UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Analysis of ordered categorical data 1988

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1988_A6_7 C42_6.pdf [ 3.16MB ]
Metadata
JSON: 1.0097667.json
JSON-LD: 1.0097667+ld.json
RDF/XML (Pretty): 1.0097667.xml
RDF/JSON: 1.0097667+rdf.json
Turtle: 1.0097667+rdf-turtle.txt
N-Triples: 1.0097667+rdf-ntriples.txt
Citation
1.0097667.ris

Full Text

ANALYSIS OF ORDERED CATEGORICAL DATA By Janis Chang Sc., (Biochemistry) University of British Columbia, 1985 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF STATISTICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1988 © Janis Chang, 1988 I n p r e s en t i n g th i s thesis i n p a r t i a l f u l f i lmen t of t he requ i rement s fo r a n advanced degree at t h e U n i v e r s i t y of B r i t i s h C o l u m b i a , I agree t h a t the L i b r a r y s ha l l m a k e i t f reely a va i l ab l e fo r reference a n d study. I f u r t h e r agree t h a t p e r m i s s i o n fo r ex tens i ve c o p y i n g of th i s thes is f o r s cho la r l y purposes m a y be g r a n t e d b y the head o f m y d e p a r t m e n t or b y h i s o r her representat i ves . It is u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n of th i s thesis f o r f i n a n c i a l g a i n s ha l l not be a l l owed w i t h o u t m y w r i t t e n pe rm i s s i on . D e p a r t m e n t o f S t a t i s t i c s T h e U n i v e r s i t y of B r i t i s h C o l u m b i a 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V 6 T 1 W 5 D a t e : Abstract M e t h o d s o f t e s t i n g for a l o c a t i o n sh i f t be tween two p o p u l a t i o n s i n a l o n g i t u d i n a l s t u d y are i n ve s t i g a t ed w h e n the d a t a o f in terest are o rde red , c a tego r i c a l a n d non - l i near . A n o n - s t a n d a r d ana ly s i s i n v o l v i n g m o d e l l i n g of d a t a over t i m e w i t h t r a n s i t i o n p r o b a b i l i t y ma t r i c e s is d i scussed. N e x t , t he r e l a t i v e eff ic iencies of s ta t i s t i c s m o r e f r e q u e n t l y used fo r t he ana ly s i s of s u ch ca tego r i c a l d a t a at a s ing le t i m e p o i n t are e x a m i n e d . T h e W i l c o x o n r a n k s u m , M c C u l l a g h , a n d 2 s amp le t s t a t i s t i c are c o m p a r e d fo r the ana ly s i s of s u c h cross s e c t i ona l d a t a u s i n g s i m u l a t i o n a n d eff icacy ca l cu l a t i on s . S i m u l a t i o n techn iques are t h e n u t i l i z e d i n c o m p a r i n g the s t r a t i f i ed W i l c o x o n , M c C u l l a g h a n d ch i s qua red - t ype s t a t i s t i c i n t h e i r eff ic iencies at d e t e c t i n g a l o c a t i o n sh i f t w h e n t he d a t a are e x a m i n e d over t w o t i m e po in t s . T h e d i s t r i b u t i o n of a ch i s qua red - t ype s t a t i s t i c based o n t he s i m p l e c o n t i n g e n c y t a b l e c o n s t r u c t e d b y me re l y n o t i n g w h e t h e r a s ub jec t i m p r o v e d , s t a y e d t he s ame or d e t e r i o r a t e d is de r i ved . A p p l i c a t i o n s o f these me thod s a n d resu l t s t o a d a t a set of M u l t i p l e Sc leros i s pa t i en t s , some of w h o m were t r e a t e d w i t h i n t e r f e r o n a n d some of w h o m rece ived a p l a cebo are p r o v i d e d t h r o u g h o u t the thesis a n d o u r f i nd ing s a re s u m m a r i z e d i n the last C h a p t e r . 11 Table of Contents Abstract ii List of Tables iv List of Figures v Acknowledgement vi 1 Introduction 1 2 Markov Analysis 3 2.1 Tests of O r d e r a n d S t a t i o n a r i t y 4 2.1.1 Tes t s of O r d e r of M a r k o v C h a i n 6 2.1.2 Tes t s o f S t a t i o n a r i t y 8 2.2 M o d e l l i n g of t he T r i d i a g o n a l s 10 2.2.1 M o d e l l i n g I n c o r p o r a t i n g V2 11 2.2.2 M o d e l l i n g W i t h o u t V2 12 3 Analysis At One Time Point 19 3.1 S i m u l a t i o n 20 3.2 E f f i c a c y C a l c u l a t i o n s 22 3.2.1 E f f i c a c y o f t he W i l c o x o n 23 3.2.2 E f f i c a c y of the T test 31 3.3 C o m p a r i s o n of the Ef f i cac ies 34 i i i k 4 Analysis at Two Time Points 36 4.1 D e s c r i p t i o n of the S i m u l a t i o n 36 4.1.1 S t r a t i f i e d W i l c o x o n 37 4.1.2 C h i - s q u a r e d S t a t i s t i c 38 4.2 S i m u l a t i o n Re su l t s 46 4.3 A p p l i c a t i o n of Tes t s t o t he M S D a t a 51 5 Conclusions 53 A Sample Transition Matrices 55 Bibliography 60 iv List of Tables 2.1 C o m p a r i s o n o f T r e a t m e n t a n d C o n t r o l G r o u p s 9 2.2 Test s o f 2nd O r d e r vs. 1st O r d e r M a r k o v P roces s 9 2.3 Tes t s o f 1st O r d e r vs. I ndependence 16 2.4 R e s u l t s of Te s t fo r S t a t i o n a r i t y 16 2.5 T r e a t m e n t vs. C o n t r o l ( genera l t r i d i a g o n a l m o d e l l i n g ) 16 2.6 F i t o f G e n e r a l T r i d i a g o n a l M o d e l s 17 2.7 T r e a t m e n t vs. C o n t r o l ( genera l t r i d i a g o n a l - no P 2 ) 17 2.8 F i t o f G e n e r a l M o d e l - no P2 17 2.9 C o m p a r i s o n o f Spec i f i c a n d G e n e r a l M o d e l s ( t r e a t m e n t ) 17 2.10 C o m p a r i s o n of Spec i f i c a n d G e n e r a l M o d e l s 18 4.1 R e s u l t s of M c C u l l a g h A n a l y s i s 51 v List of Figures 3.1 S i m u l a t i o n of D a t a at O n e T i m e P o i n t 21 4.1 S i m u l a t i o n R u n 1 (3 = 0.1, p = 0.8) 47 4.2 S i m u l a t i o n R u n 2(8 = 0.1, p = 0.6) 48 4.3 S i m u l a t i o n R u n 3(3 = 0.3, p = 0.8) 49 4.4 S i m u l a t i o n R u n 4(8 = 0.3, p = 0.6) 50 vi Acknowledgement I w o u l d l i k e t o t h a n k m y superv i so r , D r . N .E . H e c k m a n for he r c o n t i n u a l gu idance, s u p p o r t a n d m a n y i n v a l uab l e suggest ions. A l s o , I a m g r a t e f u l t o D r . A . J . P e t k a u adv i ce a n d c a r e f u l r e a d i n g of th i s wo rk . I n a d d i t i o n , the encou ragement of D r . J . B e r k o w i t z a n d e xpe r t c o m p u t i n g a d v i c e of P e t e r S c h u m a c h e r a re g r a te f u l l y a cknow ledged . I a lso w i t h t o t h a n k the M S C l i n i c at the U B C H o s p i t a l f o r t h e i r s uppo r t . T h i s re search was f u n d e d b y the M u l t i p l e Sc leros is S o c i e t y of C a n a d a t h r o u g h the U B C M S C l i n i c . vii Chapter 1 Introduction O n e h u n d r e d M u l t i p l e Sc leros is pa t i en t s f r o m the pa t i en t p o p u l a t i o n of t he M S C l i n i c at t he U B C H o s p i t a l p a r t i c i p a t e d i n a r a n d o m i z e d d o u b l e - b l i n d c l i n i c a l t r i a l t o deter - m i n e t h e effect iveness of t r e a t m e n t w i t h i n te r fe ron . M u l t i p l e Sc leros i s is a progress ive disease w h i c h a t t a c k s the nervous s y s t e m a n d o f ten resu l t s i n loss o f v i s i on , m o t o r c o o r d i n a t i o n a n d / o r sensory pe r cep t i on . T h e sever i t y of the s y m p t o m s var ies a m o n g pa t i en t s . T h e sub jec t s i n th i s s t u d y a re ch r on i c progress ive, t h a t is, t h e i r c o n d i t i o n dete r i o ra te s p rog re s s i ve l y over t ime . I n th i s t r i a l , f i f t y pa t i en t s were as s igned r a n d o m l y t o c o n t r o l a n d t r e a t m e n t groups. S ub jec t s i n the c o n t r o l g r oup were g i v en i n jec t i on s of a p l a cebo a n d those i n the t r e a t m e n t g roup were g i v en i n jec t i on s o f i n t e r f e r on for s i x m o n t h s . T h e pa t i en t s were m o n i t o r e d d u r i n g the s i x m o n t h s of t r e a t m e n t a n d fo r e i gh teen m o n t h s o f f o l l o w - u p t o t he subsequent t e r m i n a t i o n of t he t r e a t m e n t . N o s t a n d a r d q u a n t i t a t i v e m e t h o d of m e a s u r i n g t he leve l of t he disease exists. I n th i s s tudy , mea su remen t s o f s y m p t o m s such as m o b i l i t y o r numbnes s were u sed t o assess t he seve r i t y o f t he s ub jec t ' s c o n d i t i o n a n d th i s i n f o r m a t i o n was u sed to p r o d u c e the K u r t z k e e x t e n d e d d i s a b i l i t y s ta tus scale ( E D S S ) . T h e K u r t z k e E D S S , re fe r red t o here as K u r t z k e score, was chosen as the means o f t r a c i n g t he s ub jec t s ' c ond i t i on s over t ime . T h e K u r t z k e score is o rde red a n d ca tego r i c a l , t a k i n g o n va lues of 0 ( no rma l ) to 10 (dead) i n i n c r emen t s o f 0.5. It is a l so non l i nea r , so t h a t , f o r e x a m p l e , a change i n score of 1 t o 2 is no t as severe as a change of 5 t o 6. T h e n o n l i n e a r i t y a n d ca tego r i c a l n a t u r e of the scores makes i t i n a p p r o p r i a t e t o t reat 1 Chapter 1. Introduction 2 t h e m as con t i nuou s va r i ab le s i n t he assessment of the ex tent of the disease. I n p a r t i c - u l a r , any s t a t i s t i c w h i c h requ i res the a s s u m p t i o n of n o r m a l l y d i s t r i b u t e d observat ions ( s uch as t he 2 s a m p l e t s t a t i s t i c ) is no t s u i t ab l e for a n a l y z i n g th i s t y p e o f d a t a . O n the o t h e r h a n d , s t a n d a r d c a tego r i c a l d a t a ana ly s i s c o u l d b e done i g no r i n g the o r d i n a l n a t u r e of t he da t a . T h i s t y p e of ana ly s i s is no t a p p r o p r i a t e here as i n f o r m a t i o n o n the degree of i m p r o v e m e n t o r d e t e r i o r a t i o n o f the p a t i e n t ' s c o n d i t i o n w o u l d b e lost. O n e m e t h o d of ana ly s i s , de s c r i bed i n C h a p t e r 2, is t o cons ide r t he categor ies as states a n d t h e movemen t s o f the sub jec t s f r o m ca tego ry t o ca tego ry over t i m e as t r a n - s i t ions f r o m s ta te t o s tate. T h e t r e a t m e n t a n d c o n t r o l g roups were m o d e l l e d b y d i f - ferent t r a n s i t i o n p r o b a b i l i t y ma t r i ce s w h i c h were c o m p a r e d t o d e t e r m i n e t he effect of the i n t e r f e r o n i n jec t i on s . M o d e l s a s s u m i n g s t a t i ona r i t y , the M a r k o v p r o p e r t y a n d o the r r e s t r i c t i on s o n t he t r a n s i t i o n p r obab i l i t i e s were f i t t o t he d a t a t o d e t e r m i n e w h e t h e r a s i m p l e m a t r i x c o u l d be u sed t o desc r ibe the t r an s i t i on s of the pa t i en t s . I n C h a p t e r 3, t he W i l c o x o n s t a t i s t i c , M c C u l l a g h m o d e l a n d 2 s a m p l e t test were c o m p a r e d t o assess t h e i r r e l a t i v e ef f ic iency i n d e t e c t i n g a l o c a t i o n sh i f t be tween t he two g roups w h e n t he scores are a s sumed to have a n u n d e r l y i n g con t i nuou s d i s t r i b u t i o n . T h e M c C u l l a g h m o d e l , a m o d i f i c a t