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Survival and growth curve analyses applied to a barnacle data set 1983

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SURVIVAL AND GROWTH CURVE ANALYSES APPLIED TO A BARNACLE DATA SET By VIVIEN FREUND B . A . , The U n i v e r s i t y o f Durham, 1981 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES ( I n s t i t u t e of A p p l i e d M a t h e m a t i c s and S t a t i s t i c s ) 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 THE UNIVERSITY OF BRITISH COLUMBIA December 1983 © V i v i e n F r e u n d , 1983 I n 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 o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e 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 s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C olumbia 1956 Main Mall V a n c ouver, Canada V6T 1Y3 Date DE-6 (3/81) ABSTRACT A d a t a s e t r e l a t i n g t o the s u r v i v a l and growth of b a r n a c l e s i s e x a m i n e d . S u r v i v a l d i s t r i b u t i o n s a r e compared by means of t h r e e n o n p a r a m e t r i c t e s t s . The e x p o n e n t i a l model i s t h e n f i t t e d f o r the s u r v i v a l d i s t r i b u t i o n s and a random e f f e c t s model i s d e v e l o p e d f o r the s l o p e . P o l y n o m i a l growth c u r v e s a r e f i t t e d and v a r i o u s h y p o t h e s e s r e l a t i n g t o the p a r a m e t e r s a r e t e s t e d u s i n g f i r s t l y the model of P o t h o f f and Roy and s e c o n d l y the model of Rao . Owing to the n a t u r e of t h e g rowth d a t a , w h i c h i s no t l o n g i t u d i n a l , t h i s p r e s e n t s v a r i o u s s t a t i s t i c a l p rob lems w h i c h a r e d i s c u s s e d . A/j John P e t k a u T h e s i s S u p e r v i s o r - i i i - TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i LIST OF TABLES i v LIST OF FIGURES v ACNOWLEDGEMENTS v i 1. INTRODUCTION 1 1.1 The E x p e r i m e n t 3 1.2 The D a t a 5 2. EXAMINATION OF THE SURVIVAL DATA 6 2.1 D e s c r i p t i o n of the S u r v i v a l D a t a 8 2.2 E x p o n e n t i a l Mode l s 19 2.3 I n f e r e n c e on the B 's 31 2.4 Random E f f e c t s M o d e l 36 2.5 I n c o r p o r a t i n g N e t t i n g and D o w l i n g L i n e s i n t o the M o d e l . . 46 3. EXAMINATION OF THE GROWTH DATA 52 3.1 Growth D a t a 52 3.2 Growth Curves f o r Items 54 3 .3 The Growth Curve M o d e l s 56 3.4 A p p l i c a t i o n of the Mode l s 64 3.5 System One O y s t e r L i n e s 67 3 .6 System Two O y s t e r L i n e s 76 3.7 A p p l i c a t i o n of the M o d e l t o a l l S e v e n t e e n L i n e s 83 3 .8 E x t e n s i o n of the M o d e l t o A l l o w D i f f e r e n t C o v a r i a n c e M a t r i c e s f o r D i f f e r e n t I tems 91 3 .9 C o l l a p s e Over L i n e s W i t h i n each S y s t e m - t y p e C o m b i n a t i o n . 95 3 .10 Misusing °ata 98 3.11 Growth C u r v e s f o r I n d i v i d u a l B a r n a c l e s 101 3.12 D i s c u s s i o n 107 CONCLUSION 114 BIBLIOGRAPHY 116 - i v - L I S T OF TABLES Page T a b l e 1 S u r v i v a l Da ta 7 T a b l e 2 R e s u l t s of F i t t i n g the E x p o n e n t i a l Mode l S t a r t i n g a t t = 0 23 T a b l e 3 R e s u l t s of F i t t i n g the E x p o n e n t i a l M o d e l S t a r t i n g a t t = 5 25 - v - L I S T OF FIGURES Page F i g u r e 1 Growth Curves f o r the System One O y s t e r L i n e s 73 F