d i s t r i b u t i o n f u n c t i o n (c.d.f .) c a n n o t

be e x p r e s s e d e x p l i c i t l y b u t i t s i n v e r s e c .d . f . i s g i v e n a s

f o l l o w s :

As shown i n Ramberg a n d Schmei se r (1972) when X1 = 0 , Xg = X4 = -1

a n d X 2 = -3.0674, t h e RST Lambda D i s t r i b u t i o n a p p r o x i m a t e s t h e

Cauchy d i s t r i b u t i o n . % i s p a r t i c u l a r c h o i c e of RST Lambda

D i s t r i b u t i o n was u s e d t o g e n e r a t e l o g V i l a n d l o g Vi2 .

To g e n e r a t e t h e c e n s o r i n g random v a r i a b l e , t h e Uniform [O,B]

d i s t r i b u t i o n was u t i l i z e d f o r t h e l o g i s t i c a n d d o u b l e e x p o n e n t i a l

c a s e s , w h i l e t h e n a t u r a l l o g a r i t h m o f t h e Uniform [O,B]

d i s t r i b u t i o n was u s e d i n t h e normal and RST Lambda c a s e s . The

c h o i c e of B was made i n e a c h c a s e t o e n s u r e t h a t , u n d e r Ho, t h e

p r o p o r t i o n o f uncensored p a i r s was a p p r o x i m a t e l y 75% o f t h e t o t a l

sample s i z e w h i l e a p p r o x i m a t e l y 2 0 % of t h e t o t a l sample s i z e

c o n s i s t e d o f p a i r s i n which e x a c t l y one member of a p a i r was

uncensored . Consequen t ly , a p p r o x i m a t e l y 5 % of t h e p a i r s were n o t

u t i l i z e d s i n c e t h e y were p a i r s i n which b o t h members were

c e n s o r e d .

The powers o f t h e t e s t s of Ho b a s e d on t h e f i v e s t a t i s t i c s

were e s t i m a t e d u t i l i z i n g LOO0 random samples of s i z e n = 50 f rom

t h e d i s t r i b u t i o n of xo - xP2 = 0 + l o g V i l - l o g V i a i 1

c o r r e s p o n d i n g t o s e v e r a l c o m b i n a t i o n s o f a v a l u e of f3 and a


2052 P O P O V I C H AND R A O

d i s t r i b u t i o n a l fo rm f o r l o g Vil - l o g Vi2. I n o r d e r t o a s s e s s t h e

s t a t i s t i c a l s i g n i f i c a n c e o f t h e e s t i m a t e d d i f f e r e n c e i n powers ,

t h e s t a n d a r d e r r o r of t h e e s t i m a t e d d i f f e r e n c e was a l s o c a l c u l a t e d

f o r e a c h p a i r o f s t a t i s t i c s . I n a d d i t i o n , t o c o n s e r v e s p a c e we

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

More d e t a i l s c a n b e o b t a i n e d f r o m t h e f i r s t a u t h o r .

Recause t h e r e s u l t s f o r l o g i s t i c a n d norma l d i s t r i b u t i o n s

were v e r y s i m i l a r , we s h a l l l e a v e o u t t h e normal d i s t r i b u t i o n f r o m

o u r d i s c u s s i o n . 'Ihe s t a t i s t i c s T-STD a n d T-SS a l s o w i l l be l e f t

o u t b e c a u s e t h e power of t h e s e t e s t s were lower t h a n o t h e r s i n

a l m o s t a l l c a s e s e x c e p t t h a t T-SS performed s l i g h t l y b e t t e r t h a n

T-SQB f o r t h e RST Lambda D i s t r i b u t i o n .

Tab le 4.1 shows a summary o f r e s u l t s f o r t h r e e d i s t r i b u t i o n s ,

l o g i s t i c , d o u b l e e x p o n e n t i a l a n d RST Lambda a n d t h r e e s t a t i s t i c s ,

T-WL, T-EQ a n d T-SQR. In e a c h c a s e t h e h y p t h e s i s t e s t e d was Ho: 0

= 0 a g a i n s t Ha: 0 > 0. Based o n t h e a s y m p t o t i c d i s t r i b u t i o n o f

t h e t e s t s t a t i s t i c s , t h e c r i t i c a l v a l u e was chosen t o b e 1.645 i n

e a c h c a s e i n o r d e r t o have a n a p p r o x i m a t e .05 l e v e l t e s t .

