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An examination of the missing data problem in experiments on the effect of drugs on coronary ligation… Crépeau, Hélène 1983

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EXAMINATION OF THE MISSING DATA PROBLEM IN EXPERIMENTS ON THE EFFECT OF DRUGS ON CORONARY LIGATION IN RATS by HELENE CREPEAU B . S c , U n i v e r s i t e L a v a l , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d s THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1983 © Helene Crepeau, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Mathematics The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: A p r i l 11, 1983 i i A b s t r a c t A data set from ongoing experiments on r a t s t o study the e f f e c t of c e r t a i n drugs on c l i n i c a l l y induced heart a t t a c k s i s i n v e s t i g a t e d . A f t e r d e s c r i p t i o n of the experiment and the data, p r e l i m i n a r y analyses examine the r e l a t i o n s h i p between some v a r i a b l e s . The problem of m i s s i n g data inherent to t h i s type of experiment i s then approached. Two s t a t i s t i c a l t e s t s d e a l i n g with t h i s problem are proposed and a p p l i e d . F i n a l l y l o g - l i n e a r models are c o n s i d e r e d to analyse some c a t e g o r i c a l v a r i a b l e s . Nancy M. Reid T h e s i s s u p e r v i s o r i i i T able of Contents A b s t r a c t i i L i s t of t a b l e s i v L i s t of f i g u r e s v Acknowledgements v i 1 . INTRODUCTION 1 2. HISTORY OF THE DATA 3 2. 1 Experiment 3 2.2 Data 7 2.3 Problems r e l a t e d t o the data 10 3. PRELIMINARY ANALYSES OF THE DATA 11 3.1 Can we p r e d i c t death? 11 3.2 Is OZ a confounding v a r i a b l e ? 17 3.3 Are arrhythmias r e l a t e d t o time of m o r t a l i t y ? ... 21 4. EXAMINATION OF TIME-RESPONSE VARIABLES 24 4.1 Manova approach 25 4.11 D e s c r i p t i o n of the score t e s t 25 4.12 A p p l i c a t i o n of the score t e s t 27 4.2 Non-parametric approach 35 4.21 D e s c r i p t i o n of the m u l t i v a r i a t e rank t e s t .. 35 4.22 A p p l i c a t i o n of the m u l t i v a r i a t e rank t e s t .. 38 5. LOG-LINEAR MODELS 44 6. CONCLUSION 50 BIBLIOGRAPHY 52 APPENDIX - DERIVATION OF THE SCORE STATISTIC AND THE MAXIMUM LIKELIHOOD STATISTIC IN THE MANOVA SETTING ... 54 i v L i s t of t a b l e s 1. Log of t o t a l PVC at 4 hours 22 2. T o t a l number of arrhythmias at 4 hours 22 3. Q-wave occurence 22 4. Blood pressure data f o r a l l f i v e treatments at time p o i n t s 3 t o 7 31 5. Blood pressure data f o r the four treatments e x c l u d i n g halothane at 2.0% at time p o i n t s 3 t o 11 .... 32 6. DSTR data f o r a l l f i v e treatments at time p o i n t s 3 to 7 40 7. DSTR data f o r the four treatments e x c l u d i n g halothane at 2.0% at time p o i n t s 3 to 11 41 8. D i s t r i b u t i o n of the arrhythmia scores a f t e r 4 h r s by treatment groups 45 9. Adjusted r e s i d u a l s r e s u l t i n g from f i t t i n g the homogeneity model to the data i n t a b l e 8 46 10. Observed f r e q u e n c i e s and a d j u s t e d r e s i d u a l s of the in c i d e n c e of i r r e v e r s i b l e VF by treatment groups 47 11. Observed f r e q u e n c i e s of the i n c i d e n c e of i r r e v e r s i b l e VF by treatments and s i z e of occluded zone 48 V L i s t of F i g u r e s 1. P l o t of mean BP vs time f o r r a t s that d i e d and r a t s that s u r v i v e d 13 2. P l o t of mean HR vs time f o r r a t s that d i e d and r a t s that s u r v i v e d 13 4. P l o t of mean DSTRS vs time f o r r a t s that d i e d and r a t s t h a t s u r v i v e d 14 5. P l o t of mean DSTRS vs time f o r r a t s that d i e d and r a t s that s u r v i v e d with one r a t d e l e t e d 15 6. Normal p l o t of BP at time p o i n t 3 f o r the c o n t r o l groups data 28<x 7. Normal p l o t of DSTR at time p o i n t 3 f o r the c o n t r o l groups data 28b 8. Normal p l o t of BP at time p o i n t 3 f o r the halothane data 2 9 a , 9. Normal p l o t of DSTR at time p o i n t 3 f o r the halothane data 29 b Acknowledgements I would l i k e to express my g r a t i t u d e to Dr. Nancy Reid f o r her h e l p f u l guidance and a s s i s t a n c e i n the producing of t h i s t h e s i s . I am indebted to Dr. J.A. K o z i o l f o r b r i n g i n g t o my a t t e n t i o n h i s work on growth curves and f o r g r a c i o u s l y running on our data h i s computer program f o r the score s t a t i s t i c . I would l i k e to thank Dr. Walker of U.B.C.'s Department of Pharmacology who p r o v i d e d the data set and the medical d e t a i l s f o r i t s study. F i n a l l y , I would a l s o l i k e t o thank Dr. J.A. Petkau f o r h i s c a r e f u l r e a d i n g of t h i s t h e s i s and Dr. J . Zidek f o r suggesting t h i s t o p i c and p r o v i d i n g some f i n a n c i a l support. 1 1. INTRODUCTION A frequent problem with which the s t a t i s t i c i a n i s co n f r o n t e d i n p r a c t i c e i s t h a t of m i s s i n g data. In some cases t h i s problem can be circumvented and the standard s t a t i s t i c a l techniques a p p l i e d . However the data a re o f t e n c o l l e c t e d or the experiments o f t e n designed i n such a way that m i s s i n g v a l u e s a re unavoidable. T h i s was the case f o r the data submitted by Dr. Walker of U.B.C.'s Department of Pharmacology to the S t a t i s t i c a l C o n s u l t i n g S e r v i c e . Dr. Walker and h i s r e s e a r c h group are i n t e r e s t e d i n the e f f e c t of drugs on the outcome of myocardial i n f a r c t i o n . They have developed an experimental method which they t h i n k w i l l l e a d to the d e t e c t i o n of drugs capable of a m e l i o r a t i n g adverse responses to ischaemia and i n f a r c t i o n . The experiment and the data c o l l e c t e d are d e s c r i b e d i n S e c t i o n 2. In S e c t i o n 3, r e s u l t s of some p r e l i m i n a r y a n a l y s e s undertaken to examine the r e l a t i o n s h i p between the responses are presented. B r i e f l y , t h e i r method c o n s i s t s of induci n g a heart a t t a c k in r a t s exposed to v a r i o u s drugs and then measuring d i f f e r e n t responses d u r i n g a c e r t a i n p e r i o d of time. However, many r a t s d i e d u r i n g the experiment making i m p o s s i b l e the measurement of some responses. One n a t u r a l approach to the m i s s i n g data problem i s to omit a l l v e c t o r s of o b s e r v a t i o n that are incomplete. T h i s i s u n s a t i s f a c t o r y , e s p e c i a l l y i f few experimental u n i t s are a v a i l a b l e and i f many v a r i a b l e s are known fo r an incomplete v e c t o r of o b s e r v a t i o n . In S e c t i o n 4, two 2 s t a t i s t i c a l t e s t s which use a l l the i n f o r m a t i o n a v a i l a b l e on each experimental u n i t are proposed and r e s u l t s of t h e i r a p p l i c a t i o n s t o Dr. Walker's data are g i v e n . F i n a l l y , i n the l a s t s e c t i o n , l o g - l i n e a r models are suggested f o r the a n a l y s i s of arrhythmias. 3 2. HISTORY OF THE DATA Dr. Walker's study c o n c e n t r a t e d upon the p h a r m a c o l o g i c a l and p a t h o l o g i c a l a spects of responses to i r r e v e r s i b l e myocardial ischaemia and subsequent i n f a r c t i o n . From an experimental p o i n t of view, i r r e v e r s i b l e ischaemia can most e a s i l y be a chieved by l i g a t i o n of a coronary a r t e r y . In order to study the e f f e c t s of drugs on responses to such ischaemia, a method of producing l i g a t i o n i n c o n s c i o u s r a t s was developed. The experiment i n v o l v e d measuring c a r d i o v a s c u l a r , arrhythmic, ECG and m o r t a l i t y responses and c a r d i a c t i s s u e l o s s r e s u l t i n g from l i g a t i o n i n the prepared r a t s exposed to v a r i o u s drugs. T h i s s e c t i o n d e s c r i b e s the two p a r t s of the experiment, s u r g i c a l p r e p a r a t i o n and l i g a t i o n technique, the data c o l l e c t e d , and enumerates some p o t e n t i a l problems r e l a t e d t o the d a t a . 2.1 Experiment 1 Experiments, which are s t i l l ongoing, were performed on over 300 male r a t s s i n c e 1979. Approximately h a l f of the r a t s were of the W i s t a r ( C h a r l e s R i v e r farms) s t r a i n w h i l e , more r e c e n t l y , a Sprague-Dawley(Charles River Farm & Taconic) s t r a i n has been used. In order to be a b l e to produce l i g a t i o n i n r a t s , they had to undergo s u r g i c a l p r e p a r a t i o n . Rats weighted between 200 and 300 g at the time of o p e r a t i o n . 1 T h i s s e c t i o n was adapted from Jang et a l . (1983), Johnston et a l . (1983a,1983b), Macleod et a l . (1983). 4 S u r g i c a l P r e p a r a t i o n Operations were performed under c l e a n c o n d i t i o n s by one of four t e c h n i c i a n s . Rats were a n a e s t h e t i z e d with halothane and the c hest opened by a l e f t l a t e r a l i n c i s i o n at the f o u r t h or f i f t h i n t e r c o s t a l space. An o c c l u d e r which c o n s i s t e d of a (5-0) p o l y p r o p y l e n e suture was passed around the l e f t a n t e r i o r descending(LAD) coronary a r t e r y and e x t e r i o r i z e d i n a polythene guide(PE50) i n the mid-scapula r e g i o n of the neck. The s i t e of l i g a t i o n of the LAD coronary a r t e r y was l o c a t e d approximately 3mm from the a o r t i c r o o t . The suture was passed through the myocardium so as to make a loop around the a r t e r y such that t r a c t i o n on suture at the e x t e r i o r i z e d end of the guide produced complete l i g a t i o n of the a r t e r y . Permanent ECG l e a d s were implanted with a sub-dermal needle t r o c a r i n t o the p e c t o r a l i s muscle o v e r l y i n g the chest i n c i s i o n , i n the l e f t l e g and i n both arms. The f r e e ends of leads were e x t e r i o r i z e d together with the o c c l u d e r . In the e a r l y experiments, a permanent c a t h e t e r was p l a c e d i n the v e n t r a l t a i l a r t e r y and e x t e r i o r i z e d at the neck under halothane a n a e s t h e s i a one day p r i o r the experiment. In l a t e r animals, at the time of the thoracotomy, permanent venous and j u g u l a r cannulae were implanted i n abdominal a o r t a and e x t e r n a l j u g u l a r v e i n by a technique due to Weeks 2. The cannulae were a l s o e x t e r i o r i z e d with the o c c l u d e r and the ECG w i r e s . 2 See Macleod et a l . (1983) f o r f u r t h e r r e f e r e n c e s . 5 O p e r a t i v e m o r t a l i t y was below one percent with t r a i n e d t e c h n i c i a n s . Deaths g e n e r a l l y o c c u r r e d s h o r t l y a f t e r surgery and were due to blood l o s s . Animals were allowed to recover f o r at l e a s t 6 days p r i o r to l i g a t i o n . L i g a t i o n Technique On the day of the l i g a t i o n , r a t s were randomly s e l e c t e d and kept i n t h e i r home cage. Cannulae and ECG le a d s were a p p r o p r i a t e l y connected to a Grass Polygraph and i n f u s i o n pump, and a continuous r e c o r d made f o r 30 minutes p r i o r to l i g a t i o n . A p p r o x i m a t i v e l y seven percent of animals were r e j e c t e d f o r s t u d i e s s i n c e they showed changes i n t h e i r ECG's s u g g e s t i v e of c a r d i a c damage, or had blocked cannulae. L i g a t i o n was achieved w i t h i n seconds by e x e r t i n g s u f f i c i e n t t r a c t i o n on the pro l y p o p y l e n e suture ' and was made permanent by heat s e a l i n g i t to the polythene guide. V a r i o u s drug treatments were a d m i n i s t e r e d to l i g a t e d r a t s . They i n c l u d e d i n f u s i o n s begun p r e - l i g a t i o n , i ntravenous dosing, o r a l d o s i n g e t c . Animals were c o n t i n u o u s l y monitored f o r 4 hours b e f o r e being d i s c o n n e c t e d and retu r n e d to the animal house. During r e c o r d i n g of blood pressure(BP) and ECG, a l l arrhythmias were noted. I f w i t h i n the four hours post l i g a t i o n o b s e r v a t i o n p e r i o d a severe v e n t r i c u l a r arrhythmia o c c u r r e d and d i d not spontaneously r e v e r t w i t h i n 10 seconds, an attempt was made to convert the arrhythmia to si n u s rhythm by re p e a t e d l y tapping the r a t ' s c h e s t . 