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Isotonic Regression Analysis and Additivity
TRENDS lh' MATHEMATICAL PSYCHOLOGY E . Depeef and I . Van flu genhaui (editors) 0 Elsevier Science Publisfers B. I.' (North-Holland), 1984
239
ISOTONIC REGRESSION ANALYSIS AND ADDITIVITY
Ranald R . Flacdonald Deoartment o f Psychology I!ni v e r s i t,y o f S t i r l i n ?
T h i s naner reviews tl7e 1iterat.ut-e on i s o t o n i c r e g r e s s i o n w i t h a view t o f i n d i n g t e s t s f o r a d d i t i v i t y i n d a t a whose p o n u l a t i o n values a r e unioue o n l y up t o a monotonic t r a n s f o r m a t i o n .
Several
anproaches a r e c o n s i d e r e d and a s e q u e n t i a l one i s recommended
-
f i r s t t e s t i n g m o n o t o n i c i t y , and then
t e s t i n g t h e double c a n c e l l a t i o n axiom.
Two t e s t s
a r e p r e s e n t e d f o r t e s t i n g m o n o t o n i c i t y and an a l g o r i thm f o r d e t e c t i n g s i g n i f i c a n t denartures from double c a n c e l l a t i o n i s o u t l i n e d .
1. INTRODUCTION For a number o f y e a r s mathematical p s y c h o l o g i s t s have s t u d i e d a x i o m a t i c approaches t o a d d i t i v i t y .
Under t h e heading o f c o n j o i n t measurement t h e y
have been concerned w i t h whether t h e combined e f f e c t s o f a p a i r o f independ e n t l y m a n i p u l a t e d v a r i a b l e s a r e c o m p a t i b l e w i t h an u n d e r l y i n g a d d i t i v e r e presentation.
To p u t i t a n o t h e r way, what a r e t h e necessary and s u f f i c i e n t
c o n d i t i o n s under which monotonic f u n c t i o n s o f each o f two v a r i a b l e s e x i s t such t h a t when added t o g e t h e r t h e i r sum i s i t s e l f a monotonic f u n c t i o n o f t h e combined e f f e c t s ?
A much q u o t e d example i s t h a t o f a s i m p l e b e t w i t h Can t h e s u b j e c t i v e v a l u e o f
some o r o b a b i l i t y o f w i n n i n g a s p e c i f e d amount.
such b e t s be expressed i n terms o f a monotonic f u n c t i o n of t h e p r o b a b i l i t y p l u s some monotonic f u n c t i o n o f t h e D o t e n t i a l w i n n i n g s ? One r e p o r t of a s o l u t i o n t o t h e p r o b l e m i s g i v e n i n t h e f i r s t paper t o be p u b l i s h e d i n t h e J o u r n a l o f Mathematical Psychology, Luce and Tukey (1964). and o t h e r accounts a r e w i d e l y a v a i l a b l e o f which t h e most comorehensive i s K r a n t z , Luce, Suppes and T v e r s k y (1971).
The s o l u t i o n t o t h e Droblem was f o r m u l a t e d i n
terms o f t h e o r d e r r e l a t i o n s which must h o l d between a l l p o s s i b l e combined
243
R.R. MdcDonuld
e f f e c t s o f the variables. hold o r not.
I d i t h r e a l d a t a these c o u l d be f o u n d e i t h e r t o
A p a r t f r o m a numher o f u n t e s t a b l e t e c h n i c a l axioms t h e d a t a
must conform t o t y o o r d e r r e s t r i c t i o n s . t o t h e independent v a r i a b l e s .
They must be o r d e r e d w i t h r e s p e c t
T h a t i s , i f one v a r i a b l e i s h e l d c o n s t a n t ,
an i n c r e a s e i n t h e o t h e r v a r i a b l e s h o u l d l e a d t o an i n c r e a s e i n t h e v a l u e
of t h e data.
This i m n l i e s t h a t a d d i t i v e data are monotonically r e l a t e d t o
t h e indenendent v a r i a b l e s and t h ? r e s t r i c t i o n w i l l be r e f e r r e d t o as t h e monotonicity r e s t r i c t i o n .
I n a d d i t i o n t o t h e m o n o t o n i c i t y r e s t r i c t i o n Luce
and Tukey (1964) r e o u i r e t h a t t h e double c a n c e l l a t i o n axiom be met. double c a n c e l l a t i o n axiom s t a t e s t h a t if a ' , a " , a"' dependent v a r i a b l e A and b ' , b " , b " ' variable
The
a r e l e v e l s of t h e i n -
are l e v e l s o f the o t h e r independent
then i f i n e n u a l i t i e s (1) and ( 2 ) :?olrl, so s \ ~ u ! d ( 3 )
where uab i s t h e v a l u e o f t h e d a t a c o r r e s p o n d i n g t o l e v e l a o f A and b of R.
The double c a n c e l l a t i o n axiom a l s o i s r e q u i r e d t o h o l d w i t h > r e p l a c e d
by < .
T h i s i s o f l i m i t e d use when d e a l i n g w i t h t h e f a l l i b l e d a t a o f expe-
rimentalists.
I n such cases i t would b e u s e f u l t o know i f any v i o l a t i o n s
i n t h e o r d e r r e s t r i c t i o n s c o u l d be e x n l a i n e d by t h e i n h e r e n t v a r i a b i l i t y o f the data o r whether they represent r e a l effects. There i s a n o t h e r p r o b l e m i n t e s t i n g i f t h e above o r d e r r e s t r i c t i o n s h o l d i n r e a l data.
As t h e number o f l e v e l s o f t h e independent v a r i a b l e s s t u d i e d i s
f i n i t e , one cannot check t h e double c a n c e l l a t i o n axiom f o r a l l p o s s i b l e l e v e l s of A and B .
I n f i n i t e cases double c a n c e l l a t i o n may be s a t i s f i e d
w h i l e more complex c a n c e l l a t i o n s a r e n o t .
Any v i o l a t i o n o f a c a n c e l l a t i o n
no m a t t e r how complex r e n d e r s an a d d i t i v e r e p r e s e n t a t i o n i m p o s s i b l e .
Ar-
b u c k l e and L a r i m e r (1976) and M c C l e l l a n d (1977) p r o v i d e o r o b a b i l i t i e s o f complex c a n c e l l a t i o n s b e i n g v i o l a t e d g i v e n t h a t l r o n o t o n i c i t v and double c a n c e l l a t i o n h o l d i n randomly o r d e r e d m a t r i c e s . crease as t h e m a t r i c e s i n c r e a s e i n s i z e . c e l l a t i o n s would be a hideous t a s k .
a D a r t i a l way o u t .
However, Luce and Tukey (1964) p r o v i d e
B a r e s e p a r a t e d by equal aB f o r m a d u a l s t a n d a r d sequence) m o n o t o n i c i t y and
Where t h e l e v e l s o f A and
mounts ( t e c h n i c a l l y A,
These a r o b a b i l i t i e s i n -
T e s t i n g a l l a o s s i b l e complex can-
24 1
kotoiiic repenion arlalysis and additivity
double c a n c e l l a t i o n a r e necessary and s u f f i c i e n t conc'itions t o ensure an a d d i t i v e r e r J r e s e n t a t i o n o v e r t h e l e v e l s of A and B s t u d i e d .
Thus
i t i s d e s i r a b l e t o design experiments so t h a t t h i s c o n d i t i o n i s approxima-
t e l y met.
I f t h e l e v e l s of A and B do form a dual s t a n d a r d sequence, t h e n
i t i s o n l y necessary t o t e s t m o n o t o n i c i t y and double c a n c e l l a t i o n ;
i n 3:s
case complex c a n c e l l a t i o n s may be i g n o r e d .
I t i s w o r t h d i s t i n g u i s h i n g between t h e p r e s e n t c o n c e p t i o n o f a d d i t i v i t y w h i c h concerns t h e e x i s t e n c e o f a d d i t i v i t y h o l d i n g i n some monotonic t r a n s f o r m a t i o n o f t h e d a t a and a d d i t i v i t y which h o l d s i n u n t r a n s f o m d data. S t a n d a r d t e s t s o f i n t e r a c t i o n s i n a n a l y s i s o f v a r i a n c e , T u k e y ' s (1964) add i t i v i t y t e s t an6 "anc'el's (1971) work on n o n a d d i t i v i t y a r e a l l concerned w i t h t h e second sense o f a d d i t i v i t y .
The c o n j o i n t measurement Droblem i s
concerned w i t h t h e f i r s t sense and i f a d d i t i v i t y e x i s t s i n t h i s sense i t
w i l l a l s o do so a f t e r any monotonic t r a n s f o r m a t i o n has been a o p l i e d .
For
t h e r e s t o f t h e d i s c u s s i o n a d d i t i v i t y w i l l be used o n l y i n t h e more g e n e r a l sense.
A l s o i n t h e a n p l i c a t i o n s t o be c o n s i d e r e d i t w i l l be assumed t h a t
t h e o r d e r i n g o f t h e l e v e l s o f t h e indeoendent v a r i a b l e s i s known i n advance. Bartholomew (1961) p r e s e n t e d a method f o r s t a t i s t i c a l i n f e r e n c e under o r d e r r e s t r i c t i o n s which he c a l l e d i s o t o n i c r e g r e s s i o n a n a l y s i s .
More r e c e n t r e -
views o f t h e area a r e g i v e n i n Barlow, B a r t h o l o w w , E r e m e r and Brunk (1972) and Smith and Macdonald (1983).
A major a p o l i c a t i o n o f i s o t o n i c regression
a n a l y s i s i s t o p r o v i d e t e s t s o f d i f f e r e n c e s between means s u b j e c t t o c e r t a i n order restrictions.
I t would seem l i k e l y t h a t t h e s t a t i s t i c a l l i t e -
r a t u r e on i s o t o n i c r e g r e s s i o n w o u l d be r e l e v a n t t o t h e oroblems in a p p l y i n g c o n j o i n t measurement t o f a l l i b l e d a t a .
T h i s oaper i s designed t o examine
t h e e x t e n t t o which t h i s i s t r u e and t o suggest some s t a t i s t i c a l t e s t s f o r f i t t i n o an a d d i t i v e c o n j o i n t model. The aims o f t h e s t a t i s t i c i a n s w o r k i n g on i s o t o n i c r e g r e s s i o n a r e somewhat d i f f e r e n t f r o m those o f t h e c o n j o i n t neasurers.
Tests developed i n i s o t o n i c
r e g r e s s i o n a r e designed t o i n c r e a s e t h e power o f t h e t e s t when t h e o r d e r i n g o f t h e means can be p r e d i c t e d i n advance.
C o n j o i n t measurement i s concer-
ned w i t h whether t h e r e a r e any d i f f e r e n c e s between t h e means i n c o m p a t i b l e w i t h a d d i t i v i t y r a t h e r than showing whether an a d d i t i v e model g i v e s a b e t t e r f i t t h a n no model a t a l l .
242
R.R.MacDonald
2. NOTATION
The f o l l o w i n g n o t a t i o n a n o l i e s t o a two way a n a l y s i s o f variance, and t o a one way a n a l y s i s when the s u h s c r i p t k i s omitted:
J
i s the number o f l e v e l s o f v a r i a b l e A
i s the number o f l e v e l s o f v a r i a b l e B
I(
Yijk
n
i s the i t h o b s e r v a t i o n i n the j k t h treatment
i s the number o f observations i n each jk t h treatment (assumed equal)
N
i s t h e t o t a l number o f observations
u
i s the t h e o r e t i c a l o v e r a l l mean
ir
i s the observed o v e r a l l mean
ujk
i s the t h e o r e t i c a l mean i n the j k t h treatment
ir .jk ir
i s the observed mean i n the j t h l e v e l of A
irk
i s the observed mean i n the k t h l e v e l o f R
j
i s the observed mean i n the j k t h treatment
i-u
i s the i s o t o n i c regression estimate o f u . jk Jk' i*ujk i s derived w i t h respect t o some order r e s t r i c t i o n s ; u j , k , < u ~ , , ~f,o,r
some s p e c i f i e d j ' , j " , k ' , k " .
Barlow e t a1 ( 1 9 7 2 ) show t h a t the i s o t o n i c
regression e s t i m a t e i s the maximum l i k e l i h o o d e s t i m a t e o f u
w i t h these r e s t r i c t i o n s
jk
.
consistent
L i s the number o f d i f f e r e n t values taken by the i s o t o n i c regression e s t i mates i-u. Jk. Considering f i r s t the u n i v a r i a t e case, the usual s t a t i s t i c used t o t e s t f o r e q u a l i t y o f treatment means i s : 1 n ( D j-O)
F(J-1,N-J)
ij
F =
" / ( J-1)
= j
z
I n analysis o f variance n o t a t i o n :
CI
(4) ( Y -G.)*/(PI-J)
"
SSbet/dfbet - MSbet SSw/dfw -
ww
Eartholomew (1961) proposes the s t a t i s t i c E: z n(i-u.-D) 2 J E J2 = j 1 (Yij-D)' ij
(5)
243
Isotonic regression analysis m d additivity
I n o u r n o t a t i o n t h i s i s expressed:
2 SSmon EJ = E
(7)
2 J d i f f e r s f r o m F i n two i m p o r t a n t r e s p e c t s .
F i r s t l y the i s o t o n i c regres-
s i o n e s t i m a t e s a r e used t o compute t h e between t r e a t m e n t v a r i a n c e through SSmon.