i g u r e 2 Growth C u r v e s f o r the System Two O y s t e r L i n e s 73 F i g u r e 3 Growth Curves f o r the Dowl ing L i n e s 87 F i g u r e 4 Growth Curves f o r the N e t t i n g L i n e s 87 - v i - ACKNOWLEDGEMENTS I would l i k e to e x p r e s s my g r a t i t u d e to D r . John P e t k a u f o r a l l t h e t ime he spent i n h e l p i n g t o p r o d u c e t h i s t h e s i s . I am i n d e b t e d to D r . Nancy R e i d f o r h e r sympathy and encouragement a l o n g the way, and f o r r e a d i n g the t h e s i s . Many t h a n k s a l s o to He lene C repeau who a lways had t ime t o h e l p . The f i n a n c i a l s u p p o r t of the C a n a d i a n Commonwealth S c h o l a r s h i p and F e l l o w s h i p Committee i s g r a t e f u l l y a c k n o w l e d g e d . 1. INTRODUCTION The e x a m i n a t i o n of a p a r t i c u l a r d a t a s e t l e a d s , d u r i n g the c o u r s e i o f t h i s t h e s i s , t o the d i s c u s s i o n and a p p l i c a t i o n of a number o f s t a t i s t i c a l t e c h n i q u e s . These t e c h n i q u e s f a l l under the two g e n e r a l h e a d i n g s o f s u r v i v a l a n a l y s i s and growth c u r v e s . The d a t a s e t w h i c h was i n v e s t i g a t e d c o n s i s t e d of s u r v i v a l and growth d a t a f o r b a r n a c l e s . B a r n a c l e s a r e l i t t l e c r e a t u r e s which l i v e i n the o c e a n . The f i r s t p a r t of t h e i r l i v e s i s spent l o o k i n g f o r a p l a c e to l i v e , w h i c h w i l l be some o b j e c t i n the ocean such as a r o c k , a boat or a w h a l e . They u s u a l l y choose a p l a c e where t h e r e are o t h e r b a r n a c l e s . I t i s i m p o r t a n t t h a t they choose a p l a c e w i t h s t r o n g c u r r e n t s to b r i n g i n p l a n k t o n f o r f o o d . Once i t has chosen a r o c k , or some o t h e r o b j e c t , the b a r n a c l e a t t a c h e s i t s e l f f i r s t u s i n g a r e l a t i v e l y m i l d g l u e but s u b s e q u e n t l y u s i n g a d u l t g l u e w h i c h i s e x t r e m e l y s t r o n g . The b a r n a c l e can t h e n n e v e r a g a i n move. The g l u e w h i c h they use i s so s t r o n g t h a t , f o r example , i t i s e a s i e r to c h i p the r o c k t h a t the b a r n a c l e i s a t t a c h e d t o , than t o p u l l o f f the b a r n a c l e . I t has to be s t r o n g because of the f o r c e s t h a t the b a r n a c l e s have t o s t a n d up to i n the ocean s u r f . I f the b a r n a c l e c h o o s e s a bad p l a c e , where t h e r e i s not much f o o d , i t w i l l s t a r v e to d e a t h . I n any case i t might be e a t e n by p r e d a t o r s . The a d u l t b a r n a c l e l o o k s a l i t t l e l i k e a c lam c o v e r e d w i t h a p r o t e c t i v e s h e l l and w i t h a " s t a l k " pr "neck" w i t h which i t a t t a c h e s i t s e l f t o the o b j e c t . The s t a l k s e r v e s to keep the r e s t of the b a r n a c l e away f rom the r o c k . Organs which look a l i t t l e l i k e f e e t , p r o t r u d e f rom - 2 - the bottom of the s h e l l . It i s with these that the barnacle eats. The adult barnacle i s approximately 4-5 cm long. Mr. Harry Goldberg, for his master's thesis i n bioresource engineering at U.B.C. conducted an experiment r e l a t i n g to the s u r v i v a l and growth of barnacles. The barnacles