6 Twenty-four hours a f t e r l i g a t i o n , cannulae and leads were reconnected and the r a t s monitored f o r a f u r t h e r 30 minutes. The animals were then k i l l e d by stunning and e x s a n g u i n a t i o n , and the heart removed. Hearts were a l s o removed from animals dying before 24 hours had e l a p s e d . -For a l l r a t s , h e a r t s with o c c l u d e r i n t a c t were p e r f u s e d with Kreb's s o l u t i o n a t 37 C and 100mm Hg pr e s s u r e f o r 5 min to remove a l l blood. A bolus of 2.0 ml of c a r d i o - g r e e n dye(1.Omg/ml) was used to d i f f e r e n t i a t e perfused(green) from underperfused, or oc c l u d e d t i s s u e ( p i n k ) . The underperfused r e g i o n was immediately cut out and weighed to g i v e a measurement of occluded zone as a percentage of t o t a l v e n t r i c u l a r weight. For r a t s t h at s u r v i v e d 24 hours, the heart t i s s u e was then s l i c e d l o n g t i t u d i n a l l y i n t o 1.0 mm s e c t i o n s and incubated i n t e t r a z o l i u m dye at 37 C f o r 30-45 minutes. A f t e r i n c u b a t i o n , a l l s e c t i o n s were p l a c e d i n 10% formaldehyde ( i n normal s a l i n e ) f o r two days before the undyed(white) i n f a r c t e d t i s s u e was d i s s e c t e d from v i a b l e t i s s u e ( p u r p l e ) . I n f a r c t e d t i s s u e was weighted and expressed as a percentage of t o t a l v e n t r i c u l a r t i s s u e weight g i v i n g a measurement of the i n f a r c t zone. In other words, the occluded zone r e p r e s e n t s the p r o p o r t i o n of h e a r t t i s s u e f o r which the blood supply was i n t e r r u p t e d and the i n f a r c t zone the p r o p o r t i o n of dead t i s s u e . 7 2.2 Data The v a r i a b l e s measured d u r i n g the experiment were: s y s t o l i c and d i a s t o l i c p r e s s u r e , heart r a t e , ECG, arrhythmias, s i z e of occluded zone, s i z e of i n f a r c t e d zone and m o r t a l i t y . The small bore of the a o r t i c cannulae o f t e n f i l t e r e d the p r e s s u r e wave l e a d i n g to a poor r e c o r d i n g of the s y s t o l i c p r e s s u r e s . A c c o r d i n g t o Dr. Walker the mean of s y s t o l i c and d i a s t o l i c r e p r e s e n t s a more ac c u r a t e measure of blood p r e s s u r e ( B P ) . Heart r a t e was taken from BP or ECG t r a c e s u s i n g a ratemeter. The ECG was used to d e t e c t arrhythmia and f o r measurement of the s i z e of the major complex(R and RS), time t o Q-wave and the height of the S-T segment above the i s o - e l e c t r i c l i n e . The s i z e of the occluded and i n f a r c t e d zone were recorded as d e s c r i b e d i n the p r e v i o u s s e c t i o n . The v a r i a b l e s can be regrouped i n t h r e e c a t e g o r i e s a c c o r d i n g to t h e i r time measurement. The f i r s t category c o n s i s t s of c o n t i n u o u s l y monitored responses recorded at twelve d i f f e r e n t , time p o i n t s (-15min, -1min, +1min, +5min, +I0min, +I5min, +30min, +1hr, +2hr, +3hr, +4hr, +24hr) 3. F o l l o w i n g i s a l i s t of these v a r i a b l e s , as found i n the computer f i l e s . SYS: s y s t o l i c p r essure DIAS: d i a s t o l i c p r e s s u r e HR: h e a r t r a t e ST: he i g h t of the S-T segment R: height of the R-wave RS: h e i g h t of the RS segment 3 Where - and + mean before and a f t e r l i g a t i o n r e s p e c t i v e l y . 8 DST: ST at time t minus ST a t -15min. ( S T ( t ) - S T ( - l 5 ) ) 4 DSTR: ST c o r r e c t e d f o r R. ( S T ( t ) ( R ( - 1 5 ) / R ( t ) ) - S T ( - 1 5 ) ) 4 DSTRS: ST c o r r e c t e d f o r RS. ( S T ( t ) ( R S ( - 1 5 ) / R S ( t ) ) - S T ( - 1 5 ) ) 4 The second category of v a r i a b l e s i n c l u d e s responses of arrhythmias measured 30 minutes and 4 hours a f t e r l i g a t i o n and m o r t a l i t y measured 4 hours and 24 hours a f t e r l i g a t i o n . Arrhythmias c o n s i s t of premature v e n t r i c u l a r c o n t r a c t i o n s ( P V C ) , v e n t r i c u l a r t a c h y c a r d i a f l u t t e r ( V T ) , and v e n t r i c u l a r f i b r i l l a t i o n ( V F ) . Four or more PVC o c c u r r i n g c o n s e c u t i v e l y are co n s i d e r e d t o be VT. V e n t r i c u l a r t a c h y c a r d i a and v e n t r i c u l a r f i b r i l l a t i o n are d i s t i n g u i s h e d from each other by l o o k i n g at the p a t t e r n appearing i n the ECG t r a c e and the f a l l i n blood p r e s s u r e . In order t o a d j u s t arrhythmias f o r d u r a t i o n and m o r t a l i t y by i r r e v e r s i b l e VF a new v a r i a b l e was c r e a t e d . T h i s v a r i a b l e combines the number and d u r a t i o n of VT, VF,PVC and the i n c i d e n c e of i r r e v e r s i b l e VF using a 0-8 s c o r i n g s c a l e f o r the 0-30 min or 0-4 hr p o s t - l i g a t i o n p e r i o d s . The value 0 was given f o r 0-50 PVC with no VT or VF over the o b s e r v a t i o n p e r i o d ; 1, f o r 50-500 PVC on l y ; 2, f o r 500 PVC or more,or one episode of spontaneously r e v e r s i b l e VT or VF; 3, f o r more than one episode of spontaneously r e v e r s i b l e VT and/or VF or, one or more episodes of non-spontaneously r e v e r s i b l e VT and/or VF l a s t i n g l e s s than 60 sec; 4, f o r r e v e r s i b l e VT and/or VF episodes l a s t i n g 60-120 sec; 5, f o r VT and/or VF episodes l a s t i n g more than 120 sec; 6, 4 Note that these v a r i a b l e s do not have v a l u e s at time t=-15 min. 9 f o r i r r e v e r s i b l e VF c a u s i n g death w i t h i n 15-240 min of l i g a t i o n ; 7,for f a t a l VF w i t h i n 4-15 min and f i n a l l y 8, f o r f a t a l VF w i t h i n 4 min. T h i s score v a r i a b l e c a l l e d a r r h y t m i a score measures the s e v e r i t y of the arrhythmias. The v a r i a b l e s i n c l u d e d i n t h i s category are as f o l l o w s : AS30: arrhythmic score at 30 min. AS4h: arrhythmic s c o r e a t 4 h r . IVF#30: i r r e v e r s i b l e vent. f i b . 30 min. IVF#4: i r r e v e r s i b l e vent. f i b . 4 h r . MORT4: m o r t a l i t y a f t e r 4 h r . MORT24: m o r t a l i t y a f t e r 24 h r . and s e v e r a l measurements 30 min. and 4 h r s . a f t e r l i g a t i o n on l e n g t h and numbers of spontaneous and non-spontaneous VT and VF epi s o d e s . The l a s t c ategory c o n s i s t s of v a r i a b l e s measured on l y once d u r i n g the experiment. These v a r i a b l e s a r e : OZ: s i z e of occluded zone IZ: s i z e of i n f a r c t e d zone QWO: Q-wave occurrence (Y/N) QWT: Q-wave time ,time of f i r s t appearance (=240 i f >240) As a l r e a d y mentioned, the experiments have been conducted f o r some time and are s t i l l ongoing. The data f o r seven such experiments are a v a i l a b l e . Each data set i n c l u d e s s e v e r a l groups of 7 to 13 r a t s which were a d m i n i s t e r e d d i f f e r e n t drug treatments. Only f i v e out of the seven experiments have a c o n t r o l group i . e . a group of r a t s t h a t were l i g a t e d but d i d not r e c e i v e drugs. Our main concern i s the d e r i v a t i o n of s t a t i s t i c a l t echniques to d e a l with problems r e l a t e d to the data, hence throughout t h i s t h e s i s we c o n c e n t r a t e our an a l y s e s on the data 10 combining a l l the c o n t r o l groups (67 r a t s ) and the data of the halothane dose response experiment (55 r a t s ) . T h i s experiment was c a r r i e d out i n 1981 to i n v e s t i g a t e the e f f e c t of halothane at d i f f e r e n t c o n c e n t r a t i o n s . 2.3 Problems r e l a t e d to data Approximately 50% of the r a t s d i e d d u r i n g the experiments. T h i s c e n s o r i n g i s a major problem i n the study of the da t a . The o b s e r v a t i o n of the v a r i a b l e s i n the f i r s t c ategory i s r e s t r i c t e d by the value of the m o r t a l i t y v a r i a b l e s i n c e no measurements were made a f t e r the r a t ' s death. T h i s causes problems e s p e c i a l l y i f the valu e s of the e a r l y measurements ( i . e . before m o r t a l i t y ) c o n t a i n i n f o r m a t i o n about m o r t a l i t y s t a t u s . Apart from t h i s c e n s o r i n g , we a l s o have other m i s s i n g d ata. In some cases, the measurements of some v a r i a b l e s , e s p e c i a l l y BP, were not p o s s i b l e . F u r t h e r , the valu e s of v a r i a b l e s such as the number of VT or VF, Q-wave occurence and t o t a l PVC are a l s o l i m i t e d by the value of the m o r t a l i t y v a r i a b l e s i n c e these v a r i a b l e s are measured f o r a p e r i o d of time and not at one p o i n t i n time. The arrhythmia s c o r i n g system attempts to compensate f o r t h i s . 11 3. PRELIMINARY ANALYSES OF THE DATA P r e l i m i n a r y a n a l y ses of the c o n t r o l groups (67 r a t s ) were undertaken i n order t o ga i n understanding of the r e l a t i o n s h i p s among the v a r i a b l e s . A c c o r d i n g t o Dr. Walker the experiments were performed q u i t e u n i f o r m l y except f o r one experiment where the e l e c t r o d e f o r the ECG was t i e d d i r e c t l y on the heart l e a d i n g to a bigger s i g n a l . For t h i s group of r a t s the v a l u e s of the ECG v a r i a b l e s were d e l e t e d from the a n a l y s e s . We examined the data u s i n g two s t a t i s t i c a l packages (Midas .& BMDP) and keeping i n mind important q u e s t i o n s which might give some i n s i g h t i n t o the problems mentioned e a r l i e r . F i r s t , i s i t p o s s i b l e t o p r e d i c t the value of a v a r i a b l e a t l a t e time p e r i o d s from e a r l y measurements? For i n s t a n c e , i s i t p o s s i b l e to p r e d i c t death from heart r a t e s e r i e s ? Second, i s OZ a confounding v a r i a b l e which accounts f o r d i f f e r e n c e i n treatment groups? T h i r d , i s there a r e l a t i o n s h i p e x i s t i n g between the in c i d e n c e of VT, VF, PVC and the time of m o r t a l i t y ? In t h i s s e c t i o n , r e s u l t s of these p r e l i m i n a r y a n a l y s e s are gi v e n . 3.1 Can we p r e d i c t death ? For t h i s purpose, the r a t s were d i v i d e d i n t o two groups, those that s u r v i v e d (34) the experiment and those that d i e d (33). The mean of BP,HR,ST, R, RS, DST, DSTR, DSTRS, f o r each p o i n t i n time were p l o t t e d f o r the two groups. The graphs of 12 BP, HR,DSTR and DSTRS are given i n f i g u r e s 1 to 4. I t i s important to note t h a t f o r the group of r a t s that d i e d , the sample s i z e decreases with time. For example at the l a s t time p o i n t 4 ( i . e . 4 hrs a f t e r l i g a t i o n ) there are only f i v e r a t s l e f t i n the sample. For the ECG measurements (ST, R, RS ...etc) the p l o t s f o r both groups appear to have the same p a t t e r n except f o r DSTRS where at time p o i n t 3 ( i . e . 