Secondly, t h e between t r e a t m e n t v a r i a t i o n u n s o e c i f i e d by t h e o r d e r
r e s t r i c t i o n s ( w h i c h i s r e f e r r e d t o as SSres = SSbet t o t h e e r r o r term.
-
SSmon) c o n t r i b u t e s
The second d i f f e r e n c e has t h e a f f e c t o f i n c r e a s i n g t h e
) when t h e d i f f e r e n c e s between t h e j" when j ' < j " ) . Idhen means a r e i n t h e o r d e r i n g s p e c i f i e d i n H1(uj, < u j" power o f t h e t e s t a g a i n s t Ho(uj,=u
t h e r e a r e d i f f e r e n c e s between t h e means which a r e n e i t h e r s p e c i f i e d n o r
p r e c l u d e d i n H1 a t e s t u s i n g MSw as t h e e r r o r t e r m may be more p o w e r f u l 2 than EJ. The e x t e n s i o n t o t h e b i v a r i a t e case i n n o m a l analyses o f v a r i a n c e i s t o p a r t i t i o n SSbet i n t o t h a t due t o v a r i a b l e A ( S S A ) , t h a t due t o v a r i a b l e B (SSB) and a r e s i d u a l c a l l e d t h e i n t e r a c t i o n ( S S A B ) .
The degrees o f f r e e -
dom a s s o c i a t e d w i t h each a r e known and s e p a r a t e F values a r e used t o t e s t
each o f t h e components o f t h e between v a r i a t i o n a g a i n s t t h e e r r o r term, see Winer (1971). The a p p l i c a t i o n o f i s o t o n i c r e g r e s s i o n t o t h e two way a n a l y s i s i s consider e d b y Barlow e t a1 (1972) f o l l o w i n g Shorack (1967). 2
They recommend u s i n g
EJ as d e f i n e d i n e q u a t i o n (8) r a t h e r than ( 7 ) .
T h i s i s t o t e s t Ho
uj,
= u
jl'
a g a i n s t HI
ujl
u
j"
f o r some s p e c i f i e d jl and
j " . The ( A ) a f t e r SSmn denotes t h a t t h e o r d e r r e s t r i c t i o n s i n v o l v e o n l y
l e v e l s o f A.
S i m i l a r t e s t s a r e developed i n v o l v i n g t h e B v a r i a b l e b u t no
t e s t s o f d i f f e r e n c e s between means i n t h e i n t e r a c t i o n a r e d i s c u s s e d i n t h i s context, 3. UNIVARIATE TESTS OF MONOTONICITY
L e t us f i r s t c o n s i d e r t h e u n i v a r i a t e case.
T h i s i s n o t e n t i r e l y degenerate
s i n c e i f m o n o t o n i c i t y were v i o l a t e d t h e d a t a c o u l d n o t be r e p r e s e n t e d by an a d d i t i v e model.
A demonstration o f nonmonotonicity orecludes the possibi-
lity o f a d d i t i v i t y
.
R . R . MacDonald
144
I n u n i v a r i a t e i s o t o n i c r e g r e s s i o n t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s Tu. J . . J are found u s i n g t h e 0001 a d j a c e n t
v i t h r e s p e c t t o t h e o r d e r 1, 2,.
v i o l a t o r s a l g o r i t h m d e s c r i b e d i n B a r l o w e t a? (1972). M o n o t o n i c v a r i s t i o n 2 i n t h e means i s t h e n demonstrated u s i n g EJ as d e f i n e d i n e q u a t i o n ( 6 ) . I n t h e a p p l i c a t i o n o f c o n j o i n t measurement we a r e concerned i n d e m o n s t r a t i n g I t seems n a t u r a l t o t a k e SSres = z n(ir i-u.)' as 2 j- J Nonmonotonic j v a r i a t i o n w i l l an e s t i m a t e o f t h e p o D u l a t i o n v a r i a n c e u nonmonotonic v a r i a t i o n .
.
i n c r e a s e t h e v a l u e o f SSres and i f i t i s s i g n i f i c a n t l y g r e a t e r t h a n w o u l d be e x p e c t e d f r o m t h e w i t h i n c o n d i t i o n v a r i a n c e , t h e nonmonotonic v a r i a t i o n has been demonstrated.
f o r a l l j ' and j " , i s t r u e then
I f Ho, uj,=uj,, 2 2
SSres i s d i s t r i b u t e d as no
A p r o b l e m a r i s e s when t h e n u l l hypo-
t h e s i s t o be t e s t e d i s H i : e i t h e r u j , = uj,, o r u j , < uj,,,
f o r .i'< j " .
Suppose u < u i t i s s t i l l possible t h a t b., > and as such t h e J j'" j ' j'" and c o u l d c o n t r i b u t e t o SSres. The e x p e c t e d v a r i a t i o n between iI j' J v a l u e o f such v a r i a n c e w o u l d be l e s s t h a n t h a t i f u = u Thus SSres i s 2 2 j' j". d i s t r i b u t e d as l e s s t h a n n u x ( ~ - ~ ) T. h i s argument i s developed more f o r m a l l y by H a r t i g a n (1967) who e s t a b l i s h e s f o r SSres t h e u p p e r bound 2 2 nc x ( ~ - ~ ) .Thus an a p p r o p r i a t e t e s t s t a t i s t i c f o r cigClt be F I L as de::,I,
f i n e d i n e q u a t i o n ( 9 ) . though i t w i l l be c o n s e r v a t i v e t o an unknown e x t e n t f o r reasons g i v e n above.
F I L (J-L,N-JK)
=
SSres / (J-L)
(9)
MSw
Bartholomew (1961) c o n s i d e r e d u s i n g a F I L s t a t i s t i c as a t e s t f o r a d d i t i v i He r e j e c t e d t h i s approach as L i s a random v a r i a b l e w h i c h i n c r e a s e s as 2 t h e p r o b a b i l i t y o f Ho decreases and H1 i n c r e a s e s . EJ t h e recommended t e s t ,
ty.
incorporates L i n t o the t e s t s t a t i s t i c .
I n t h e monotonic case n o n - l i m i t i n g
values o f L w i l l be d e t e r m i n e d m a i n l y by t h e monotonic v a r i a t i o n .
The non-
monotonic t e s t i s concerned w i t h t e s t i n g t h e s i g n i f i c a n c e o f whatever nonmonotonic
v a r i a t i o n e x i s t s i n d e p e n d e n t l y o f t h e s i z e o f t h e monotonic
L
variation.
I t i s n o t t h e r e f o r e a p p r o p r i a t e t o use
statistic.
S i m i l a r l y i t i s a p p r o p r i a t e t o use SSw as t h e e r r o r t e r m as i t
as p a r t o f t h e t e s t
i s d e s i r e d t o show t h a t t h e nonmonotonic v a r i a t i o n i s g r e a t e r t h a n t h e e s t i mate o f t h e w i t h i n c o n d i t i o n v a r i a t i o n .
T h i s p o i n t has been d i s c u s s e d above.
The c o n s e r v a t i s m o f t h e p r o p c s e d t e s t ( e q u a t i o n ( 9 ) ) i s due t o t h e presence o f s m a l l amounts o f v a r i a t i o n between means w h i c h a r e t r u l y m o n o t o n i c .
It
245
Isototiic regression unulysis und additivity
w o u l d seem p r e f e r a b l e t o use a t e s t which gave g r e a t e r w e i g h t t o t h e l a r ger nonmnotonic v a r i a t i o n .
Such a t e s t i s t h e s t u d e n t i z e d range t e s t -see
b l i n e r (1971). To use t h i s t e s t we f i n d t h e r e s i d u a l e s t i m a t e s o f t h e t r e a t ment means ( r i i ~ . ) as d e f i n e d i n e q u a t i o n (10).
J
n'bi. =. 0j
-
Ti. J t G
(10)
The l a r g e s t and s m a l l e s t o f these a r e s u b s t i t u t e d i n t o t h e t e s t s t a t i s t i c q as i n e q u a t i o n (11). q =
n-ularFest
"smallest MSwIn
(11)
The s t u d e n t i z e d range t e s t i s designed t o d e t e c t d i f f e r e n c e s b e b e e n k t r e a t m e n t means f r o m t h e d i f f e r e n c e between t h e l a r g e s t and s m a l l e s t mean. Since i n t h e p r e s e n t case t h e r e a r e J t r e a t m e n t s and
L different isotonic
e s t i m a t e s , t h e a p p r o p r i a t e v a l u e t o use f o r k when l o o k i n g up t h e t a b l e s i s J-Ltl. F o r reasons analogous t o those a p p l y i n g t o FIL t h e range t e s t
w i l l a l s o be c o n s e r v a t i v e b u t s i n c e i t g i v e s most w e i g h t i n g t o t h e l a r g e s t d i f f e r e n c e i t s h o u l d be m r e p o w e r f u l t h a n F ( L .
The r e l a t i v e powers o f t h e two t e s t s and t h e e x t e n t o f t h e i r c o n s e r v a t i s m cannot he a s c e r t a i n e d
i n general, s i n c e i t w i l l depend on t h e amount o f t r u e monotonic v a r i a t i o n p r e s e n t i n t h e p o p u l a t i o n means. I n d e e d t h e c o n s e r v a t i s m o f these t e s t s w i l l depend on t h e t r u e values o f t h e p o p u l a t i o n means and t h e p r o b a b i l i t y t h a t p o p u l a t i o n means i n t h e p r e d i c t e d o r d e r w i l l g i v e r i s e t o sample means o u t o f o r d e r .
I f t h e values o f
t h e i s o t o n i c e s t i m a t e s o f t h e p o p u l a t i o n means a r e t r e a t e d as p o p u l a t i o n values, t h e e x t e n t o f t h e c o n s e r v a t i s m c o u l d be e s t i m a t e d by s i m u l a t i o n . From r e p e a t e d s i m u l a t i o n s t h e e x p e c t e d values o f SSres and L c o u l d be found where t h e r e i s no nonmonotonic v a r i a t i o n . equation (13) t o a l l o w f o r t h i s .
C o r r e c t i o n s c o u l d be made t o
However, t h e r e q u i r e m e n t t o r u n a simu-
l a t i o n on each t e s t r u l e s t h i s o u t as a r o u t i n e procedure. 4. BIVARIATE TESTS OF MONOTONICITY T h i s s e c t i o n a t t e m p t s t o g e n e r a l i s e these r e s u l t s t o b i v a r i a t e m o n o t o n i c i t y . The s i m p l e s t approach i s t o r e l y on o n l y t h e o v e r a l l means f o r each of t h e l e v e l s o f A and 5. T h i s i s e s s e n t i a l l y t h e aoproach suggested by Shorack
(1967) mentioned above, who t r e a t e d t h e two way a n a l y s i s as i f i t were two
246
R . R . AfacUuiiald
indenendent one way analyses. found w j t h r e s p e c t t o 1, 2,
2
The i s o t o n i c r e g r e s s i o n e s t i m a t e s i-u. a r e
..., J
J
and monotonic v a r i a t i o n w i t h r e s p e c t
t o A t e s t e d u s i n g EJ as d e f i n e d i n e q u a t i o n ( 8 ) .
To t e s t t h e presence o f
nonmonotonic v a r i a t i o n w i t h r e s p e c t t o A one c o u l d t h e r e f o r e use fined i n equation ( 9 ) . t o variable B.
FIL as de-
The t e s t i n g procedure can be r e n e a t e d w i t h r e s p e c t
The two t e s t s a r e m u t u a l l y independent when t h e number o f
o b s e r v a t i o n s i n each o f t h e c o n d i t i o n s i s e q u a l .
Barlow e t a1 (1972) d i s -
cuss t h e case where t h e numbers o f o b s e r v a t i o n s a r e unequal. James (1961). comnenting on Partholomew's work, suggested ways t h a t i s o t o n i c r e g r e s s i o n m i g h t be extended t o i n c l u d e i n t e r a c t i o n s .
Rartholomew
(1961) r e p l i e d t h a t he had c o n s i d e r e d t h e p r o b l e m b u t f o u n d i t f o r b i d d i n g . H i r o t s u (1978) f o l l o w e d
UD
these i d e a s by c o n s i d e r i n g v a r i o u s comDarisons
o f t r e a t m e n t means which c o n t r i b u t e d t o t h e i n t e r a c t i o n , e . p .
(l.,')
I f a l l such comparisons a r e n o s i t i v e ( o r n e g a t i v e ) t h e d a t a a r e monotonic. This i s a stronger r e s t r i c t i o n r e q u i r i n g n o t o n l y t h a t the data monotonicall y i n c r e a s e s w i t h t h e indenendent v a r i a b l e s b u t t h a t t h e i n c r e a s e w i t h each
successive l e v e l o f an independent v a r i a b l e i s i t s e l f m o n o t o n i c a l l y i n c r e a s i n g ( o r decreasina).
Another way o f p u t t i n g i t i s t h a t t h e change due t o
+he independent v a r i a b l e s i s a c c e l e r a t i n g e i t h e r n o s i t i v e l y o r n e g a t i v e l y . Consequently any t e s t which showed a l l such comparisons t o be p o s i t i v e o r n e g a t i v e would i n d e e d e s t a b l i s h m o n o t o n i c i ty ( t h o u g h n o t double c a n c e l l a t i o n ) , b u t a n e g a t i v e r e s u l t would l e a v e open t h e p o s s i b i l i t y o f monotonicity.