An i n s p e c t i o n o f T a b l e 4.1 shows t h a t b a s e d on t h e i r l a r g e

s a m p l e (n=50) power, none o f t h e t h r e e s t a t i s t i c s emerges a s t h e

b e s t s t a t i s t i c f o r a l l c a s e s c o n s i d e r e d . F o r t h e l o g i s t i c

d i s t r i b u t i o n , T-WL a n d T-EQ have s i m i l a r powers a t a l t e r n a t i v e s

c l o s e t o Ho, b u t T-WL i s b e t t e r a t l a r g e r v a l u e s o f 0. F o r t h e

d o u b l e e x p o n e n t i a l d i s t r i b u t i o n , T-EQ a n d T-S QR behave s i m i l a r l y

w i t h b o t h h a v i n g b e t t e r power t h a n T-WL. F o r t h e RST Lambda

D i s t r i b u t i o n , t h e t e s t based o n T-SQR i s d e f i n i t e l y s u p e r i o r .

T h e r e f o r e , o n t h e b a s i s of t h e i r powers , i t a p p e a r s t h a t t h e

performance of t h e s t a t i s t i c T-SQK g e t s p r o g r e s s i v e l y b e t t e r

r e l a t i v e t o t h e o t h e r s a s one moves f r o m l i g h t e r t a i l e d

d i s t r i b u t i o n s t o h e a v i e r - t a i l e d o n e s .

5. A N EXAMPLE

To i l l u s t r a t e t h e u s e o f t h e s t a t i s t i c s d e f i n e d i n S e c t i o n 3

t h e d a t a s e t c o n s i d e r e d by Woolson a n d Lachenbruch (1980) w i l l b e


TABLE 4.1 1000 X DWFERENCL IN POWER . . t

DISTRIBUTION LOGISTIC DEXPO RST-LAbBDA w

e 0.00 0.100 0.250 0.500 0.00 0.100 0.250 0.500 0.00 0.40 0.60 0.80 5 r n

T-WL 48 167 555 9 50 4 5 257 704 993 4 3 200 366 527

(T-WL)-(T-EQ) -4 4 36* 5 0" -1 -17* -17* - 2 6 6 2 12

(T-WL)-(T-SQR) -4 2 4* 6 8* 5 I* - 5 -15* -29* -2 -2 -32* -37" -54%

(T-EQ)-(T-SQR) 0 2 0* 3 2* 1 -4 2 -1 2 0 - 8 -38" -39* -66*

Notes: ( 2 ) 7he t o p row g i v e s 1000 X e s t i m a t e d power o f T-WL

(2 ) The v a l u e of a2 e q u a l s t h e v a r i a n c e o f t h e d i s t r i b u t i o n which e q u a l s

n2 /3 f o r t h e L o g i s t i c d i s t r i b u t i o n a n d 2 f o r t h e Double E x p o n e n t i a l

d i s t r i b u t i o n .

( 3 ) A s t e r i s k n e x t t o a number i n d i c a t e s a d i f f e r e n c e g r e a t e r t h a n o r e q u a l t o t w i c e t h e s t a n d a r d e r r o r .



c o n s i d e r e d i n t h i s s e c t i o n . The d a t a s e t i s reproduced below.

TABLE 5.1

DIFFERENCE OF THE LOGARITHMS I N SURVIVAL TIME OF THE SKIN GRAFTS

P a t i e n t 1 2 3 4 5 6

'il- 'i2 0.2436 0.3795 1.3550+ 1.2745 0,3747 0.2578

P a t i e n t 7 8 9 10 I I

Xil- Xi2 -0.2624 -0.1542 0.3819 0.6592 0.4055+

Note: + d e n o t e s a r igh t -censored d i f f e r e n c e .

As can b e s e e n from t h e d a t a , N i = 9, N2L = 0, N2R = 2, and

N2 = 2. It f o l l o w s by some e lementa ry c a l c u l a t i o n s t h a t

Tln = 40, T2n = 2, ~ ; ~ ( 9 ) = 2.073, a n d ~ i ~ ( 2 ) = 1.414.

The t e s t s t a t i s t i c Tn(9,2) c a n b e c a l c u l a t e d i f Ln i s

known. For t h e purpose of i l l u s t r a t i o n we s h a l l t a k e Ln = .5,

which i m p l i e s t h a t t h e t e s t w i l l be based on T-EQ. Now,

Comparing t h i s r e s u l t w i t h t h e c r i t i c a l v a l u e s a p p e a r i n g i n t h e

t a b l e s g iven i n Popovich (1983) i t can be s e e n t h a t

s o t h a t t h e observed v a l u e o f 2.466 i s s i g n i f i c a n t a t t h e .O1

l e v e l . A c t u a l l y t h e e x a c t l e v e l a t t a i n e d i s .00488. As shown i n

Woolson and Lachenbruch (1980) t h e W-L t e s t s t a t i s t i c w i t h

l o g i s t i c s c o r e s y i e l d s t h e v a l u e Z = 2.49, which i n d i c a t e s

s i g n i f i c a n c e a t t h e .O1 l e v e l . Also, t h e e x a c t l e v e l f o r t h e W-L

t e s t s t a t i s t i c based on t h e p e r m t a t i o n d i s t r i b u t i o n o f t h e



l o g i s t i c s c o r e s i s 11 /2O48 o r 0.00537, a g a i n showing s i g n i f i c a n c e

a t t h e .O1 l e v e l . Note t h a t t h e v a l u e of 8/2048 a s g i v e n i n

Woolson and Lachenbruch (1980) i s i n c o r r e c t due t o a minor

computa t iona l e r r o r .