1 min a f t e r l i g a t i o n ) the d i f f e r e n c e between the two groups i s very l a r g e . However, t h i s i s due t o the extreme value (-7.5 compared to an average of .2) of DSTRS on one r a t . In f a c t when t h i s r a t i s d e l e t e d the curves are q u i t e s i m i l a r (see f i g u r e s 4 and 5 ) . I t seems that f o r r a t s t h a t d i e d , BP e x h i b i t s a g r e a t e r decrease a f t e r l i g a t i o n than f o r r a t s that s u r v i v e d , and then remains lower. We compared (usual t - t e s t ) the decrease i n BP from the second to the t h i r d time p o i n t between the two groups and found a p-value of .07. Heart r a t e f o r the two groups have the same p a t t e r n f o r the f i r s t time p o i n t s . At l a t e r time p o i n t s , HR goes down s t e a d i l y f o r r a t s t h a t d i e d while i t remains constant f o r the other group. An important p o i n t t o v e r i f y i s the homogeneity of the two groups before the experiment. T h i s can be done by t e s t i n g the values of the v a r i a b l e s f o r the f i r s t two time p o i n t s . Using H o t e l l i n g ' s T-square or the t - s t a t i s t i c when a p p r o p r i a t e , i t 13 FIGURE t. Plot of mean BP vs lime for rats that died and rats that survived Legend A DIED X SURVIVED TIME POINT FIGURE 2. Plot of mean HR vs time for rats that died and rats that survived 420-410-400-390-380-370-I 360-z < LLJ 350-340-330-320-310-300-290-Legend A DIED X SURVIVED 4 5 6 TIME POINT 14 FIGURE 3. Plot of mean DSTR vs time for rats that died and rats that survived 0.31 Legend A PIED X SURVIVED 4 5 6 7 TIME POINT FIGURE 4. Plot of mean DSTRS vs time for rats that died and rats that survived Legend A DIED X SURVIVED T 1 1 1 1 r 0 1 2 5 4 5 6 7 TWE POINT 1 5 FIGURE 5. Plot of mean DSTRS vs time for rats that died and rats that survived with one rat deleted 0.437-= 0.397-0.357-0.317-I/O 0.277^ cr (— 0.237-o :AN 0.197-0.157-0.117-0.077^ 0.037--0 .003 J - i * 1 1 1 \ 1 r 0 1 2 3 4 5 6 7 8 TIME POINT L e g e n d A DIED X SURVIVED 10 11 16 appeared that the two groups were q u i t e homogeneous f o r a l l v a r i a b l e s except f o r blood pressure where the p-value was .004. We c o u l d not f i n d any reason to e x p l a i n t h i s unexpected low p-v a l u e . The comparison of means at time p o i n t 3 g i v e s the f o l l o w i n g r e s u l t s : v a r i a b l e s ! BP HR ST R RS DST DSTR DSTRS p-values ! .03 .73 .78 .50 .29 .39 .32 .21 The p-values i n d i c a t e that the means are not s i g n i f i c a n t l y d i f f e r e n t between the two groups of r a t s except f o r BP. At t h i s stage, i t seems t h a t death cannot be p r e d i c t e d with the ECG v a r i a b l e s and with the e a r l y measurements of heart r a t e . There i s s m a l l evidence t h a t the drop i n blood p r e s s u r e j u s t a f t e r l i g a t i o n might be h e l p f u l t o p r e d i c t death. However the r e l i a b i l i t y of t h i s r e s u l t i s q u e s t i o n a b l e , s i n c e a c c o r d i n g to H o t e l l i n g ' s T-square t e s t the two groups were not s i m i l a r before the experiment. The r e s u l t s on blood p r e s s u r e are not very c o n c l u s i v e . To o b t a i n more i n f o r m a t i o n on the r e l a t i o n s h i p between blood p r e s s u r e and m o r t a l i t y , a l o g i s t i c model was f i t t e d t o the p r o p o r t i o n of deaths. The l o g i s t i c f u n c t i o n i s P = exp U/ ( 1 + exp U ) 17 where P i s the p r e d i c t e d p r o p o r t i o n of death and U i s a l i n e a r f u n c t i o n of the v a r i a b l e s BP a t d i f f e r e n t time p o i n t s . Using BMDP, we performed a stepwise l o g i s t i c r e g r e s s i o n on the d ata. The f o l l o w i n g p-values were ob t a i n e d f o r the s i g n i f i c a n c e of the c o e f f i c i e n t s of the BP v a r i a b l e a t time p o i n t s 1 to 5. v a r i a b l e s BP1 BP2 BP3 BP4 BP5 p-values .42 .59 .25 .66 .70 From these r e s u l t s , blood p r e s s u r e does not seem to be a good p r e d i c t o r f o r the p r o b a b i l i t y of death. 3.2 Is OZ a confounding v a r i a b l e ? As a f i r s t step, the r e l a t i o n s h i p of OZ with the other v a r i a b l e s was examined. A c o r r e l a t i o n a n a l y s i s showed that the arrhythmia score and the s i z e of i n f a r c t e d zone are the only v a r i a b l e s which are s i g n i f i c a n t l y c o r r e l a t e d with the s i z e of o c c luded zone. The c o r r e l a t i o n of OZ with AS & IZ are .42 and .44 r e s p e c t i v e l y . The f a c t that OZ i s not c o r r e l a t e d with the v a r i a b l e s i n the f i r s t c ategory ( i . e . BP,HR, ST,...etc) i s a l s o confirmed by the s c a t t e r p l o t of these v a r i a b l e s and OZ except i n the case of BP. There seems to be a l i n e a r r e l a t i o n s h i p between BP and OZ that i s not d e t e c t e d by r e g r e s s i o n because of t h r e e o u t l i e r s ( r a t s ) which have u n u s u a l l y low v a l u e s f o r OZ. 18 In f a c t , when these three r a t s are d e l e t e d the s i g n i f i c a n c e l e v e l of the r e g r e s s i o n goes from .1052 to .0002. Acco r d i n g t o Dr. Walker, these u n u s u a l l y low value s f o r OZ are an i n d i c a t i o n t h a t something went wrong i n the experiment. Hence, we are not i n t r o d u c i n g any b i a s when d e l e t i n g these three v a l u e s from the da t a . I t appears t h a t the s i z e of occluded zone v a r i e s with the s t r a i n of the r a t s . A p-value of .0002 i s ob t a i n e d when comparing the means us i n g the usual t - t e s t . However, the d i f f e r e n c e i n OZ between the two s t r a i n s of r a t s i s not too important s i n c e i n any one experiment the same s t r a i n was used throughout. However when combining data from many experiments a c o v a r i a t e r e p r e s e n t i n g the s t r a i n s of the r a t s should be i n c l u d e d i n the model c o n s i d e r e d . A small d i f f e r e n c e i n the s i z e of occluded zone i s a l s o d e t e c t e d between r a t s t h at d i e d and r a t s that s u r v i v e d . A comparison of the mean of OZ between these two groups g i v e s a p-value of .06. In the preceeding s e c t i o n , we found that at the t h i r d time p o i n t the mean blood pressure i n the two groups of r a t s were a l s o d i f f e r e n t . I t would then be i n t e r e s t i n g to see i f t h i s d i f f e r e n c e c o u l d be e x p l a i n e d by the d i f f e r e n c e i n OZ. In the f o l l o w i n g t a b l e , we compare the r e s u l t s of the a n a l y s i s of v a r i a n c e of BP with the r e s u l t s of the c o v a r i a n c e a n a l y s i s of BP u s i n g OZ as a concomitant v a r i a b l e . 19 ! a l l r a t s J 3 r a t s d e l e t e d i d i e d s u r v i v e d ! d i e d s u r v i v e d means j 88.98 101.82 j 90.77 102.82 ANOVA | p-value| .03 j .04 a d j u s t e d | means j 89.81 100.95 j 94.09 99.57 ANOCOV j p-valueJ .07 j .35 i i From t h i s t a b l e , we can see th a t the d i f f e r e n c e i n blood p r e s s u r e between the two groups of r a t s can be a t t r i b u t e d to the confounding e f f e c t of OZ. I t should be p o i n t e d out that i n both cases the assumption about the e q u a l i t y of s l o p e ( i . e . the c o e f f i c i e n t of OZ) was not v i o l a t e d . I t i s now important to determine i f d i f f e r e n c e s i n treatments c o u l d a l s o be e x p l a i n e d by the d i f f e r e n c e i n the s i z e of occluded zone. For t h i s purpose, we used the data of the halothane dose response experiment. In t h i s experiment, halothane was ad m i n i s t e r e d at c o n c e n t r a t i o n s of O ( c o n t r o l ) , 0.25, 0.50, 1.0 and 2.0 pe r c e n t . In t h i s experiment, the r e g r e s s i o n of BP on OZ i s l e s s s i g n i f i c a n t than w i t h the data on c o n t r o l groups. The e f f e c t of halothane seems to t a n g l e the e f f e c t of OZ on BP. The sl o p e of the r e g r e s s i o n i n c r e a s e s s t e a d i l y from -2.13 a t dose 0 t o .48 a t dose 2%. In f a c t , at dose 1% and 2% the r e g r e s s i o n of BP on OZ 20 i s not s i g n i f i c a n t at the 5 percent l e v e l . A c c o r d i n g to an a n a l y s i s of v a r i a n c e , the mean of BP appears t o be s i g n i f i c a n t l y d i f f e r e n t among treatments and there i s a l s o a s l i g h t d i f f e r e n c e i n the mean of OZ between treatments. However as opposed t o the r e s u l t s found f o r the c o n t r o l groups, a c o v a r i a n c e a n a l y s i s on BP using OZ as a concomitant v a r i a b l e d i d not reduce the d i f f e r e n c e i n BP between treatments. T h i s might be due to the f a c t that d i f f e r e n c e i n OZ between treatments i s not very s i g n i f i c a n t and a l s o that the r e l a t i o n between BP and OZ i s not the same i n each treatment. The r e s u l t s are shown i n the f o l l o w i n g treatments t a b l e . c o n t r o l 0. 25% 0.50% 1.0% 2. 0% means 101.82 103 .75 89.09 82.05 50 .46 ANOVA p-value < .00005 a d j u s t e d means 104.85 102 .87 84.18 83.42 51 .76 ANOCOV p-value < .00005 mean of OZ ANOVA p-value 35.95 29. 42 25.63 .0925 33.79 33 .71 A c o v a r i a n c e a n a l y s i s was a l s o performed on HR, ST...etc at 21 time p o i n t 3 with OZ as a concomitant v a r i a b l e . The c o n c l u s i o n i s that i n t h i s experiment the s i z e of occluded zone i s not a s i g n i f i c a n t concomitant v a r i a b l e and does not account f o r d i f f e r e n c e i n treatments. 3.3 Are arrhythmias r e l a t e d to time of m o r t a l i t y ? We a l r e a d y mentioned that the number of VT or VF, Q-wave occurrence and t o t a l PVC were l i m i t e d by the value of the m o r t a l i t y s i n c e they were measured over a p e r i o d of time. To v e r i f y t h i s a s s e r t i o n , t a b l e s 1-3 i l l u s t r a t i n g the r e l a t i o n between those v a r i a b l e s and the time of m o r t a l i t y are shown below. In the f i r s t t a b l e , the mean of l o g PVC i n c r e a s e s with the time of death. However the mean f o r the r a t s t h a t s u r v i v e d and the mean f o r those that d i e d between 3 and 24 hours are about the same. T h i s i s probably an a r t i f a c t s i n c e the number of t o t a l PVC are recorded up t o 4 hours o n l y . The l o g t r a n s f o r m a t i o n i s used t o make the d i s t r i b u t i o n of PVC more normal. In the second t a b l e , the t o t a l number of arrhythmias i n c r e a s e s with the time of death and i n the t h i r d t a b l e the longer a r a t s u r v i v e s the l a r g e r i s the chance that a Q-wave oc c u r s . 22 Table 1. Log of t o t a l PVC at 4 hours. time of J death | mean ! SD (1) b e f o r e | 2. 75 [ 1 .31 10 min j ! (2) between! 5. 21 ! i .73 10min&3hrJ j (3) between) 6. 96 ! o .90 3hr&24hrj ! (4) s u r v i v e ] 6 .67 ! 1 .06 Table 2. T o t a l number of arrhythmias at 4 hours. time of j death [ 0 I 1 t o t a l # ! 2 to of 4 arrhytmias 5 to 9 j 10 and + t o t a l ( D i 7 3 ! 4 1 [ 0 15 (2) i o 0 j 1 2 i 4 7 (3) | o 1 ! o 4 i 4 9 (4) | 2 4 i 8 3 i 1 1 28 t o t a l ! 9 8 i 13 10 j 19 59 Table 3. Q-wave occurrence Q-wave occurrence before 1 0 m i n j b e t time of 10min&4hr death ! bet 4hr&24hr[ s u r v i v e j yes ! o ! i 9 i i i ! 5 ! 1 | 31 i 1 no i 16 j 4 i j 1 o ! 1 2 ! t o t a l 16 ; 13 i • 5 ! 