Unless t h e r e i s reason t o e x p e c t an a c c e l e r a t i n g e f f e c t o f t h e i n -
dependent v a r i a b l e s , t h i s approach i s p r o b a b l y n o t r e l e v a n t . Shorack's procedures o u t l i n e d above a r e r e l a t i v e l y s t r a i g h f o t w a r d b u t t h e y ignore p o t e n t i a l l y r e l e v a n t v a r i a t i o n c o n t r i b u t i n g t o the i n t e r a c t i o n .
In
o t h e r words, u n l e s s t h e n o n m n o t o n i c v a r i a t i o n i n one v a r i a b l e i s p r e s e n t averaged o v e r a l l l e v e l s o f t h e o t h e r , t h i s anproach w i l l f a i l t o d e t e c t the nonmonotonicity.
I s o t o n i c r e g r e s s i o n e s t i m a t e s i u can be found w i t h
respect t o the p a r t i a l o r d e r j , k
< j',k'
i f j < j ' and k < k ' .
Several
algorithms t o f i n d i s o t o n i c regression estimates w i t h respect t o p a r t i a l o r d e r s a r e r e p o r t e d i n B a r l o w e t a1 (1972).
The p o o l a d j a c e n t v i o l a t o r s
a l g o r i t h m which worked w i t h simpl'e o r d e r s f a i l s i n t h e more g e n e r a l case, As i s o t o n i c r e g r e s s i o n e s t i m a t e s can be found t h e SSbet can be s p l i t i n t o
247
Isotonic regression malysis and additiuity
t h e sums o f squares between t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s SSmon and SSres, a r e s i d u a l due t o nonmonotonic v a r i a t i o n .
Again u s i n g t h e upper
bound f o r SSres ( H a r t i g a n ( 1 9 6 7 ) ) an o v e r a l l t e s t based on F I L i s p o s s i b l e This i s presented i n equation (13). FIL(JK-L,N-JK)
=
S i m i l a r l y t h e m u l t i p l e range t e s t can be a p p l i e d t o t h e b i v a r i a t e case. Again t h e r e s i d u a l ticnn!onotcni c e s t i v a t e s cC ttse t r e a t m e n t means can he found and t h e l a r g e s t and s m a l l e s t s u b s t i t u t e d i n t o e q u a t i o n ( 1 1 ) . v a l u e f o r k i n t h i s case i s JK-L+1.
The
5 . EXAMP1.F The t e s t s o u t l i n e d above w i l l be i l l u s t r a t e d u s i n g a r t i f i c i a l d a t a produced by Cunningham (1982) and g i v e n i n t a b l e 1 below.
I t w i l l be assumed t h a t
t h e raw d a t a a r e t h e means o f t e n independent o b s e r v a t i o n s ( n = 10). Isotonic
Data
Estimates
B 3 7
6
9
6.5
6.5
9.0
B 2 3
1
8
2.57
2.=7
Q.?
BIZ
4
5
2.0
2.67
5.0
A1 A2
A1
A3
A2
A3
Table 1 F i c t i t i o u s example f r o m Cunningham (1982) A p p l y i n g S h o r a c k ' s approach t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t T h i s r e s u l t s i n an SSmon (A) o f 140
t o A ( t h e i u ' s ) a r e 3.83,
3.83,
and a SSres (A) o f 0.66.
N o n a d d i t i v i t y i n A w o u l d be t e s t e d by F(1.81) =
0.66/MSw.
7.33.
I t w o u l d r e q u i r e tlsw t o be l e s s t h a n .166 t o i n d i c a t e s i g n i f i c a n t
n o n a d d i t i v i t y a t t h e 0.05 l e v e l ,
T h i s approach w o u l d produce no evidence
o f n o n a d d i t i v i t y i n B, since a l l t h e v a r i a t i o n i s monotonic and SSres (B) e q u a l s zero. Usin? t t e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o t h e p a r t i a l o r d e r UjlkI <
j ",k"
for j ' , j " and k ' , k " g i v e n i n t a b l e 1,
SSmon and SSres
248
R.R. MucDonald
come o u t as 549 and 51. /MSw.
N o n a d d i t i v i t y c c u l d be t e s t e d b y F(3,81)
= (51/3)
A MSw o f l e s s t h a n 6.83 w o u l d be r e q u i r e d f o r t h e r e s u l t t o be s i p I f t h e s t u d e n t i z e d range t e s t were a p p l i e d
n i f i c a n t a t t h e 0.05 l e v e l .
c r l e s s , w o u l d be r e q u i r e d f o r s i g n i f i c a n c e
o = 9.48/MSw and a Ffsw of 7.€8
a t t h e 0.05 l e v e l . 6 . DOUBLE CANCELLATION A X I O M
! l o n o t o n i c d a t a a r e n o t n e c e s s a r i l y a d d i t i v e as t h e example due t o K r u s k a l i n S c h e f f e (1959) shows.
Here m a t r i x A g i v e s t h e r a n k o r d e r o f t h e d a t a
from smallest t o l a r g e s t .
A i s m n o t o n i c and R and C a r e m a t r i c e s formed
by a s u b s e t o f t h e elements i n A.
A
B
C
[: :] I f t h e d a t a a r e assumed t o be a d d i t i v e , f r o m
from C l s t + 8 t h = 4th+2nd.
m a t r i x B l s t + 7 t h = 3 r d + d t h and
The l e f t hand s i d e o f t h e f i r s t e q u a t i o n i s a l -
ways l e s s t h a n t h e l e f t hand s i d e o f t h e second under any monotonic t r a n s formation o f the data.
The r e v e r s e i s t r u e o f t h e r i g h t hand s i d e .
l y t h e a d d i t i v i t y assumption i s u n t e n a b l e .
Clear-
Data must be shown n o t t o de-
D a r t from m o n o t o n i c i t y and n o t t o v i o l a t e double c a n c e l l a t i o n b e f o r e we can conclude t h e y a r e a d d i t i v e . Having c o n s i d e r e d t e s t s o f m o n o t o n i c i t y we now l o o k a t ways o f t e s t i n g double c a n c e l l a t i o n assuming t h a t t h e d a t a a r e monotonic.
I n i t s most ge-
n e r a l form we must t e s t a l l t h e 3 x 3 m a t r i c e s f o r i n e q u a l i t i e s (1) t o ( 3 ) . I f we assume m n o t o n i c i t y many fewer m a t r i c e s need t o be examined.
3 x 3 m a t r i c e s where a ' < a " < a"'
and b ' < b " < b"'
Only i n
i s m o n o t o n i c i t y compa-
t i b l e w i t h i n e q u a l i t i e s ( 1 ) and ( 2 ) b e i n q t r u e and ( 3 ) b e i n q f a l s e o r v i c e vprsa.
S t i l l f u r t h e r r e d u c t i o n s on t h e number o f 3 x 3 m a t r i c e s which need
t o be examined a r e n o s s i b l e .
An a l a o r i t h m f o r f i n d i n n a l l t h e v i o l a t i o n s o f
double c a n c e l l a t i o n i n a s e t o f d a t a which a r e c o m p a t i b l e w i t h m o n o t o n i c i t y i s g i v e n i n t h e appendix. Procedures f o r f i n d i n g i s o t o n i c r e g r e s s i o n e s t i m a t e s r e q u i r e a p r i o r i o r d e r i n g o f t h e c o n d i t i o n s even i f i t i s o n l y p a r t i a l .
The double c a n c e l l a -
t i o n axiom imposes a r e s t r i c t i o n c o n d i t i o n a l on o t h e r v a l u e s o f t h e d a t a .
249
Isotonic regression amlysis and additivity
As t h i s i s n o t an a p r i o r i o r d e r i n g t h e e s t i m a t i o n a l g o r i t h m s g i v e n i n
Barlow q t a1 (1972) would have t o be extended i n o r d e r f o r t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s t o be found,
Ift h e a l g o r i t h m s were so extended t h e
r e s u l t i n g procedure f o r t e s t i n g f o r n o n - i s o t o n i c v a r i a t i o n would be t h e t e s t s g i v e n i n e q u a t i o n s ( 9 ) and (11).
I n t h e absence o f such e x t e n s i o n s one
must l o o k f o r s t a t i s t i c a l t e s t s o f double c a n c e l l a t i o n u s i n g a more conv e n t i o n a l approach. Having t e s t e d m o n o t o n i c i t y u s i n g one of t h e techniques g i v e n above and f a i l e d t o f i n d a s i g n i f i c a n t v i o l a t i o n , i t i s now assumed t h a t t h e d a t a a r e monotonic.
The i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o b i v a r i a t e
m o n o t o n i c i t y a r e f e d i n t o t h e a l g o r i t h m i n t h e appendix.
This gives a l l
t h e 3 x 3 submatrices f r o m t h e complete d a t a m a t r i x which v i o l a t e t h e double c a n c e l l a t i o n axiom.
F o r t h e v i o l a t i o n t o be s t a t i s t i c a l l y s i g n i f i -
c a n t each c f t h e t h r e e i n e q u a l i t i e s c o n t r i b u t i n g t o t h e v i o l a t i o n s h o u l d be s i g n i f i c a n t i n i t s own r i g h t .
Various methods e x i s t f o r e s t a b l i s h i n g
t h e c o n f i d e n c e i n t e r v a l s f o r t h e d i f f e r e n c e s between means a f t e r an a n a l y s i s o f v a r i a n c e (see Winer ( 1 9 7 1 ) ) .
Statistically significant violations
o f t h e double c a n c e l l a t i o n axiom would have been demonstrated i f t h e confidence i n t e r v a l s o f t h e d i f f e r e n c e i n a l l three i n e q u a l i t i e s c o n t r i b u t i n g t o any v i o l a t i o n d i d n o t i n c l u d e z e r o , Indeed, t h e a l g o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o u l d be r e w r i t t e n f o r l o o k i n g f o r s t a t i s t i c a l l y s i g n i f i c a n t v i o l a t i o n . w o u l d r e q u i r e t h a t t h e i n e q u a l i t y s i g n s < a n d > be r e p l a c e d by <
A l l this
s and
> s
i n d i c a t i n g s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s and t h e i n e q u a l i t i e s b e i n g t r e a t e d as equal when t h e d i f f e r e n c e s were n o t s i g n i f i c a n t . 7. CONCLUSIONS The o v e r a l l t e s t i n g procedure f o r a d d i t i v i t y which seems t o make most use o f t h e data i s a s e q u e n t i a l one.
The e x p e r i m e n t s h o u l d b e designed so t h a t
t h e e f f e c t s o f s u c c e s s i v e l e v e l s o f t h e independent v a r i a b l e s a r e a p p r o x i mately equal.
A t e s t f o r monotonicity using e i t h e r FIL (equation (13)) o r
t h e s t u d e n t i z e d range t e s t ( e q u a t i o n (11)) s h o u l d n e x t be performed.
If
t h i s r e s u l t s i n a s i g n i f i c a n t e f f e c t m o n o t o n i c i t y and hence a d d i t i v i t y have been v i o l a t e d .
I f no s i g n i f i c a n t e f f e c t s a r e found,
t h e double c a n c e l l a -
t i o n axiom i s t e s t e d assuming m o n o t o n i c i t y as o u t l i n e d above.
Ifn e i t h e r
o f t h e above t e s t s produces s i g n i f i c a n t e f f e c t s , t h e d a t a have n o t been shown t o s i g n i f i c a n t l y d e n a r t f r o r a d d i t i v i t y .
3 0
?F FF PF Ft CF
?r\uc'.!e,
:.,
ar?8
!.ariwr, ?.,
l'-e
rl.i+?r
o c t v o wav t a h l e s s a t i s f y i n g
c e r t a i n addi t i v i t'/ axioms, J o u r n a l o f r!athematical Psycholooy, 13 (?.-:-,),
P9-100. Rartholomew, D.J., Bremner, J.M. and Brunk, H.D.,
Barlow, R.E.,
Sta-
t i s t i c a l I n f e r e n c e under Order R e s t r i c t i o n s , W i l e y , New York ( 1 9 7 2 ) . Partholomew, D.J., A t e s t o f homogeneity o f means under r e s t r i c t e d a l t e r n a t i v e s , J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23
( 1 9 6 1 ) , 239-281. Cunningham, J . P . ,
M u l t i o l e monotone r e g r e s s i o n , Psych. B u l l . , 92 (1982)
7917800. H a r t i g a n , J.A.,
D i s t r i b u t i o n o f t h e r e s i d u a l sums o f squares i n f i t t i n g
i n e q u a l i t i e s , R i o m e t r i k a , 54 ( 1 9 6 7 ) , 69-84. Hirotsu, C.,
Ordered a l t e r n a t i v e s f o r i n t e r a c t i o n e f f e c t s , B i o m e t r i k a ,
65 ( 1 9 7 8 ) , 561-570. C o n t r i b u t i o n t o d i s c u s s i o n o f t h e naper by D.J. B a r t h o l o -
James, G.S.,
mew, J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23 ( 1 9 6 1 ) ,
278-279. Krantz, D . H . ,
Luce,
R.D., SuDpes, P. and Tversky, ',., Fcundations o f
wasurewiit. ka+tvic
P r c c s , I'el-1 Yor!< (1971.).
Luce, R.D. and Tukey, J.I.I., Simultaneous c o n j o i n t measurement: a new t y p e o f fundamental measurement, J o u r n a l o f Math. P s y c h o l . ,1 ( 1 9 6 4 ) ,
1-27. Mandel, J . , A new a n a l y s i s o f v a r i a n c e model f o r n o n - a d d i t i v e d a t a , Technometrics,
13 ( 1 9 7 1 ) , 1-18.