were attached to l i n e s of three d i f f e r e n t types of material. These l i n e s were arranged i n two systems. Barnacles were dying due to lack of food and because of predators. At each of a number of times data was c o l l e c t e d on these barnacles, namely the number dead on each l i n e , s h e l l length, neck length, s h e l l weight and neck weight. Mr. Goldberg was interested i n modelling the s u r v i v a l d i s t r i b u t i o n and i n comparing the d i s t r i b u t i o n s for the three d i f f e r e n t types of l i n e and for the two systems. He was also interested i n f i t t i n g growth curves for each l i n e for s h e l l length and neck length and i n comparing the curves obtained across systems and material types. F i n a l l y he was interested i n the length-weight r e l a t i o n s h i p for s h e l l and neck. From our point of view the s t a t i s t i c a l methodology i s of more in t e r e s t than t h i s p a r t i c u l a r data set, so the length-weight r e l a t i o n s h i p , which involves straightforward regression, w i l l not be discussed. Also growth curves w i l l be f i t t e d only for s h e l l length, not for neck length. Everything that i s done for s h e l l length could be done i n an i d e n t i c a l way for neck length. Due to the l i m i t e d nature of the growth data - l o n g i t u d i n a l data was not c o l l e c t e d - i t was not straightforward to f i t meaningful growth curves and this presented i n t e r e s t i n g s t a t i s t i c a l problems. The remainder of Section 1 i s devoted to a f u l l e r d e s c r i p t i o n of the experiment and the data. In Section 2 the s u r v i v a l data i s - 3 - e x a m i n e d : f i r s t l y the s u r v i v a l d i s t r i b u t i o n s f o r v a r i o u s d i f f e r e n t l i n e s a r e compared u s i n g n o n p a r a m e t r i c t e s t s . Then the e x p o n e n t i a l model i s f i t t e d f o r each l i n e and the s l o p e s of the r e s u l t i n g c u r v e s a r e compared by means of the l i k e l i h o o d r a t i o t e s t . F i n a l l y random e f f e c t s mode ls a r e d e v e l o p e d f o r the s l o p e s of the c u r v e s . I n S e c t i o n 3 growth c u r v e s a r e f i t t e d f o r s h e l l - l e n g t h f o r each of the s e v e n t e e n l i n e s and v a r i o u s h y p o t h e s e s about the p a r a m e t e r s a re t e s t e d . In t h i s way i t i s d e t e r m i n e d whether sys tem and m a t e r i a l t ype a r e i m p o r t a n t f a c t o r s i n d e t e r m i n i n g growth c h a r a c t e r i s t i c s . However because of the l i m i t e d d a t a a v a i l a b l e , s t r o n g a s s u m p t i o n s have t o be made i n o r d e r to be a b l e t o a p p l y the growth cu rve m o d e l s . W i t h i n e a c h of the s e v e n t e e n l i n e s , t h e r e a r e a number of " i t e m s " ( w h i c h w i l l be d e s c r i b e d l a t e r ) . Growth c u r v e s a r e t h e n f i t t e d s e p a r a t e l y f o r each i t e m w i t h i n each l i n e . An a t tempt i s made to t e s t f o r d i f f e r e n c e s between the i t ems w i t h i n a l i n e , but t h i s i s found t o be i m p o s s i b l e w i t h the d a t a a v a i l a b l e . 