33 j 23 In summary, i t does not seem p o s s i b l e t o p r e d i c t the death of a r a t from e a r l y measurements of v a r i a b l e s such as BP, HR, ST, e t c . T h i s statement i s stonger i n the cases of HR and ECG measurements than i n the case of BP. A l s o , the d i f f e r e n c e s i n treatments i n the halothane dose response experiment c o u l d not be e x p l a i n e d by the s i z e of occluded zone. However, t h i s i s not the case when comparing BP between groups of r a t s t h a t d i e d and s u r v i v e d , i n d i c a t i n g that OZ i s a very important v a r i a b l e to c o n s i d e r when a n a l y s i n g t h i s type of data. F i n a l l y , as expected, the arrhythmias are r e l a t e d t o the time of m o r t a l i t y . In the next s e c t i o n s we analyse the set of repeated BP and DSTR measurements u s i n g manova types models and a n a l y s e the d e r i v e d v a r i a b l e arrhythmia score u s i n g l o g - l i n e a r models. 24 4. EXAMINATION OF TIME-RESPONSE VARIABLES The time-response v a r i a b l e s recorded d u r i n g the experiment can be a n a l y s e d u s i n g growth curves model without r e s t r i c t i o n on growth i . e . on the slope of the c u r v e s . A v a r i e t y of methods fo r the comparison of growth curves has been developed (e.g. Box (1950), P o t t h o f f and Roy (1964), C R . Rao (1965,1966), G r i z z l e and A l i e n (1969)). U n f o r t u n a t e l y most of these methods are not a p p r o p r i a t e f o r our data s i n c e they r e q u i r e complete data. K o z i o l et a l . (1981) proposed a d i s t r i b u t i o n - f r e e s t a t i s t i c a l methodology f o r the comparison of growth curves t h a t may be used with incomplete data. Furthermore K o z i o l and Yuh (1982) developed a score t e s t , as d e f i n e d by Cox and H i n k l e y (1974), f o r t e s t i n g group d i f f e r e n c e s when o b s e r v a t i o n s are m i s s i n g . Both of these methods assumed t h a t m i s s i n g data are m i s s i n g at random. In other words, the assumption i s t h a t the v a r i a b l e s are m i s s i n g without regard to v a l u e s that would have been observed. The r e s u l t s of the p r e l i m i n a r y analyses seem to i n d i c a t e t h a t t h i s assumption i s not v i o l a t e d i n our d a t a . In t h i s s e c t i o n , the two methods suggested above are d e s c r i b e d and the r e s u l t s of t h e i r a p p l i c a t i o n s u s i n g the a p p r o p r i a t e data are presented. 25 4.1 Manova approach The score s t a t i s t i c d e r i v e d by K o z i o l and Yuh i s based on a m u l t i v a r i a t e a n a l y s i s of v a r i a n c e model which a l l o w s f o r m i s s i n g o b s e r v a t i o n s . T h i s t e s t i s d e s c r i b e d below. 4.1.1 D e s c r i p t i o n of the score t e s t In g e n e r a l , s e v e r a l measurements (p) are c o l l e c t e d f o r each experimental u n i t i n a s e t of k treatments. Those measurements are assumed t o be independent o b s e r v a t i o n s on p-dimensional m u l t i v a r i a t e normal v a r i a t e s with mean v e c t o r s u , ... , u 1 k and a common unknown c o v a r i a n c e matrix $2. Moreover, i t i s assumed that the p a t t e r n of m i s s i n g v a l u e s i s such that the data matrix has a monotonic or nested p a t t e r n : f o r each l i n e r e p r e s e n t i n g an experimental u n i t when one o b s e r v a t i o n i s m i s s i n g a l l the other o b s e r v a t i o n s t o i t s r i g h t are a l s o m i s s i n g . I t should be p o i n t e d out t h a t t h i s s p e c i a l case of m i s s i n g data a l l o w s us, as shown i n the Appendix, t o f i n d e x p l i c i t e x p r e s s i o n s f o r the maximum l i k e l i h o o d e s t i m a t e s of the mean, v a r i a n c e s and c o v a r i a n c e s of the u n d e r l y i n g normal p o p u l a t i o n . In the framework of t h i s model, we would l i k e t o determine whether the mean v e c t o r s d i f f e r among the treatments groups. For t h i s purpose the f o l l o w i n g form of the score s t a t i s t i c was d e r i v e d by K o z i o l and Yuh 26 w = z (f - £)'sr1s ~ 1n~1 (Y" - y) u i = 1 i (4.1) where * n i / ~ i ~ 1 S = (n ) - 2 2 / Q 0 i i j=1 joo \ 0 0 Here, M and fl~1are the maximum l i k e l i h o o d e s t i m a t e s under the n u l l h y p o t h e s i s of the mean v e c t o r and the v a r i a n c e -c o v a r i a n c e matrix r e s p e c t i v e l y , 0 i s the matrix of the joo elements of H corres p o n d i n g t o the v a r i a b l e s observed f o r the th i j u n i t , ? i s the estimate of the mean v e c t o r f o r treatment i , and n i s the number of r a t s i n treatment i . i More d e t a i l s c o n c e r n i n g the n o t a t i o n used are presented i n Appendix. In a d d i t i o n , the maximum l i k e l i h o o d (ML) s t a t i s t i c and the score s t a t i s t i c f o r complete data and the ML s t a t i s t i c f o r incomplete data are d e r i v e d i n the appendix. I t i s found that with complete data the ML and the score s t a t i s t i c s are e x a c t l y the same. With incomplete data the ML s t a t i s t i c cannot be reduced t o a simple form so that the comparison of the two s t a t i s t i c s i s not s t r a i g h t f o w a r d . However these s t a t i s t i c s are a s y m p t o t i c a l l y e q u i v a l e n t and e i t h e r c o u l d be used f o r our a n a l y s i s . 27 The t e s t we used i n t h i s s e c t i o n i s based on the score s t a t i s t i c . Under the n u l l h y p o thesis that the treatment means u are equal, W has approximately a c h i - s q u a r e d i s t r i b u t i o n i u with p ( k - l ) degrees of freedom. Hence l a r g e v a l u e s of W u compared to a x 2(pk-p) w i l l i n d i c a t e t h at the treatment means v e c t o r s are s i g n i f i c a n t l y d i f f e r e n t . We can now apply t h i s t e s t t o our data. 4.1.2 A p p l i c a t i o n of the score t e s t F i r s t we should v e r i f y i f m u l t i v a r i a t e n o r m a l i t y of the v a r i a b l e s i s a reasonable assumption f o r our time-response data. We only checked f o r normal marginals assuming that we c o u l d extend i t to the j o i n t d i s t r i b u t i o n . For t h i s purpose ,normal p r o b a b i l i t y p l o t s of BP, HR, R, RS, DSTR 1 f o r the f i r s t four time p o i n t s were produced. The p l o t s suggested that the assumption of n o r m a l i t y seems a c c e p t a b l e only f o r BP and HR. Some t r a n s f o r m a t i o n s t o n o r m a l i t y were attempted on the other v a r i a b l e s but without success. The normal p l o t s of BP and DSTR f o r the c o n t r o l groups and the halothane data at time p o i n t 3 are presented i n f i g u r e s 6 to 9 r e s p e c t i v e l y . The a n a l y s i s r e p o r t e d here i s f o r the data on blood p r e s s u r e . There are f i v e treatment groups; one c o n t r o l group and four groups with halothane a d m i n i s t e r e d at a c o n c e n t r a t i o n 1 The v a r i a b l e s DST,DSTR,DSTRS repr e s e n t d i f f e r e n t c o r r e c t i o n s to ST and DSTR i s the one Dr. Walker u s u a l l y c o n s i d e r s . 28a Figure 6. Normal p l o t of BP at time point 3 f o r the control groups data. NORMAL P L O T OF V A R I A B L E 2 . 5 + COUNT 6 4 . + . . . . + MEAN S T . D E V . 9 5 . 2 3 4 2 3 . 1 9 0 + . . . . • . . . . + . . . . + . . . . + . - 2 . 5 + - J 3 7 . 5 0 5 2 . 5 0 6 7 . 5 0 8 2 . 5 0 9 7 . 5 0 1 1 2 . 5 1 2 7 . 5 1 4 2 . 5 3 0 . 0 0 4 5 . 0 0 6 0 . 0 0 7 5 . 0 0 9 0 . 0 0 1 0 5 . 0 1 2 0 . 0 1 3 5 . 0 2 8 b Figure 7. Normal p l o t of DSTR at time point 3 f o r the control groups data. NORMAL P L O T OF V A R I A B L E 2 . 5 + COUNT 5 8 . + . . . . • MEAN S T . D E V . 0 . 1 8 3 0 . 1 4 3 • . . . . + . . . . + . . . . • . E X P E C T E D N 0 R M A L V A L U E 2 . 0 + 1 . 5 1 . 0 . 5 0 0 . 0 - . 5 0 • 1 . 0 1 . 5 - 2 . 0 - 2 . 5 0 . 0 29a Figure 8. Normal plot of BP at time point 3 for the halothane data. N O R M A L P L O T O F V A R I A B L E 2.5 + 2 . 0 1.5 C O U N T M E A N S T . D E V . 54 - 0 . 0 0 0 20 .358 .•....•....•....•....•....•....•....+....+....•....+. E X P E C T E D 1 .0 . 50 C C C C C C C c C C C C C C N 0 R M A L V A L U E 0 . 0 - . 5 0 1 .0 1 .5 -2 .0 C C C C C C C C C C C C C C -2.5 . - . • . . . . • . . . . • . . . . • • • . . . . • . . . . + . . . . • . . . . • . . . . • . . . . • . . . . • . . . . + . . . . • . . . . • . . . . • . , . , • . . , - 5 2 . 5 - 3 7 . 5 - 2 2 . 5 - 7 . 5 0 7 .50 2 2 . 5 3 7 . 5 5 2 ! s -60 .0 - 4 5 . 0 - 3 0 . 0 - 1 5 . 0 0 . 0 0 1 5 . 0 3 0 . 0 4 5 . 0 29b Figure 9. Normal p l o t of DSTR at time point 3 for the halothane data. N O R M A L P L O T O F V A R I A B L E 2.5 + COUNT MEAN ST .DEV . 55 O O O O 0 . 1 3 8 .•....•....•....+....•....•....•. C c cc c cc c cc cc c cc c c cc cc cc c c c c c c -2 .0 - 2 . 5 - . 3 0 " . - . 1 8 - . 0 6 - . 2 4 - .12 0 . 0 .06 . 18 .30 .42 .54 . 12 .24 .36 .48 .60 30 of 0.25, 0.50, 1.0 and 2 p e r c e n t . Some c h a r a c t e r i s t i c s of these data are worth mentioning. In the experiment, a l l the r a t s t h a t were a d m i n i s t e r e d halothane a t 2% c o n c e n t r a t i o n d i e d b e f o r e the end of the experiment and none of them s u r v i v e d more than one hour a f t e r l i g a t i o n . For t h i s reason the s c o r e t e s t was performed on two data s e t s s e p a r a t e l y . The f i r s t one i n c l u d e d BP a t time p o i n t s 3 to 7 ( i . e . 1 to 30 min a f t e r l i g a t i o n ) f o r a l l f i v e treatments and the second i n c l u d e d BP a t time p o i n t s 3 to 11 only f o r the f i r s t f our treatments. The measurements 24 hours a f t e r l i g a t i o n were not r e l i a b l e and t h e r e f o r e were not i n c l u d e d . In those two data s e t s , one r a t had m i s s i n g v a l u e s other than by death. In order t h a t the data matrix had a nested p a t t e r n , t h i s r a t was d e l e t e d . The two data s e t s are given i n t a b l e s 4 and 5; the r e s u l t s o b t a i n e d from K o z i o l ' s program are summarized below. 31 Table 4. Blood pressure data f o r a l l f i v e treatments at time p o i n t s 3 to 7. t r e a t time p o i n t s # 3 4 5 6 7 1 112. 50 100. 50 102. 50 102 .50 107. 50 1 92. 50 102. 50 105. 00 100 .00 1 10. 00 1 132. 50 125. 00 1 15. 00 112 .50 1 10. 00 1 110. 00 1 10. 00 1 122. 50 127. 50 1 . 102. 50 107. 50 107. 50 102 .50 90. 00 1 42. 50 42. 50 1 107. 50 80. 00 1 110. 00 130. 00 1 15. 00 105 .00 112. 50 1 97. 50 97. 50 80. 00 82 .50 82. 50 1 90. 00 70. 00 85. 00 85 .00 92. 50 2 1 15. 00 1 15. 00 107. 50 107 .50 112. 50 2 120. 00 2 125. 00 125. 00 120. 00 120 .00 117. 50 2 95. 00 90. 00 95. 00 90 .00 100. 00 2 97. 50 70. 00 2 87. 50 65. 50 85. 00 90 .00 105. 00 2 90. 00 87. 50 97. 50 95 .00 100. 00 2 97. 50 92. 50 57. 50 55 .00 90. 00 2 107. 50 107. 50 145. 00 110 .00 105. 00 2 102. 50 130. 00 85. 00 80 .00 127. 50 3 107. 50 107. 50 102. 50 102 .50 102. 50 3 67. 50 20. 00 3 97. 50 108. 50 94. 50 102 .50 102. 50 3 105. 00 105. 00 3 85. 00 60. 00 3 100. 00 105. 00 105. 00 105 .00 1 10. 00 3 95. 00 95. 00 90. 00 100 .00 100. 00 3 85. 00 92. 50 92. 50 92 .50 90. 00 3 82. 50 77. 50 75. 00 65 .50 65. 00 3 92. 50 75. 00 40. 00 35 .00 3 62. 50 75. 00 1 15. 00 110 .00 100. 00 4 70. 00 67. 50 67. 50 77 .50 77. 50 4 45. 00 37. 50 45. 00 45 .00 47. 50 4 52. 50 22. 50 90. 00 65 .00 60. 00 4 100. 00 100. 00 100. 00 100 .00 97. 50 4 47. 50 30. 00 4 102. 50 90. 00 4 115. 00 1 10. 00 100. 00 110 .00 105. 00 4 97. 50 97. 50 97. 50 105 .00 95. 00 4 95. 00 125. 00 130. 00 125 .00 1 15. 00 4 72. 50 87. 50 65. 00 57 .50 92. 50 4 105. 00 105. 00 105. 00 105 .00 102.50 5 52. 50 52. 50 57. 50 52 .50 44. 00 5 15. 00 10. 50 5 17. 50 17. 50 14. 