i o n o f the l og i s t i c regress ion m o d e l , i n co rpo ra te s t he o r d i n a l n a t u r e o f t he scores. I n th i s ana ly s i s t he d a t a at o n l y one t i m e p o i n t was used. T h e c o m p a r i s o n of the tests was ba sed on a s y m p t o t i c ef f icacy c a l cu l a t i on s a n d s imu l a t i o n s . I n C h a p t e r 4, the s t r a t i f i ed W i l c o x o n , M c C u l l a g h a n d c h i s qua red - t ype s t a t i s t i c were c o m p a r e d u s i n g s i m u l a t i o n w h e n a n a l y z i n g the d a t a be tween t w o t i m e po in t s . T h e ch i s qua red - t y p e s t a t i s t i c c a l c u l a t e d was the u s u a l ana ly s i s i n v o l v i n g c o n t i n g e n c y tab les whe re t he cel ls were t he n u m b e r of pa t i en t s i n t r e a t m e n t a n d c o n t r o l w h o i m p r o v e d , s t ayed t he same or worsened. T h e d i s t r i b u t i o n o f th i s s t a t i s t i c unde r t he hypothes i s of n o d i f ference b e t w e e n t he g roups is d i scussed. Chapter 2 Markov Analysis S ince pa t i en t s i n th i s s t u d y m o v e d f r o m s ta te t o s ta te over t i m e , t he movement s f r o m one s ta te t o a n o t h e r were m o d e l l e d fo r each g r oup u s i n g t r a n s i t i o n p r o b a b i l i t y m a - tr ices . C o m p a r i s o n of the t w o groups t h e n i n v o l v e d a n a l y z i n g the ma t r i c e s e s t i m a t e d for each g roup. T h i s a p p r o a c h does not r equ i re the a s s u m p t i o n of a p a r a m e t r i c f o r m fo r t he t r a n s i t i o n p r obab i l i t i e s , however some mode l s a s s u m i n g a spec i f i c f o r m for the p r o b a b i l i t i e s were fit t o the d a t a t o d e t e r m i n e i f t he n u m b e r o f pa r amete r s r equ i r ed to de sc r i be t he d a t a c o u l d be reduced . I n i t i a l l y , t he d a t a was a n a l y z e d i n i ts o r i g i n a l f o rm . T r a n s i t i o n p r o b a b i l i t y ma t r i ce s were p r o d u c e d fo r each g r oup ( t r e a tmen t a n d con t ro l ) f r o m 0 m o n t h s t o each of the o t h e r t i m e p o i n t s (1,3,6,9,12,18,24 mon th s ) . M o s t observat ions were o n o r nea r the d iagona l s , i n d i c a t i n g t h a t the p a t i e n t s ' scores d i d no t change m u c h f r o m one t i m e p e r i o d t o t he nex t . A s t he d imens i on s of the ma t r i ce s were 21 x 21 a n d the re were o n l y f i f t y pa t i en t s i n each g roup, m a n y cel ls were empty . F o r th i s reason, t he K u r t z k e scores were g r o u p e d i n t o five categor ies, chosen t o ensure at least f ou r peop le i n each ca tego ry at 0 m o n t h s . T h e categor ies were scores of 0-4, 4.5-5.5, 6.0, 6.5, a n d 7-10, w i t h 0-4 b e c o m i n g s ta te 1, 4.5-5.5 b e c o m i n g s ta te 2, etc. T r a n s i t i o n ma t r i ce s of the pa t i en t s f r o m 0 m o n t h s t o 24 m o n t h s are d i s p l a yed i n A p p e n d i x A . A l l f u r t h e r c a l cu l a t i on s i n th i s c hap te r are ba sed u p o n th i s co l l ap sed d a t a . I n S e c t i o n 2.1, l i k e l i h o o d r a t i o s ta t i s t i c s were used t o d e t e r m i n e i f a second o rder M a r k o v s t r u c t u r e fit a p p r e c i a b l y b e t t e r t h a n a f i r s t o rder m o d e l . A test fo r s t a t i o n a r i t y 3 Chapter 2. Markov Analysis 4 was a l so a p p l i e d t o t he d a t a . S o m e m o d e l l i n g o f t he d a t a i n t he t r i d i a g o n a l po s i t i on s is d i s cus sed i n S e c t i o n 2.2. 2.1 Tests of Order and Stationarity T h e f o l l o w i n g tests were based o n those de sc r i bed b y B h a t [1]. S o m e n o t a t i o n used i n the r e m a i n d e r of th i s c h a p t e r is: z — n u m b e r o f states t = t i m e po i n t s s t ud i ed , t = 0 , 1 , . . . , 7 c o r r e s p o n d i n g t o t h e t imes 0, 1, 3, 6, 9, 12, 18, 24 m o n t h s Xt = K u r t z k e score at t i m e t. U s i n g the co l l ap sed d a t a , t r a n s i t i o n p r o b a b i l i t y ma t r i c e s were c a l c u l a t e d fo r t r a n - s i t ions f r o m 0 m o n t h s t o each of the o the r t i m e po i n t s s t ud i ed . M a x i m u m l i k e l i h o o d e s t imates fo r PT a n d Pc, t he t r a n s i t i o n ma t r i ce s of the t r e a t m e n t a n d c o n t r o l g r oup were c o m p u t e d , as w e l l as fo r P, t he t r a n s i t i o n m a t r i x u n d e r Ho : PT = P°. A l i k e l i - h o o d r a t i o s t a t i s t i c was c o m p u t e d sepa ra te l y fo r each t i m e p o i n t t = 0 , 1 , . . . , 7 to test w h e t h e r t h e t w o g roup s ' t r a n s i t i o n p r o b a b i l i t i e s were s im i l a r . U n d e r t he n u l l h y p o t h e - sis t h a t t he t w o g roups c a n b e m o d e l l e d u s i n g t he same t r a n s i t i o n p r o b a b i l i t y m a t r i x , i t c a n b e s h o w n [6] t h a t the l og l i k e l i h o o d f u n c t i o n l n L ( P ) is: InL(P) = B + J2 I > S + < l n ^ ) - ( 2 - 1 ) i=l j=l where Chapter 2. Markov Analysis B = a t e r m i ndependen t of the p 4 j ' s nf- = n u m b e r of obse rva t ions where Xt = j a n d X o = i i n the t r e a t - m e n t g r o u p ng = n u m b e r o f observat ions i n t he c o n t r o l g r oup whe re Xt = j a n d XQ = i P i j = Pv[Xt = j \ X 0 = i]. U n d e r t h e a l t e r na te hypothes i s , Hi : P T ^ P c , t he l o g l i k e l i h o o d f u n c t i o n is: l n L ( P c , P T ) = B + ± ± (nj,lnpg + ng I n p g ) «'=1 j=l where pfj — P r [ X ( = j | Xo = i i n t r e a t m e n t group] pfj = Pr[Xt = j \ Xo = i in. c o n t r o l group]. T h e l o g l i k e l i h o o d s t a t i s t i c , G2, is t h e n G2 = -2 [ l n L ( P ) - l n L ( P c , P r ) ] = 2 ± ± [(nl l n p g + ng I n p g ) - (nj + ng)(lnp,,)] . t'=i j=i where a l l t he p r o b a b i l i t i e s were e s t i m a t e d u s i n g m a x i m u m l i k e l i h o o d techn iques : l n L ( P ) = the log likelihood at P Pij = nIj + n% nf.+nf. Pij = "S Pij = nj. <. = Chapter 2. Markov Analysis 6 U n d e r t h e n u l l hypothes i s , G2 has a n a s y m p t o t i c ch i - squa red d i s t r i b u t i o n w i t h degrees of f r e e d o m e q u a l t o z[z — 1) — yp where yp is t he n u m b e r o f zero entr ies i n P. Pea r s on ' s c h i s qua red s t a t i s t i c , X2, was a l so c a l c u l a t ed : *2 = ££ + i=l j=l L mIj mij (2 .3 ) where mJ- = nj.pij, a n d mfj = nf.pij are the e x p e c t e d n u m b e r o f observat ions i n the t r e a t m e n t a n d c o n t r o l g r oup w h o m o v e d f r o m s ta te i t o s ta te j b e tween t i m e 0 a n d t i m e t. R e s u l t s f r o m t he seven tests a re g i v en i n T a b l e 2.1 a n d t he t r a n s i t i o n ma t r i ce s be tween t i m e 0 a n d 24 m o n t h s are d i s p l a yed i n A p p e n d i x A . T h e p-values r e p o r t e d were based o n t he x 2 a p p r o x i m a t i o n . T h e p-values were sma l le s t ( a l t h o u g h not s i gn i f i cant ) at s i x a n d n i ne m o n t h s , w h i c h is w h e n t he t r e a t m e n t was d i s con t i nued . T h i s suggests t h a t t he t r e a t m e n t m a y have h a d a n effect at th i s t i m e , w h i c h wo re off at e ighteen mon th s . S i nce t he s ta t i s t i c s c a l c u l a t e d o n l y measu re ab so l u te di f ferences i t was no t poss ib le t o d e t e r m i n e f r o m t h e m whe the r t he t r e a t m e n t g r o u p d i d b e t t e r o r worse t h a n the c o n t r o l g roup. N o obv iou s t r e n d c o u l d be seen f r o m e x a m i n a t i o n o f the e s t i m a t e d t r a n s i t i o n mat r i ce s , b u t p rev i ou s w o r k [7] i n d i c a t e d t h a t i n fact t he t r e a t m e n t g r oup regressed ( r e l a t i ve t o t he cont ro l s ) f r o m 0 t o 6 m o n t h s . 2 . 1 . 1 Tests of Order of M a r k o v C h a i n Second o rde r a n d first o rde r M a r k o v C h a i n mode l s were fit t o the d a t a a n d the re su l t an t e s t imates were c o m p a r e d . E a c h g r oup was tes ted separate ly . A l i k e l i h o o d r a t i o s t a t i s t i c Chapter 2. Markov Analysis 7 was u sed t o test H0 : c h a i n is l s < o rder M a r k o v vs. Hi : c h a i n is 2nd o rde r M a r k o v . T h i s test was c a r r i e d out sepa ra te l y f o r eve ry th ree con secu t i ve t i m e po i n t s i n the s tudy, i.e. at 0-1-3 mon th s , 1-3-6 m o n t h s , etc. If t he d a t a a p p e a r e d t o act as a 2nd o rder M a r k o v c h a i n t h e n t h a t w o u l d i m p l y t h a t a p a t i e n t ' s score w o u l d d e p e n d o n h i s score b o t h at t h e p rev i ou s t i m e p o i n t a n d t he t i m e p o i n t before t ha t . I n th i s p r o b l e m , t he G2 s t a t i s t i c becomes: i=i j=i k=i nijk x n). inh- x n)k where n\-k — n u m b e r o f observat ions i n w h i c h Xt-2 = h Xt-i = j, Xt = k njk = n u m b e r o f observat ions i n w h i c h Xt_i = j, Xt = k n). — Yli=i ntjk nij. — J2t=i n\jk t = 2,...,z. G\ has a n a p p r o x i m a t e ch i s qua red d i s t r i b u t i o n w i t h J2j=i(z ~ rj — 1) X — Cj — 1) degrees o f f r e e d o m whe re z is t h e n u m b e r of categor ies, Cj a n d rj are t he n u m b e r of zero rows a n d c o l u m n s r e spec t i ve l y i n t he two d i m e n s i o n a l t r a n s i t i o n m a t r i x con s i s t i ng o f t he t r a n s i t i o n p r o b a b i l i t i e s p\jk (i = 1 , . . . , z, j is f i x e d a n d k = 1 , . . . , z ) . T h e resu l t s f r o m these tests are i n T a b l e 2.2. O n l y t he s t a t i s t i c fo r t he t r ea tmen t g r o u p i n the t i m e i n t e r v a l s i x t o twe lve m o n t h s was s i gn i f i cant at a n a l e ve l of 0.05, w h i c h i n d i c a t e d t ha t m o d e l l i n g w i t h a Is* o rder M a r k o v c h a i n was reasonab le f o r mos t of t he t i m e pe r i od s . T h e coun t s i n these mat r i ce s were ve r y l ow, so the s ta t i s t i c s c a l c u l a t e d m a y be m i s l e ad i n g . S i m i l a r tests were a p p l i e d t o t he 1 s t o rde r M a r k o v t r a n s i t i o n ma t r i ce s to d e t e r m i n e w h e t h e r t h e f i n a l response depended o n the p rev i ou s score (1st o rde r M a r k o v ) o r not Chapter 2. Markov Analysis 8 ( i ndependence ) . T h e a p p r o p r i a t e G2 s t a t i s t i c becomes: G ! = 2 £ 5 X . l n z z (2.4) whe re T h e s t a t i s t i c s c a l c u l a t e d u s i n g (2.4) a re i n T a b l e 2.3. A l l of the s ta t i s t i c s were ve ry l a rge i n c o m p a r i s o n w i t h t he degrees of f r eedom at a l l t i m e po i n t s w h i c h m e a n t t h a t t he r e d u c t i o n f r o m a 1ST o rde r M a r k o v c h a i n m o d e l t o a n i ndependence m o d e l was not reasonable. 