1.1 The E x p e r i m e n t S e v e n t e e n l i n e s of t h r e e d i f f e r e n t t y p e s of m a t e r i a l were c o n s t r u c t e d as f o l l o w s : i . O y s t e r l i n e s - r o p e s , e a c h w i t h 10 or 11 o y s t e r s h e l l s a t t a c h e d . i i . D o w l i n g l i n e s - ropes each w i t h 20 p i e c e s of wood ( d o w l i n g ) a t t a c h e d . - 4 - i i i . N e t t i n g l i n e s - l o n g c y l i n d r i c a l p i e c e s of n e t t i n g each w i t h 10 compartments and a p i e c e of ha rd r u b b e r i n e a c h compartment . The s e v e n t e e n l i n e s were a r r a n g e d i n two s y s t e m s , the number of l i n e s i n e a c h sys tem b e i n g g i v e n be low: O y s t e r D o w l i n g N e t t i n g System 1 6 2 1 System 2 6 1 1 The l i n e s were t a k e n out t o the o c e a n , where b a r n a c l e s became a t t a c h e d to the i t e m s on each l i n e . The l i n e s were l a t e r r e t r i e v e d and se t up a l o n g the c o a s t l i n e i n a n a t u r a l e n v i r o n m e n t . At t h i s s t a g e no new b a r n a c l e s c o u l d become a t t a c h e d and t h o s e a l r e a d y a t t a c h e d c o u l d no t move. I n t h i s e n v i r o n m e n t the b a r n a c l e s were d y i n g due t o l a c k of f ood and because of p r e d a t o r s . A r e c o r d was made of the number of b a r n a c l e s i n i t i a l l y (a t t=0) on each l i n e . T h e r e were a p p r o x i m a t e l y 200 b a r n a c l e s i n i t i a l l y on each sys tem one o y s t e r l i n e , a p p r o x i m a t e l y 500 on each sys tem two o y s t e r l i n e , a p p r o x i m a t e l y 1000 on each d o w l i n g l i n e and on the system one n e t t i n g l i n e , and 1600 on the sys tem two n e t t i n g l i n e . The a c t u a l numbers a r e g i v e n i n T a b l e 1. At a number o f subsequent t imes d a t a was c o l l e c t e d on the b a r n a c l e s . - 5 - 1.2 The Data S u r v i v a l Data A t t = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 14 and 17 weeks the number of b a r n a c l e s found dead on each l i n e was r e c o r d e d . The dead b a r n a c l e s were t h e n removed. So f o r example the b a r n a c l e s found dead at t = 7 weeks were t h o s e t h a t had d i e d between t = 6 and t = 7 weeks . No b a r n a c l e s d i e d b e f o r e t = 5 weeks . More d e t a i l e d i n f o r m a t i o n , namely the number of dea ths at each t ime on each i t e m w i t h i n a l i n e , was not a v a i l a b l e t o us a l t h o u g h i t was p r o b a b l y r e c o r d e d at some s t a g e d u r i n g the e x p e r i m e n t . The s u r v i v a l d a t a i s g i v e n i n T a b l e 1. Growth Data At each of the above t i m e s a h a p h a z a r d sample of l i v e b a r n a c l e s was s e l e c t e d f rom each i t e m on each l i n e . I t was assumed t h a t w i t h i n each i t e m each b a r n a c l e had e q u a l p r o b a b i l i t y of b e i n g s a m p l e d . F o r each sampled b a r n a c l e the f o l l o w i n g measurements were t a k e n : ( i ) neck l e n g t h ( i i ) s h e l l l e n g t h The d a t a was r e c o r d e d i n c e n t i m e t r e s . T h e s e b a r n a c l e s were not removed f rom the i t e m s . The sample s i z e was 5 e x c e p t i n the cases of samples f rom i t e m s on n e t t i n g l i n e s where i t was 10. So the t o t a l number of b a r n a c l e s measured at each t ime was: From o y s t e r l i n e s : 2 x 6 x 10 ( o r 11) x 5 = 600 From d o w l i n g l i n e s : 3 x 20 x 5 = 3 0 0 From n e t t i n g l i n e s : 2 x 10 x 10 = 200 