50 10 .50 5 25. 00 27. 50 5 52. 50 47. 50 5 65. 00 65. 50 65. 50 47 .50 27. 50 5 75. 00 62. 50 5 42. 50 27. 50 40. 00 5 65. 00 67. 50 67. 50 20 .00 5 45. 00 15. 00 5 100. 00 97. 50 62. 50 32 Table 5. Blood Pressure data f o r the four treatments e x c l u d i n g halothane at 2.0% at time p o i n t s 3 to 11. t r e a t time p o i n t s # 3 4 5 6 7 8 9 1 0 11 1 1 12. 50 100. 50 102. 50 102. 50 107. 50 107. 50 95. 00 102. 50 100. 50 1 92. 50 102. 50 105. 00 100. 00 1 10. 00 117. 50 97. 50 102. 50 112. 50 1 132. 50 125. 00 1 15. 00 112. 50 110. 00 1 10. 00 127. 50 1 110. 00 1 10. 00 1 122. 50 127. 50 1 102. 50 107. 50 107. 50 102. 50 90. 00 112. 50 107. 50 1 10. 00 1 12. 50 1 42. 50 42. 50 1 107. 50 80. 00 1 110. 00 130. 00 115. 00 105. 00 112. 50 110. 00 1 15. 00 102. 50 92. 50 1 97. 50 97. 50 80. 00 82. 50 82. 50 102. 50 100. 00 95.00 95. 00 1 90. 00 70. 00 85. 00 85. 00 92. 50 97. 50 107. 50 97. 50 90. 00 2 115. 00 1 15. 00 107. 50 107. 50 112. 50 107. 50 112. 50 107. 50 107. 50 2 120. 00 2 125. 00 125. 00 120. 00 120. 00 117. 50 125. 00 122. 50 120. 00 120. 00 2 95. 00 90. 00 95. 00 90. 00 100. 00 107. 50 100. 00 100. 00 92. 50 2 97. 50 70. 00 2 87. 50 65. 50 85. 00 90. 00 105. 00 90. 00 85. 00 87. 50 100. 00 2 90. 00 87. 50 97. 50 95. 00 100. 00 95. 00 102. 50 2 97. 50 92. 50 57. 50 55. 00 90. 00 97. 50 1 10. 00 1 15. 00 105. 00 2 107. 50 107. 50 145. 00 110. 00 105. 00 112. 50 2 102. 50 130. 00 85. 00 80. 00 127. 50 97. 50 117. 50 102. 50 127. 50 3 107. 50 1 07. 50 102. 50 1 02. 50 102. 50 97. 50 98. 50 102. 50 92. 50 3 67. 50 20. 00 3 97. 50 108. 50 94. 50 102. 50 102. 50 107. 50 117. 50 112. 50 3 105. 00 105. 00 3 85. 00 60. 00 3 100. 00 105. 00 105. 00 105. 00 1 10. 00 1 10. 00 1 15. 00 107. 50 1 05. 00 3 95. 00 95. 00 90. 00 100. 00 100. 00 100. 00 95. 00 90. 00 100. 00 3 85. 00 92. 50 92. 50 92. 50 90. 00 1 10. 00 100. 00 102. 50 87. 50 3 82. 50 77. 50 75. 00 65. 50 65. 00 72. 50 72. 50 67. 50 67. 50 3 92. 50 75. 00 40. 00 35. 00 3 62. 50 75. 00 115. 00 110. 00 100. 00 100. 00 4 70. 00 67. 50 67. 50 77. 50 77. 50 77. 50 72. 50 65. 00 55. 00 4 45. 00 37. 50 45. 00 45. 00 47. 50 45. 00 50. 00 45. 00 50. 00 4 52. 50 22. 50 90. 00 65. 00 60. 00 65. 50 52. 50 47. 50 57. 50 4 100. 00 100. 00 100. 00 100. 00 97. 50 92. 50 4 47. 50 30. 00 4 102. 50 90. 00 4 115. 00 110. 00 100. 00 110. 00 105. 00 105. 00 105. 00 105. 00 105. 00 4 97. 50 97. 50 97. 50 105. 00 95. 00 92. 50 92. 50 92. 50 92. 50 4 95. 00 125. 00 130. 00 125. 00 1 15. 00 1 17. 50 110. 00 105. 00 102. 50 4 72. 50 87. 50 65. 00 57. 50 92. 50 82. 50 57. 50 50. 00 50.00 4 105. 00 105. 00 105. 00 105. 00 102. 50 100. 00 95. 00 92. 50 87. 50 33 Maximum l i k e l i h o o d estimate of the mean v e c t o r f o r each treatment data set # 1 j data set # 2 time c o n t r o l 0.25% 0.50% 1 . 0% 2.0% ! c o n t r o l 0.25% 0.50% 1.0% 3 101.82 103.7b 89.09 82 .05 50.45 |101.77 103.75 89.09 82.05 4 99.36 100.23 83.73 79 .32 44.64 | 99.36 100.38 83.72 79.32 5 97.31 98.50 85.24 85 .52 49.84 | 98.04 98.47 86.18 86.52 6 94.42 93.78 84.10 83 .64 38.25 j 95.36 93.65 85.18 84.92 7 97.28 105.40 86.52 84 .52 43.42 | 98.35 105.34 88.84 85.98 8 |103.80 103.41 92.22 85.13 9 |103.00 106.34 91 .57 80.15 10 |100.70 103.57 89.24 76.33 11 | 99.68 105.83 87.63 76.35 R e s u l t s of the score t e s t data set #1 data set #2 X 2(20) = 51.445 X 2(27) = 30.854 p-value = .00014 p-value = .277 These r e s u l t s suggest t h a t no drugs and small dose ( i . e . < 2%) of halothane have no d i f f e r e n t e f f e c t on blood p r e s s u r e f o r r a t s t h a t have a heart a t t a c k . However, when ad m i n i s t e r e d at 2% c o n c e n t r a t i o n halothane c l e a r l y lowered down the blood p r e s s u r e . T h i s was expected from i n s p e c t i o n of the raw data. Other a n a l y s e s such as p a i r w i s e comparisons of treatments would probably p r o v i d e more i n f o r m a t i o n on the e f f e c t of halothane on BP. I t would a l s o be i n t e r e s t i n g to see i f the e f f e c t of halothane on BP i s the same whether the r a t has a heart a t t a c k or not. U n f o r t u n a t e l y those analyses c o u l d not be performed because of the i n a c c e s s i b i l i t y of the program. 34 Another approach that i s not r i g o r o u s but might be an a l t e r n a t i v e to the score t e s t has been c o n s i d e r e d and the r e s u l t s compared to those o b t a i n e d with the score t e s t . The method c o n s i s t s of r e p l a c i n g the mi s s i n g v a l u e s by t h e i r r e g r e s s i o n e s t i m a t e s . T h i s was done using the BMDP:PAM program where each m i s s i n g value f o r a v a r i a b l e i s estimated by r e g r e s s i n g i t on a l l v a r i a b l e s t h a t have non-missing v a l u e s i n the case. Once a l l the mi s s i n g v a l u e s are r e p l a c e d we can perform a one-way m u l t i v a r i a t e a n a l y s i s of v a r i a n c e on the new data s e t u s i n g any program or s t a t i s t i c a l package a v a i l a b l e . The r e s u l t s obtained f o r the BP data using t h i s method were as f o l l o w s . data set #1 F(20,150) = 3.28 p-value < .00005 MAXROOT = .6239 p-value < .00005 data s et #2 F(27,91) = 1.36 2 p-value = .1426 MAXROOT = .5301 3 p-value = .0250 The p-values o b t a i n e d with t h i s t e s t appear to be too s m a l l . T h i s was expected s i n c e with t h i s method we d i d not take i n t o account the l o s s of degrees of freedom due to e s t i m a t i n g the m i s s i n g data from the observed data. U n f o r t u n a t e l y the formulas f o r the estimated degrees of freedom of the F s t a t i s t i c 2 F corresponds to a t r a n s f o r m a t i o n of Wilk's lambda that can be compared with the F d i s t r i b u t i o n (Rao 1973). 3 MAXROOT i s Roy l a r g e s t root s t a t i s t i c (Morrison,1976). 35 a r e such t h a t no s t r a i g h t f o r w a r d c a l c u l a t i o n c o u l d be made t o a d j u s t them f o r m i s s i n g v a l u e s . The s i g n i f i c a n c e l e v e l of the app r o x i m a t e F - t e s t i s c l o s e r t o the s i g n i f i c a n c e l e v e l of t h e s c o r e t e s t . T h i s may not be s u r p r i s i n g s i n c e b o t h s t a t i s t i c s a r e d e v e l o p e d through the l i k e l i h o o d r a t i o p r i n c i p l e . R i g o r o u s h y p o t h e s i s t e s t i n g cannot be performed w i t h t h i s a p p roach. N e v e r t h e l e s s , i t s t i l l p r o v i d e d some good i n d i c a t i o n s about the s i g n i f i c a n c e of t r e a t m e n t s . 4.2 Non-p a r a m e t r i c approach The t e s t d e s c r i b e d i n the p r e c e e d i n g s e c t i o n assumes m u l t i v a r i a t e n o r m a l i t y of t h e v a r i a b l e s which as we have seen i s not a r e a s o n a b l e assumption f o r some v a r i a b l e s (R, RS, DSTR). For any a n a l y s i s which i n c l u d e s v a r i a b l e s we th e n need a t e s t which i n v o l v e s no e x p l i c i t a s s u m p t i o n s about t h e d i s t r i b u t i o n of the o b s e r v a t i o n s . K o z i o l e t a l . (1981) have d e v e l o p e d a t e s t , based on a m u l t i v a r i a t e rank s t a t i s t i c f o r the comparison of growth c u r v e s , which i s d i s t r i b u t i o n - f r e e . T h i s t e s t may be used w i t h i n c o m p l e t e o b s e r v a t i o n s and i n s i t u a t i o n s where p a r a m e t r i c models f o r growth may not be a p p r o p r i a t e . T h i s p r o c e d u r e i s a p p r o p r i a t e f o r the v a r i a b l e s mentioned above. 4.2.1 D e s c r i p t i o n of the m u l t i v a r i a t e rank t e s t L e t F denote the p - v a r i a t e c o n t i n u o u s c u m u l a t i v e i t h d i s t r i b u t i o n f u n c t i o n f o r the i group. The h y p o t h e s i s of 36 i n t e r e s t i s H : F = F f o r a l l i=1,2,...,k o i a g a i n s t the a l t e r n a t i v e t h a t F < F f o r some i , j = 1,2,...,k; i j that i s , F (• ) <, F (•) f o r each t=1,2,...,p, with at l e a s t one i t j t s t r i c t i n e q u a l i t y . The t e s t proposed by K o z i o l et a l . (1981) f o r the d e t e c t i o n of t h i s kind of a l t e r n a t i v e i . e . s t o c h a s t i c o r d e r i n g of d i s t r i b u t i o n s , i s analogue t o the rank t e s t s suggested f o r randomized block and makes use of the b a s i c rank permutation p r i n c i p l e . A c c o r d i n g to t h i s p r i n c i p l e a l l permutations of the N v e c t o r s of o b s e r v a t i o n s are e q u a l l y l i k e l y under the n u l l h y p o t h e s i s (see P u r i and Sen 1971). Now l e t n be the i n i t i a l number of experimental u n i t s i k ( i > ( r a t s i n our case) i n group i with N = I n . Let y denote i=1 i t j th the value of the time-response v a r i a b l e of the j r a t i n group ( i > i at time t . Note that y may not be observed so t h a t i n t j g e n e r a l only n of the n r a t s i n group i y i e l d o b s e r v a t i o n s a t i t i ( i ) ( i ) time t . I f y i s observed, l e t R denote i t s rank among the t j t j k n = E n a v a i l a b l e v a l u e s at time t . D e f i n e f o r i=1,...,k .t i=1 i t 37 n i , 1 , S = (n ) _ 1 Z a ( R ) , i t i t j =1 t t j here the a , t=1,2,...,T are the u n i v a r i a t e score f u n c t i o n s , t Without l o s s of g e n e r a l i t y , the score f u n c t i o n s are chosen such _ n.t < i , t h a t a = Z a (k) = 0. When y i s not observed then t k =1 t t j ( i ) a (R ) = 0. t t j From the rank permutation p r i n c i p l e , i t can be shown that E( S | P ) = 0 i t N and cov(S ,S |P )=(n n )" 1n {5 -(n / N ) } ( N - 1 ) _ 1 i t ms N i t ms i im m k n i , i , , i , x Z Z a (R )a (R ) (4.2) i=1j=1 t t j s s j where 8 denotes the Kronecker d e l t a : 6 =1 i f i=m and 0 im im otherwise, and P r e p r e s e n t s the permutation p r o b a b i l i t y measure N k generated by N!/ II n ! p o s s i b l e d i s t i n c t permutations of the i =1 i observed data v e c t o r s . The t e s t proposed f o r t e s t i n g the h y p o t h e s i s formulated p r e v i o u s l y i s based on the f o l l o w i n g m u l t i v a r i a t e s t a t i s t i c 38 T M = S V~ S (4.3) N N N N P where S = (S ,S ,...,S ) with S = L S and V denotes a N 1 . 2. I . i . i = 1 i t N g e n e r a l i z e d i n v e r s e of V , the c o v a r i a n c e matrix of S as N N c a l c u l a t e d from (4.2). Under P and H , the permutation d i s t r i b u t i o n of M i s N o N d i s t r i b u t i o n - f r e e . To c a r r y out the t e s t , we would need to k study a l l the N!/ II n ! d i s t i n c t v a l u e s of M . Except f o r i =1 i N small v a l u e s of N and p an exact a p p l i c a t i o n of the t e s t i s d i f f i c u l t because of the l a r g e amount of computation. N e v e r t h e l e s s , t h i s permutation t e s t procedure can be s i m p l i f i e d i n l a r g e samples. I t can be proved that under r a t h e r general c o n d i t i o n s the j o i n t c o n d i t i o n a l asymptotic d i s t r i b u t i o n of the S i s m u l t i v a r i a t e normal. Using t h i s f a c t i t can be shown i t t h a t the q u a d r a t i c form M has approximately a c h i - s q u a r e N d i s t r i b u t i o n with degrees of freedom equal t o the rank of V N Hence l a r g e v a l u e s of M r e l a t i v e to x 2(k) would l e a d t o the N r e j e c t i o n of H . In the s p e c i a l case where n = n f o r a l l t o i t 1 the rank of V i s k-1. N 39 4.2.2 A p p l i c a t i o n of the m u l t i v a r i a t e rank t e s t The data on DSTR i n the halothane experiment was chosen f o r the a p p l i c a t i o n of t h i s t e s t (see t a b l e s 6 and 7 ) . As b e f o r e , two data s e t s were analysed s e p a r a t e l y ; the f i r s t data set c o n t a i n e d o b s e r v a t i o n s at time p o i n t s 3 t o 7 f o r the 5 treatments and the second c o n t a i n e d o b s e r v a t i o n s at time p o i n t s 3 t o 11 f o r the f i r s t four groups. The Wilcoxon score was chosen f o r a ; t a ( j ) = j - (n + l ) / 2 , j=1,2,...,n t=1,2,...,p. t . t . t With the f i r s t data s e t , i t was found t h a t S =(14.69,27.36,-26.87,-9.33,-4.49) N permutation c o v a r i a n c e matrix with c o r r e s p o n d i n g V = N 333.93 -74.25 -73.88 -72.49 -134.81 -74.25 264.83 -65.83 -64.61 -119.14 -73.89 -65.83 261.85 -64.25-118.78 -72.49 -64.61 -64.25 252.25 -116.39 •134.81 -119.14 -118.78 -116.39 889.00 Hence from (3.3) M = 5.52, g i v i n g an approximate p-value of N .36. 