McClelland, G., A Note on A r b u c k l e and L a r i w r , " T h e number o f two-way t a b l e s s a t i s f y i n g c e r t a i n a d d i t i v i t y axiom;:
J o u r n a l o f Math. Psychol .,
1 5 , ( 1 9 7 7 ) , 292-295. The A n a l y s i s o f Variance, W i l e y , New York ( 1 9 5 9 ) .
Scheffe, H., Shorack, G . R . ,
T e s t i n g a g a i n s t o r d e r e d a l t e r n a t i v e s i n model 1 a n a l y -
s i s o f v a r i a n c e : normal t h e o r y and n o n p a r a m e t r i c , Ann. Math. S t a t i s t . ,
38 ( 1 9 6 7 ) , 1740-1753. Smith, P.T.
and Macdonald, R.R.,
Methods f o r i n c o r p o r a t i n g o r d i n a l i n -
formation i n t o analysis o f variance: g e n e r a l i z a t i o n s o f o n e - t a i l tests, B r i t i s h J o u r n a l o f Mathematical and S t a t i s t i c a l PsycholoSy, 36 ( 1 ? 8 3 ) ,
1-21. Tukey, J.W..
One degree o f freedom f o r n o n a d d i t i v i t y , B i o m e t r i c s , 5
( 1949) , 232-242.
Isotonic
251
regression analysis and additivity
[I51 V i n e r y R.J., S t a t i s t i c a l P r i n c i p l e s i n Experimental Design, PlcGrawH i l l (1971). APPEND1 X A l q o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o n s i s t e n t w i t h monotoni c i t y
In what f o l l o w s i n e q u a l i t i e s a r e taken t o be t r u e ifand only i f t h e y h o l d I n o t h e r words i f b o t h s i d e s o f an i n e q u a l i t y a r e
i n t h e s t r i c t sense.
e v a l u a t e d and found t o be equal t h e i n e q u a l i t y w i l l be d e s c r i b e d as b o t h " n o t t r u e " and " n o t f a l s e " . L e t A be t h e complete J x K m a t r i x
c o n t a i n i n g a l l t h e d a t a and l e t A j k be
t h e s u b m a t r i x formed f r o m A w i t h columns j , j t l and j + 2 and rcws k, k - 1 L e t us s t a r t by c o n s i d e r i n g t h e s u b m a t r i x A j k and t e s t i n e q u a l i -
and k-2.
t i e s (I\,(?) and ( 3 ) i n t h i s m a t r i x .
(I:) a r e t r u e or n o t ( l ) , n o t
I f and o n l y i f e i t h e r !1),(2)
( 2 ) and ( 3 ) a r e t r u e i s douhle c a n c e l l a t i o n v i o -
''e n e x t c o n s i d e r a l l t h e ' x 3 s u h r a t r i c e s f r o m
lated.
and ?, and rows Y,
pl
and n o t
fib w i t h
and n (k-rw). Tte t r u t h o f ( 1 ) , ( 2 )
c o l u m s l , 2,
and ( 3 )
;7
has
i m p l i c a t i o n s f o r t h e i r t r u t h i n t h e submatrices we a r e c o n s i d e r i n g as spec i f i e d i n (14) t o ( 1 6 ) below: '2,K
'1,K-1 '1,K
> '3,K-1
=. d
> '2,K-2 > '3,K-2
'2,K
=9
' '3,m
'1,K-1 u
l,K
> '2,n
> u
3,n
m
<
K-1
n <. K-2
(15
n <. K-2
W i t h these r e s t r i c t i o n s i n mind we s h a l l c o n s i d e r a l l p o s s i b l e s t a t e s f o r i n e q u a l i t i e s ( l ) a n d ( 3 ) i n A. State
1 : ( l ) a n d ( 3) t r u e .
I n e q u a l i t i e s ( l q a n d (16) guarantee t h a t i n e q u a l i t i e s (1) and (3) w i l l b e t r u e f o r a l l t h e submatrices under c o n s i d e r a t i o n .
Thus t h e double c a n c e l -
l a t i o n axiom i s n e v e r v i o l a t e d . State
2 :(1) f a l s e and ( 3 ) t r u e .
From (16) i n e q u a l i t y (3) w i l l be t r u e i n a l l t h e submatrices under d i s c u s sion.
Because o f m o n o t o n i c i t y and i n e q u a l i t y ( 3 ) we have:
252
R . R . MdcDorurld
'2,K
> '1,K
> '3.K-2
This means t h a t i n e o u a l i t y ( 1 ) w i l l be t r u e f o r m = K-2 and hence I I ~ C R U o fS ~ p o n o t o n i c i t y f o r a l l v a l u e s o f m n o t equal t o K - 1 .
V i o l a t i o n s o f double
c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when i n e q u a l i t y ( 2 ) i s f a l s e a t
m = K-I.
IJe t h e r e f o r e t e s t i n e q u a l i t y ( 2 ) a t
1;1
=
K-1 f o r successively
d e c r e a s i n g values o f n s t a r t i n g a t n = K-2 u n t i l i n e o u a l i t y ( 2 ) i s found t o be n o t f a l s e .
M o n o t o n i c i t y quarantees t h a t i n e q u a l i t y w i l l c o n t i n u e n o t t o
he f a l s e f o r a l l s m a l l e r values o f n.
Where i n e q u a l i t y ( 2 ) has been f o u n d
t o be f a l s e , v i o l a t i o n s o f double c a n c e l l a t i o n have o c c u r r e d . State
3:
(1) t r u e and ( 3 ) f a l s e .
By ( 1 4 ) i n e q u a l i t y ( 1 ) w i l l be t r u e f o r a l l t h e s u b m a t r i c e s and v i o l a t i o n s
w i l l o n l y o c c u r where i n e q u a l i t y ( 3 ) i s f a l s e and ( 2 ) i s t r u e .
The values
o f n f o r w h i c h i n e q u a l i t y ( 3 ) i s f a l s e a r e found by t e s t i n g t h i s i n e q u a l i t y
for s u c c e s s i v e l y d e c r e a s i n g v a l u e s o f n s t a r t i n g a t K-2 u n t i l i t i s n o l o n I n e q u a l i t y (2) s h o u l d now be t e s t e d f o r each v a l u e o f n f o r
oer false.
which i n e q u a l i t y ( 3 ) i s f a l s e .
T h i s i s done f o r a l l p o s s i b l e values o f m
d e c r e a s i n g from K - 1 u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r t r u e .
Each t i m e i n -
e q u a l i t y (2) i s found t o be t r u e , a v i o l a t i o n o f double c a n c e l l a t i o n has occurred. State
4:
b o t h (1) and ( 3 ) f a l s e .
To b e g i n w i t h , i n e q u a l i t y
(1) i s t e s t e d f o r d e c r e a s i n g values o f m u n t i l a t
m = c ' i t i s f o u n d t o be n o t f a l s e and a t m = c " i t i s f o u n d t o be t r u e :
(c',~").
A t t h i s p o i n t we know we know t h a t i n e q u a l i t y ( 3 ) must be f a l s e
a t n = c ' + l (18) and s i n c e (1) i s a l s o f a l s e double c a n c e l l a t i o n w i l l h o l d f o r n u r e a t e r than c '
.
We n e x t t e s t i n e q u a l i t y ( 3 ) a t n = c ' .
Ift h i s i s t r u e i n e q u a l i t y (1) i s
f a l s e f o r a l l values o f m from K - 1 t o c ' + l and t h e procedures adopted i n state
2
f o r m = K - 1 s h o u l d be r e a p p l i e d h e r e f o r a l l v a l u e s o f m f r o m
K-1 t o c ' + l .
No v i o l a t i o n o f t h e t y p e 1: 2 and n o t 3 i s o o s s i b l e s i n c e 1
and 3 a r e f a l s e f o r a l l values o f n g r e a t e r t h a n c ' by m o n o t o n i c i t y and 3 i s t r u e f o r a l l values o f n g r e a t e r t h a n c ' .
I f c ' does n o t equal c " i n e q u a l i t y (1) w i l l be equal f o r values o f n f r o m
253
Isotonic regression analysis and additivity
Double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e c ' i n these cases.
c"t1 to c'.
I f however, i n e q u a l i t y ( 3 ) was found t o be f a l s e a t n = c ' no v i o l a t i o n
Occurs as i n e q u a l i t y (1) i s a l s o f a l s e f o r a l l values o f m g r e a t e r t h a n c ' .
A v i o l a t i o n i s P o s s i b l e a t m = c " (where i n e q u a l i t y (1) i s t r u e ) i f i n e q u a l i t y ( 3 ) i s f a l s e f o r n = c"-1. ? should
!3e
Here t h e Procedures adonted i n s t a t e
amlied.
There s t i l l remains t h e p o s s i b i l i t y of a v i o l a t i o n o f double c a n c e l l a t i o n i f i n e q u a l i t y (1) i s f a l s e f o r a l l values o f m.
Here by r e a s o n i n g s i m i l a r -
l y t o ( 1 7 ) i n e q u a l i t y ( 3 ) w i l l be f a l s e f o r n g r e a t e r than 1.
I f however,
i n e q u a l i t y ( 3 ) were found t o be t r u e a t n = 1, i n e q u a l i t y ( 2 ) s h o u l d be checked f r o m m = 2 u n t i l K - 1 o r i n e q u a l i t y ( 2 ) i s found t o be t r u e .
Double
c a n c e l l a t i o n w i l l have been v i o l a t e d f o r e v e r y f a l s i f i c a t i o n of i n e q u a l i t y
(2)
*
S t a t e 5 : one o r more o f i n e q u a l i t i e s (1) and ( 3 ) a r e e y a l . I f e i t h e r i n e q u a l i t y ( 1 ) o r ( 3 ) i s t r u e o r b o t h a r e equal t h e s i t u a t i o n i s like state
I n e q u a l i t i e s (14) and (16) a r e t r u e i n t h e weak f o r m (where
1.
Thus
t h e g r e a t e r t h a n s i a n s a r e reDlaced by g r e a t e r t h a n o r equal s i g n s ) .
i n t h i s s i t u a t i o n double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e d as n e i t h e r i n e q u a l i t y (1) n o r (3) can be f a l s e . I f i n e q u a l i t y (1) i s f a l s e and i n e q u a l i t y ( 3 ) i s equal t h e n o s i t i o n i s s i milar t o state
2
.
The weak v e r s i o n o f i n e q u a l i t y ( 1 7 ) guarantees inequa-
l i t y ( 1 ) i s n o t f a l s e f o r m = K-2. V i o l a t i o n s o f double c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when m = K-1, i n e q u a l i t y ( 3 ) i s t r u e and i n e q u a l i t y
( 2 ) f a l s e f o r some v a l u e o f n. I n e q u a l i t y ( 3 ) s h o u l d be t e s t e d f o r m = K-1 f o r s u c c e s s i v e l y d e c r e a s i n g values o f n u n t i l i n e q u a l i t y (3) i s f o u n d t o be true a t n
I
c ' a f t e r w h i c h i t w i l l be t r u e f o r a l l s m a l l e r values.
Inequa-
l i t y ( 2 ) s h o u l d then be e v a l u a t e d a t m = K - 1 f o r values o f n d e c r e a s i n a from n = c ' u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r f a l s e .
Each f a l s i f i c a t i o n
o f i n e q u a l i t y ( 2 ) corresponds t o a v i o l a t i o n o f t h e double c a n c e l l a t i o n axiom. The f i n a l p o s s i b i l i t y i s where i n e q u a l i t y (1) i s e o u a l and ( 3 ) i s f a l s e . Here t h e procedure i s as i n s t a t e
4 where c ' and c " a r e d i f f e r e n t .
Ine-
q u a l i t y (1) i s t e s t e d f o r s u c c e s s i v e l y d e c r e a s i n g values o f m u n t i l a t rn = c " i t i s t r u e .
I n e q u a l i t y ( 3 ) i s e v a l u a t e d a t n = c"-1 and i f f a l s e
t h e Drocedure f o r s t a t e
3 i s adonted.
224
R.R. MdcDonald
Ile have now checked f o r double c a n c e l l a t i o n i n a l l t h e 3 x 3 r r a t r i c e s w i t h columns 1, 2 and 3 and rows K, F, and n. p and rows K , K-1 and
lues o f o
c
o < J.
Each m a t r i x w i t h columns 1, o and
K-2 can be t r e a t e d i n t h e same way as A, f o r a l l va-
T h i s w i l l e n a b l e t h e t e s t i n a o f double c a n c e l l a t i o n i n
a l l m a t r i c e s w i t h rows 1, o and p and columns K,
pl
and n.
T h i s comoletes
uIK.
I n o r d e r t o c o n s i d e r e v e r y poss i b l e v i o l a t i o n t h e nrocedures s h o u l d be r e o e a t e d f o r a l l s u b m a t r i c e s A j k , t e s t i n q a l l 3 x 3 matrices i n c l u d i n n
j
,I-1 and k
2.
>
ACKN@'*lLF:DGEIIE NTS The a u L ' @ rwishes t o exnress h i s thanks t o D r . P.T. the m r u c c r i n t .
rmit':
f o r comments on
239
ISOTONIC REGRESSION ANALYSIS AND ADDITIVITY
Ranald R . Flacdonald Deoartment o f Psychology I!ni v e r s i t,y o f S t i r l i n ?