2.1.2 Tests o f S ta t iona r i ty A s s u m i n g the d a t a h a d the M a r k o v P r o p e r t y , a no t he r l i k e l i h o o d r a t i o s t a t i s t i c was c o m p u t e d , t e s t i n g f o r s t a t i ona r i t y . W e as sume t ha t t he same t r a n s i t i o n m a t r i x c o u l d be used t o de s c r i be t he p a t i e n t s ' movement s be tween a l l of t he t i m e in te rva l s measu red , i n d e p e n d e n t o f t he a c t u a l r e a l - t i m e l e n g t h o f t h a t t i m e i n te r va l . A t r a n s i t i o n m a t r i x was e s t i m a t e d u s i n g d a t a f r o m a l l t i m e po in t s . T h i s was c o m p a r e d t o the seven mat r i ce s e s t i m a t e d w i t h the count s f o r every t w o con secu t i ve t i m e p o i n t s i n the s t u d y (0-1 m o n t h , 1-3 mon th s , etc.) . T h e G2 s t a t i s t i c here takes o n t he f o r m : z z 7 n\. x riij t=:i j=i t=i where Chapter 2. Markov Analysis 9 T a b l e 2.1: C o m p a r i s o n of T r e a t m e n t a n d C o n t r o l G r o u p s T i m e P e r i o d G 2 p va lue X2 p va lue degrees of ( m o n t h s ) s t a t i s t i c ( G 2 ) s t a t i s t i c ( X 2 ) f r e e d o m 0-1 12.386 0.19 10.718 0.30 9 0-3 15.086 0.13 12.745 0.24 10 0-6 22.050 0.08 17.387 0.24 14 0-9 17.985 0.08 14.931 0.19 11 0-12 17.502 0.13 14.239 0.29 12 0-18 12.621 0.32 10.683 0.47 11 0-24 14.072 0.44 11.825 0.62 14 T a b l e 2.2: Tes t s of 2nd O r d e r vs. 1st O r d e r M a r k o v r r o c e s s T i m e P e r i o d degrees of p -va lue (mon th s ) s t a t i s t i c f r e e d o m 0 - » 1 -> 3 25.400 19 0.15 1 - » 3 -• 6 22.141 16 0.14 T r e a t m e n t 3 -> 6 -4 9 20.006 15 0.17 G r o u p 6 -> 9 -> 12 21.161 11 0.03 9 -»• 12 -> 18 3.589 12 0.99 12 18 -»• 24 29.795 24 0.19 0 1 ^ 3 18.429 12 0.10 1 ^ 3 - ^ 6 5.997 12 0.92 C o n t r o l 3 -> 6 -> 9 10.104 9 0.34 G r o u p 6 9 12 19.101 13 0.12 9 '-»• 12 - » 18 11.920 11 0.37 12 18 -> 24 11.251 17 0.84 Chapter 2. Markov Analysis 10 T h e resu l t s are i n T a b l e 2.4. T h e s ta t i s t i c s fo r b o t h g roups were nons i gn i f i cant (at a = 0.05) w h i c h i m p l i e d t h a t th i s m o d e l was not un rea sonab le f o r the t i m e p e r i o d s t ud i ed . 2.2 Modelling of the Tridiagonals Spec i f i c m o d e l l i n g of t he t r i d i a gona l s was done as mos t t r an s i t i on s were m a d e t o ne i gh - b o u r i n g K u r t z k e scores. I n th i s sec t i on , a l l t r an s i t i on s m a d e t o n o n - n e i g h b o r i n g states were i g no r ed a n d t r a n s i t i o n p r o b a b i l i t i e s no t i n a t r i d i a g o n a l p o s i t i o n were a s sumed t o be zero; t h a t is, p^ = 0 i f | i — j |> 2. I n t he f o l l o w i n g c a l cu l a t i on s , t he t r a n s i t i o n ma t r i c e s were a s sumed t o be s t a t i o n a r y a n d f i r s t o rder M a r k o v . T h e obse rva t i on s o n the t r i d i a gona l s were m o d e l l e d f i r s t u s i n g a genera l f o r m where each r o w h a d d i f ferent ent r ies , t h e n u s i n g a m o r e spec i f ic f o r m w h i c h was suggested b y t he d a t a . I n th i s s ec t i on , t he u n e v e n spac i ng of the t i m e po i n t s w i l l be t a k e n i n t o account . T h e d a t a at the one m o n t h t i m e p o i n t were o m i t t e d so t h a t t he r e m a i n i n g t i m e po i n t s (0,3,6,9,12,18,24) were sepa ra ted b y in te rva l s w h i c h were m u l t i p l e s of th ree month s . T h i s a l l o w e d m o d e l l i n g o f t r an s i t i on s ove r a th ree m o n t h p e r i o d u s i n g t he m a t r i x , V. T h u s , t r an s i t i o n s over a s ix m o n t h p e r i o d c o u l d be m o d e l l e d u s i n g V2. T o ad ju s t fo r th i s om i s s i on , redef ine t = 0,1,..., 6 c o r r e s pond i n g t o 0, 3, 6, 9, 12, 18, a n d 24 month s . S u b s e c t i o n 2.2.1 d iscusses m o d e l l i n g t he t r i d i agona l s u s i n g t he mat r i ce s V a n d V2 to de sc r i be t r an s i t i o n s o c c u r r i n g over th ree a n d s i x m o n t h per iods respect ive ly . In S e c t i o n 2.2.2, mode l s are f i t t e d a s s u m i n g t h a t t he same t r a n s i t i o n m a t r i x , P , c a n b e used t o m o d e l b o t h th ree a n d s i x m o n t h per iods . T h e en t i r e t i m e i n t e r v a l a n d t ha t d u r i n g t r e a t m e n t (0, 3, 6 m o n t h s ) a n d af te r t r e a tmen t (6, 9, 12, 18, 24 m o n t h s ) were m o d e l l e d sepa ra te l y so t h a t t he es t imates c o u l d be compa red . S o m e o the r obse rva t ions were o m i t t e d f r o m the ana ly s i s because t he t r an s i t i on s were Chapter 2. Markov Analysis 11 l a rge r t h a n those a l l o w e d b y t he m o d e l . I n th i s case, d a t a f r o m the sub jec t i n que s t i on was used u n t i l t he v i o l a t i o n o ccu r red . A sub jec t w i t h a m i s s i n g o b s e r v a t i o n was dea l t w i t h s im i l a r l y . A p p r o x i m a t e l y 2 5 % of the sub jec t s i n each g r oup h a d a m i s s i n g va lue at some t i m e p o i n t a f te r s i x month s . O n l y s i x pa t i en t s i n t he t r e a t m e n t g r oup a n d th ree i n t he c o n t r o l g r oup h a d t r an s i t i on s w h i c h were l a rge r t h a n those a l l owed by the m o d e l . 2.2.1 Modelling Incorporating V2 T r a n s i t i o n ma t r i c e s f o r t r e a t m e n t (VT) a n d c o n t r o l g roups (Vc) were e s t i m a t e d a n d c o m p a r e d t o a m a t r i x e s t i m a t e d unde r t he n u l l hypo thes i s V° = VT. T h e d a t a were f i r s t m o d e l l e d u s i n g t he genera l m a t r i x , V: Pll 1 - p n 0 0 0 P21 P22 1 — P21 — P22 0 0 0 P32 P33 1 - P32 - P33 0 0 0 P43 P44 1 - P43 ~ 0 0 0 1 -P55 P55 T h e l i k e l i h o o d f u n c t i o n L(V) was as fo l lows: L(V) = Kx n , ^ " . ( P 2 ) * , • where pij was redef ined as: P i j = P r [Xt = j | Xt-i = i] fo r t = 1, 2, 3,4 a n d Nij = Ei=5 n\j Pi = Pr[Xt = j | Xt_x = i] fo r t = 5,6 K = a con s tan t i ndependen t of Pi/s. Chapter 2. Markov Analysis 12 T o c a l c u l a t e t he m.l.e.'s, i t was necessary t o so lve e ight n o n l i n e a r equat ions . T h i s was a c c o m p l i s h e d u s i ng P o w e l l ' s m e t h o d , de s c r i bed i n [5]. T h e G2 s t a t i s t i c c a l c u l a t e d to c o m p a r e t he t w o g roups was: <3 = £ E t=i j=i (2.5) where njj, nfj, Nf-, Nf- we re def ined t o c o r r e spond t o count s i n t he t r e a t m e n t a n d c o n t r o l g roups i n t he obv iou s way. Re su l t s of t he test are i n T a b l e 2.5. I n t he t i m e p e r i o d d u r i n g t he t r ea tmen t , G\ was large r e l a t i v e t o the degrees of f reedom. T h e l o w p va lues ( c a l c u l a t ed u s ing the \ 2 a p p r o x i m a t i o n ) i n d i c a t e d tha t , d u r i n g th i s t i m e p e r i o d , t he t w o g roup s ' t r a n s i t i o n p r o b a b i l i t y ma t r i c e s were d i f ferent. I n t he f o l l o w u p p e r i o d a f te r t r e a t m e n t , a n d the ent i re t w o year p e r i o d , the re was no ev idence t h a t t he two g roups behaved d i f ferent ly . G2 s t a t i s t i c s were c a l c u l a t e d t o d e t e r m i n e h o w w e l l t he mode l s fit t he d a t a (see T a b l e 2.6). T h e s e s t a t i s t i c s i n d i c a t e d t h a t t he fit of t he m o d e l was reasonab le at a l l of t he t i m e in te rva l s m o d e l l e d . 2.2.2 Modelling Without V2 S ince t h e " s t a t i o n a r i t y " test seemed to i n d i c a t e t h a t a l l consecut i ve i n te rva l s i n the s t u d y c o u l d be m o d e l l e d u s i n g the same m a t r i x , the above ana ly s i s was r epea ted unde r th i s a s s u m p t i o n w i t h t he d a t a ga the red at one m o n t h a ga i n o m i t t e d . T h e p e r i o d d u r i n g t r e a t m e n t d i d no t i n c l u d e a n y s i x m o n t h i n te rva l s so es t imates a re t he same as i n t he p rev i ou s sec t i on . G|, the l o g l i k e l i h o o d s t a t i s t i c c o m p a r i n g t he two groups, was s i m i l a r t o t h a t c a l c u l a t e d i n (2.3). T h e resu l t s , d i s p l a yed i n T a b l e 2.7, show t ha t the g roups c o u l d be m o d e l l e d r ea sonab l y a c c o r d i n g t o the same t r a n s i t i o n p r o b a b i l i t y m a t r i x a f te r t r e a t m e n t . O v e r t he t w o year p e r i o d , t he s ta t i s t i c s i n d i c a t e d t ha t t he t r a n s i t i o n p r o b a b i l i t y ma t r i ce s fo r t he two g roups were d i f ferent, due p r o b a b l y t o the differences w h i c h were de tec ted i n t he first s i x m o n t h s of the s t u d y (see T a b l e 2.5). Chapter 2. Markov Analysis 13 G2 s t a t i s t i c s f o r t he p e r i o d a f te r t r e a t m e n t a n d t he who le t i m e s t u d i e d a re d i s p l a yed i n T a b l e 2.8. T h e s e show t ha t t he p red i c t i on s f r o m the mode l s ag reed rea sonab l y we l l w i t h the a c t u a l d a t a . A f t e r f u r t h e r e x a m i n a t i o n of the d a t a i t was no ted t ha t mos t o f t he d i a g o n a l ele- men t s were s im i l a r . A p a t t e r n a m o n g the off d i a g o n a l e lements was n o t i c e d also. T o d e t e r m i n e w h e t h e r t he d a t a c o u l d be m o d e l l e d w i t h f ou r pa r amete r s i n s t ead of the e ight used above, the p rev i ou s c a l cu l a t i on s were r epea ted , th i s t i m e u s i n g a t r a n s i t i o n p r o b a b i l i t y m a t r i x suggested b y the d a t a , name l y : Pll 1 - Pn 0 0 0 P21 P22 1 - P21- P22 0 0 0 P32 Pll 1 ~ P32 ~ Pll 0 0 0 1 ~P32 ~ Pll Pll P32 0 0 0 1 - Pll Pll T h e e s t imates c o m p u t e d fo r t he spec i f ic a n d genera l t r i d i a g o n a l mode l s were c o m p a r e d for each g r o u p sepa ra te l y u s i ng G2: G 2 7 = 2 £ £ < > i=l j=l 'A ,Pfj where pfj = m.l.e. o f pij u n d e r t he hypothes i s o f a genera l t r i d i a g o n a l m a t r i x pfj = m.l.e. of pij u n d e r t he hypothes i s of the spec i f ic t r i d i a g o n a l m a t r i x above. Chapter 2. Markov Analysis 14 T h e resu l t s are s h o w n i n T a b l e 2.9 t o 2 . 