So t h e r e i s an enormous amount of growth d a t a . The growth d a t a and the ways i n w h i c h i t i s i n c o m p l e t e , a r e d i s c u s s e d more f u l l y i n S e c t i o n 3 . - 6 - 2. EXAMINATION OF THE SURVIVAL DATA In o r d e r t o o b t a i n the l e n g t h - w e i g h t d a t a , a p p r o x i m a t e l y 40 b a r n a c l e s were removed f rom each type o f m a t e r i a l a t each of a number of t i m e s (on a t o t a l of f i v e o c c a s i o n s ) . In a l l a n a l y s e s r e l a t i n g to the s u r v i v a l d a t a , t h i s f a c t was n e g l e c t e d . I t was assumed t h a t the remova l of t h e s e b a r n a c l e s would not s i g n i f i c a n t l y a f f e c t the a n a l y s e s . T h i s a s s u m p t i o n seemed r e a s o n a b l e s i n c e the b a r n a c l e s removed were chosen a t random and the number removed was v e r y s m a l l compared t o the t o t a l number: i n t o t a l a p p r o x i m a t e l y 200 b a r n a c l e s were removed f rom each type of m a t e r i a l , whereas the t o t a l number of b a r n a c l e s i n i t i a l l y on r e s p e c t i v e l y o y s t e r , d o w l i n g and n e t t i n g l i n e s was 6561, 2969 and 2552. The t o t a l number of b a r n a c l e s on each of t h e s e t y p e s of l i n e at the end o f the s tudy was r e s p e c t i v e l y 1668, 1200 and 1121. The i n i t i a l number of b a r n a c l e s on each l i n e and the number found dead a t each t ime a r e g i v e n i n T a b l e 1. No b a r n a c l e s d i e d b e f o r e t = 5 so the f i r s t t ime at w h i c h b a r n a c l e s were found dead was t = 6 . More d e t a i l e d i n f o r m a t i o n , namely the number of d e a t h s on each i t e m w i t h i n each l i n e was not a v a i a b l e to u s . The f i r s t q u e s t i o n of i n t e r e s t was whether the s u r v i v a l d i s t r i b u t i o n s f o r l i n e s of the same type w i t h i n the same sys tem were e s s e n t i a l l y the same. I f t h i s were found t o be the case i t would be r e a s o n a b l e to c o l l a p s e a c r o s s the s i x o y s t e r l i n e s w i t h i n each sys tem and a c r o s s the two d o w l i n g l i n e s i n sys tem 1. Then the s i t u a t i o n would be s i m p l e r - we would have t h r e e " t r e a t m e n t s " ( t y p e s of m a t e r i a l ) w i t h i n - 7 - Table 1 - Survival Data I n i t i a l Number Found Dead at Week # 6 7 8 9 10 11 14 17 01 204 31 32 33 3 6 4 7 5 02 187 27 30 22 0 8 3 11 9 03 134 22 31 15 1 3 18 1 3 04 137 27 21 19 5 6 7 5 5 05 187 34 26 29 9 6 10 4 8 06 306 40 27 25 9 7 14 11 14 DI 937 128 122 113 32 29 41 44 45 D2 817 127 168 118 37 32 16 50 37 Nl 952 75 226 119 116 19 75 10 7 07 530 30 42 69 34 35 26 39 20 08 180 9 18 12 7 14 7 9 17 09 579 24 40 39 47 78 49 37 17 010 155 14 16 28 9 9 1 7 4 011 677 20 61 80 35 62 30 67 16 012 470 7 22 33 28 38 21 33 25 D3 1215 50 89 126 77 96 61 99 32 N2 1600 19 23 118 216 175 110 75 48 - 8 - each of two systems instead of the present more complicated s i t u a t i o n of l i n e s within treatment within system. The l i n e e f f e c t would be known to be i n s i g n i f i c a n t . Three non-parametric rank tests were performed i n order to determine whether differences existed between the s u r v i v a l d i s t r i b u t i o n s of various groups of l i n e s . 