T h i s naner reviews tl7e 1iterat.ut-e on i s o t o n i c r e g r e s s i o n w i t h a view t o f i n d i n g t e s t s f o r a d d i t i v i t y i n d a t a whose p o n u l a t i o n values a r e unioue o n l y up t o a monotonic t r a n s f o r m a t i o n .
Several
anproaches a r e c o n s i d e r e d and a s e q u e n t i a l one i s recommended
-
f i r s t t e s t i n g m o n o t o n i c i t y , and then
t e s t i n g t h e double c a n c e l l a t i o n axiom.
Two t e s t s
a r e p r e s e n t e d f o r t e s t i n g m o n o t o n i c i t y and an a l g o r i thm f o r d e t e c t i n g s i g n i f i c a n t denartures from double c a n c e l l a t i o n i s o u t l i n e d .
1. INTRODUCTION For a number o f y e a r s mathematical p s y c h o l o g i s t s have s t u d i e d a x i o m a t i c approaches t o a d d i t i v i t y .
Under t h e heading o f c o n j o i n t measurement t h e y
have been concerned w i t h whether t h e combined e f f e c t s o f a p a i r o f independ e n t l y m a n i p u l a t e d v a r i a b l e s a r e c o m p a t i b l e w i t h an u n d e r l y i n g a d d i t i v e r e presentation.
To p u t i t a n o t h e r way, what a r e t h e necessary and s u f f i c i e n t
c o n d i t i o n s under which monotonic f u n c t i o n s o f each o f two v a r i a b l e s e x i s t such t h a t when added t o g e t h e r t h e i r sum i s i t s e l f a monotonic f u n c t i o n o f t h e combined e f f e c t s ?
A much q u o t e d example i s t h a t o f a s i m p l e b e t w i t h Can t h e s u b j e c t i v e v a l u e o f
some o r o b a b i l i t y o f w i n n i n g a s p e c i f e d amount.
such b e t s be expressed i n terms o f a monotonic f u n c t i o n of t h e p r o b a b i l i t y p l u s some monotonic f u n c t i o n o f t h e D o t e n t i a l w i n n i n g s ? One r e p o r t of a s o l u t i o n t o t h e p r o b l e m i s g i v e n i n t h e f i r s t paper t o be p u b l i s h e d i n t h e J o u r n a l o f Mathematical Psychology, Luce and Tukey (1964). and o t h e r accounts a r e w i d e l y a v a i l a b l e o f which t h e most comorehensive i s K r a n t z , Luce, Suppes and T v e r s k y (1971).
The s o l u t i o n t o t h e Droblem was f o r m u l a t e d i n
terms o f t h e o r d e r r e l a t i o n s which must h o l d between a l l p o s s i b l e combined
243
R.R. MdcDonuld
e f f e c t s o f the variables. hold o r not.
I d i t h r e a l d a t a these c o u l d be f o u n d e i t h e r t o
A p a r t f r o m a numher o f u n t e s t a b l e t e c h n i c a l axioms t h e d a t a
must conform t o t y o o r d e r r e s t r i c t i o n s . t o t h e independent v a r i a b l e s .
They must be o r d e r e d w i t h r e s p e c t
T h a t i s , i f one v a r i a b l e i s h e l d c o n s t a n t ,
an i n c r e a s e i n t h e o t h e r v a r i a b l e s h o u l d l e a d t o an i n c r e a s e i n t h e v a l u e
of t h e data.
This i m n l i e s t h a t a d d i t i v e data are monotonically r e l a t e d t o
t h e indenendent v a r i a b l e s and t h ? r e s t r i c t i o n w i l l be r e f e r r e d t o as t h e monotonicity r e s t r i c t i o n .
I n a d d i t i o n t o t h e m o n o t o n i c i t y r e s t r i c t i o n Luce
and Tukey (1964) r e o u i r e t h a t t h e double c a n c e l l a t i o n axiom be met. double c a n c e l l a t i o n axiom s t a t e s t h a t if a ' , a " , a"' dependent v a r i a b l e A and b ' , b " , b " ' variable
The
a r e l e v e l s of t h e i n -
are l e v e l s o f the o t h e r independent
then i f i n e n u a l i t i e s (1) and ( 2 ) :?olrl, so s \ ~ u ! d ( 3 )
where uab i s t h e v a l u e o f t h e d a t a c o r r e s p o n d i n g t o l e v e l a o f A and b of R.
The double c a n c e l l a t i o n axiom a l s o i s r e q u i r e d t o h o l d w i t h > r e p l a c e d
by < .
T h i s i s o f l i m i t e d use when d e a l i n g w i t h t h e f a l l i b l e d a t a o f expe-
rimentalists.
I n such cases i t would b e u s e f u l t o know i f any v i o l a t i o n s
i n t h e o r d e r r e s t r i c t i o n s c o u l d be e x n l a i n e d by t h e i n h e r e n t v a r i a b i l i t y o f the data o r whether they represent r e a l effects. There i s a n o t h e r p r o b l e m i n t e s t i n g i f t h e above o r d e r r e s t r i c t i o n s h o l d i n r e a l data.
As t h e number o f l e v e l s o f t h e independent v a r i a b l e s s t u d i e d i s
f i n i t e , one cannot check t h e double c a n c e l l a t i o n axiom f o r a l l p o s s i b l e l e v e l s of A and B .
I n f i n i t e cases double c a n c e l l a t i o n may be s a t i s f i e d
w h i l e more complex c a n c e l l a t i o n s a r e n o t .
Any v i o l a t i o n o f a c a n c e l l a t i o n
no m a t t e r how complex r e n d e r s an a d d i t i v e r e p r e s e n t a t i o n i m p o s s i b l e .
Ar-
b u c k l e and L a r i m e r (1976) and M c C l e l l a n d (1977) p r o v i d e o r o b a b i l i t i e s o f complex c a n c e l l a t i o n s b e i n g v i o l a t e d g i v e n t h a t l r o n o t o n i c i t v and double c a n c e l l a t i o n h o l d i n randomly o r d e r e d m a t r i c e s . crease as t h e m a t r i c e s i n c r e a s e i n s i z e . c e l l a t i o n s would be a hideous t a s k .
a D a r t i a l way o u t .
However, Luce and Tukey (1964) p r o v i d e
B a r e s e p a r a t e d by equal aB f o r m a d u a l s t a n d a r d sequence) m o n o t o n i c i t y and
Where t h e l e v e l s o f A and
mounts ( t e c h n i c a l l y A,
These a r o b a b i l i t i e s i n -
T e s t i n g a l l a o s s i b l e complex can-
24 1
kotoiiic repenion arlalysis and additivity
double c a n c e l l a t i o n a r e necessary and s u f f i c i e n t conc'itions t o ensure an a d d i t i v e r e r J r e s e n t a t i o n o v e r t h e l e v e l s of A and B s t u d i e d .
Thus
i t i s d e s i r a b l e t o design experiments so t h a t t h i s c o n d i t i o n i s approxima-
t e l y met.
I f t h e l e v e l s of A and B do form a dual s t a n d a r d sequence, t h e n
i t i s o n l y necessary t o t e s t m o n o t o n i c i t y and double c a n c e l l a t i o n ;
i n 3:s
case complex c a n c e l l a t i o n s may be i g n o r e d .
I t i s w o r t h d i s t i n g u i s h i n g between t h e p r e s e n t c o n c e p t i o n o f a d d i t i v i t y w h i c h concerns t h e e x i s t e n c e o f a d d i t i v i t y h o l d i n g i n some monotonic t r a n s f o r m a t i o n o f t h e d a t a and a d d i t i v i t y which h o l d s i n u n t r a n s f o m d data. S t a n d a r d t e s t s o f i n t e r a c t i o n s i n a n a l y s i s o f v a r i a n c e , T u k e y ' s (1964) add i t i v i t y t e s t an6 "anc'el's (1971) work on n o n a d d i t i v i t y a r e a l l concerned w i t h t h e second sense o f a d d i t i v i t y .
The c o n j o i n t measurement Droblem i s
concerned w i t h t h e f i r s t sense and i f a d d i t i v i t y e x i s t s i n t h i s sense i t
w i l l a l s o do so a f t e r any monotonic t r a n s f o r m a t i o n has been a o p l i e d .
For
t h e r e s t o f t h e d i s c u s s i o n a d d i t i v i t y w i l l be used o n l y i n t h e more g e n e r a l sense.
A l s o i n t h e a n p l i c a t i o n s t o be c o n s i d e r e d i t w i l l be assumed t h a t
t h e o r d e r i n g o f t h e l e v e l s o f t h e indeoendent v a r i a b l e s i s known i n advance. Bartholomew (1961) p r e s e n t e d a method f o r s t a t i s t i c a l i n f e r e n c e under o r d e r r e s t r i c t i o n s which he c a l l e d i s o t o n i c r e g r e s s i o n a n a l y s i s .
More r e c e n t r e -
views o f t h e area a r e g i v e n i n Barlow, B a r t h o l o w w , E r e m e r and Brunk (1972) and Smith and Macdonald (1983).
A major a p o l i c a t i o n o f i s o t o n i c regression
a n a l y s i s i s t o p r o v i d e t e s t s o f d i f f e r e n c e s between means s u b j e c t t o c e r t a i n order restrictions.
I t would seem l i k e l y t h a t t h e s t a t i s t i c a l l i t e -
r a t u r e on i s o t o n i c r e g r e s s i o n w o u l d be r e l e v a n t t o t h e oroblems in a p p l y i n g c o n j o i n t measurement t o f a l l i b l e d a t a .
T h i s oaper i s designed t o examine
t h e e x t e n t t o which t h i s i s t r u e and t o suggest some s t a t i s t i c a l t e s t s f o r f i t t i n o an a d d i t i v e c o n j o i n t model. The aims o f t h e s t a t i s t i c i a n s w o r k i n g on i s o t o n i c r e g r e s s i o n a r e somewhat d i f f e r e n t f r o m those o f t h e c o n j o i n t neasurers.
Tests developed i n i s o t o n i c
r e g r e s s i o n a r e designed t o i n c r e a s e t h e power o f t h e t e s t when t h e o r d e r i n g o f t h e means can be p r e d i c t e d i n advance.
C o n j o i n t measurement i s concer-
ned w i t h whether t h e r e a r e any d i f f e r e n c e s between t h e means i n c o m p a t i b l e w i t h a d d i t i v i t y r a t h e r than showing whether an a d d i t i v e model g i v e s a b e t t e r f i t t h a n no model a t a l l .
242
R.R.MacDonald
2. NOTATION
The f o l l o w i n g n o t a t i o n a n o l i e s t o a two way a n a l y s i s o f variance, and t o a one way a n a l y s i s when the s u h s c r i p t k i s omitted:
J
i s the number o f l e v e l s o f v a r i a b l e A
i s the number o f l e v e l s o f v a r i a b l e B
I(
Yijk
n
i s the i t h o b s e r v a t i o n i n the j k t h treatment
i s the number o f observations i n each jk t h treatment (assumed equal)
N
i s t h e t o t a l number o f observations
u
i s the t h e o r e t i c a l o v e r a l l mean
ir
i s the observed o v e r a l l mean
ujk
i s the t h e o r e t i c a l mean i n the j k t h treatment
ir .jk ir
i s the observed mean i n the j t h l e v e l of A
irk
i s the observed mean i n the k t h l e v e l o f R
j
i s the observed mean i n the j k t h treatment
i-u
i s the i s o t o n i c regression estimate o f u . jk Jk' i*ujk i s derived w i t h respect t o some order r e s t r i c t i o n s ; u j , k , < u ~ , , ~f,o,r
some s p e c i f i e d j ' , j " , k ' , k " .
Barlow e t a1 ( 1 9 7 2 ) show t h a t the i s o t o n i c
regression e s t i m a t e i s the maximum l i k e l i h o o d e s t i m a t e o f u
w i t h these r e s t r i c t i o n s
jk
.
consistent
L i s the number o f d i f f e r e n t values taken by the i s o t o n i c regression e s t i mates i-u. Jk. Considering f i r s t the u n i v a r i a t e case, the usual s t a t i s t i c used t o t e s t f o r e q u a l i t y o f treatment means i s : 1 n ( D j-O)
F(J-1,N-J)
ij
F =
" / ( J-1)
= j
z
I n analysis o f variance n o t a t i o n :
CI
(4) ( Y -G.)*/(PI-J)
"
SSbet/dfbet - MSbet SSw/dfw -
ww
Eartholomew (1961) proposes the s t a t i s t i c E: z n(i-u.-D) 2 J E J2 = j 1 (Yij-D)' ij
(5)
243
Isotonic regression analysis m d additivity
I n o u r n o t a t i o n t h i s i s expressed:
2 SSmon EJ = E
(7)
2 J d i f f e r s f r o m F i n two i m p o r t a n t r e s p e c t s .
F i r s t l y the i s o t o n i c regres-
s i o n e s t i m a t e s a r e used t o compute t h e between t r e a t m e n t v a r i a n c e through SSmon.
Secondly, t h e between t r e a t m e n t v a r i a t i o n u n s o e c i f i e d by t h e o r d e r
r e s t r i c t i o n s ( w h i c h i s r e f e r r e d t o as SSres = SSbet t o t h e e r r o r term.