1 0 a n d e s t imates fo r the spec i f ic a n d genera l ma t r i ce s are p r o v i d e d i n the A p p e n d i x . T h e s ta t i s t i c s fo r t he c o n t r o l g r oup i nd i c a t e t h a t t he genera l m o d e l does no t f i t s i g n i f i c an t l y b e t t e r (at a n a l e ve l of 0.05) t h a n t he spec i f ic m o d e l ove r t he two years s t ud i ed . T h e spec i f ic m a t r i x e s t i m a t e d fo r the t r e a t m e n t g r o u p however, was o n l y reasonab le fo r the fo l l ow u p p e r i o d . A s i t was of in te re s t w h e t h e r t he t r e a t m e n t a n d c o n t r o l g r oup c o u l d be m o d e l l e d u s i n g t he same m a t r i x a f t e r t r e a t m e n t , th i s m a t r i x was c o m p a r e d t o t h a t fo r the c o n t r o l g r oup (over t he t w o y e a r p e r i o d ) a n d they were f o u n d to be s i g n i f i c an t l y d i f ferent . T h i s d i f ference m a y be due t o t he fac t t h a t the spec i f i c m a t r i x fo r the c o n t r o l g r o u p does no t fit the d a t a p a r t i c u l a r l y we l l . A ch i - s qua red s t a t i s t i c fo r the goodness of f i t was 6.056 (p=0.195). F r o m a n e x a m i n a t i o n o f t he mat r i ce s , i t appea r s t h a t i n t he c o n t r o l g roup, those pa t i en t s i n s tates o the r t h a n t w o were m o r e l i k e l y t o s tay i n t h a t s tate. If t h e y were i n s ta te two, however, t h e y were m o r e l i k e l y to m o v e to s ta te one. T h i s was a lso t r ue for pa t i en t s i n the t r e a t m e n t g r oup a f te r s i x mon th s . D u r i n g a d m i n i s t r a t i o n of the d r ug , t r e a t m e n t s ub jec t s were m o r e l i k e l y t o m o v e t o a h i ghe r s ta te (i.e. t o b e c o m e s icker) t h a n t h e c o n t r o l g roup . A l l of these ana ly ses i n d i c a t e t h a t the progress of the disease i n the t r e a t m e n t g r oup was no t t he same as i n the c o n t r o l g r oup i n the first s i x m o n t h s of the s tudy. I n t he f o l l ow up p e r i o d , the re was no ev idence t o i n d i c a t e t h a t a n y o f t he groups was wor se off t h a n the o t h e r a n d over t he ent i re t i m e p e r i o d s t u d i e d there seemed to be l i t t l e d i f fe rence be tween the two. A m o d e l a s s um ing the t r a n s i t i o n p r o b a b i l i t y m a t r i x was first o rde r M a r k o v a n d s t a t i o n a r y seemed t o fit t he d a t a r ea sonab l y we l l . T h e m o d e l l i n g o f t he t r i d i a g o n a l e lements revea led t h a t a spec i f ic m o d e l i n w h i c h a l l d i a g o n a l e lements were the same except fo r the second, a p p e a r e d t o fit t he da t a . F r o m the m a t r i c e s e s t i m a t e d u s i n g th i s m o d e l , i t is seems t h a t pa t i en t s w i t h scores of 4.5 Chapter 2. Markov Analysis 15 t o 5.5 are m o r e l i k e l y t o m o v e to a n adjacent score t h a n pa t i en t s i n a n y o f the o the r categor ies used. Howeve r , i t s h o u l d be n o t e d t ha t t he co l l ap sed categor ies used i n th i s ana ly s i s were s omewha t a r b i t r a r y a n d m a y p r o d u c e m i s l e a d i n g resu l t s . A s we l l , m a n y of the cel ls i n t he t r a n s i t i o n ma t r i ce s te s ted were zero, w h i c h c o u l d p r o d u c e s ta t i s t i c s t h a t do not ref lect t he d a t a accura te l y . Chapter 2. Markov Analysis 16 T a b l e 2.3: Test s of 1 s t O r d e r vs. I ndependence T i m e P e r i o d Gl _ degrees of (month s ) s t a t i s t i c f r e e d o m 0 -> 1 52.025 16 1 -»• 3 42.851 16 T r e a t m e n t 3 ^ 6 53.937 16 G r o u p 6 -> 9 46.017 16 9 - » 12 59.955 16 12 -»• 18 51.898 16 18 - * 24 49.019 16 0 -> 1 58.278 16 1 -> 3 44.748 16 C o n t r o l 3 -> 6 55.247 16 G r o u p 6 ^ 9 45.811 16 9 ^ 12 52.629 16 12 -»• 18 51.747 16 18 ->• 24 63.797 16 T a b l e 2.4: Re su l t s of Tes t fo r S t a t i o n a r i t y G r o u p G\ d f p va lue t r e a t m e n t c o n t r o l 88.921 65.198 84 78 0.34 0.85 T a b l e 2.5: T r e a t m e n t vs. C o n t r o l ( genera l t r i d i a g o n a l m o d e l l i n g ) T i m e (mon th s ) 0-3-6 6-9-12-18-24 0-3-6-9-12-18-24 G\ 19.250 4.283 11.415 p-va lue 0.01 0.83 0.18 Chapter 2. Markov Analysis 17 T a b l e 2.6: F i t o f G e n e r a l T r i d i a g o n a l M o d e l s T i m e T r e a t m e n t degrees P C o n t r o l degrees P ( i n m o n t h s ) G r o u p o f va lue G r o u p of v a l ue f r e e d o m f r e e d o m 0-3-6 1.22 8 0.99 3.73 8 0.88 6-9-12-18-24 14.75 24 0.93 25.91 24 0.36 0-3-6-9-12-18-24 43.76 40 0.32 42.80 40 0.35 T a b l e 2.7: T r e a t m e n t vs. C o n t r o l ( genera l t r i d i a g o n a l - n o P2) T i m e ( i n m o n t h s ) 6-9-12-18-24 0-3-6-9-12-18-24 3.456 9.642 p-va lue 0.484 0.047 T a b l e 2.8: F i t of G e n e r a l M o d e l - n o P2 T i m e T r e a t m e n t degrees P C o n t r o l degrees P ( i n m o n t h s ) G r o u p of f r e e d o m va lue G r o u p of f r e e d o m va l ue 6-9-12-18-24 16.74 8 0.86 27.66 24 0.27 0-3-6-9-12-18-24 44.08 40 0.30 43.55 ) 40 0.32 T a b l e 2.9: C o m p a r i s o n of Spec i f i c a n d G e n e r a l M o d e l s ( t r e a t m e n t ) T i m e T r e a t m e n t ( i n mon th s ) G2? degrees p -va lue of f r e e d o m 0-3-6 11.815 4 0.018 6-9-12-18-24 3.919 4 0.417 0-3-6-9-12-18-24 14.7211 4 0.005 Chapter 2. Markov Analysis 18 T a b l e 2.10: C o m p a r i s o n of Spec i f i c a n d G e n e r a l M o d e l s T i m e C o n t r o l ( i n m o n t h s ) G? degrees p -va lue of f r e e d o m 0-3-6 6.531 3 0.088 6-9-12-18-24 6.017 4 0.197 0-3-6-9-12-18-24 5.565 4 0.234 Chapter 3 Analysis At One Time Point A t y p i c a l ana l y s i s i nvo lves c o m p a r i n g the scores o f t he pa t i en t s i n the t r e a t m e n t a n d c o n t r o l g r oup at one t i m e po i n t . C o m m o n l y used for th i s are t he 2 s amp le t s t a t i s t i c ( a l t h o u g h i n a p p r o p r i a t e ) a n d the W i l c o x o n r ank s u m s ta t i s t i c . Here , these s ta t i s t i c s were c o m p a r e d t o a M c C u l l a g h m o d e l s t a t i s t i c de s c r i bed i n [4]. T h e M c C u l l a g h m o d e l c a n be used t o a n a l y z e d a t a w i t h o rde red , c a t ego r i c a l responses. T h e f o r m used i n th i s c h a p t e r is t he p r o p o r t i o n a l odds m o d e l : l og 7 j ( g ) 1 - 7 j ( £ ) . 9j -f3Tx . . — (3.6) whe re j = 1,2, ...,z z = t o t a l n u m b e r of categor ies x = v e c t o r o f covar iates 7j(af) = p r o b a b i l i t y of b e i n g i n ca tego ry j o r l ower g i ven cova r i a te x 9j = c u t p o i n t j , t he ( u n k n o w n ) p o i n t s epa r a t i n g the categor ies j a n d j + 1 f3 = a v e c t o r of u n k n o w n pa ramete r s K = a sca le pa r amete r . T h e r e l a t i v e eff ic iencies of the th ree tests were c o m p a r e d u s i ng s i m u l a t i o n a n d ef- ficacy c a l cu l a t i on s . D a t a fo r the t r e a t m e n t a n d c o n t r o l g r oup were s i m u l a t e d b y first g ene r a t i n g n o r m a l d a t a w i t h d i f ferent l o c a t i o n pa ramete r s . T h i s con t i nuou s d a t a was 19 Chapter 3. Analysis At One Time Point 20 used t o c reate o rde red c a tego r i c a l d a t a a c c o r d i n g t o a chosen set of c u t p o i n t s . Tests f o r d i f ferences i n the t w o groups were t h e n c a r r i e d out u s i n g the 2 s a m p l e t, W i l c o x o n r a n k s u m a n d M c C u l l a g h s t a t i s t i c . T h e 2 s amp le t test was a lso c a l c u l a t e d o n the unca tego r i z ed d a t a so t h a t a test w h i c h d i d not lose i n f o r m a t i o n due t o t he ca tego r i z a - t i o n c o u l d b e c o m p a r e d t o t he others. The se resu l t s a re g i ven i n S e c t i o n 3.1. E f f i c a c y c a l c u l a t i o n s were a l so m a d e t o d e t e r m i n e h o w di f ferent t he tests were a s y m p t o t i c a l l y . T h e eff icacies of t he W i l c o x o n r a n k s u m test are c a l c u l a t e d i n S e c t i o n 3.2.1, a n d for the t test c a l c u l a t e d on the c a tego r i c a l d a t a i n Sec t i on 3.2.2. T h e s e eff icacies are c o m p a r e d to t he t test c a l c u l a t e d o n the u n d e r l y i n g con t i nuous d a t a i n S e c t i o n 3.3. 3.1 Simulation A s i m u l a t i o n was r u n t o c o m p a r e the powers of the W i l c o x o n r a n k s u m s t a t i s t i c , b o t h 2 s a m p l e t tests a n d M c C u l l a g h m o d e l e s t imates . T h e c o n t r o l a n d t r e a t m e n t g r oup were s i m u l a t e d u s i n g i V ( 0 , l ) a n d i V ( A , l ) d i s t r i b u t i o n s r e spec t i ve l y t o p r o d u c e u n d e r l y i n g con t i nuou s responses. T h e s e responses were t h e n ca tego r i zed w i t h cu t p o i n t s chosen so t h a t t he p r o b a b i l i t y of b e i n g i n a n y ca tego ry was 0.2 i f t he d a t a c ame f r o m a i V ( 0 , 1 ) d i s t r i b u t i o n . F i f t y observat ions ( the same n u m b e r as were i n t he M u l t i p l e Sc leros i s d a t a ) were genera ted for e a ch g roup. T h e s imu l a t i o n s i n v o l v e d one t h o u s a n d r ep l i c a t i on s a t each va lue o f A . T h e A values used were those f r o m 0 t o 1 i n i n c rement s o f 0.1. T h e d a t a were genera ted w i t h o u t c o n d i t i o n i n g o n t he n u m b e r o f obse rva t ions i n each ca tego ry as th i s was t he f o r m of t he M u l t i p l e Sc leros i s d a t a . P o w e r cu rves were c a l c u l a t e d fo r a one s i ded 0.05 l eve l test o f the n u l l hypothes i s H0 : A = 0 vs. t he a l t e r n a t e hypothes i s Hi : A > 0. T h e resu l t s of the s i m u l a t i o n a p p e a r i n F i g u r e 3.1. A s expec ted , t he 2 s amp le t test o n t he u n d e r l y i n g con t i nuous d a t a was t he mos t p o w e r f u l because n o i n f o r m a t i o n was lost t h r o u g h c o l l a p s i n g t he Chapter 3. Analysis At One Time Point F i g u r e 3.1: S i m u l a t i o n of D a t a at O n e T i m e P o i n t Chapter 3. Analysis At One Time Point 22 d a t a . T h e W i l c o x o n r a n k s u m s ta t i s t i c a n d the 2 s a m p l e t test o n t he ca tego r i zed d a t a were ve ry s i m i l a r a t a l l po in t s . F o r the resu l t s s u m m a r i z e d i n F i g u r e 3.1, K, t he sca le p a r a m e t e r f o r the M c C u l l a g h ana lys i s was set t o one. T h e cova r i a t e x was a n i n d i c a t o r v a r i ab l e d i s t i n g u i s h i n g t he t r ea tmen t f r o m the c o n t r o l obse rvat ions . I n th i s case t he test ba sed o n the M c C u l l a g h s t a t i s t i c was as p o w e r f u l as the o t h e r tests on the c a t ego r i z ed d a t a . I f K was i n c l u d e d as a p a r a m e t e r d e p e n d i n g o n cova r i a t e x, i t was not s i g n i f i c an t l y d i f fe rent f r o m 1 ~ 9 2 % o f the t ime. Howeve r , i n c l u d i n g th i s p a r a m e t e r dec reased the p o w e r o f t he M c C u l l a g h s t a t i s t i c f o r va lues of A la rger t h a n 1.5. D i f fe rences i n t he s i m u l a t e d power be tween any of t he mode l s were no t ve ry large. T h e m a x i m u m d i f ference was ~ 0.05 be tween the 2 s amp le t test c a l c u l a t e d u s ing the c o n t i n u o u s d a t a a n d the M c C u l l a g h m o d e l at A = 0.7, whe re the powe r o f t h e t test is a b o u t 0.95. G i v e n t h a t the s t a n d a r d e r ro r of th is d i f ference was ~ 0.02, t h e t test was s i g n i f i c a n t l y m o r e p o w e r f u l t h a n the M c C u l l a g h m o d e l at th i s t i m e po i n t . 