40 Table 6. DSTR f o r a l l f i v e treatments and at time p o i n t s 3 to 7. t r e a t time p o i n t s # 3 4 5 6 7 ! .82 1 .08 1 .02 .98 .98 1 .17 .24 .22 .23 .22 1 .08 .28 .22 .27 .24 1 .32 .40 1 .13 .33 1 .35 .20 .25 .24 .24 1 .09 .12 1 .29 .31 1 .41 .49 .49 .45 .42 1 .00 .08 .15 .13 .15 1 . 1 1 .34 .34 .38 .37 2 .12 .23 . 1 1 .03 .18 2 .16 .08 .10 .10 .01 2 .20 2 .23 .33 .33 .33 .39 2 .36 .33 .32 .34 .31 2 .23 .38 2 .26 .51 .52 .52 .53 2 .17 .27 .32 .35 .31 2 .19 .27 .24 .29 .28 2 .31 .39 .37 .41 .37 2 .28 .34 .61 .51 .57 3 .24 .39 .34 .33 .29 3 -.02 .20 3 -.01 .07 .07 .09 .10 3 .18 .21 3 .14 .20 .27 .26 .28 3 .21 .27 .29 .28 .28 3 .25 .32 .31 .32 .30 3 .19 .23 .23 .22 .23 3 .18 .18 .18 .18 .18 3 .06 .14 .15 .17 3 .04 .10 . 1 1 .12 .13 4 .16 .25 .31 .30 .29 4 .33 .56 .81 .47 .48 4 .06 .09 .25 .37 .31 4 .20 .24 .25 .26 .26 4 .21 .25 4 .16 .21 4 .06 .06 .06 .06 .06 4 .15 .21 .24 .26 .27 4 .32 .32 .32 .31 .32 4 .02 .10 .19 .21 .24 4 .18 .22 .23 .23 .23 5 .52 .84 .88 .87 .96 5 -.03 .17 5 .02 .08 .13 .09 5 .08 .10 5 .18 .29 5 .01 .02 .03 .03 .03 5 .16 .28 5 .20 .23 .28 5 .35 .35 .35 .35 5 .20 .27 5 ' .17 .28 .46 Table 7. DSTR f o r the four treatments e x c l u d i n g halothane 2.0% f o r the time p o i n t s 3 to 11. t r e a t time p o i n t s # 3 4 5 6 7 8 9 10 1 1 ! .82 1 .08 1 .02 .98 .98 .97 .99 .98 1.01 1 .17 .24 .22 .23 .22 .22 .21 .22 .22 1 .08 .28 .22 .27 .24 ;22 .25 1 .32 .40 1 .13 .33 1 .35 .20 .25 .24 .24 .22 .20 .21 .20 1 .09 .12 1 .29 .31 1 .41 .49 .49 .45 .42 .42 .36 .39 .43 1 .00 .08 .15 .13 .15 .08 . 1 1 . 1 1 .10 1 . 1 1 .34 .34 .38 .37 .37 .35 .33 .40 2 .12 .23 . 1 1 .03 .18 . 1 1 -.01 .03 -.05 2 .16 .08 .10 .10 .01 .04 -.01 .04 .16 2 .20 2 .23 .33 .33 .33 .39 .33 .33 .38 .35 2 .36 .33 .32 .34 .31 .32 .33 .35 .30 2 .23 .38 2 .26 .51 .52 .52 .53 .54 .52 .54 .49 2 .17 .27 .32 .35 .31 .34 .34 2 .19 .27 .24 .29 .28 .26 .23 .23 .23 2 .31 .39 .37 .41 .37 .35 2 .28 .34 .61 .51 .57 .53 .57 .53 .53 3 .24 .39 .34 .33 .29 .30 .31 .29 .22 3 -.02 .20 3 -.01 .07 .07 .09 .10 .09 .03 .03 3 .18 .21 3 .14 .20 .27 .26 .28 .28 .29 .28 .28 3 .21 .27 .29 .28 .28 .28 .28 .28 .28 3 .25 .32 .31 .32 .30 .30 .31 .29 .29 3 .19 .23 .23 .22 .23 .23 .22 .23 .22 3 .18 .18 .18 .18 .18 .18 .18 .18 .18 3 .06 .14 .15 .17 3 .04 .10 . 1 1 .12 .13 .13 .13 .12 .12 4 .16 .25 .31 .30 .29 .28 .29 .28 .30 4 .33 .56 .81 .47 .48 .47 .50 .62 .61 4 .06 .09 .25 .37 .31 .37 .39 .37 .37 4 .20 .24 .25 .26 .26 .25 4 .21 .25 4 .16 .21 4 .06 .06 .06 .06 .06 .06 .06 .06 .06 4 .15 .21 .24 .26 .27 .25 .27 .27 .28 4 .32 .32 .32 .31 .32 .32 .32 .32 .32 4 .02 .10 .19 .21 .24 .23 .22 .16 .17 4 .18 .22 .23 .23 .23 .23 .23 .23 .23 42 S i m i l a r l y , i t was found f o r the second data set that S =(15.21,31.00,-38.64,-5.99) and N V = N 763.32 -217.89 -218.20 -209.47 •217.89 560.35 -186.99 -179.56 •218.20 -186.99 561.65 -179.77 •209.47 -179.56 -179.77 519.89 With these, M = 4.71, y i e l d i n g an approximate p-value of .32. N T h i s a n a l y s i s seems to i n d i c a t e t h a t the a d m i n i s t r a t i o n of halothane at d i f f e r e n t doses when r a t s have a heart a t t a c k has no e f f e c t on the values of DSTR measured by the ECG. U n l i k e blood p r e s s u r e , DSTR does not seem t o be a f f e c t e d by a 2% c o n c e n t r a t i o n of halothane. For comparison with the two methods proposed i n the preceeding s e c t i o n , the m u l t i v a r i a t e rank s t a t i s t i c was a l s o computed f o r the blood p r e s s u r e d a t a . The r e s u l t s are summarized below. data set #1 data set #2 M = 20.42 M = 6.89 N N p-value = .001 p-value = .141 43 In data set #1, the s i g n i f i c a n t d i f f e r e n c e between treatments i s l e s s marked than u s i n g any of the other t e s t s d e s c r i b e d e a r l i e r . On the other hand, f o r data set #2 the s i g n i f i c a n c e l e v e l of t h i s t e s t , which i s comparable to the one of the approximate F - t e s t , i s lower than the s i g n i f i c a n c e l e v e l of the score t e s t . In c o n c l u s i o n , the score t e s t should be used, f o r the a n a l y s i s of the type of data we have at hand, when the assumption of m u l t i v a r i a t e n o r m a l i t y i s reasonable. However i f the a p p l i c a t i o n of t h i s t e s t i s not p o s s i b l e , one c o u l d estimate the m i s s i n g v a l u e s and perform an o r d i n a r y manova. T h i s method i s not r i g o r o u s so we must be c a u t i o u s when i n t e r p r e t i n g the r e s u l t s . On the other hand, when the assumption of n o r m a l i t y i s not a c c e p t a b l e the m u l t i v a r i a t e rank t e s t should be used. T h i s t e s t c o u l d a l s o be performed with normal v a r i a t e s but the t e s t would not be as s e n s i t i v e as the s c o r e t e s t . 44 5. LOG-LINEAR MODELS An important o b j e c t i v e of the experiments i s to i n v e s t i g a t e the a n t i a r r h y t h m i c a c t i o n of d i f f e r e n t drugs. The responses measured f o r t h i s purpose are composed of the number and the d u r a t i o n of v e n t r i c u l a r t a c h y c a r d i a and v e n t r i c u l a r f i b r i l a t i o n (spontaneous or non-spontaneous), the t o t a l number of premature v e n t r i c u l a r c o n t r a c t i o n s and death by i r r e v e r s i b l e v e n t r i c u l a r f i b r i l a t i o n . As shown i n s e c t i o n 3.3 the o b s e r v a t i o n of these v a r i a b l e s i s censored by the death of the r a t s . Hence comparison of means of these v a r i a b l e s t o study the e f f e c t of treatments on arrhythmia c o u l d be m i s l e a d i n g . Dr. Walker and h i s group p a r t i a l l y s o l v e d t h i s problem by c o n s t r u c t i n g the arrhythmia score v a r i a b l e d e s c r i b e d i n s e c t i o n 2.2. T h i s v a r i a b l e combined the i n f o r m a t i o n given by the v a r i a b l e s mentioned above, to measure the s e v e r i t y of the arrhythmia. The scores were d e f i n e d such that the s c a l e c o u l d be regarded as l i n e a r . C o n s i d e r i n g the v a r i a b l e as continuous and a n a l y s i n g i t u s i n g s t a t i s t i c a l methods developed f o r continuous v a r i a b l e s i s not very e f f i c i e n t although i t i s probably not harmful. These data can be set up i n the form of a contingency t a b l e with the f i r s t dimension being the treatments and the second dimension the arrhythmia s c o r e s as shown i n t a b l e 8. I t i s then more a p p r o p r i a t e to use l o g - l i n e a r models to analyse t h i s two dimensional contingency t a b l e . 45 Table 8. D i s t r i b u t i o n of the arrhythmia s c o r e s a f t e r 4 hrs by treatment groups. AS4 t r e a t | 0 1 2 3 4 5 6 7 8 | t o t a l c o n t r o l j 0 0 0 0 2 4 1 1 3 ! 11 0.25% j 1 1 0 1 1 4 1 1 1 I 11 0.50% j 4 1 1 0 2 2 0 1 0 j 11 1.0% i 4 0 0 4 1 1 0 1 0 j 11 2.0% i 4 0 0 0 2 1 0 4 0 I 11 t o t a l | 13 2 1 5 8 12 2 8 4 | 55 The homogeneity model ( i . e . the model with no i n t e r a c t i o n ) was f i t t e d to these observed counts. The goodness of f i t t e s t (GOF) u s i n g the Pearson c h i - s q u a r e s t a t i s t i c y i e l d e d a p-value of .058. The a d j u s t e d r e s i d u a l s are given i n t a b l e 9. The GOF t e s t seems to i n d i c a t e that the d i s t r i b u t i o n of arrhythmia scores i s d i f f e r e n t among the treatment groups. In a d d i t i o n , the a d j u s t e d r e s i d u a l s show that the l a c k of f i t i s caused by the c o n t r o l group which suggests t h a t halothane might have an a n t i a r r h y t h m i c a c t i o n . However t h i s c o n c l u s i o n should be t r e a t e d with c a u t i o n because of a l l the empty c e l l s found i n the t a b l e . In f a c t the r e s u l t s obtained when f i t t i n g a l o g -l i n e a r model to a t a b l e with so many empty c e l l s are not very meaningful. 46 Table 9. Adjusted r e s i d u a l s r e s u l t i n g from f i t t i n g the homogeneity model to the data i n t a b l e 8. AS4 t r e a t | 0 1 2 3 4 5 6 7 8 c o n t r o l | -2.1 -0.7 -0.5 -1.2 0.4 1.3 1 .1 -0.6 2.9 0.25% j -1.3 1 .1 -0.5 0.0 -0.6 1.3 1 . 1 -0.6 0.3 0.50% j 1 .1 1 • 1 2.0 -1.2 0.4 -0.3 -0.7 -0.6 -1.0 1.0% j 1 .1 -0.7 -0.5 3.5 -0.6 -1 .1 -0.7 -0.6 -1.0 2.0% i 1 .1 -0.7 -0.5 -1.2 0.4 -1 .1 -0.7 2.3 -1.0 To overcome t h i s problem one might c o l l e c t more data o r , i f not p o s s i b l e , reduce the number of c e l l s e i t h e r by c o l l a p s i n g them or c l a s s i f y i n g a c c o r d i n g to another v a r i a b l e with l e s s c a t e g o r i e s . The f o l l o w i n g way to combine the scores was suggested by Dr. Walker : 0 and 1, 2 and 3, 4, 5, and 6 to 9. Apparently t h i s c a t e g o r i z a t i o n i s r e p r e s e n t a t i v e of the s c o r i n g system o r i g i n a l l y c o n s t r u c t e d by Dr Walker. The frequency t a b l e a s s o c i a t e d with these new c a t e g o r i e s can be obtained from t a b l e 8. In s p i t e of the empty c e l l s the homogeneity model was a l s o f i t t e d t o these d a t a . I t appeared that the homogeneity model f i t s b e t t e r than b e f o r e . The Pearson c h i - s q u a r e was equal to 23.49 with 16 degrees of freedom g i v i n g a p-value of .101. However as i n the f i r s t a n a l y s i s , there was a l s o a p a t t e r n i n the a d j u s t e d r e s i d u a l s of the c o n t r o l group. More data would 47 c e r t a i n l y be necessary before reaching any c o n c l u s i o n . Another s o l u t i o n proposed i s to c l a s s i f y a c c o r d i n g to v a r i a b l e s with few c a t e g o r i e s . For example, a two-way contingency t a b l e can be c o n s t r u c t e d with the f i r s t v a r i a b l e being the treatments and the second v a r i a b l e r e p r e s e n t i n g the i n c i d e n c e of i r r e v e r s i b l e VF. Table 10 shows the observed f r e q u e n c i e s and the a d j u s t e d r e s i d u a l s o b t a i n e d when f i t t i n g the homogeneity model t o t h i s frequency t a b l e . Table 10. Observed f r e q u e n c i e s and a d j u s t e d r e s i d u a l s ( i n p a r e n t h e s i s ) of the i n c i d e n c e of i r r e v e r s i b l e VF by treatment groups. i r r e v e r s i b l e VF t r e a t no yes c o n t r o l 5 6 (-2.7) (2.7) 0.25% 8 3 (-0.3) (0.3) 0.50% 1 1 0 (2.1) (-2.1) 1 .0% 10 1 (1.3) (-1.3) 2.0% 8 3 (-0.3) (0.3) t o t a l | 42 13 The Pearson c h i - s q u a r e obtained was 10.68 with 4 degrees of freedom g i v i n g a p-value of .03. T h i s model does not f i t the data very w e l l suggesting that the a d m i n i s t r a t i o n of halothane 48 reduces the number of deaths by i r r e v e r s i b l e v e n t r i c u l a r f i b r i l a t i o n . L o g - l i n e a r models a l s o allow us to analyse m u l t i d i m e n s i o n a l contingency t a b l e s . For example, as shown i n t a b l e 11, a t h i r d v a r i a b l e r e p r e s e n t i n g the s i z e of the occluded zone c o u l d be added. T h i s would permit the study of the i n t e r a c t i o n among the thr e e v a r i a b l e s and maybe p r o v i d e more i n f o r m a t i o n about the e f f e c t of treatments on i r r e v e r s i b l e VF. Table 11. Observed f r e q u e n c i e s of the i n c i d e n c e of i r r e v e r s i b l e VF by treatments and the s i z e of the occluded zone. i r r e v e r s i b l e VF t r e a t OZ j no yes | t o t a l c o n t r o l sma11 j 0 1 | 1 medium | 2 4 ! 