-
SSmon) c o n t r i b u t e s
The second d i f f e r e n c e has t h e a f f e c t o f i n c r e a s i n g t h e
) when t h e d i f f e r e n c e s between t h e j" when j ' < j " ) . Idhen means a r e i n t h e o r d e r i n g s p e c i f i e d i n H1(uj, < u j" power o f t h e t e s t a g a i n s t Ho(uj,=u
t h e r e a r e d i f f e r e n c e s between t h e means which a r e n e i t h e r s p e c i f i e d n o r
p r e c l u d e d i n H1 a t e s t u s i n g MSw as t h e e r r o r t e r m may be more p o w e r f u l 2 than EJ. The e x t e n s i o n t o t h e b i v a r i a t e case i n n o m a l analyses o f v a r i a n c e i s t o p a r t i t i o n SSbet i n t o t h a t due t o v a r i a b l e A ( S S A ) , t h a t due t o v a r i a b l e B (SSB) and a r e s i d u a l c a l l e d t h e i n t e r a c t i o n ( S S A B ) .
The degrees o f f r e e -
dom a s s o c i a t e d w i t h each a r e known and s e p a r a t e F values a r e used t o t e s t
each o f t h e components o f t h e between v a r i a t i o n a g a i n s t t h e e r r o r term, see Winer (1971). The a p p l i c a t i o n o f i s o t o n i c r e g r e s s i o n t o t h e two way a n a l y s i s i s consider e d b y Barlow e t a1 (1972) f o l l o w i n g Shorack (1967). 2
They recommend u s i n g
EJ as d e f i n e d i n e q u a t i o n (8) r a t h e r than ( 7 ) .
T h i s i s t o t e s t Ho
uj,
= u
jl'
a g a i n s t HI
ujl
u
j"
f o r some s p e c i f i e d jl and
j " . The ( A ) a f t e r SSmn denotes t h a t t h e o r d e r r e s t r i c t i o n s i n v o l v e o n l y
l e v e l s o f A.
S i m i l a r t e s t s a r e developed i n v o l v i n g t h e B v a r i a b l e b u t no
t e s t s o f d i f f e r e n c e s between means i n t h e i n t e r a c t i o n a r e d i s c u s s e d i n t h i s context, 3. UNIVARIATE TESTS OF MONOTONICITY
L e t us f i r s t c o n s i d e r t h e u n i v a r i a t e case.
T h i s i s n o t e n t i r e l y degenerate
s i n c e i f m o n o t o n i c i t y were v i o l a t e d t h e d a t a c o u l d n o t be r e p r e s e n t e d by an a d d i t i v e model.
A demonstration o f nonmonotonicity orecludes the possibi-
lity o f a d d i t i v i t y
.
R . R . MacDonald
144
I n u n i v a r i a t e i s o t o n i c r e g r e s s i o n t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s Tu. J . . J are found u s i n g t h e 0001 a d j a c e n t
v i t h r e s p e c t t o t h e o r d e r 1, 2,.
v i o l a t o r s a l g o r i t h m d e s c r i b e d i n B a r l o w e t a? (1972). M o n o t o n i c v a r i s t i o n 2 i n t h e means i s t h e n demonstrated u s i n g EJ as d e f i n e d i n e q u a t i o n ( 6 ) . I n t h e a p p l i c a t i o n o f c o n j o i n t measurement we a r e concerned i n d e m o n s t r a t i n g I t seems n a t u r a l t o t a k e SSres = z n(ir i-u.)' as 2 j- J Nonmonotonic j v a r i a t i o n w i l l an e s t i m a t e o f t h e p o D u l a t i o n v a r i a n c e u nonmonotonic v a r i a t i o n .
.
i n c r e a s e t h e v a l u e o f SSres and i f i t i s s i g n i f i c a n t l y g r e a t e r t h a n w o u l d be e x p e c t e d f r o m t h e w i t h i n c o n d i t i o n v a r i a n c e , t h e nonmonotonic v a r i a t i o n has been demonstrated.
f o r a l l j ' and j " , i s t r u e then
I f Ho, uj,=uj,, 2 2
SSres i s d i s t r i b u t e d as no
A p r o b l e m a r i s e s when t h e n u l l hypo-
t h e s i s t o be t e s t e d i s H i : e i t h e r u j , = uj,, o r u j , < uj,,,
f o r .i'< j " .
Suppose u < u i t i s s t i l l possible t h a t b., > and as such t h e J j'" j ' j'" and c o u l d c o n t r i b u t e t o SSres. The e x p e c t e d v a r i a t i o n between iI j' J v a l u e o f such v a r i a n c e w o u l d be l e s s t h a n t h a t i f u = u Thus SSres i s 2 2 j' j". d i s t r i b u t e d as l e s s t h a n n u x ( ~ - ~ ) T. h i s argument i s developed more f o r m a l l y by H a r t i g a n (1967) who e s t a b l i s h e s f o r SSres t h e u p p e r bound 2 2 nc x ( ~ - ~ ) .Thus an a p p r o p r i a t e t e s t s t a t i s t i c f o r cigClt be F I L as de::,I,
f i n e d i n e q u a t i o n ( 9 ) . though i t w i l l be c o n s e r v a t i v e t o an unknown e x t e n t f o r reasons g i v e n above.
F I L (J-L,N-JK)
=
SSres / (J-L)
(9)
MSw
Bartholomew (1961) c o n s i d e r e d u s i n g a F I L s t a t i s t i c as a t e s t f o r a d d i t i v i He r e j e c t e d t h i s approach as L i s a random v a r i a b l e w h i c h i n c r e a s e s as 2 t h e p r o b a b i l i t y o f Ho decreases and H1 i n c r e a s e s . EJ t h e recommended t e s t ,
ty.
incorporates L i n t o the t e s t s t a t i s t i c .
I n t h e monotonic case n o n - l i m i t i n g
values o f L w i l l be d e t e r m i n e d m a i n l y by t h e monotonic v a r i a t i o n .
The non-
monotonic t e s t i s concerned w i t h t e s t i n g t h e s i g n i f i c a n c e o f whatever nonmonotonic
v a r i a t i o n e x i s t s i n d e p e n d e n t l y o f t h e s i z e o f t h e monotonic
L
variation.
I t i s n o t t h e r e f o r e a p p r o p r i a t e t o use
statistic.
S i m i l a r l y i t i s a p p r o p r i a t e t o use SSw as t h e e r r o r t e r m as i t
as p a r t o f t h e t e s t
i s d e s i r e d t o show t h a t t h e nonmonotonic v a r i a t i o n i s g r e a t e r t h a n t h e e s t i mate o f t h e w i t h i n c o n d i t i o n v a r i a t i o n .
T h i s p o i n t has been d i s c u s s e d above.
The c o n s e r v a t i s m o f t h e p r o p c s e d t e s t ( e q u a t i o n ( 9 ) ) i s due t o t h e presence o f s m a l l amounts o f v a r i a t i o n between means w h i c h a r e t r u l y m o n o t o n i c .
It
245
Isototiic regression unulysis und additivity
w o u l d seem p r e f e r a b l e t o use a t e s t which gave g r e a t e r w e i g h t t o t h e l a r ger nonmnotonic v a r i a t i o n .
Such a t e s t i s t h e s t u d e n t i z e d range t e s t -see
b l i n e r (1971). To use t h i s t e s t we f i n d t h e r e s i d u a l e s t i m a t e s o f t h e t r e a t ment means ( r i i ~ . ) as d e f i n e d i n e q u a t i o n (10).
J
n'bi. =. 0j
-
Ti. J t G
(10)
The l a r g e s t and s m a l l e s t o f these a r e s u b s t i t u t e d i n t o t h e t e s t s t a t i s t i c q as i n e q u a t i o n (11). q =
n-ularFest
"smallest MSwIn
(11)
The s t u d e n t i z e d range t e s t i s designed t o d e t e c t d i f f e r e n c e s b e b e e n k t r e a t m e n t means f r o m t h e d i f f e r e n c e between t h e l a r g e s t and s m a l l e s t mean. Since i n t h e p r e s e n t case t h e r e a r e J t r e a t m e n t s and
L different isotonic
e s t i m a t e s , t h e a p p r o p r i a t e v a l u e t o use f o r k when l o o k i n g up t h e t a b l e s i s J-Ltl. F o r reasons analogous t o those a p p l y i n g t o FIL t h e range t e s t
w i l l a l s o be c o n s e r v a t i v e b u t s i n c e i t g i v e s most w e i g h t i n g t o t h e l a r g e s t d i f f e r e n c e i t s h o u l d be m r e p o w e r f u l t h a n F ( L .
The r e l a t i v e powers o f t h e two t e s t s and t h e e x t e n t o f t h e i r c o n s e r v a t i s m cannot he a s c e r t a i n e d
i n general, s i n c e i t w i l l depend on t h e amount o f t r u e monotonic v a r i a t i o n p r e s e n t i n t h e p o p u l a t i o n means. I n d e e d t h e c o n s e r v a t i s m o f these t e s t s w i l l depend on t h e t r u e values o f t h e p o p u l a t i o n means and t h e p r o b a b i l i t y t h a t p o p u l a t i o n means i n t h e p r e d i c t e d o r d e r w i l l g i v e r i s e t o sample means o u t o f o r d e r .
I f t h e values o f
t h e i s o t o n i c e s t i m a t e s o f t h e p o p u l a t i o n means a r e t r e a t e d as p o p u l a t i o n values, t h e e x t e n t o f t h e c o n s e r v a t i s m c o u l d be e s t i m a t e d by s i m u l a t i o n . From r e p e a t e d s i m u l a t i o n s t h e e x p e c t e d values o f SSres and L c o u l d be found where t h e r e i s no nonmonotonic v a r i a t i o n . equation (13) t o a l l o w f o r t h i s .
C o r r e c t i o n s c o u l d be made t o
However, t h e r e q u i r e m e n t t o r u n a simu-
l a t i o n on each t e s t r u l e s t h i s o u t as a r o u t i n e procedure. 4. BIVARIATE TESTS OF MONOTONICITY T h i s s e c t i o n a t t e m p t s t o g e n e r a l i s e these r e s u l t s t o b i v a r i a t e m o n o t o n i c i t y . The s i m p l e s t approach i s t o r e l y on o n l y t h e o v e r a l l means f o r each of t h e l e v e l s o f A and 5. T h i s i s e s s e n t i a l l y t h e aoproach suggested by Shorack
(1967) mentioned above, who t r e a t e d t h e two way a n a l y s i s as i f i t were two
246
R . R . AfacUuiiald
indenendent one way analyses. found w j t h r e s p e c t t o 1, 2,
2
The i s o t o n i c r e g r e s s i o n e s t i m a t e s i-u. a r e
..., J
J
and monotonic v a r i a t i o n w i t h r e s p e c t
t o A t e s t e d u s i n g EJ as d e f i n e d i n e q u a t i o n ( 8 ) .
To t e s t t h e presence o f
nonmonotonic v a r i a t i o n w i t h r e s p e c t t o A one c o u l d t h e r e f o r e use fined i n equation ( 9 ) . t o variable B.
FIL as de-
The t e s t i n g procedure can be r e n e a t e d w i t h r e s p e c t
The two t e s t s a r e m u t u a l l y independent when t h e number o f
o b s e r v a t i o n s i n each o f t h e c o n d i t i o n s i s e q u a l .
Barlow e t a1 (1972) d i s -
cuss t h e case where t h e numbers o f o b s e r v a t i o n s a r e unequal. James (1961). comnenting on Partholomew's work, suggested ways t h a t i s o t o n i c r e g r e s s i o n m i g h t be extended t o i n c l u d e i n t e r a c t i o n s .
Rartholomew
(1961) r e p l i e d t h a t he had c o n s i d e r e d t h e p r o b l e m b u t f o u n d i t f o r b i d d i n g . H i r o t s u (1978) f o l l o w e d
UD
these i d e a s by c o n s i d e r i n g v a r i o u s comDarisons
o f t r e a t m e n t means which c o n t r i b u t e d t o t h e i n t e r a c t i o n , e . p .
(l.,')
I f a l l such comparisons a r e n o s i t i v e ( o r n e g a t i v e ) t h e d a t a a r e monotonic. This i s a stronger r e s t r i c t i o n r e q u i r i n g n o t o n l y t h a t the data monotonicall y i n c r e a s e s w i t h t h e indenendent v a r i a b l e s b u t t h a t t h e i n c r e a s e w i t h each
successive l e v e l o f an independent v a r i a b l e i s i t s e l f m o n o t o n i c a l l y i n c r e a s i n g ( o r decreasina).
Another way o f p u t t i n g i t i s t h a t t h e change due t o
+he independent v a r i a b l e s i s a c c e l e r a t i n g e i t h e r n o s i t i v e l y o r n e g a t i v e l y . Consequently any t e s t which showed a l l such comparisons t o be p o s i t i v e o r n e g a t i v e would i n d e e d e s t a b l i s h m o n o t o n i c i ty ( t h o u g h n o t double c a n c e l l a t i o n ) , b u t a n e g a t i v e r e s u l t would l e a v e open t h e p o s s i b i l i t y o f monotonicity.
Unless t h e r e i s reason t o e x p e c t an a c c e l e r a t i n g e f f e c t o f t h e i n -
dependent v a r i a b l e s , t h i s approach i s p r o b a b l y n o t r e l e v a n t . Shorack's procedures o u t l i n e d above a r e r e l a t i v e l y s t r a i g h f o t w a r d b u t t h e y ignore p o t e n t i a l l y r e l e v a n t v a r i a t i o n c o n t r i b u t i n g t o the i n t e r a c t i o n .