3.2 Efficacy Calculations A n o t h e r w a y of c o m p a r i n g the tests is to ca l cu l a te t he i r a s y m p t o t i c r e l a t i v e eff ic iencies. T h e m e t h o d o f c a l c u l a t i o n used is t ha t i n L e h m a n n [3]. L e t Vjv, V}f b e two sequences of s ta t i s t i c s ba sed o n N observat ions . A s s u m e the d i s t r i b u t i o n s o f b o t h Vpi a n d d e p e n d o n a r ea l v a l u e d p a r a m e t e r 9. L e t the n u l l h ypo the s i s b e 9 = 90 a n d the a l t e rna te hypothes i s be 9 > 8Q as i n the s i m u l a t i o n . D e f i n e BN t o be t he p o w e r of the test w h i c h rejects HQ i f Vff ~ /*(flo) C/v(#0) > CAT (3.7) a n d 8'N t o be the p o w e r o f the test based o n w h i c h rejects HQ i f cr'N(90) (3.8) Chapter 3. Analysis At One Time Point 23 where pi(90), CJV(# 0) a n d fi'(90), cr'N(60) a re n o r m a l i z i n g cons tan t s w h i c h m a y be the e x p e c t a t i o n a n d s t a n d a r d d e v i a t i o n of VN a n d respect i ve ly , a n d c/v a n d c'N are sequences o f c r i t i c a l va lues. A s s u m e 9^ is a sequence of a l t e rna t i ve s , conve rg i ng to 90 i n s u ch a way t ha t 9^ = 90 + F o r mos t c o m m o n l y used tests, th i s c o n d i t i o n is suf f ic ient t o ensure t ha t PN(9N) —• B 0 , 0 < 0O < 1. De f i ne N' t o be t he s a m p l e s ize r equ i r ed f o r V^, t o ach ieve t he s ame l i m i t i n g p o w e r as Vjv aga ins t t he same sequence of a l t e r na t i v e s 9^. If c/y, c^i —> za ( t he (1 — a)th q u a n t i l e of t he s t a n d a r d n o r m a l d i s t r i b u t i o n ) as N —> oo, t h e n the P i t m a n ef f ic iency o f VN r e l a t i v e t o V^, is l imjv-foo (N'/N). Suppo se t ha t wheneve r 9^ = 90 + 0N(9N) - $ ( c A - za) a n d 3'N(9N) $ ( c ' A - * „ ) . T h e n c a n d c ' a re ca l l ed t he eff icacies o f t he tests ba sed o n Vpj a n d V^, a n d t he P i t m a n ef f ic iency of Vjv r e l a t i v e t o is: ( c / c ' f 3.2.1 Efficacy of the Wilcoxon Suppo se the o rde red c a t e go r i c a l v a r i ab l e takes o n values gi < g2 < • • • < gz, where z is t he n u m b e r of categor ies. I n th i s ana ly s i s , observat ions t i ed i n a ca tego ry were ass igned m i d r a n k s , so the W i l c o x o n r a n k s u m s t a t i s t i c ( w i t h t ies) was: W = ^(iVa + l)/2 + T2(NX + (N2 + l)/2) + • • • + T , ( £ AT,- + (Nz + l)/2) (3.9) Chapter 3. Analysis At One Time Point 24 where Tj = n u m b e r of ob se rva t i on s i n the t r e a t m e n t g r oup i n ca tego ry j Cj = n u m b e r of obse rva t ions i n t he c o n t r o l g r oup i n ca tego ry j a n d Nj = Tj + Cj. L e t qj = P r [ i n ca tego ry j | i n c o n t r o l group] Pj = P r f i n c a tego r y j | i n t r e a t m e n t group] i V = t o t a l n u m b e r of observat ions . T h e test ba sed o n t he W i l c o x o n i n v o l v e d n o r m a l i z i n g W as fo l lows: W-E0(W) yjvar*0{W) ' where E0(W) is t he e x p e c t a t i o n o f the W i l c o x o n u n d e r H0 : qj = pj V j , a n d varl(W) is the v a r i ance of t he W i l c o x o n unde r HQ c o n d i t i o n a l o n the n u m b e r of t i e d obse rva t ions i n each category. T h e p o w e r o f t he test at a spec i f i c a l t e r n a t i v e c o u l d have been c a l c u l a t e d e i ther u n c o n d i t i o n a l l y o r c o n d i t i o n a l l y o n the t i e d observat ions . I n th i s case, u n c o n d i t i o n a l p o w e r was used as i t s i m p l i f i e d t he eff icacy ca l cu l a t i on s of the t test a n d e l i m i n a t e d the need t o generate a f i x ed n u m b e r of observat ions i n each ca tego ry fo r t he s imu la t i on s . T h e u n c o n d i t i o n a l p o w e r of t he test u s i n g the W i l c o x o n s t a t i s t i c was: f3N(p,6)=?r where W - E0(W) > z, Jvar*(W) a Chapter 3. Analysis At One Time Point 25 E*(W) = E0(W) = ^ ^ - v a r : ( W ) = _ ^ ^ ^ N f ^ N j ) 8 = y/N{q-p) m = t o t a l n u m b e r of observat ions i n the c o n t r o l g r o u p n = t o t a l n u m b e r of observat ions i n t he t r e a t m e n t g r oup JV = t o t a l n u m b e r of observat ions i n t he s t u d y = n + m. Theorem 3.2.1 Suppose ( T i , T 2 , . . . , Tz) ~ multinomial(n, pi, p2, • • •, pz) independent of ( C i , C2, • • •, Cz) ~ multinomial(m,qi,q2,. .. ,qz) and N —> oo in such a way that n/N -> a, m/N -> b, 0 < a < 1. Then, as N —> oo, MP) - $ ( / ( £ £ ) - za) ( 3 . i o ) where * V ^E ; : ; *J+E£ E £ P * - E £ E ^ + 1 W * ) V^;=i r>p>-Ei=i E,= i '•i'-iPiPj rj = 1 + Ef=j+i P i — Ei=i defining E/_1 Pi ^° & e 0 /or aw?/ integer I. I n t he s i m u l a t i o n r u n of Sec t i on 3.1 px = p2 = • • • = pz a n d t he u n d e r l y i n g c o n t i n u - ous d i s t r i b u t i o n was k n o w n . T h e f o l l o w i n g co ro l l a r ie s s ta te t he efficacies u n d e r those c ond i t i o n s a n d t he p r o o f of T h e o r e m 3.2.1 fo l lows. —» —* Corollary 3.2.1 Assume that the multinomial vectors T and C are generated from XT± , XT2, •. •, Xxn ~ Hd N(A, 1) and XGl, Xc2,..., Xcm ~ iid i V ( 0 , 1 ) in the usual way. That is, an observation is in category i if and only if the X value is between cutpoints di and 9^ where 90 = —oo and 9Z = oo. Then under the assumptions of Theorem 3.2.1, and with A = 3N(A) -> $ ( c • £ - za) Chapter 3. Analysis At One Time Point 26 where \/a6 vEi=i ryPi-E,'=i Ej=i - / or j = 1,.. . , z - 1 probability density function of the standard normal distribution at 6j. Corol lary 3.2.2 I n i/ie case where —A, - X r 2 — A , . . . , XT„ —A anc? > -^c*2> • • • > -^ "c m are zzc? observations with a continuous density, h, and p1 = p2 = • • • = pz then, under the assumptions of Theorem 3.2.1., the efficacy of the Wilcoxon becomes: Va~b^-\(z - j)e3 where A = (5/x/iV e j =h(0j)-h(6j-1)j = l,2,...,z-l Oj = outpoint j. The above results do not depend upon the scores associated with the categories. Proof of Theorem 3.2.1: PN(p,6) = P r = P r W - Ea(W) y/var*(W) W - E(W) > zD var*(W) E(W) - Ea(W) y/mnN V mnN y/mnN It w i l l be shown that i f N —» oo in such a way that jj —> a, ^ —» 6, then 1- W^^ 0'^) ^ = i [ £ £ } r ? P i - E*=i EJ=1 r t T i P i P i ] r,- = 1 + Ej=J+i Pj ~ Ej^i Pj Chapter 3. Analysis At One Time Point 27 „ vart(W) -, 3 . w n - ^ w - * ffat). 1. T o s i m p l i f y t h e ca l cu l a t i on s , s u b s t i t u t e m — £;=i Q f o r C 2 a n d n — Ti fo r T 2 i n t he expre s s i on fo r the W i l c o x o n ( refeq:wi lc ) t o o b t a i n : W - E(W) _ 1 \J mnN 2 V m n N mn 2 - 1 2 - 1 2 - 1 2 - 1 n Y J C i - m Y J T j + Y ^ T j d - £ E TJCi !=1 j=l j=l t'=l j=l «=j+l 2 - 1 2 - 1 j - l Z - l 2 - 1 E ^ + E E ^ - E E j=i«=j+i L e t c; Cj - mqj Ti — npi T h e n , W - E(W) _ 1 yj mnN 2 2 - 1 »=i 2 - 1 - l i - i £ E rtc: - ̂ E w + ̂  E E (27c; - ir q ) S ince T * , C j " converge i n d i s t r i b u t i o n t o n o r m a l r a n d o m var iab le s w i t h m e a n 0 a n d va r i ance P j ( l — Pj), as iV —> 0 0 , I 2 - 1 j - i T h u s as N —> 0 0 , W - E(W) y/mnN where 2 - 1 J - I ri = 1 + E Pi - E Chapter 3. Analysis At One Time Point 28 2-1 t-1 »t = i + Yl 9j - £ 9; j=i+l j=l de f i n i n g E ^ f c = 0 a n d EfJmPi = 0- S i nce cot;(C;,C;) = -qiqj a n d cot;(T*,T*) = —piPj, wjJ^^ converges t o a n o r m a l l y d i s t r i b u t e d r a n d o m va r i ab l e w i t h m e a n 0 a n d variance a w- ^ / z—1 z—1z—1 uw = j\YL r)pj - Y Yl ririPiPj * \j=i t=i j=i 2. R e w r i t i n g t he r a t i o as var*0(W) _ var*(W) var0(W) mnNa2v var0(W) mnNo~w' i t w i l l be s h o w n t h a t b o t h ra t i o s o n the r i gh t h a n d s ide converge t o 1. F o r t he first r a t i o , var0(W) = E0[var*0(W)] + var0[E*0(W)] = E0[var*0(W)] s ince E*(W) is a con s tan t . So var*0(W) _ var*0(W) var0(W) mnNo-w E0[var*(W)] mnNa^' B u t varn W Jf^arliW) 1 N~* ab 12 ab 12 mn(N + 1) ™ Y(N3-N) 12N(N - 1) jr?"3 j ) as N —> oo, and 12 m. i - E N Chapter 3. Analysis At One Time Point 29 ab 12 ab 12 1 - £ ( aPi + Hjf 3=1 under HQ. T h u s , varl (^f) converges t o a cons tant , say y, a n d is b o u n d e d a n d po s i t i ve . Hence , E var* ^-^==)j a l so converges t o y, a n d so var*0(W) N o w we need t o p r o v e t h a t E[var*(W)] var0(W) mnNaw 1. R e c a l l i n g t h a t u n d e r H0, pj = qj V j , a n d so rj = Sj V j i t c a n b e s h o w n t h a t z-l z-l (W-E(W)\ 1 var0 . —• - V VmnN J 4 z-l £ r]pj - £ £ rtrjp;pj t=i j=i So var, „2 crw. o(W) mnNa^ as N —> oo. 3. F i n a l l y , u s i n g pa r t s 1 a n d 2, the eff icacy t e r m c a n be s h o w n t o converge t o a cons tant : E(W) - E0(W) r2w yj mnNa\ KM. So n o w we have t h a t 3N(pJ) - P r [JV(0,1) < /(p,<?) - za\ . Chapter 3. Analysis At One Time Point 30 P r o o f of C o r o l l a r y 3.2.1. N o t e t h a t , P j = - A) - - A). Hence, as N —> oo, S u b s t i t u t i o n o f th i s exp re s s i on i n t o (3.10) y ie ld s t he resu l t of C o r o l l a r y 3.2.1. P r o o f o f C o r o l l a r y 3.2.2. F r o m T h e o r e m 3.2.1, t he eff icacy t e r m is: F o r t he case pj = 1/zV j, y/XJZl fjPi - E & 1 E g r , T i P i P i (2—1 2—1j—1 2—1 2—1 £ ^ + £ £ M * ' - £ £ Vab = — Vao" z = —y/ab -sz + - £ - ; -1) - - E ( J - 2-2 j=2 1 2 - z 2 — 2 + - E - 2 i ^ ) + (*i - ^ - 0 Z j=2 Z 2-1 2-1 * £ * ; + £ ( * - 2 j ) $ > 3=1 3=1 2*£(z-j)6e since ^ Si = 0 »=i S ince i t c a n eas i l y be seen f r o m the above a r g u m e n t t h a t Si ~ VNA[h(8j) - h(0j..i)] = Se, Chapter 3. Analysis At One Time Point 31 fo r any con t i nuou s den s i t y h. N o t e t h a t w h e n pj = 1/z, z-l 12 ri = z-l 3=1 g r ? = 2(> - 1 ) ( 2 * - 1) 3=1 3 3z S u b s t i t u t i o n of these express ions i n t o the d e n o m i n a t o r of the ef f icacy t e r m produces : z-l z - l z - l \ 12 rjPj ~1212 nrjPiPj = \ j=i i=i j=i a n d i t c a n eas i l y b e seen t h a t t he eff icacy t e r m is: vV-i)/i2 3.2.2 Efficacy of the T test ?2 - 1 L e t t he response v a r i ab l e be as i n Sec t i on 3.2.1. T h e n the 2 s amp le t s t a t i s t i c c a n be w r i t t e n as: T = Z]Zl(9i ~ 9z)(mT3 - nCj) nmS\Jl/m + 1/n where S = p o o l e d s t a n d a r d d e v i a t i o n of the 2 groups ( t r e a t m e n t a n d c o n t r o l ) . A s N -> oo, ( w i t h n/N -> a, m/N —• b), a Z 2—12—1 12(dj ~ 9z)2Pj ~ 12 12(9j - 9z)(9i ~ 9z)PiPj j=l 3=1 *=1 Z 2—1 2—1 - 9z)2Pj - 12 12(9j - 9z)(gi - 9z)qi9j j=l 3=1 i=l + a n d , u n d e r HQ, z-l z - l z - l S2 -* a\ = Y(9j ~ 9z)2Pj - 12 12(9j ~ 9z)(9i ~ 9z)p, Pj- Chapter 3. Analysis At One Time Point T h e p o w e r of the 2 s a m p l e t test, I I ^ (p ) is nN(p) = Pr^[T>ta}. T h e o r e m 3.2.2 Under the assumptions of Theorem 3.2.1 r —* IIJV(P) $ [w(p, 8) - za where ™IP, ~ - Pj Corol lary 3.2.3 If XT and Xc are as in Section 3.2.1 then, as N oo, njv(A) = $(c • 8 - za) where dj and 8 are as in the previous section and vya&££}(fl,i - 9z)dj c = \JHU\(9j - 9z)2PJ - £̂ =1 ELiO - 9z)(g, - 9z)PiPj where A = 6/y/W and n/N —• a, m/N —* b. Corol lary 3.2.4 When P l = p2 = • • • = pz = \, (XTl - A,XTi - A , . . . , XTz and (Xcx, Xc2, • • •, Xcz) are as in Corollary 3.2.2 and gi oc i, the efficacy term 2 sample t test becomes where ej is de f i ned as p rev ious l y . P r o o f of T h e o r e m 2.2.2: HN(p) = Prp^[T>ta} Chapter 3. Analysis At One Time Point 33 L e t 7 = £jZ\(9j ~ 9z)(mTj - nCj). T h e n ILN(p) = P r f t ? 