6 l a r g e j 3 1 ! 4 0.25% small | 2 1 ! 3 medium j 4 1 ! 5 l a r g e | 2 1 ! 3 0.50% small ! 7 0 ! 1 medium | 3 0 ! 3 l a r g e j 1 0 ! 1 1.0% small j 3 1 I 4 medium | 2 0 ! 2 l a r g e | 5 0 ! 5 2.0% small j 2 1 ! 3 medium | 2 1 ! 3 l a r g e j 4 1 ! 5 t o t a l ! 42 13 | 55 The a n a l y s i s of t h i s t a b l e using l o g - l i n e a r models was not 49 c a r r i e d out s i n c e there are not enough data to get meaningful r e s u l t s . In c o n c l u s i o n , l o g - l i n e a r models c o u l d be a u s e f u l t o o l t o analyse arrhythmia i f more data were a v a i l a b l e . I t should be p o i n t e d out that t h i s technique when used with the arrhythmia score v a r i a b l e does not c o n s i d e r the c e n s o r i n g problem caused by r a t s that d i e d of heart f a i l u r e . 50 6. CONCLUSION The data s et pro v i d e d by Dr. Walker gave us the o p p o r t u n i t y of a p p l y i n g v a r i o u s s t a t i s t i c a l techniques and being c o n f r o n t e d with the common problem of mis s i n g d a t a . In a f i r s t s t e p, i t has been found that death of the r a t s c o u l d not be p r e d i c t e d by e a r l y measurements of the responses. T h i s i m p l i e d t h a t the m i s s i n g data c o u l d be c o n s i d e r e d as "missing at random" so that the process c a u s i n g m i s s i n g data c o u l d be ignored when making s t a t i s t i c a l i n f e r e n c e s . I t was a l s o found that the s i z e of the occluded zone c o u l d be a confounding v a r i a b l e when s t u d y i n g the e f f e c t of drugs on other responses. Furthermore we showed that the arrhythmia responses were censored by death of the r a t s . In a second step, a score t e s t and a m u l t i v a r i a t e rank t e s t were proposed f o r the a n a l y s i s of time-response v a r i a b l e s with incomplete o b s e r v a t i o n s . The score t e s t assumed m u l t i v a r i a t e n o r m a l i t y of the o b s e r v a t i o n s while the m u l t i v a r i a t e rank t e s t made no assumption on the d i s t r i b u t i o n of the responses. An a l t e r n a t i v e to the score t e s t c o n s i s t i n g of r e p l a c i n g the mi s s i n g v a l u e s by an t h e i r r e g r e s s i o n e s t i m a t e s was a l s o p r e s e n t e d . T h i s t e s t l e a d to a p-value which appeared t o be too small compared t o the p-value of the score t e s t and which c o u l d probably be e x p l a i n e d by the l o s s of degrees of freedom due to e s t i m a t i n g the m i s s i n g data from the observed d a t a . The a p p l i c a t i o n of these t e s t s on the halothane data showed that only halothane at a c o n c e n t r a t i o n of 2% s i g n i f i c a n t l y lowered 51 the blood p r e s s u r e while the value of DSTR d i d not seem to be a f f e c t e d by the a d m i n i s t r a t i o n of halothane. In a t h i r d s t ep, l o g - l i n e a r models were c o n s i d e r e d t o analyse c a t e g o r i c a l v a r i a b l e s such as the arrhythmia s c o r e s and the i n c i d e n c e of i r r e v e r s i b l e VF. U n f o r t u n a t e l y the r e s u l t s obtained are not very r e l i a b l e s i n c e there was not enough data f o r the a p p l i c a t i o n of t h i s technique. An i n c r e a s e i n the s i z e of the treatment groups would be necessary to a s c e r t a i n the e f f e c t of drugs on arrhythmias. F i n a l l y i t should be p o i n t e d out t h a t when l o g - l i n e a r models are a p p l i e d to the arrhythmia score v a r i a b l e , i t does not c o n s i d e r the c e n s o r i n g caused by r a t s t h a t d i e d of heart f a i l u r e . A more a p p r o p r i a t e technique which would take i n t o account t h i s c e n s o r i n g problem i s s t i l l to be developed. 52 BIBLIOGRAPHY Anderson, T.W. (1957). Maximum l i k e l i h o o d estimate f o r a m u l t i v a r i a t e normal d i s t r i b u t i o n when some o b s e r v a t i o n s are m i s s i n g . J . Amer. S t a t i s t . Assoc. 52, 200-203. Beale, E.M.L. and L i t t l e , R.J.A. (1975). M i s s i n g v a l u e s i n m u l t i v a r i a t e a n a l y s i s . J . R. S t a t i s t . Soc. B 37, 129-145. Bishop, Y.M.M., Fie n b e r g , S.E. and H o l l a n d , P.W. (1975). D i s c r e t e m u l t i v a r i a t e a n a l y s i s ; theory and p r a c t i c e . Cambridge, Mass., MIT P r e s s . Cox, D.R. and H i n k l e y , D.V. (1974). T h e o r i t i c a l s t a t i s t i c s . London: Chapman and H a l l . F i e n b e r g , S.E. (1977). The a n a l y s i s of c r o s s - c l a s s i f i e d  c a t e g o r i c a l d ata. Cambridge, Mass., MIT P r e s s . Haberman, S.J. (1978). A n a l y s i s of q u a l i t a t i v e d a ta. v o l 1: i n t r o d u c t o r y t o p i c s . New york: Academic P r e s s . Haberman, S.J. (1978). A n a l y s i s of q u a l i t a t i v e data, v o l 2: new developments. New york: Academic P r e s s . Jang, T.L., Macleod, B.A. and Walker, M.J.A. (1983). The e f f e c t s of halogenated hydrocarbon a n e s t h e t i c s on responses to l i g a t i o n of a coronary a r t e r y i n c h r o n i c a l l y prepared r a t s . Department of Pharmacology, U n i v e r s i t y of B r i t i s h Columbia. (Submitted to A n a s t h e s i o l o g y ) Johnston, K.M., Macleod, B.A. and Walker, M.J.A. (1983a). E f f e c t of A s p i r i n and P r o s t a c y l i n on responses t o l i g a t i o n of a coronary a r t e r y and on i n f a r c t s i z e . Br.  J . Pharmac. 78, 029-037. Johnston, K.M., Macleod, B.A. and Walker, M.J.A. (1983b). Response to l i g a t i o n of a coronary a r t e r y i n c o n s c i o u s r a t s and the a c t i o n s of a n t i a r r h y t h m i c s . Department of pharmacology, U n i v e r s i t y of B r i t i s h Columbia. (Submitted to the Canadian J o u r n a l of Phy s i o l o g y and pharmacology) K o z i o l , J.A., Maxwell, D.A., Fukushima, M., Colmerauer, M.E.M. and P i l c h , Y.H. (1981). A d i s t r i b u t i o n - f r e e approach to tumor growth curve a n a l y s e s with a p p l i c a t i o n to an animal tumor immunotherapy experiment. B i o m e t r i c s . 37, 383-390. K o z i o l , J.A. and Yuh, Y. (1982). An a p p l i c a t i o n of m i s s i n g data techniques t o growth curve a n a l y s e s . T e c h n i c a l r e p o r t . Departments of Mathematics and Medecine, 53 U n i v e r s i t y of San Diego. Macleod, B.A., Augerau, P. and Walker, M.J.A. (1983). E f f e c t s of halothane a n e s t h e s i a compared with f e n t a n y l a n e s t h e s i a and no a n e s t h e s i a d u r i n g coronary l i g a t i o n i n r a t s . A n e s t h e s i o l o g y , Vol.58, No.1, 44-52. Mor r i s o n , D.F. (1976). M u l t i v a r i a t e s t a t i s t i c a l methods. New York: McGraw-Hill. Orchard, T. and Woodbury, M.A. (1972). A m i s s i n g i n f o r m a t i o n p r i n c i p l e : theory and a p p l i c a t i o n s . Proc. 6th Berkeley  Symposium on Math. S t a t i s t , and Prob. 1 , 697-715. P u r i , M.L. and Sen, P.K. (1971). Nonparametric methods  in m u l t i v a r i a t e a n a l y s i s . New York: Wiley. Rubin, D.B. (1976). Inference and mi s s i n g data . Biometrika 63, 581-592. Rao, C R . (1973). L i n e a r s t a t i s t i c a l i n f e r e n c e and i t s a p p l i c a t i o n s . New York: Wiley. Rao, C R . (1947). Large sample t e s t s of s t a t i s t i c a l hypotheses concerning s e v e r a l parameters with a p p l i c a t i o n s to problem of e s t i m a t i o n . Proc. Camb. P h i l . Soc. 44, 50-57. Wald, A. (1943). T e s t s of s t a s t i s t i c a l hypotheses concerning s e v e r a l parameters when the number of o b s e r v a t i o n s i s l a r g e . Trans. Amer. Math. Soc. 54, 426-482. 54 APPENDIX DERIVATION OF THE MAXIMUM LIKELIHOOD STATISTIC AND THE SCORE STATISTIC IN THE MANOVA SETTING In t h i s appendix, the maximum l i k e l i h o o d (ML) s t a t i s t i c and the score s t a t i s t i c w i l l be d e r i v e d under the m u l t i v a r i a t e normal model. The case of complete data and the case of incomplete data w i l l be c o n s i d e r e d s e p a r a t e l y . I. COMPLETE DATA In a one-way c l a s s i f i c a t i o n m u l t i v a r i a t e a n a l y s i s of v a r i a n c e , the model assumes t h a t there are k independent samples from a p-dimensional m u l t i v a r i a t e normal d i s t r i b u t i o n s with mean v e c t o r s m f..., M and a common unknown c o v a r i a n c e matrix $2 . k We f i r s t assume that there are no m i s s i n g o b s e r v a t i o n s . The q u e s t i o n of i n t e r e s t i s whether or not the treatment means are a l l e q u a l . The n o t a t i o n f o r the model i s the f o l l o w i n g : Y Y i . i . d . N ( u , £2 ) i = 1,...,k i1 i n i p i The n u l l h y p o t h e s i s i s H 0 : Mi = 1*2 - • • • = M k 5 5 The a l t e r n a t i v e hypothesis i s : a not a l l equal, The l i k e l i h o o d f u n c t i o n can be w r i t t e n L ( J I , , . . . , M , 0 ; Y ) = k k n i -p/2 -1/2 n n (2TT) |Q| exp{-l/2(y -a ) T n _ 1 ( y -u )} i=1j=1 i j i i j i and the l o g l i k e l i h o o d i s k n i 1 = C-N / 2(log| 0 | )-l/2{ Z Z (y —u ) TJ2- 1 (y -u )} (1) 1=13=1 i ] 1 i ] 1 k where N = Z n i =1 i The f o l l o w i n g n o t a t i o n f o r the v e c t o r score f u n c t i o n and the in f o r m a t i o n matrix w i l l be used throughout t h i s appendix. U U , 0 ; Y ) = E [ - ( 3 / 9 M ) 1 ] • M i (u,Q) = E [ - ( 9 2 / 3 2 M ) 1 ] MM i (M,0) = E[-(d2/dndSl) 1] M® i = E [ - O 2 / 9 2 O ) 1] o n 56 A. D e r i v a t i o n of the maximum l i k e l i h o o d s t a t i s t i c In g e n e r a l , we wish to t e s t the composite h y p o t h e s i s H 0: $ = t//0 a g a i n s t H i : $ * \l>0 , without s p e c i f y i n g any value f o r the nuisance parameter X . The ML s t a t i s t i c d e f i n e d i n Cox and Hi n k l e y (1974) and a l s o proposed by Wald (1943) i s W = (£-tfo>T i.U:X) (2) e where i . ( ^ : X ) = i U,X) - i U, X) i " 1 ( + , X) i U,X) (3) ^t// vpX XX X^ and 4/" , " x are the maximum l i k e l i h o o d e stimates (mle) of ^ and X. T h i s t e s t i s d e r i v e d by Cox and H i n k l e y (1974, example 9.23) i n the m u l t i v a r i a t e one-way a n a l y s i s of v a r i a n c e case. The d e t a i l s of the d e r i v a t i o n are giv e n here. For t h i s problem Cox and H i n k l e y d e f i n e i/> = ( M 2 " M I , . . . , M - M i ) T k X = ( j i l f O ) T . The n u l l h y p o t h e s i s i s then \f/ = 0 wit h the a l t e r n a t i v e $ * 0. The mle of ^ i s R e p l a c i n g u by V +u^ i n equation (1) we f i n d t h a t i i hence U (<//,X;Y) n i o~1 i (y - M I ) ; j=1 i j i i 0,X) = n fi~1 , i i ( i ^ , X ) = 0 i U , X ) = 0 , i 0,X) = n J i " 1 , and \piu1 i i U , X ) = NJT 1 . M1M1 l ^ S , no i x ) = = i($2), then the i n f o r m a t i o n matrix i s n 2 n _ 1 0 . . . 0 n 2 0 _ 1 0 . . . 0 0 n 3 n _ 1 • • . . . 0 • I n 3 r r 1 * 0 . . . 0 • • • • • » • • • • n 0~ 1 k • • | n fi"' k • • • • 0 . . . • • 0 . n 0 ' 1 k | NO- 1 0 . . . 