In
o t h e r words, u n l e s s t h e n o n m n o t o n i c v a r i a t i o n i n one v a r i a b l e i s p r e s e n t averaged o v e r a l l l e v e l s o f t h e o t h e r , t h i s anproach w i l l f a i l t o d e t e c t the nonmonotonicity.
I s o t o n i c r e g r e s s i o n e s t i m a t e s i u can be found w i t h
respect t o the p a r t i a l o r d e r j , k
< j',k'
i f j < j ' and k < k ' .
Several
algorithms t o f i n d i s o t o n i c regression estimates w i t h respect t o p a r t i a l o r d e r s a r e r e p o r t e d i n B a r l o w e t a1 (1972).
The p o o l a d j a c e n t v i o l a t o r s
a l g o r i t h m which worked w i t h simpl'e o r d e r s f a i l s i n t h e more g e n e r a l case, As i s o t o n i c r e g r e s s i o n e s t i m a t e s can be found t h e SSbet can be s p l i t i n t o
247
Isotonic regression malysis and additiuity
t h e sums o f squares between t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s SSmon and SSres, a r e s i d u a l due t o nonmonotonic v a r i a t i o n .
Again u s i n g t h e upper
bound f o r SSres ( H a r t i g a n ( 1 9 6 7 ) ) an o v e r a l l t e s t based on F I L i s p o s s i b l e This i s presented i n equation (13). FIL(JK-L,N-JK)
=
S i m i l a r l y t h e m u l t i p l e range t e s t can be a p p l i e d t o t h e b i v a r i a t e case. Again t h e r e s i d u a l ticnn!onotcni c e s t i v a t e s cC ttse t r e a t m e n t means can he found and t h e l a r g e s t and s m a l l e s t s u b s t i t u t e d i n t o e q u a t i o n ( 1 1 ) . v a l u e f o r k i n t h i s case i s JK-L+1.
The
5 . EXAMP1.F The t e s t s o u t l i n e d above w i l l be i l l u s t r a t e d u s i n g a r t i f i c i a l d a t a produced by Cunningham (1982) and g i v e n i n t a b l e 1 below.
I t w i l l be assumed t h a t
t h e raw d a t a a r e t h e means o f t e n independent o b s e r v a t i o n s ( n = 10). Isotonic
Data
Estimates
B 3 7
6
9
6.5
6.5
9.0
B 2 3
1
8
2.57
2.=7
Q.?
BIZ
4
5
2.0
2.67
5.0
A1 A2
A1
A3
A2
A3
Table 1 F i c t i t i o u s example f r o m Cunningham (1982) A p p l y i n g S h o r a c k ' s approach t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t T h i s r e s u l t s i n an SSmon (A) o f 140
t o A ( t h e i u ' s ) a r e 3.83,
3.83,
and a SSres (A) o f 0.66.
N o n a d d i t i v i t y i n A w o u l d be t e s t e d by F(1.81) =
0.66/MSw.
7.33.
I t w o u l d r e q u i r e tlsw t o be l e s s t h a n .166 t o i n d i c a t e s i g n i f i c a n t
n o n a d d i t i v i t y a t t h e 0.05 l e v e l ,
T h i s approach w o u l d produce no evidence
o f n o n a d d i t i v i t y i n B, since a l l t h e v a r i a t i o n i s monotonic and SSres (B) e q u a l s zero. Usin? t t e i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o t h e p a r t i a l o r d e r UjlkI <
j ",k"
for j ' , j " and k ' , k " g i v e n i n t a b l e 1,
SSmon and SSres
248
R.R. MucDonald
come o u t as 549 and 51. /MSw.
N o n a d d i t i v i t y c c u l d be t e s t e d b y F(3,81)
= (51/3)
A MSw o f l e s s t h a n 6.83 w o u l d be r e q u i r e d f o r t h e r e s u l t t o be s i p I f t h e s t u d e n t i z e d range t e s t were a p p l i e d
n i f i c a n t a t t h e 0.05 l e v e l .
c r l e s s , w o u l d be r e q u i r e d f o r s i g n i f i c a n c e
o = 9.48/MSw and a Ffsw of 7.€8
a t t h e 0.05 l e v e l . 6 . DOUBLE CANCELLATION A X I O M
! l o n o t o n i c d a t a a r e n o t n e c e s s a r i l y a d d i t i v e as t h e example due t o K r u s k a l i n S c h e f f e (1959) shows.
Here m a t r i x A g i v e s t h e r a n k o r d e r o f t h e d a t a
from smallest t o l a r g e s t .
A i s m n o t o n i c and R and C a r e m a t r i c e s formed
by a s u b s e t o f t h e elements i n A.
A
B
C
[: :] I f t h e d a t a a r e assumed t o be a d d i t i v e , f r o m
from C l s t + 8 t h = 4th+2nd.
m a t r i x B l s t + 7 t h = 3 r d + d t h and
The l e f t hand s i d e o f t h e f i r s t e q u a t i o n i s a l -
ways l e s s t h a n t h e l e f t hand s i d e o f t h e second under any monotonic t r a n s formation o f the data.
The r e v e r s e i s t r u e o f t h e r i g h t hand s i d e .
l y t h e a d d i t i v i t y assumption i s u n t e n a b l e .
Clear-
Data must be shown n o t t o de-
D a r t from m o n o t o n i c i t y and n o t t o v i o l a t e double c a n c e l l a t i o n b e f o r e we can conclude t h e y a r e a d d i t i v e . Having c o n s i d e r e d t e s t s o f m o n o t o n i c i t y we now l o o k a t ways o f t e s t i n g double c a n c e l l a t i o n assuming t h a t t h e d a t a a r e monotonic.
I n i t s most ge-
n e r a l form we must t e s t a l l t h e 3 x 3 m a t r i c e s f o r i n e q u a l i t i e s (1) t o ( 3 ) . I f we assume m n o t o n i c i t y many fewer m a t r i c e s need t o be examined.
3 x 3 m a t r i c e s where a ' < a " < a"'
and b ' < b " < b"'
Only i n
i s m o n o t o n i c i t y compa-
t i b l e w i t h i n e q u a l i t i e s ( 1 ) and ( 2 ) b e i n q t r u e and ( 3 ) b e i n q f a l s e o r v i c e vprsa.
S t i l l f u r t h e r r e d u c t i o n s on t h e number o f 3 x 3 m a t r i c e s which need
t o be examined a r e n o s s i b l e .
An a l a o r i t h m f o r f i n d i n n a l l t h e v i o l a t i o n s o f
double c a n c e l l a t i o n i n a s e t o f d a t a which a r e c o m p a t i b l e w i t h m o n o t o n i c i t y i s g i v e n i n t h e appendix. Procedures f o r f i n d i n g i s o t o n i c r e g r e s s i o n e s t i m a t e s r e q u i r e a p r i o r i o r d e r i n g o f t h e c o n d i t i o n s even i f i t i s o n l y p a r t i a l .
The double c a n c e l l a -
t i o n axiom imposes a r e s t r i c t i o n c o n d i t i o n a l on o t h e r v a l u e s o f t h e d a t a .
249
Isotonic regression amlysis and additivity
As t h i s i s n o t an a p r i o r i o r d e r i n g t h e e s t i m a t i o n a l g o r i t h m s g i v e n i n
Barlow q t a1 (1972) would have t o be extended i n o r d e r f o r t h e i s o t o n i c r e g r e s s i o n e s t i m a t e s t o be found,
Ift h e a l g o r i t h m s were so extended t h e
r e s u l t i n g procedure f o r t e s t i n g f o r n o n - i s o t o n i c v a r i a t i o n would be t h e t e s t s g i v e n i n e q u a t i o n s ( 9 ) and (11).
I n t h e absence o f such e x t e n s i o n s one
must l o o k f o r s t a t i s t i c a l t e s t s o f double c a n c e l l a t i o n u s i n g a more conv e n t i o n a l approach. Having t e s t e d m o n o t o n i c i t y u s i n g one of t h e techniques g i v e n above and f a i l e d t o f i n d a s i g n i f i c a n t v i o l a t i o n , i t i s now assumed t h a t t h e d a t a a r e monotonic.
The i s o t o n i c r e g r e s s i o n e s t i m a t e s w i t h r e s p e c t t o b i v a r i a t e
m o n o t o n i c i t y a r e f e d i n t o t h e a l g o r i t h m i n t h e appendix.
This gives a l l
t h e 3 x 3 submatrices f r o m t h e complete d a t a m a t r i x which v i o l a t e t h e double c a n c e l l a t i o n axiom.
F o r t h e v i o l a t i o n t o be s t a t i s t i c a l l y s i g n i f i -
c a n t each c f t h e t h r e e i n e q u a l i t i e s c o n t r i b u t i n g t o t h e v i o l a t i o n s h o u l d be s i g n i f i c a n t i n i t s own r i g h t .
Various methods e x i s t f o r e s t a b l i s h i n g
t h e c o n f i d e n c e i n t e r v a l s f o r t h e d i f f e r e n c e s between means a f t e r an a n a l y s i s o f v a r i a n c e (see Winer ( 1 9 7 1 ) ) .
Statistically significant violations
o f t h e double c a n c e l l a t i o n axiom would have been demonstrated i f t h e confidence i n t e r v a l s o f t h e d i f f e r e n c e i n a l l three i n e q u a l i t i e s c o n t r i b u t i n g t o any v i o l a t i o n d i d n o t i n c l u d e z e r o , Indeed, t h e a l g o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o u l d be r e w r i t t e n f o r l o o k i n g f o r s t a t i s t i c a l l y s i g n i f i c a n t v i o l a t i o n . w o u l d r e q u i r e t h a t t h e i n e q u a l i t y s i g n s < a n d > be r e p l a c e d by <
A l l this
s and
> s
i n d i c a t i n g s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s and t h e i n e q u a l i t i e s b e i n g t r e a t e d as equal when t h e d i f f e r e n c e s were n o t s i g n i f i c a n t . 7. CONCLUSIONS The o v e r a l l t e s t i n g procedure f o r a d d i t i v i t y which seems t o make most use o f t h e data i s a s e q u e n t i a l one.
The e x p e r i m e n t s h o u l d b e designed so t h a t
t h e e f f e c t s o f s u c c e s s i v e l e v e l s o f t h e independent v a r i a b l e s a r e a p p r o x i mately equal.
A t e s t f o r monotonicity using e i t h e r FIL (equation (13)) o r
t h e s t u d e n t i z e d range t e s t ( e q u a t i o n (11)) s h o u l d n e x t be performed.
If
t h i s r e s u l t s i n a s i g n i f i c a n t e f f e c t m o n o t o n i c i t y and hence a d d i t i v i t y have been v i o l a t e d .
I f no s i g n i f i c a n t e f f e c t s a r e found,
t h e double c a n c e l l a -
t i o n axiom i s t e s t e d assuming m o n o t o n i c i t y as o u t l i n e d above.
Ifn e i t h e r
o f t h e above t e s t s produces s i g n i f i c a n t e f f e c t s , t h e d a t a have n o t been shown t o s i g n i f i c a n t l y d e n a r t f r o r a d d i t i v i t y .
3 0
?F FF PF Ft CF
?r\uc'.!e,
:.,
ar?8
!.ariwr, ?.,
l'-e
rl.i+?r
o c t v o wav t a h l e s s a t i s f y i n g
c e r t a i n addi t i v i t'/ axioms, J o u r n a l o f r!athematical Psycholooy, 13 (?.-:-,),
P9-100. Rartholomew, D.J., Bremner, J.M. and Brunk, H.D.,
Barlow, R.E.,
Sta-
t i s t i c a l I n f e r e n c e under Order R e s t r i c t i o n s , W i l e y , New York ( 1 9 7 2 ) . Partholomew, D.J., A t e s t o f homogeneity o f means under r e s t r i c t e d a l t e r n a t i v e s , J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23
( 1 9 6 1 ) , 239-281. Cunningham, J . P . ,
M u l t i o l e monotone r e g r e s s i o n , Psych. B u l l . , 92 (1982)
7917800. H a r t i g a n , J.A.,
D i s t r i b u t i o n o f t h e r e s i d u a l sums o f squares i n f i t t i n g
i n e q u a l i t i e s , R i o m e t r i k a , 54 ( 1 9 6 7 ) , 69-84. Hirotsu, C.,
Ordered a l t e r n a t i v e s f o r i n t e r a c t i o n e f f e c t s , B i o m e t r i k a ,
65 ( 1 9 7 8 ) , 561-570. C o n t r i b u t i o n t o d i s c u s s i o n o f t h e naper by D.J. B a r t h o l o -
James, G.S.,
mew, J o u r n a l o f t h e Royal S t a t i s t i c a l S o c i e t y , S e r i e s B, 23 ( 1 9 6 1 ) ,
278-279. Krantz, D . H . ,
Luce,
R.D., SuDpes, P. and Tversky, ',., Fcundations o f
wasurewiit. ka+tvic
P r c c s , I'el-1 Yor!< (1971.).
Luce, R.D. and Tukey, J.I.I., Simultaneous c o n j o i n t measurement: a new t y p e o f fundamental measurement, J o u r n a l o f Math. P s y c h o l . ,1 ( 1 9 6 4 ) ,
1-27. Mandel, J . , A new a n a l y s i s o f v a r i a n c e model f o r n o n - a d d i t i v e d a t a , Technometrics,
13 ( 1 9 7 1 ) , 1-18.