7 mnSyJl/m + 1/n P r p,5 7 £(7)' mntaSvyJl/m + 1/n - E(j) > \/var(y) I n o rde r t o reduce th i s exp re s s i on t o the f o r m i n (3.2.2) the f o l l o w i n g w i l l be shown: 1. ^±^N(0,1) y/var(-y) 2. var( i 7 „ ) —> <7 yJl~F" yf var(y) mntaSWl/m+l/n 3. / . . • zc Eh) w(p,6) yj var{i) P a r t 1 is t r ue b y t he C e n t r a l L i m i t T h e o r e m . T o p rove pa r t 2, no te t ha t : u a r ( 7 ) = var J^(9j ~ 9z)(mTj - nC3) v=1 mn z z—1z—X 12(9 i - 9zf [mpj + nq3] - ̂  12(9 j ~ 9z)(9i ~ 9z) [mpiPj + nqiq3} j=l j=l i=l A s N —> oo, p3 —> qj a n d hence, (z-l z-lz-l Y(9j - 9z)2Pj - 12 12(9j - 9z)(9i ~ 9z)piPj j=l j=l i=l . var(- 7 N o w ' y/mnN' mntaS^l/m + 1/n _ taSy/mnN sfmr^y) taS ,/uar(-7=2==) V v VmnN' Chapter 3. Analysis At One Time Point 34 T h e ef f icacy t e r m c a n t h e n be f o u n d to converge t o a con s tan t : Vob'ZjZiigz - 9 y/Z&9j ~ 9z)2Pi - Ei=i£f=i(0.- - g,)(gj - g,)PiPj 3.3 Comparison of the Efficacies T h e 2 s amp le t test is o f ten u sed i n a n a l y z i n g o rde red c a t e go r i c a l d a t a a l t h o u g h i t is no t a p p r o p r i a t e . F o r th i s reason, the r e l a t i v e ef f ic iency of t he W i l c o x o n a n d t he t test is of some in teres t . U n d e r cond i t i on s l i s t ed be low, as the n u m b e r o f obse rva t ions becomes large, the ef f icacy t e rms fo r the W i l c o x o n a n d t h e t tests a p p l i e d t o s u c h d a t a converge t o t he s ame express ion: Suf f i c ient c ond i t i o n s fo r (3.11) t o h o l d are: 1. the p r o b a b i l i t i e s of the responses are equa l i.e. p1 = p2 = • • • = pz a n d 2. t he scores f o r t he categor ies are p r o p o r t i o n a l t o t he ca tego ry number . 3. t he scores are genera ted b y a con t i nuou s d i s t r i b u t i o n . T h e ef f icacy fo r t he t test a p p l i e d t o the con t i nuous d a t a is c a l c u l a t e d i n L e h m a n n [3]. It is c = whe re Chapter 3. Analysis At One Time Point 35 c is the s t a n d a r d d e v i a t i o n of t he d i s t r i b u t i o n of the d a t a a = n/N, t he p r o p o r t i o n of t he s amp le i n the t r e a t m e n t g r oup b = m/N, t he p r o p o r t i o n o f t he s a m p l e i n t he c o n t r o l g roup. T h e ef f ic iency o f the W i l c o x o n a p p l i e d t o the ca tego r i c a l d a t a r e l a t i v e t o the the t test a p p l i e d t o the u n d e r l y i n g con t i nuou s d a t a was of in te res t as th i s i n d i c a t e d the decrease i n p o w e r due t o t he c a t e go r i z a t i o n of t he con t i nuous da t a . U n d e r t he c o n d i t i o n of e q u a l p/s a n d d a t a genera ted f r o m a con t i nuou s d i s t r i b u t i o n , the P i t m a n eff ic iency o f t he W i l c o x o n t o t he t test was: ,2 1 ewilc,t(cut) - [ 1 2 ( 7 2 ( E - l ^ _ I ) E . ) 2 So, f o r e x a m p l e i f t he c o n t r o l g roup ' s scores were u n i f o r m l y d i s t r i b u t e d o n [0,1] a n d t he t r e a t m e n t g roup ' s scores were u n i f o r m l y d i s t r i b u t e d o n [A , 1 + A ] t h e n the eff ic iency of t he t ( u n c u t ) test t o t he W i l c o x o n ( on scores ca tego r i zed as i n t he s i m u l a t i o n ) is: c - ( z - i y ( A l t h o u g h t he u n i f o r m den s i t y is no t con t i nuou s at 0 a n d 1, C o r o l l a r y 3.2.2 c a n be m o d i f i e d f o r th i s case.) It is eas i l y seen f r o m th i s t h a t as z, t he n u m b e r o f categor ies becomes large, c —> 1, w h i c h is w h a t w o u l d be expec ted . T h e s i m u l a t i o n i n S e c t i o n 3.1 was r u n u n d e r c ond i t i on s 1 a n d 2 w i t h the n o r m a l as its u n d e r l y i n g con t i nuou s d i s t r i b u t i o n . A l t h o u g h there were o n l y 100 observat ions i n each r u n , t he curves i n F i g u r e 3.1 are v e r y close. U s i n g t he ef f icacy te rms c a l c u l a t e d , t he a s y m p t o t i c P i t m a n ef f ic iency of t he W i l c o x o n w i t h respect t o t he t test o n the con t i nuou s d a t a was c a l c u l a t e d a n d f o u n d t o b e 0.89. T h i s means t h a t t he t test requ i res ~ 8 9 % of t he observat ions needed b y the W i l c o x o n i n o rde r t o a t t a i n the same l i m i t i n g p o w e r u n d e r t he same a l t e r na te hypotheses . The se c a l cu l a t i on s seem t o agree w i t h t he s i m u l a t i o n r u n i n S e c t i o n 3.1. Chapter 4 Analysis at Two Time Points M o s t pa t i en t s i n t he M u l t i p l e Sc leros is t r i a l s were e x a m i n e d at a l l of t he t i m e po i n t s 0, 1, 3, 6, 9, 12, 18, 24 m o n t h s d u r i n g the s tudy. A b e t t e r i n d i c a t i o n of t he differences be tween t he t r e a t m e n t a n d c o n t r o l g r oup c o u l d be ga ined b y a n a l y z i n g h o w the scores of t he pa t i en t s change b e t w e e n t w o t i m e po in t s . T h e p o w e r of some s ta t i s t i c s f r equen t l y u sed t o a n a l y z e o r de red d a t a over t i m e were c o m p a r e d u s i ng s i m u l a t i o n . T h e s e s t a t i s - t i c s were t he s t r a t i f i e d W i l c o x o n , t he M c C u l l a g h , a n d a c h i s qua red s t a t i s t i c . S e c t i on 4.1 con ta i n s a d e s c r i p t i o n of the s i m u l a t i o n runs. T h e W i l c o x o n a n d the ch i - squa red s t a t i s t i c a re de s c r i bed m o r e f u l l y i n Sect ions 4.1.1 a n d 4.1.2. I n t he l a t t e r , t he d i s - t r i b u t i o n o f t he ch i - s qua red s t a t i s t i c is de r i ved u n d e r the hypo the s i s o f n o d i f ference b e t w e e n t he t r e a t m e n t a n d c o n t r o l groups. I n S e c t i on 4.2, t he resu l t s of the s i m u l a t i o n are d i scussed. 4.1 Description of the Simulation A n i n i t i a l a n d final score were genera ted for each sub jec t s i m u l a t i n g his c o n d i t i o n at t w o t i m e po in t s . T h e va lues of t he u n d e r l y i n g con t i nuou s r a n d o m va r i ab le s g i v i n g r ise t o these scores were d e n o t e d b y (Rf, Y?) o r (Rf,Yf) fo r those i n the t r e a t m e n t or c o n t r o l g roup . T h e s e d a t a were ca tego r i zed u s i n g the same cu t po i n t s as i n the p rev ious c h a p t e r t o p r o d u c e scores. T h e d i s t r i b u t i o n o f these pa i r s was: / ^ \ / r n 1 r, . I N Y? J I 0 1 P BVN \ . £ + A . i . P 1 . 36 Chapter 4. Analysis at Two Time Points 37 a n d \ / T h e pa r amete r s 3 a n d A represented the changes i n the scores due to t he progres s ion o f t he d isease a n d the t r e a t m e n t respect ive ly . P o w e r curves were c o m p u t e d fo r a leve l a =0 .05 test of t he n u l l hypo thes i s A = 0 vs. t he a l t e r na te hypo the s i s A ^ 0. A s i m u l a t i o n r u n gene ra ted t w o groups ( f i f t y ob se rva t i on s/g roup ) at A va lues of 0 t o 1.2 i n i n c r emen t s of 0.1. E a c h cu r ve was c a l c u l a t e d u s i n g one t h o u s a n d s i m u l a t i o n runs. T h e s t r a t i f i e d W i l c o x o n , ch i - squa red a n d M c C u l l a g h m o d e l s t a t i s t i c s were c a l c u - l a t e d f o r each set of d a t a generated. T h e M c C u l l a g h m o d e l s t a t i s t i c was c o m p u t e d u s i n g t he p l u m so f tware i n a w a y s i m i l a r t o t h a t i n C h a p t e r 2, except t h a t t he co- v a r i a t e x h a d m u l t i p l e componen t s . O n e c o m p o n e n t was a v a r i ab l e i n d i c a t i n g whe the r t he ob se r v a t i on was f r o m the t r e a t m e n t o r c o n t r o l g roup. T h e o t h e r c o m p o n e n t s were i n d i c a t o r va r i ab le s f o r each pos s ib le i n i t i a l score. T h e scale p a r a m e t e r was set to one. S ince, i n t he s i m u l a t i o n , t he con t i nuous obse rva t ions were k n o w n , t he con t i nuous response v a r i a b l e was regressed u s i ng a n i n d i c a t o r f o r the t r e a t m e n t / c o n t r o l g roups a n d t he i n i t i a l va lues fo r covar iates . T h e n u l l hypothes i s H0 : A = 0 was re jected i f the e s t ima te fo r t he t r e a t m e n t / c o n t r o l v a r i ab l e was s i gn i f i c an t l y d i f fe rent f r o m zero. T h i s s t a t i s t i c was i n c l u d e d so t h a t the decrease i n powe r due to t he loss of i n f o r m a t i o n f r o m c a t e g o r i z a t i o n c o u l d be de te rm ined . 4.1.1 Stratified Wilcoxon T h e s t r a t i f i e d W i l c o x o n r a n k s u m s ta t i s t i c , de s c r i bed i n [3] (p. 132), is a n ex ten s i on o f t h a t e m p l o y e d i n the p rev i ou s chapte r . T h e sub jec t s i n t he t w o g roups were first s t r a t i f i e d a c c o r d i n g t o t he i r i n i t i a l score. W i t h i n each s t r a t a , s, t he W i l c o x o n s t a t i s t i c / Rc \ I - BVN 0 0 1 P P 1 Chapter 4. Analysis at Two Time Points 38 Ws was c o m p u t e d , as i n C h a p t e r 3, u s i ng the f i n a l scores as the basis fo r the r a n k i n g p r o cedu re a n d s u m m i n g the t r e a t m e n t ranks . E a c h of t he s ta t i s t i c s , Ws, was n o r m a l i z e d b y s u b t r a c t i o n o f i t s m e a n t o p r o d u c e W*: w; = wa- E(w8) where E(W.) = 2 ^ ± H Ts — n u m b e r of observat ions i n t r e a t m e n t g r oup w i t h i n i t i a l score s Ns = t o t a l n u m b e r of obse rva t ions w i t h i n i t i a l score s. T h e o ve r a l l W i l c o x o n , W * , was t h e n c a l c u l a t e d as: WT T h e va r i ance o f W* was: ~se(W*) [se(W*)f = £ where Ni + 1 z = n u m b e r of s t r a t a se(Wj) ~ s t a n d a r d d e v i a t i o n of W * c a l c u l a t e d i n the u sua l way. T h e test was t h e n re jec ted o r a c cep ted u s i ng the n o r m a l a p p r o x i m a t i o n : W* se(W*) 4.1.2 Chi-squared Statistic JV(0,1). A n a t u r a l test of the n u l l hypothes i s t h a t a s ub jec t ' s progress is i ndependen t of t r ea t - m e n t rece i ved is based o n a s i m p l e c on t i n gency t ab l e ana lys i s . T h e r e su l t i n g s ta t i s t i c , re fe r red t o as xL/o ^ s D a s e d o n the n u m b e r o f sub jec t s whose c o n d i t i o n i m p r o v e d , de- t e r i o r a t e d o r r e m a i n e d the same over the two t i m e po in t s . T o ca l cu l a t e Xca/ci ^ n e d a t a were co l l ap sed i n t o a 3 x 2 c on t i n gency tab le: Chapter 4. Analysis at Two Time Points T r e a t m e n t C o n t r o l T o t a l s c-n~ rpo C° n° T+ C+ n+ n