0 0 0 1 o » • • 1 « i ( f l ) • 0 0 • 1 o From t h i s matrix and u s i n g (3) , we can c a l c u l a t e the matrix 58 i . (vt>: X) ; i . U : X ) = ( n 2 - n 2 2 / N ) n _ 1 - ( n 2 n 3 / N ) n _ 1 ... - ( n 2 n / N ) 0 - 1 k - ( n 2 n / N ) f i ~ 1 - ( n 3 n /N ) 8 ~ 1 ... (n - n 2 / N ) f l ~ 1 k k k k Then s u b s t i t u t i n g t h i s m a t r i x i n ( 2 ) we g e t W = Z (n - n 2 / N ) ( Y - Y ) T n _ 1 ( Y - Y ) e i = 2 i i i . 1. i . 1 . - Z {(Y - Y ) T ( n / N ) n _ 1 Z n (Y - Y )} i = 2 i . 1 . i j ^ i j j . 1 • where Si i s t h e mle o f $2. But Z n (Y - Y ) = N(Y - Y ) - n (Y - Y ) j * i j j . 1 . . . 1 . i i . 1 . Hence W = Z n (Y - Y ) T f i ~ 1 ( Y - Y ) e i = 2 i i . 1. i . 1 . - N(Y - Y ) T f i " 1 ( Y - Y ) . 1 • • • 1* 5 9 F u r t h e r i f we r e p l a c e ( Y - Y ) b y ( Y - Y - Y + Y ) and i • 1* i • •• 1 • • * s i m p l i f y , W reduces to the e x p r e s s i o n found i n Cox and H i n k l e y e (equation 63) ; k _ _ ^ _ W = Z n (Y - Y ) T Q _ 1 ( Y - Y ) . (4) e i=1 i i . .. i . We would l i k e to p o i n t out t h a t u s i n g the f o l l o w i n g p a r a m e t r i z a t i o n yfj = (ni~n,... ,u -u)J ( 5 ) k k X = ( J U , 0 ) t , where a - ( N ) " 1 Z n u i = 1 i i the same r e s u l t would be obtained (more e a s i l y ) . B. D e r i v a t i o n of the score s t a t i s t i c In the same coritext as the ML s t a t i s t i c , Cox and H i n k l e y d e f i n e the score s t a t i s t i c as f o l l o w s : W = U T Uo,X 0) i ( * o X ) U < t f 0 X > (6) u .\p where \ Q is the mle of X under H 0, and 6 0 i = i _ 1 + i - 1 i ( i - i i _ 1 i ) _ 1 i i " 1 (7) T h i s t e s t was a l s o proposed by Rao (1947). Using the score f u n c t i o n and the i n f o r m a t i o n m a t r i c e s determined i n the preceeding s e c t i o n i t can be shown that i (V 'o/Xo) = J2 (1 /n, +1 /n 2 ) J2/n, ... Q/n, — » . — v fi/n, ... fl/n, O/n, B(1/n,+1/n ) k and u (\^0fX0) = ( n _ , n 2 ( y - Y ), ... f n _ 1 n (Y - Y ) ) T .\p 2. k k. Using these r e s u l t s we t h e r e f o r e can c a l c u l a t e W : u W = Z (n /n,)(n,+n )(Y - Y ) T f i ~ 1 ( Y - Y ) u i=2 i i i . i . + Z [(n /n,)(Y - Y ) T f T 1 Z n (Y - Y )] . i=2 i i . .. j * i j j . Going through the same kind of a l g e b r a as f o r the ML s t a t i s t i c we f i n d t h a t the score s t a t i s t i c i s the same as the ML s t a t i s t i c 61 k _ _ ^ _ _ W = I n (Y - Y ) T J T 1 (Y - Y ) . u i=1 i i . .. i . Again we should p o i n t out t h a t the same r e s u l t would be ob t a i n e d u s i n g the p a r a m e t r i z a t i o n d e f i n e d i n ( 5 ) . In f a c t , under t h i s p a r a m e t r i z a t i o n we would f i n d t h a t ( i - i i ~ 1 i ) - 1 i i - 1 = 0 XX X^ '/'X hence from (7) i (^,X) = i _ 1 ( ^ , X ) . T h i s makes the c a l c u l a t i o n s to f i n d W much e a s i e r , u Cox and H i n k l e y argue that t e s t s based on W and W are e u a s y m p t o t i c a l l y e q u i v a l e n t i n r e g u l a r problems. We have shown that f o r the m u l t i v a r i a t e a n a l y s i s of v a r i a n c e they are a l s o e q u i v a l e n t f o r f i n i t e samples. I I . INCOMPLETE DATA In t h i s s e c t i o n , the ML s t a t i s t i c and the score s t a t i s t i c w i l l be d e r i v e d under the same model as bef o r e but now assuming that we have incomplete data. The p a t t e r n of m i s s i n g data i s assumed t o be nested so that e x p l i c i t equations f o r the mle of the mean and the v a r i a n c e - c o v a r i a n c e matrix are a v a i l a b l e . Indeed, e x p l i c i t equations f o r the estimates are ex t e n s i o n s of Anderson's (1957) methodology. th Taking i n t o account the m i s s i n g o b s e r v a t i o n s the j t h o b s e r v a t i o n from the i sample proceeds as f o l l o w s : 62 Y = j IS \ DO i DO Z = / 0 jm j = 1 ,... ,n where Y has a p-dimensional multi-normal d i s t r i b u t i o n with mean D i i i a and c o v a r i a n c e Q. Y i s the observed p o r t i o n of Y and Y , i j o j jm the m i s s i n g p o r t i o n . For purposes of n o t a t i o n and d e r i v a t i o n the s u p e r s c r i p t i w i l l be dropped t e m p o r a r i l y . We then have Y = Z +Z ~ N (u,Sl) j j o jm p j=1 r...n, For each Y , u and 0 can be p a r t i t i o n e d a p p r o p r i a t e l y ; D E[Z ] = 6 jo j o DO E [ Z ] = e = / o jm jm jm/ u = 6 + 6 jo jm cov(Y ) = 0 cov(Y ) = 8 , and jo joo jm jmm 63 o = / 0 0 \ j o o j o m y j m o jmmy Note that Y |Y ~ N (M + 0 J2~1 ( Y -M ) , V ) jm jo jm jmo joo jo jo j ( 8 ) where v = n - n n _ 1 n j jmm jmo joo jom The score f u n c t i o n s and the i n f o r m a t i o n m a t r i c e s w i l l be d e r i v e d with incomplete data u s i n g equation (2.13) of Orchard and Woodbury (1972): U U,S2;Y ) = E [ U (M,J2;Y) |Y ] . . M O . u O In the case of complete data we know that hence n u U,S2;Y) = n ~ i z (Y -n) .u j=1 j n u (M,$2;Y ) = J2"1 Z E ( Y -M|Y ) .n o j=i j o 64 N / 1 v / I T 1 Z j»1 Y -u j o jo \ • / $2 n - 1 (Y -M ) \ jmo joo jo jo / Z $2 - 1/ I \ j-1 \ jmo j o o / (Y - M ) . jo j o ( 9 ) Dropping the s u b s c r i p t j t e m p o r a r i l y , $2 - 1 mo oo o _ 1 n v _ 1 n Q - - 1 oo oo om mo oo - v - 1 n a - 1 mo oo o v " 1 oo om $2 J2 _ 1 \ mo oo j £2" oo Then 65 U U,0;Y ) = n / j=1 a- 1 ( Y - M ) joo j o jo (10) Next i (u,Q) = E[U U , f i;Y )U ( M , 0 ; Y ) T ] UU .a O . u O n = E{[ Z j = 1| fa"1 (Y -u )] [ Z to"1 (Y -a \V} joo j o j o j=1 joo jo jo n = z / r r 1 o j = 1| joo 0 0 (11) F u r t h e r i = E[-9/3J2 U (/u,fl;Y ] R e p l a c i n g U (JI,°,;Y ) by e x p r e s s i o n (9) we get . u o i u,f i ) = z - [ ( ( a / a n ) n - 1 / I \ ) E ( Y -u )] D = 1 = 0 . 1 , DO 3 0 \ jmo joo/ These r e s u l t s are s u f f i c i e n t t o d e r i v e the ML and score s t a t i s t i c s . 66 A. D e r i v a t i o n of the ML s t a t i s t i c The ML s t a t i s t i c i s giv e n by (2) and ( 3 ) . The p a r a m e t r i z a t i o n d e f i n e d i n (5) i s e a s i e r to work with and t h e r e f o r e w i l l be used. Going through the same d e r i v a t i o n s as above with \fj = a - a and i i \ - ( u , Q ), we get the f o l l o w i n g r e s u l t s . n i U U,X;Y ) = ST 1 I i|d o j = 1 IJ j o j o j o \ i (*,X) n i Z j-1 i s ' A joo 0 i (<//,X) = 0 \j/i\j/s i#s , i U,X) DWi" 1 . \ Z j = 1 3 0 0 0 k n i il i - i \ / o n \ i (i|/,X) = Z Z 0 0 un i = l j = l joo \ 0 0 / , and 67 i (<//,A) = i v>,X) = 0 Using equation (3) we can show that i.(vfr:X) = A - A BA -A BA ... -A BA n, n , n , n, n 2 n, nk -A BA n 2 n, -A BA nk n, A - A BA nk nk nk where and n i A = Z n i j = 1 / i " 1 Q 0 joo \ B = k n i Z Z i=1j=1 ii-< \ Q 0 joo r 1 Note the s i m i l a r i t y of t h i s e x p r e s s i o n to the e x p r e s s i o n f o r with no m i s s i n g data (page 58). The e x t r a n o t a t i o n i s necessary because the o b s e r v a t i o n s (indexed by j ) are allowed to have d i f f e r i n g numbers of m i s s i n g components i n each treatment. Now we have to compute u and ft from the pooled data. The approach suggested here i s an e x t e n s i o n of Anderson's (1957) methodology and i s d e s c r i b e d i n Morrison (1976) . 68 The d a t a a r e p o o l e d and r e a r r a n g e d such t h a t the m a t r i x has the f o l l o w i n g p a t t e r n : X X 11 12 X X 21 22 X X 1-1,1 1-1,2 X 11 X X 1,q-1 1,q X 2,q~2 where the dashes i n d i c a t e b l o c k s of m i s s i n g o b s e r v a t i o n s . I n the f o l l o w i n g n o t a t i o n X r e p r e s e n t s an r x w s u b m a t r i x of uv u v 1 o b s e r v a t i o n s w i t h Z r =N the t o t a l number of e x p e r i m e n t a l u n i t s u= 1 u and Z w = p the t o t a l number of r e s p o n s e s . v= 1 v The mean v e c t o r i s c o r r e s p o n d i n g l y p a r t i t i o n e d as M T = ( M i T , M 2 T r • • • t U T ) and the c o v a r i a n c e m a t r i x c o n s i s t s of w x q u w s u b m a t r i c e s $2 , u,v=1,2,...,q. The l i k e l i h o o d can be v uv w r i t t e n as 1 1-1 1-2 L U , n ) = { n f ( x ) } { n f ( x | x ) } { n f ( x | x , x ) } . u=1 u1 u=1 u2 u l u=1 u3 u l u2 { f ( X | X , . . . , X ) } 1,q 11 1,q-1 69 where the r i g h t - h a n d s i d e has been w r i t t e n i n term of c o n d i t i o n a l d e n s i t i e s i n s t e a d of l i k e l i h o o d to a v o i d unnecessary . complicated n o t a t i o n . To maximize L( u , tl ) , we have to maximize each f a c t o r i n the b r a c k e t s . The f i r s t f a c t o r i s the " l i k e l i h o o d " of a normal d i s t r i b u t i o n with mean M , and v a r i a n c e - c o v a r i a n c e matrix £2 11 T h e r e f o r e the mle M i and based on a l l N sampling u n i t s w i l l 11 maximize t h i s f a c t o r . The second f a c t o r i s the c o n d i t i o n a l " l i k e l i h o o d " of a normal d i s t r i b u t i o n with mean 1x2 + 0 £2~1(X -2111 1 M i and v a r i a n c e - c o v a r i a n c e matrix £2 -12 £2-1£2 . Regressing X 22 21 11 12 1 on X , w i l l g ive e s t i m a t e s of u2- £2 £2 - 1 M i , the r e g r e s s i o n 2 2111 parameters £2 £2" 1 and the c o n d i t i o n a l c o v a r i a n c e matrix £2 -£2 21 11 22 21 £2~1£2 that w i l l maximize the second term. With these and us i n g 11 12 the f i r s t e s timates £, and £2 we can e a s i l y f i n d the maximum 11 l i k e l i h o o d e stimates of u2 , $2 and $2 . We con t i n u e i n t h i s 12 22 t h f a s h i o n u n t i l the u n c o n d i t i o n a l parameters of the q set have been computed. Now t h a t we have M and J2 we then form where a n d Wi Y = Z + 2 j j o jm 3m 1 M .+ 0 0 " 1 ( Y - M ) \ jm jmo j o o j o j o j t h The c o r r e s p o n d i n g e s t i m a t e d p a r a m e t e r s , f o r t h e j t h o b s e r v a t i o n i n t h e i s a m p l e a r e o b t a i n e d by p a r t i t i o n i n g £ a n d ^ a s DO i i jm a n d = / Q j o o jom jmo jmm F o r e a c h g r o u p i , c a l c u l a t e 71 I n i l 1 = (n Z Y . i j-1 j From (8) i t can be shown that Y" i s the u n r e s t r i c t e d mle of u . i S u b s t i t u t i n g i . ( yp : X ) and the mle i n t o equation (2) we o b t a i n k i i W = Z (1 - ¥ ) T ( A -A BA ) ( ¥ - ¥ ) e i=1 n i n i n i k i j - Z [(1 - Y")TA B I A i=1 n i j * i nj (12) where A = E n i j = 1 n i / i _ 1 'a 0 joo B = k n i / i " ' \ Z Z i-1j=1 ] 0 0 0 -1 B. D e r i v a t i o n of the score s t a t i s t i c The score s t a t i s t i c w i l l be d e r i v e d u s i n g the p a r a m e t r i z a t i o n i n (5) and equations (6) and ( 7 ) . As i n the case of complete data i t can be shown that ( i - i ) _ , i i ~ 1 = 0 , XX Xi// ^  \f/\ Xi// 72 Hence i (v£/,X) = i - 1 v > , X ) S u b s t i t u t i n g the mle we have n1 L - 1 1 j = 1 A o 1 ( Y -/Lt y joo j o jo \ j=l n k / j r 1 k \ n ( Y -u joo j o jo ) ) / and i ( ^ o J ^ o ) - d i a g ni Z j = 1 n o joo 1° o / - 1 T h e r e f o r e W = Z u i=1 n i / ^ i - 1 i Z j = 1 n ( Y - M ) joo j o jo \ n i U " 1 - 1 n i / ^ i " 1 i . Z 0 Z n ( Y - M ) j = 1 joo j = 1 joo j o jo 1° \ 0 / But Y - M = / Y - u j j o jo £2 J2~ 1 ( Y - M ) jmo joo jo jo \ 73 / i \ (y - M ) jo j o jmo joo Si I Si" 1 0 joo 0 0 /y - % * jo j o 0 / ' i r 1 (y - M ) joo j o j o 0 Thus / sr 1 ( Y — M )\ = n _ 1 (Y - u) joo j o jo 0 S u b s t i t u t i n g t h i s equation i n W we f i n a l l y o b t a i n u k i ^ * i W = Z (? - M ) T n - 1 s " 1J2~ 1 (7 - M) u i= 1 i (13) where S = n i (n ) ~ 2 Z i j - l P " 1 \ ' 0 0 » joo 0 0 U n l i k e the complete data case, there does not seem to be a 74 s i m p l i f i c a t i o n of W which would make p o s s i b l e a comparison e between W and W . One way to compare them would be to use a u e r e a l data s e t and compute both s t a t i s t i c s u s i n g equations (12) and (13). T h i s was not done here s i n c e the programming of the two equations would have taken a c o n s i d e r a b l e amount of time and was beyond the scope of t h i s t h e s i s . 

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