2 .1 Description of the Non-Parametric Tests a. Logrank (Savage) Test This test can be derived either as a l i n e a r rank test or i n the context of Cox's proportional hazards model. It w i l l f i r s t be considered from the l a t t e r point of view: The proportional hazards model (Cox, 1972) for f a i l u r e time i s s p e c i f i e d by the hazard function X(t;z) = \o(t) exp (zp) where T i s f a i l u r e time, t i s the observed value of T, z ( l x p) i s a vector of covariates, B(p x 1) i s a vector of regression c o e f f i c i e n t s and Xo(t) i s an a r b i t r a r y unspecified base-line hazard function. Let U(p x 1) be the vector whose j t h element i s given by ologL j = 1 > PS where L i s the l i k e l i h o o d for 6. Let I(p x p) be the matrix whose ( j , h ) t h element i s given by j,h = 1 ,••• , p. - 9 - Then a t e s t of the n u l l h y p o t h e s i s H Q:8 = 8 0 i s based on the s c o r e s t a t i s t i c U(8o). Under Ho and o t h e r m i l d r e g u l a r i t y c o n d i t i o n s U(6o) i s a s y m p t o t i c a l l y n o r m a l w i t h mean 0 and c o v a r i a n c e m a t r i x w h i c h can be c o n s i s t e n t l y e s t i m a t e d by I(Bo). Note t h a t i s b e i n g used t o denote t h e o b s e r v e d F i s h e r i n f o r m a t i o n m a t r i x . To f i n d U and I the l i k e l i h o o d must be o b t a i n e d : L e t the d i s t i n c t d e a t h t imes be t ( ^ ) < . . . < t ( k ) * L e t dj[ be the number of dea ths a t t(±)* L e t n^ be the number a t r i s k j u s t p r i o r t o t ^ ^ . I n the case where t i e s and c e n s o r i n g a r e a l l o w e d the l i k e l i h o o d i s g i v e n by ( K a l b f l e i s c h and P r e n t i c e , 1980): k L = n i=i e x p C s ^ ) T. e x p C s ^ ) ^ R d i ( t ( i ) ) where : s^ i s the sum of the c o v a r i a t e s a s s o c i a t e d w i t h the d^ d i f a i l u r e s a t s^ = T, , 1 = (1^,... l d ) , ^ . ( t ^ ) i s the se t of a l l i n d i v i d u a l s a t r i s k at t ^ - 0 and ( t ( ^ ) ) i s t n e s e t of a l l i s u b s e t s of d^ i t e m s c h o s e n f rom R(t(£)) w i t h o u t r e p l a c e m e n t . The e x a c t p a r t i a l l i k e l i h o o d w i t h t i e s a r i s e s f rom a d i s c r e t e model s p e c i f i e d by X ( t ; z ) d t \ j ( t ) d t 1 - \ < t ; z ) d t = 1 - \ , < t ) d t e x p ( z P ) a - 1 0 - T h i s l i k e l i h o o d i s v e r y l a b o r i o u s to compute . P e t o ( 1 9 7 2 ) and B r e s l o w ( 1 9 7 4 ) s u g g e s t e d the f o l l o w i n g a p p r o x i m a t i o n to L: L = n e x p ( s , 8 ) / a i = l 1 R ( t ( i ) ) e x p ( z 8) T h i s a p p r o x i m a t i o n i s good p r o v i d e d t h a t d±/n± 1 S s m a l l , i = l , . . . , k . T h i s means t h a t a t e a c h t ime the number o f f a i l u r e s must be s m a l l compared to t h e number a t r i s k . o l o g L a > j ~ 1 , . . . , p. The t e s t s t a t i s t i c i s t h e n U ( 8 0 ) where U . ( B ) = 3 98j U(Bg) i s a s y m p t o t i c a l l y normal w i t h mean 0 and c o v a r i a n c e m a t r i x e s t i m a t e d by I ( 8 n ) where I . , (8) j n -b' l o g L 8 B j b K a ' j , h = 1 , . . . , p . In p a r t i c u l a r , to t e s t H o : 8 = 0 ( t h a t f a i l u r e t i m e s are u n r e l a t e d to the c o v a r i a t e s ) , U ( 0 ) ' V - 1 U ( 0 ) i s compared to X 2 ( p ) t a b l e s where U ( 0 ) = k i = l d i n i and V . , = I . , ( 0 ) , 3h j h I n the case where the s u r v i v a l d i s t r i b u t i o n s f o r samples f rom p + 1 d i f f