m N where rp— 1 X^z T1 c = Ef=i EjLt+i Cij C° = E ? = 1 C y rp-\- V-"*2 VM—1 rp 1 — 2^i=2 l^j=l •Lij c+ = E U E & C i , - T{j = t he n u m b e r of observat ions i n the t r e a t m e n t g r oup w i t h i n i t i a l score i a n d f i n a l score j Cij = the n u m b e r o f observat ions i n t he c o n t r o l g r oup w i t h i n i t i a l score i a n d f i n a l score j n = n u m b e r of observat ions i n the t r e a t m e n t g r oup m = n u m b e r o f obse rva t ions i n t he c o n t r o l g r oup N = t o t a l n u m b e r of observat ions i n t he study. T h e s t a t i s t i c was c a l c u l a t e d as: y 2 = (r- - ^ ) 2 (c- - ^ ) 2 X-calc nn- ~ r mn- N N ( rpo nn"^ (Vio mn° \ 1 ~ir) A c ~ —) nn" ' mn" N N Chapter 4. Analysis at Two Time Points 40 (T+ - (c+ TV T h e s t a t i s t i c , Xcaici l s u s u a l l y a s s umed t o have a l i m i t i n g x2 d i s t r i b u t i o n w i t h two degrees o f f reedom. It was not c lea r i f th i s a p p r o x i m a t i o n was v a l i d so t he l i m i t i n g n u l l d i s t r i b u t i o n of xL/c w a s de r i ved . T w o m e t h o d s o f s a m p l i n g w h i c h p r oduce d a t a i n a f a s h i o n s i m i l a r t o t h a t i n t h e M u l t i p l e Scleros is s t u d y were cons ide red , a n d l ed to d i s t i n c t l i m i t i n g n u l l d i s t r i b u t i o n s fo r Xcalc- ^ d e s c r i p t i o n of each of these s a m p l i n g schemes a n d t he d i s t r i b u t i o n of xlaic *s p receded b y some n o t a t i o n u sed i n th i s analys i s : Pij — P r [ f i n a l score = j | i n i t i a l score = i, i n t r e a t m e n t g r o u p ] qij = P r [ f i n a l score == j | i n i t i a l score = i, i n c o n t r o l g r oup ] p^j = P r [ f i n a l score = j, i n i t i a l score = i, i n t r e a t m e n t g r oup ] qfj = P r [ final score = j, i n i t i a l score = i, i n c o n t r o l g r oup ] rii = n u m b e r of t r e a t m e n t observat ions i n s t r a t a i rrii = n u m b e r of c o n t r o l observat ions i n s t r a t a i Unconditional Sampling I n u n c o n d i t i o n a l s a m p l i n g , sub jec t s are r a n d o m l y ass igned t o g roups a n d t h e n are s t r a t i f i e d a c c o r d i n g t o t he chosen covar ia te. I n th i s case, the n u m b e r s of observat ions i n each s t r a t a is r a n d o m . T h i s t y p e of s a m p l i n g is a p p r o p r i a t e w h e n i t is no t i m p o r t a n t i f s t r a t a are empty , o r there a re so m a n y sub jec t s t h a t t he e x p e r i m e n t e r is a s sured t ha t i t is u n l i k e l y t h a t a n y of the s t r a t a w i l l be empty . Suppo se t h a t ( T n , T i 2 , . . . ,TZZ) ~ multinornial(n,p"2,...,p"_J, i ndependen t of ( C n , C 1 2 , . . . , Czz) ~ multinomial(m, g n , q™2,..., q™z) u n d e r u n c o n d i t i o n a l s amp l i n g . If we def ine T t o be a vec t o r w i t h t he c o m p o n e n t s ( T + , T~,T°) a n d C t o be ( C + , C~, C ° ) Chapter 4. Analysis at Two Time Points 41 t h e n T a n d C a re i ndependen t w i t h d i s t r i b u t i o n s : T ~ multinomial(n,p+ ,p ,p°) C ~ multinomial (m, q+, q , q° ) whe re P + — Ei=2 Ej=i Pij P ~ Ei=l ^2j=i+l Pij P° = Z U P I J o+ — V* y ^ - 1 a?-y — i^i=2 Hi] 1 = £i=i £j=;+i Qtj I n th i s case, t he u s u a l a s y m p t o t i c t h e o r y fo r i ndependen t m u l t i n o m i a l s ho lds a n d u n d e r t he hypo the s i s p+ = q+, p° — q°, p~ = q~, ( a nd thu s a l so u n d e r t he m o r e r e s t r i c t i v e n u l l h ypo the s i s p^- = q^ V i,j), xlaic 1S a s y m p t o t i c a l l y d i s t r i b u t e d as a x 2 w i t h t w o degrees of f r eedom. Conditional Sampling In c o n d i t i o n a l s a m p l i n g , t he n u m b e r o f sub jec t s i n each s t r a t a is fixed at the beg i n - n i n g of t he e x p e r i m e n t . T h i s t y p e of s a m p l i n g c o u l d be used w h e n i t is expens i ve t o a l l o w m a n y sub jec t s t o take pa r t . It ensures t h a t there w i l l be observat ions i n each s t r a t a , a l t h o u g h t h e t o t a l n u m b e r of sub jec t s m a y be r e l a t i v e l y s m a l l . T h i s t y p e of s a m p l i n g is c o m m o n so i t is n a t u r a l t o ca l cu l a te t h e d i s t r i b u t i o n of xlaic c o n d i t i o n a l o n n{ a n d m t-. I n th i s case, t he d i s t r i b u t i o n o f the count s becomes (Tn,Ti2, • • • , T J 2 ) ~ multinomial(ni,pn,pi2, . . . ,Piz) a n d (Cn, C,- 2 ,..., Ciz) ~ multinomial (mi, qn, qi2,..., qiz) where i = 1,2,...,z a n d a l l m u l t i n o m i a l s are i ndependen t . S ome a s s umpt i on s m a d e to Chapter 4. Analysis at Two Time Points 42 f a c i l i t a t e c o m p u t a t i o n s were t h a t n,- = m t- V i a n d t ha t N rii/N —¥ ai. N o t e t h a t xlaic c a n ^ e r e w r i t t e n as: xLc = Kl + K\ + Kj oo i n s u ch a w a y t ha t whe re K — J m \ 1 (T0 nn°\]2 • Z X 2 V mnn" [VW V AT /J js- I f 1 / rp+ nn+ \ Theorem 4.1.1 If tii — TTij V i, then, under H0 : p,j = ^ j j , V i,j as N —• oo ane? ^2,7^) ^>MVN(0,X) with E12 S 1 3 and E23 = 1 E t = 2 ^ j = l Efc=l  aiPijPik Ef=2 E j = i fli'Pij E L i «jP 33 1 - E j = i ajPjj>' E i = l Ej=i+1 Efe=t'+1 aiPijPik Ej=: l Ej=i+1 at 'Pij vj=l ^iPiiPij E z v-»t—1 i=2 Z^j 1/2 [(E*=2 E } = \ a,-p„) ( E j = i ajP«)] — Et'=2 E j = i Efc=i+i AiPijPik (E;=2 E j = l aiPij) ( E f = i E j = ; + l Q-iPijj 1/2 E f = l E j = j - | - l aiPiiPij [(zui zUi+i aiPij) ( E j = i ajPii) 1/2 Chapter 4. Analysis at Two Time Points 43 Thus, where Zt, Z2, Z3 are iid N(0,1) and Ai,A2,A3 are the eigenvalues o / S . P r o o f o f T h e o r e m 4.1.1. L e t X 2 ~ VW\ ~N~ x3 = _ L ( T + - ^ VN \ N T o d e t e r m i n e t he l i m i t i n g d i s t r i b u t i o n of Xcaio n r s ^ n n d t he l i m i t i n g d i s t r i b u t i o n s of Xi,X2,X3. I f a n d t h e n rp* Tjj TljPij 13 V^i c. * Cij rriiqij 'mi Xx = nn ~N z i—1 EE 2* nn vwyuu N Chapter 4. Analysis at Two Time Points 44 1 ) z i—1 ^ z i—1 £ £ Tij - JJ £ £ °ij i=2 j=l i v i=2 j=\ J V 7 t=2 i=l i V i=2 i=l Therefore, under iJ, 0 5 z i—1 i=2 j=l since ra; = mt- and p,j = under HQ. Under H 0 , T*j and C*j converge to normal random variables with mean zero, and thus Xi is asymptotically normal with mean zero and variance: var(Xi) ~ var Under H 0 , n Z \t=2i=l j ( z i—1 z i—1 j—1 ^ £ E a^iiC1 - Pij) - 2 E E E aiPijPik i=2 j=l i=2 i=l fc=l , z i—1 z i—1 i—1 E E E E E diPijPik i=2j=l i=2j=lk=l iV z i—1 ~ fefeU iV + m, X iV z i—1 EE(P<'i f l i + ft'ia') i=2 j=l z i—1 = 2 £ £ < w , i=2 j=l and thus K~i is asymptotically normal with mean zero and variance, Et=2 E j - l QiPij ~ £j-2 Ej=l Efc = l AiPijPik 12i=2 Ej=l a iPij Chapter 4. Analysis at Two Time Points 45 S i m i l a r c a l c u l a t i o n s s how t h a t K2 a n d K3 are a s y m p t o t i c a l l y n o r m a l . T h e va r i ance/cova r i ance c a l cu l a t i on s are s t r a i g h t f o r w a r d . N o w (KUK2,K3) ~ MVN(0,V). If S is a po s i t i v e s em ide f i n i t e m a t r i x , 3 a d i a g o n a l m a t r i x D, w i t h t he eigenvalues ( A i , A 2 , A 3 ) of E as i t s e lements , a n d a m a t r i x P, s u ch t ha t PTP = I, a n d PHPT — D. L e t Y = PK, t h e n Y ~ N(0, D) a n d KTK = YTY ~ A i Z x + A2Z2 + A3Z3 I n mos t cases, t he eigenvalues o f S are d i f f i cu l t t o f i n d e xp l i c i t l y . Howeve r , one s imp l e case is con s i de red i n t he f o l l ow ing : Corollary 4.1.1 Suppose that n\ = n2 = ... = nz and p^ = 1/z V i,j. Under the null hypothesis, where Z U Z 2 ~ JV(0,1 ) . Thus, xlaic does not have a limiting chi-squared distribution with two degrees of freedom in this case. P r o o f o f C o r o l l a r y : U s i n g T h e o r e m 4.1.1, t he a s y m p t o t i c cova r i ance m a t r i x of (Kx, K2, K3) is: T h e eigenvalues fo r th i s cova r i ance m a t r i x are A i = 0, X2 = 1, A 3 = | — ^ . Chapter 4. Analysis at Two Time Points 46 4.2 S i m u l a t i o n Resu l t s T h e powe r curves c o m p u t e d b y the s imu l a t i o n s a p p e a r i n F i gu re s 4.1 t o 4.4. I n a l l of the figures, t he mos t p o w e r f u l test was the one based o n the regress ion pa ramete r . T h i s re su l t was e x p e c t e d as no i n f o r m a t i o n due t o c a t e go r i z a t i o n was lost i n t he c a l c u l a t i o n of t h i s s t a t i s t i c . T h e nex t mos t p o w e r f u l was t he M c C u l l a g h s t a t i s t i c w h i c h h a d a s i m i l a r p o w e r c u r v e t o the W i l c o x o n . I n a l l cases b o t h of these s ta t i s t i c s were m u c h m o r e p o w e r f u l t h a n t he ch i - squa red s ta t i s t i c . Becau se t he ch i - s qua red s t a t i s t i c d i d no t t a k e i n t o a c coun t t he i n i t i a l score o r t h e size of t he d i f ference, th i s was was not su rp r i s i n g . T h e powers o f t he tests i nc reased w i t h i n c rea s i ng p. T h i s p r o b a b l y o c c u r r e d because i n c r ea s i n g p w o u l d decrease the v a r i a t i o n i n di f ferences b e t w e e n t he i n i t i a l a n d final scores a t fixed va lues of 0 a n d A . Change s i n /? d i d no t affect t he p o w e r curves of t he W i l c o x o n , M c C u l l a g h o r regress ion s ta t i s t i c s , b u t i nc rea s i ng i t d i d decrease the p o w e r o f t he ch i - s qua red s t a t i s t i c . A s /? became large, mos t p a t i e n t s ' scores inc reased, a n d t he d i f ference be tween the t r e a t m e n t a n d c o n t r o l g roups were i n t he sizes o f these increases. S i n ce t he W i l c o x o n a n d the M c C u l l a g h s ta t i s t i c s c o m p a r e d the t r e a t m e n t a n d c o n t r o l g roups based o n t he differences o f the final scores g i v en t he i n i t a l scores, a n increase i n 0 w o u l d not be expec ted to change the powe r of these s ta t i s t i c s . However , v e r y l a rge changes i n /3 w o u l d resu l t i n a l l pa t i en t s m o v i n g t o the largest score, i n w h i c h case, none o f t he tests w o u l d detect a n y di f ference be tween the groups. T h e ch i - square s t a t i s t i c c o m p u t e d here was u n c o n d i t i o n a l as t he i n i t i a l n u m b e r of observat ions i n the s t r a t a were no t f i xed . It o n l y r eco rded w h e t h e r o r not the p a t i e n t s ' scores inc reased, decreased o r r e m a i n e d t he same, so i ts p o w e r was e x p e c t e d t o decrease as /3 i nc reased. Chapter 4. Analysis at Two Time Points delta Figure 4.1: Simulation Run 1 (8 = 0.1, p = 0.8) Figure 4.2: Simulation Run 2 (/? = 0.1, p = 0.6) Chapter 4. Analysis at Two Time Points Chapter 4. Analysis at Two Time Points delta F i g u r e 4.4: S i m u l a t i o n R u n 4 (0 = 0.3, p = 0.6) Chapter 4. Analysis at Two Time Points 51 T a b l e 4.1: Re su l t s o f M c C u l l a g h A n a l y s i s T i m e e s t ima te s t a n d a r d z-score ( i n m o n t h s ) of t r e