6. C ONCLUS IONS

The s t a t i s t i c s T-EQ a n d T-SQR proposed i n S e c t i o n 4 have

s e v e r a l advantages when a p p l i e d t o randomly r i g h t censored p a i r e d

da ta . They prov ide s imple d i s t r i b u t i o n - f r e e t e s t s based on e a s i l y

t a b u l a t e d c r i t i c a l v a l u e s ( a s opposed t o p e r n u t a t i o n t e s t s w i t h

c r i t i c a l v a l u e s depending upon t h e o b s e r v a t i o n s ) . Our s i m u l a t i o n

s t u d y i n d i c a t e s t h a t a t l a r g e sample s i z e s they have b e t t e r power

t h a n T-WL f o r medium and heavy t a i l e d d i s t r i b u t i o n s . F o r s m a l l

sample s i z e s , t h e a b i l i t y t o o b t a i n e x a c t t e s t s by r e f e r r i n g t o

e x i s t i n g t a b l e s of t h e s i g n t e s t and t h e s igned-rank t e s t would

f a v o r T-EQ and T-SQR o v e r T-WL even f o r l i g h t - t a i l e d

d i s t r i b u t i o n s . Therefore , based on t h e s i m u l a t i o n s tudy and o t h e r

c o n s i d e r a t i o n s noted i n S e c t i o n 1, f o r l i g h t - t a i l e d d i s t r i b u t i o n s

we recommend T-EQ f o r s m a l l and moderate sample s i z e s (making

e x a c t t e s t s f e a s i b l e ) a n d T-WL f o r l a r g e sample s i z e s . We

recommend T-SQR f o r moderate t o heavy- ta i l ed d i s t r i b u t i o n s . I f a

s i n g l e s t a t i s t i c is t o be recommended f o r t e s t i n g Ho, t h e n we

would f a v o r T-SQR.

7. ACKNOWLEDGEMENTS

'Ihe a u t h o r s a r e g r a t e f u l t o t h e r e f e r e e f o r h e l p f u l comments.

REFEKENCES

Anscombe, F .J. (1952). Large-Sample Theory of S e q u e n t i a l Est imation. Proc. Cambridge Phi los . Soc., 4 8 , 600-607.

Hajek, J. and S i d a k , Z. (1967). l 'heory of Rank T e s t s . Academic P r e s s , New York.

K a l b f l e i s c h , J . D . and P r e n t i c e , R.L. (1973). Plarginal L ike l ihoods Based on Cox's Regression and L i f e b d e l . Biometr ika, 6 0 , 267-278.



Popov ich , E.A. ( 1 9 8 3 ) . h o n p a r a m e t r i c A n a l y s i s of B i v a r i a t e Censored Gata. '

D i s s e r t a t i o n . U n i v e r s i t y of F l o r i d a .

Ramberg, J . S . , and bchmei se r , B.W. ( 1 4 7 2 ) . An approx in ia t e h e t h o d t o r Generating Synmetr i c Random V a r i a b l e s . Commnica t ions of t h ? A s s o c i a t i o n f o r Computing h a c h i n e r y , i n c . , 15, 987-9913.

Rand les , R.h., a n d Ldolfe, L.A. (1979) . I n t r o d u c t i o n t o t h e Theory of Nonparamet r i c S t a t i s t i c s . Wi ley , New York.

S e r f l i n g , R.J. (198b) . Approx ima t ion Theorems of h t h e m a t i c a l S t a t i s t i c s . W i l e y , Iden York.

S p r o u l e , R.N. ( 1 9 7 4 ) . Asympto t i c P r o p e r t i e s of U - S t a t i s t i c s . T rans . Am. Y ~ t h e m a t i c a l Soc . , 1 9 9 , 55-61;.

l i o o l s o n , K.F., and Lachenbruch, P.H. (1980) . Hank T e s t s f o r Censored hatchec P a i r s . B i o m e t r i k a , 6 7 , 597-606.

Woolson, R.F. , a n d Lachenbruch, P.A. (1984) . C o r r e c t i o n s t o 'Bank T e s t s f o r Censored h t c h e d P a i r s . ' B i o m e t r i k a , 7 1 , 220.

Received AugunL, 19 t i 5 ; Revbed hluy, 1 9 t i 5 .

Recommended by L . 51. W e i , Gco/rgc Wa~lungLon UvhehbLty, WuhCurzgLo~, DC


Conditional Tests for Censored Match Pairs - 1985 Popovich and Rao

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