McClelland, G., A Note on A r b u c k l e and L a r i w r , " T h e number o f two-way t a b l e s s a t i s f y i n g c e r t a i n a d d i t i v i t y axiom;:
J o u r n a l o f Math. Psychol .,
1 5 , ( 1 9 7 7 ) , 292-295. The A n a l y s i s o f Variance, W i l e y , New York ( 1 9 5 9 ) .
Scheffe, H., Shorack, G . R . ,
T e s t i n g a g a i n s t o r d e r e d a l t e r n a t i v e s i n model 1 a n a l y -
s i s o f v a r i a n c e : normal t h e o r y and n o n p a r a m e t r i c , Ann. Math. S t a t i s t . ,
38 ( 1 9 6 7 ) , 1740-1753. Smith, P.T.
and Macdonald, R.R.,
Methods f o r i n c o r p o r a t i n g o r d i n a l i n -
formation i n t o analysis o f variance: g e n e r a l i z a t i o n s o f o n e - t a i l tests, B r i t i s h J o u r n a l o f Mathematical and S t a t i s t i c a l PsycholoSy, 36 ( 1 ? 8 3 ) ,
1-21. Tukey, J.W..
One degree o f freedom f o r n o n a d d i t i v i t y , B i o m e t r i c s , 5
( 1949) , 232-242.
Isotonic
251
regression analysis and additivity
[I51 V i n e r y R.J., S t a t i s t i c a l P r i n c i p l e s i n Experimental Design, PlcGrawH i l l (1971). APPEND1 X A l q o r i t h m f o r f i n d i n g v i o l a t i o n s i n double c a n c e l l a t i o n c o n s i s t e n t w i t h monotoni c i t y
In what f o l l o w s i n e q u a l i t i e s a r e taken t o be t r u e ifand only i f t h e y h o l d I n o t h e r words i f b o t h s i d e s o f an i n e q u a l i t y a r e
i n t h e s t r i c t sense.
e v a l u a t e d and found t o be equal t h e i n e q u a l i t y w i l l be d e s c r i b e d as b o t h " n o t t r u e " and " n o t f a l s e " . L e t A be t h e complete J x K m a t r i x
c o n t a i n i n g a l l t h e d a t a and l e t A j k be
t h e s u b m a t r i x formed f r o m A w i t h columns j , j t l and j + 2 and rcws k, k - 1 L e t us s t a r t by c o n s i d e r i n g t h e s u b m a t r i x A j k and t e s t i n e q u a l i -
and k-2.
t i e s (I\,(?) and ( 3 ) i n t h i s m a t r i x .
(I:) a r e t r u e or n o t ( l ) , n o t
I f and o n l y i f e i t h e r !1),(2)
( 2 ) and ( 3 ) a r e t r u e i s douhle c a n c e l l a t i o n v i o -
''e n e x t c o n s i d e r a l l t h e ' x 3 s u h r a t r i c e s f r o m
lated.
and ?, and rows Y,
pl
and n o t
fib w i t h
and n (k-rw). Tte t r u t h o f ( 1 ) , ( 2 )
c o l u m s l , 2,
and ( 3 )
;7
has
i m p l i c a t i o n s f o r t h e i r t r u t h i n t h e submatrices we a r e c o n s i d e r i n g as spec i f i e d i n (14) t o ( 1 6 ) below: '2,K
'1,K-1 '1,K
> '3,K-1
=. d
> '2,K-2 > '3,K-2
'2,K
=9
' '3,m
'1,K-1 u
l,K
> '2,n
> u
3,n
m
<
K-1
n <. K-2
(15
n <. K-2
W i t h these r e s t r i c t i o n s i n mind we s h a l l c o n s i d e r a l l p o s s i b l e s t a t e s f o r i n e q u a l i t i e s ( l ) a n d ( 3 ) i n A. State
1 : ( l ) a n d ( 3) t r u e .
I n e q u a l i t i e s ( l q a n d (16) guarantee t h a t i n e q u a l i t i e s (1) and (3) w i l l b e t r u e f o r a l l t h e submatrices under c o n s i d e r a t i o n .
Thus t h e double c a n c e l -
l a t i o n axiom i s n e v e r v i o l a t e d . State
2 :(1) f a l s e and ( 3 ) t r u e .
From (16) i n e q u a l i t y (3) w i l l be t r u e i n a l l t h e submatrices under d i s c u s sion.
Because o f m o n o t o n i c i t y and i n e q u a l i t y ( 3 ) we have:
252
R . R . MdcDorurld
'2,K
> '1,K
> '3.K-2
This means t h a t i n e o u a l i t y ( 1 ) w i l l be t r u e f o r m = K-2 and hence I I ~ C R U o fS ~ p o n o t o n i c i t y f o r a l l v a l u e s o f m n o t equal t o K - 1 .
V i o l a t i o n s o f double
c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when i n e q u a l i t y ( 2 ) i s f a l s e a t
m = K-I.
IJe t h e r e f o r e t e s t i n e q u a l i t y ( 2 ) a t
1;1
=
K-1 f o r successively
d e c r e a s i n g values o f n s t a r t i n g a t n = K-2 u n t i l i n e o u a l i t y ( 2 ) i s found t o be n o t f a l s e .
M o n o t o n i c i t y quarantees t h a t i n e q u a l i t y w i l l c o n t i n u e n o t t o
he f a l s e f o r a l l s m a l l e r values o f n.
Where i n e q u a l i t y ( 2 ) has been f o u n d
t o be f a l s e , v i o l a t i o n s o f double c a n c e l l a t i o n have o c c u r r e d . State
3:
(1) t r u e and ( 3 ) f a l s e .
By ( 1 4 ) i n e q u a l i t y ( 1 ) w i l l be t r u e f o r a l l t h e s u b m a t r i c e s and v i o l a t i o n s
w i l l o n l y o c c u r where i n e q u a l i t y ( 3 ) i s f a l s e and ( 2 ) i s t r u e .
The values
o f n f o r w h i c h i n e q u a l i t y ( 3 ) i s f a l s e a r e found by t e s t i n g t h i s i n e q u a l i t y
for s u c c e s s i v e l y d e c r e a s i n g v a l u e s o f n s t a r t i n g a t K-2 u n t i l i t i s n o l o n I n e q u a l i t y (2) s h o u l d now be t e s t e d f o r each v a l u e o f n f o r
oer false.
which i n e q u a l i t y ( 3 ) i s f a l s e .
T h i s i s done f o r a l l p o s s i b l e values o f m
d e c r e a s i n g from K - 1 u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r t r u e .
Each t i m e i n -
e q u a l i t y (2) i s found t o be t r u e , a v i o l a t i o n o f double c a n c e l l a t i o n has occurred. State
4:
b o t h (1) and ( 3 ) f a l s e .
To b e g i n w i t h , i n e q u a l i t y
(1) i s t e s t e d f o r d e c r e a s i n g values o f m u n t i l a t
m = c ' i t i s f o u n d t o be n o t f a l s e and a t m = c " i t i s f o u n d t o be t r u e :
(c',~").
A t t h i s p o i n t we know we know t h a t i n e q u a l i t y ( 3 ) must be f a l s e
a t n = c ' + l (18) and s i n c e (1) i s a l s o f a l s e double c a n c e l l a t i o n w i l l h o l d f o r n u r e a t e r than c '
.
We n e x t t e s t i n e q u a l i t y ( 3 ) a t n = c ' .
Ift h i s i s t r u e i n e q u a l i t y (1) i s
f a l s e f o r a l l values o f m from K - 1 t o c ' + l and t h e procedures adopted i n state
2
f o r m = K - 1 s h o u l d be r e a p p l i e d h e r e f o r a l l v a l u e s o f m f r o m
K-1 t o c ' + l .
No v i o l a t i o n o f t h e t y p e 1: 2 and n o t 3 i s o o s s i b l e s i n c e 1
and 3 a r e f a l s e f o r a l l values o f n g r e a t e r t h a n c ' by m o n o t o n i c i t y and 3 i s t r u e f o r a l l values o f n g r e a t e r t h a n c ' .
I f c ' does n o t equal c " i n e q u a l i t y (1) w i l l be equal f o r values o f n f r o m
253
Isotonic regression analysis and additivity
Double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e c ' i n these cases.
c"t1 to c'.
I f however, i n e q u a l i t y ( 3 ) was found t o be f a l s e a t n = c ' no v i o l a t i o n
Occurs as i n e q u a l i t y (1) i s a l s o f a l s e f o r a l l values o f m g r e a t e r t h a n c ' .
A v i o l a t i o n i s P o s s i b l e a t m = c " (where i n e q u a l i t y (1) i s t r u e ) i f i n e q u a l i t y ( 3 ) i s f a l s e f o r n = c"-1. ? should
!3e
Here t h e Procedures adonted i n s t a t e
amlied.
There s t i l l remains t h e p o s s i b i l i t y of a v i o l a t i o n o f double c a n c e l l a t i o n i f i n e q u a l i t y (1) i s f a l s e f o r a l l values o f m.
Here by r e a s o n i n g s i m i l a r -
l y t o ( 1 7 ) i n e q u a l i t y ( 3 ) w i l l be f a l s e f o r n g r e a t e r than 1.
I f however,
i n e q u a l i t y ( 3 ) were found t o be t r u e a t n = 1, i n e q u a l i t y ( 2 ) s h o u l d be checked f r o m m = 2 u n t i l K - 1 o r i n e q u a l i t y ( 2 ) i s found t o be t r u e .
Double
c a n c e l l a t i o n w i l l have been v i o l a t e d f o r e v e r y f a l s i f i c a t i o n of i n e q u a l i t y
(2)
*
S t a t e 5 : one o r more o f i n e q u a l i t i e s (1) and ( 3 ) a r e e y a l . I f e i t h e r i n e q u a l i t y ( 1 ) o r ( 3 ) i s t r u e o r b o t h a r e equal t h e s i t u a t i o n i s like state
I n e q u a l i t i e s (14) and (16) a r e t r u e i n t h e weak f o r m (where
1.
Thus
t h e g r e a t e r t h a n s i a n s a r e reDlaced by g r e a t e r t h a n o r equal s i g n s ) .
i n t h i s s i t u a t i o n double c a n c e l l a t i o n w i l l n e v e r be v i o l a t e d as n e i t h e r i n e q u a l i t y (1) n o r (3) can be f a l s e . I f i n e q u a l i t y (1) i s f a l s e and i n e q u a l i t y ( 3 ) i s equal t h e n o s i t i o n i s s i milar t o state
2
.
The weak v e r s i o n o f i n e q u a l i t y ( 1 7 ) guarantees inequa-
l i t y ( 1 ) i s n o t f a l s e f o r m = K-2. V i o l a t i o n s o f double c a n c e l l a t i o n can t h e r e f o r e o n l y o c c u r when m = K-1, i n e q u a l i t y ( 3 ) i s t r u e and i n e q u a l i t y
( 2 ) f a l s e f o r some v a l u e o f n. I n e q u a l i t y ( 3 ) s h o u l d be t e s t e d f o r m = K-1 f o r s u c c e s s i v e l y d e c r e a s i n g values o f n u n t i l i n e q u a l i t y (3) i s f o u n d t o be true a t n
I
c ' a f t e r w h i c h i t w i l l be t r u e f o r a l l s m a l l e r values.
Inequa-
l i t y ( 2 ) s h o u l d then be e v a l u a t e d a t m = K - 1 f o r values o f n d e c r e a s i n a from n = c ' u n t i l i n e q u a l i t y ( 2 ) i s no l o n g e r f a l s e .
Each f a l s i f i c a t i o n
o f i n e q u a l i t y ( 2 ) corresponds t o a v i o l a t i o n o f t h e double c a n c e l l a t i o n axiom. The f i n a l p o s s i b i l i t y i s where i n e q u a l i t y (1) i s e o u a l and ( 3 ) i s f a l s e . Here t h e procedure i s as i n s t a t e
4 where c ' and c " a r e d i f f e r e n t .
Ine-
q u a l i t y (1) i s t e s t e d f o r s u c c e s s i v e l y d e c r e a s i n g values o f m u n t i l a t rn = c " i t i s t r u e .
I n e q u a l i t y ( 3 ) i s e v a l u a t e d a t n = c"-1 and i f f a l s e
t h e Drocedure f o r s t a t e
3 i s adonted.
224
R.R. MdcDonald
Ile have now checked f o r double c a n c e l l a t i o n i n a l l t h e 3 x 3 r r a t r i c e s w i t h columns 1, 2 and 3 and rows K, F, and n. p and rows K , K-1 and
lues o f o
c
o < J.
Each m a t r i x w i t h columns 1, o and
K-2 can be t r e a t e d i n t h e same way as A, f o r a l l va-
T h i s w i l l e n a b l e t h e t e s t i n a o f double c a n c e l l a t i o n i n
a l l m a t r i c e s w i t h rows 1, o and p and columns K,
pl
and n.
T h i s comoletes
uIK.
I n o r d e r t o c o n s i d e r e v e r y poss i b l e v i o l a t i o n t h e nrocedures s h o u l d be r e o e a t e d f o r a l l s u b m a t r i c e s A j k , t e s t i n q a l l 3 x 3 matrices i n c l u d i n n
j
,I-1 and k
2.
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ACKN@'*lLF:DGEIIE NTS The a u L ' @ rwishes t o exnress h i s thanks t o D r . P.T. the m r u c c r i n t .
rmit':
f o r comments on