a t m e n t p a r a m e t e r e r ro r 0-1 -0.82 0.418 -1.95 1-3 -0.40 0.403 -1.00 3-6 -0.17 0.424 -0.40 6-9 0.98 0.466 2.10 9-12 0.07 0.51 0.14 12-18 0.31 0.452 0.69 18-24 -0.16 0.48 -0.33 4.3 Application of Tests to the MS Data T h e s t r a t i f i e d W i l c o x o n a n d x 2 tests have been c a l c u l a t e d o n t he M S d a t a set p r ev i ou s l y [2]. B o t h s t a t i s t i c s were c a l c u l a t e d o n t he d a t a be tween 0 m o n t h s a n d each of t he o the r t imes t he pa t i en t s were observed. T h e resu l t s of the test ba sed o n t he W i l c o x o n s t a t i s t i c showed t h a t t he t r e a t m e n t g r oup regressed (p ~ 0.05) r e l a t i ve t o t he c o n t r o l g o rup i n the t i m e pe r i od s 0-1 m o n t h s a n d 0-3 mon th s . T h e r e m a i n i n g s t a t i s t i c s c a l c u l a t e d were non s i gn i f i c an t . Howeve r , the change i n s i gn i n those t i m e pe r i od s l a rge r t h a n s i x m o n t h s i nd i ca te s t h a t pa t i en t s i n the c o n t r o l g r oup m a y have f a red less w e l l t h a n those i n t he t r e a t m e n t g r oup i n t he f o l l o w up p e r i o d . S ince the x 2 is less p o w e r f u l t h a n the W i l c o x o n , i t is no t s u r p r i s i n g t h a t none of t he tests ba sed o n i t were s i gn i f i cant . T h e d a t a were fit t o a M c C u l l a g h m o d e l . U n l i k e the p rev i ou s ana ly s i s , t he co l l ap sed scores were used here s ince the d a t a were sparse. T h e covar iates i n th i s ana ly s i s were i n d i c a t o r va r i ab le s f o r each o f the po s s i b le i n i t i a l scores a n d one for t r e a t m e n t / c o n t r o l g roup . T h e d a t a were m o d e l l e d be tween consecut i ve t i m e po i n t s r a t h e r t h a n f r o m base l i ne t o t he o t h e r scores. T h e resu l t s a re s h o w n i n T a b l e 4.1. T h e v a r i a b l e fo r t he t r e a t m e n t g r oup p a r a m e t e r was set up so t h a t a nega t i ve va lue Chapter 4. Analysis at Two Time Points 52 i m p l i e d t h a t t he c o n t r o l g r oup regressed w i t h respect t o the t r e a t m e n t g roup. C o l l a p s - i n g t he d a t a has p r o d u c e d resu l t s w h i c h a re c o n t r a d i c t o r y t o t he p rev i ou s W i l c o x o n ana ly s i s i n t he 0-1 m o n t h t i m e p e r i o d . T h e W i l c o x o n a n d x2 ana ly s i s agree w i t h the ana ly s i s u s i n g the M a r k o v techn iques i n t h a t a l l t h ree show t h a t pa t i en t s i n the t r e a t m e n t g r oup m a y not have progressed at the same r a te as those i n t he c o n t r o l g r oup d u r i n g t he f i rst s i x m o n t h s of t he s tudy. I n t he case of t he M a r k o v ana ly s i s , i t is no t poss ib le t o d e t e r m i n e w h i c h g r oup i m p r o v e d r e l a t i v e t o t he other . Chapter 5 Conclusions A s the data set was small , the number of observations which had any part icular K u r t z k e score was low. F o r this reason, the data were collapsed into five categories chosen only to ensure at least four observations i n each category at zero months. Results from the M a r k o v analysis indicate that, during the administrat ion of interferon, the treatment group regressed relative to the controls. After this period, subjects i n the treatment group appeared to return to a state i n which their transit ion probabilities were not significantly different from the controls. Models which d id not incorporate information about the length of t ime intervals between observations fit the data just as well as those that d i d . T h a t is, a model which assumed that transitions between any two consecutive t ime points could be modelled using the same transit ion matr ix fit as well as a model which allowed a different matr ix for each t ime interval. M o d e l l i n g w i t h a specific tr idiagonal m a t r i x w i t h a l l diagonal elements equal, except for the second, proved to be reasonable for the control group over the entire t ime period. T h e treatment group could be modelled reasonably by this form only i n the eighteen m o n t h follow up period. T h e major differences between the two groups were i n those patients that started at K u r t z k e scores i n the range 4.5 to 5.5. C o n t r o l patients w i t h i n i t i a l scores i n this range fared better than treatment patients, during the period when the interferon was administered. Statistics commonly used to compare the groups at one t ime point were examined 53 Chapter 5. Conclusions 5 4 u s i n g s i m u l a t i o n techn iques . I f d a t a f o r t he c o n t r o l g r oup were genera ted u s i n g unde r - l y i n g iV(0,1) d i s t r i b u t i o n a n d those fo r the t r e a t m e n t g r oup b y a N(A, 1) d i s t r i b u t i o n , a n d the p r o b a b i l i t y o f b e i n g i n each ca tego ry was equa l fo r A = 0, t h e n t he W i l c o x o n , M c C u l l a g h a n d 2 s amp le t test were f o u n d t o have s i m i l a r r e l a t i v e eff ic iencies. T h e o r e t i - c a l a s y m p t o t i c c a l cu l a t i on s y i e l d e d express ions fo r P i t m a n eff ic iencies o f these s ta t i s t i c s fo r gene ra l sh i f t mode l s . U n d e r t he cond i t i on s of t he s imu l a t i on s , t he efficacies of the W i l c o x o n a n d 2 s a m p l e t test c a l c u l a t e d o n t he ca tego r i zed d a t a were f o u n d t o be equa l . T h e P i t m a n ef f ic iency of these two s ta t i s t i c s w i t h respect t o t he t test c a l c u l a t e d o n the u n d e r l y i n g con t i nuou s d a t a was 0.89. A c o m p a r i s o n o f s ta t i s t i c s used t o c o m p a r e the t w o g roup s ' p rog res s ion of disease b e t w e e n t w o t i m e p o i n t s was t h e n c a r r i e d out . S i m u l a t i o n re su l t s s howed t h a t w h e n the c a t e go r i c a l d a t a were genera ted u s i n g a b i v a r i a t e n o r m a l d i s t r i b u t i o n , w i t h equa l i n i t i a l p r o b a b i l i t i e s of b e i n g i n a n y category, t he W i l c o x o n a n d M c C u l l a g h s ta t i s t i c s were m u c h m o r e eff ic ient t h a n the c h i squa red s ta t i s t i c . 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 of t he ch i - s qua red s t a t i s t i c was de r i v ed u n d e r t he hypothes i s of no t r e a t m e n t effect. It was d e t e r m i n e d t h a t t he ch i - squa red s t a t i s t i c d i d not neces sa r i l y have a l i m i t i n g x2 d i s t r i b u t i o n w i t h two degrees o f f r e e d o m if, i n t he s a m p l e of pa t i en t s , the i n i t i a l n u m b e r of sub jec t s i n each ca tego ry was f i xed. However , i f the n u m b e r was no t f i xed , t he ch i squa red s t a t i s t i c d i d have a n a s y m p t o t i c x2 d i s t r i b u t i o n w i t h two degrees of f reedom. Appendix A Sample Transition Matrices T r a n s i t i o n M a t r i x fo r C o n t r o l G r o u p f r o m 0 to 24 m o n t h s 21 7 1 0 0 8 7 3 0 0 0 2 67 29 1 0 0 23 49 7 0 0 1 5 8 T r a n s i t i o n M a t r i x f o r T r e a t m e n t G r o u p f r o m 0 t o 24 m o n t h s 23 6 1 0 0 4 2 7 0 0 1 3 59 20 1 0 0 20 40 14 0 0 0 12 16 55 Appendix A. Sample Transition Matrices 56 C o n t r o l G r o u p - G e n e r a l T r i d i a g o n a l M o d e l l i n g 0 m o n t h s to 6 m o n t h s 0.67 0.33 0.00 0.00 0.00 0.57 0.29 0.14 0.00 0.00 0.00 0.03 0.65 0.32 0.00 0.00 0.00 0.35 0.59 0.06 0.00 0.00 0.00 1.00 0.00 6 m o n t h s t o 24 m o n t h s 0.81 0.19 0.00 0.00 0.00 0.36 0.45 0.18 0.00 0.00 0.00 0.02 0.71 0.28 0.00 0.00 0.00 0.24 0.64 0.11 0.00 0.00 0.00 0.20 0.80 0 m o n t h s to 24 m o n t h s 0.75 0.25 0.00 0.00 0.00 0.44 0.39 0.17 0.00 0.00 0.00 0.02 0.68 0.30 0.00 0.00 0.00 0.29 0.62 0.09 0.00 0.00 0.00 0.38 0.62 Appendix A. Sample Transition Matrices T r e a t m e n t g r oup - G e n e r a l M o d e l l i n g 0 m o n t h s t o 6 m o n t h s 0.75 0.25 0.00 0.00 0.00 0.13 0.13 0.75 0.00 0.00 0.00 0.07 0.57 0.37 0.00 0.00 0.00 0.35 0.32 0.32 0.00 0.00 0.00 0.58 0.42 6 m o n t h s t o 24 m o n t h s 0.82 0.18 0.00 0.00 0.00 0.60 0.20 0.20 0.00 0.00 0.00 0.02 0.81 0.17 0.00 0.00 0.00 0.21 0.70 0.09 0.00 0.00 0.00 0.31 0.69 0 m o n t h s t o 24 m o n t h s 0.79 0.21 0.00 0.00 0.00 0.31 0.15 0.54 0.00 0.00 0.00 0.04 0.72 0.24 0.00 0.00 0.00 0.27 0.54 0.19 0.00 0.00 0.00 0.43 0.57 Appendix A. Sample Transition Matrices 58 C o n t r o l G r o u p - Spec i f i c T r i d i a g o n a l M o d e l l i n g 0 m o n t h s t o 6 m o n t h s 0.61 0.39 0.00 0.00 0.00 0.57 0.29 0.14 0.00 0.00 0.00 0.04 0.61 0.35 0.00 0.00 0.00 0.35 0.61 0.04 0.00 0.00 0.00 0.39 0.61 6 m o n t h s t o 24 m o n t h s 0.71 0.29 0.00 0.00 0.00 0.36 0.45 0.18 0.00 0.00 0.00 0.05 0.71 0.24 0.00 0.00 0.00 0.24 0.71 0.05 0.00 0.00 0.00 0.29 0.71 0 m o n t h s to 24 m o n t h s 0.67 0.33 0.00 0.00 0.00 0.44 0.39 0.17 0.00 0.00 0.00 0.05 0.67 0.29 0.00 0.00 0.00 0.29 0.67 0.05 0.00 0.00 0.00 0.33 0.67 Appendix A. Sample Transition Matrices T r e a t m e n t G r o u p - T r i d i a g o n a l M o d e l l i n g 0 m o n t h s t o 6 m o n t h s 0.48 0.52 0.00 0.00 0.00 0.13 0.13 0.75 0.00 0.00 0.00 0.18 0.48 0.33 0.00 0.00 0.00 0.33 0.48 0.18 0.00 0.00 0.00 0.52 0.48 6 m o n t h s to 24 m o n t h s 0.76 0.24 0.00 0.00 0.00 0.60 0.20 0.20 0.00 0.00 0.00 0.05 0.76 0.19 0.00 0.00 0.00 0.19 0.76 0.05 0.00 0.00 0.00 0.24 0.76 0 m o n t h s to 24 m o n t h s 0.65 0.35 0.00 0.00 0.00 0.31 0.15 0.54 0.00 0.00 0.00 0.11 0.65 0.25 0.00 0.00 0.00 0.25 0.65 0.11 0.00 0.00 0.00 0.35 0.65 Bibliography [1] B h a t , N a r a y a n U., (1984). Elements of Applied Stochastic Processes 2 n d ed i t i on . J o h n W i l e y & Sons. [2] K a s t r u k o f f , L., et. a l . , (1988). u n p u b l i s h e d m a n u s c r i p t . U B C H o s p i t a l . [3] L e h m a n n , E .L . (1975). Nonparametrics: Statistical Methods Based on Ranks. H o l d e n - D a y Inc., S a n F r anc i s co , C a l i f o r n i a . [4] M c C u l l a g h , P. (1980). Reg re s s i on M o d e l s F o r O r d i n a l D a t a . J.R. Statist. Soc. B, 42, No. 2, pp. 109-142. [5] W a l s h , G .R . (1975). Methods of Optimization. J o h n W i l e y & Sons. [6] W h i t t l e , P. (1955). S ome D i s t r i b u t i o n a n d M o m e n t F o r m u l a e F o r the M a r k o v C h a i n . J.R. Statist. Soc. B, 17, pp. 235-242. [7] T h e S t a t i s t i c a l A d v i s o r y Se rv i ce (1988). M u l t i p l e Scleros is Da ta se t . u n p u b l i s h e d m a n u s c r i p t , U n i v e r s i t y of M a n i t o b a . 60

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
China 10 7
United States 3 0
Japan 2 0
City Views Downloads
Beijing 6 7
Shenzhen 4 0
Tokyo 2 0
Redmond 1 0
Mountain View 1 0
Ashburn 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}

Share

Share to:

Comment

Related Items