e r e n t p o p u l a t i o n s a re to be compared, z ( l x p ) i s an i n d i c a t o r v a r i a b l e : z^ c o n s i s t s of a one i n the j t h p o s i t i o n and z e r o s e l s e w h e r e i f the i t h o b s e r v a t i o n i s i n the j t h s a m p l e . In t h i s case t e s t i n g H Q : 6 = 0 i s e q u i v a l e n t to t e s t i n g t h a t a l l the p o p u l a t i o n s have - l i - t h e same s u r v i v a l d i s t r i b u t i o n . In t h i s case k U . ( 0 ) = E ( d . . - J i = l J n j i d i / n i ) k d . ( n . - d . ) n, ' h i and = 1 ,••• ,p where 6 j n i s the K r o n e c k e r d e l t a ( 6 j n = 1 i f j = h and 0 o t h e r w i s e ) , n-ji = number a t r i s k i n the j t h sample j u s t p r i o r t o and d j ^ = number of d e a t h s a t t ime t ( i ) i t i t h e j t h s a m p l e . T h i s t e s t can a l s o be d e r i v e d as a l i n e a r rank t e s t f o r compar ing s u r v i v a l d i s t r i b u t i o n s w h i c h d i f f e r o n l y w i t h r e s p e c t t o l o c a t i o n ( L a w l e s s , 1982) : suppose we have p + 1 d i s t r i b u t i o n s d e f i n e d by p r o b a b i l i t y d e n s i t y f u n c t i o n s ( p . d . f . ' s ) g ( y - 9 i ) , . . . g (y - 0 p ) , g ( y ) where y = l o g l i f e t i m e . We w i s h t o t e s t t h a t 01 = . . . = 0 p = 0 i . e . t h a t a l l d i s t r i b u t i o n s a r e i d e n t i c a l . L e t z ( l x p) be an i n d i c a t o r v a r i a b l e ( Z - J J = 1 i f the i t h o b s e r v a t i o n i s f rom sample j and z^-j = Q o t h e r w i s e ) . L e t 0 = ( 0 1 , . . . , 0 p ) . T h e n , g i v e n the r e g r e s s i o n v e c t o r z , the p . d . f . of y i s f ( y | z ) = g ( y - z 0 ' ) . Under t h i s model we w i s h t o t e s t 0 = 0 . We f i r s t c o n s i d e r the case w i t h no c e n s o r i n g : suppose we o b t a i n a sample y i , . . . , y n f rom t h e s e d i s t r i b u t i o n s . L e t y ( i ) ^ ••• ^ y ( n ) ^ e the o r d e r e d o b s e r v a t i o n s (assumed to be d i s t i n c t ) and be the number of o b s e r v a t i o n s f rom d i s t r i b u t i o n i I f r (1 x n) i s the rank v e c t o r based on the y-j/s, then a t e s t of 0=0 can be based on U(0) (p x 1) whose j t h e lement i s g i v e n by ( i = l , . • •, P + D . - 12 - U.(0) = & l o y> ( r ; 0 ) where p ( r ; 0) i s the p r o b a b i l i t y mass f u n c t i o n of r . I t f o l l o w s t h a t U.(0) = £ Z ( i ) . a ± j - 1 , . . . ,p where z (^) i s the i n d i c a t o r v a r i a b l e a s s o c i a t e d w i t h y(^)> z ( i ) j i s the j t h e lement of z ( j j and the a-^'s a r e s c o r e s g i v e n by - g ' ( y ( i ) ) g ^ ( i ) ) To d e f i n e a p a r t i c u l a r rank t e s t , the o^'s a r e g e n e r a t e d by c h o o s i n g a s p e c i f i c p . d . f . g ( y ) . Then i f the d a t a a c t u a l l y comes f rom t h i s p . d . f . the t e s t w i l l be a s y m p t o t i c a l l y f u l l y e f f i c i e n t r e l a t i v e t o the p a r a m e t r i c p r o c e d u r e based on the a c t u a l v a l u e s r a t h e r t h a n the r a n k s of the y±'s. I f the d a t a a r i s e s f rom a d i f f e r e n t p . d . f . the rank t e s t w i l l be more e f f i c i e n t . In the case of the l o g r a n k t e s t the s c o r e s a r e g e n e r a t e d by l e t t i n g g ( y ) = e x p ( y - e v ) , the extreme v a l u e d i s t r i b u t i o n . The mean and c o v a r i a n c e m a t r i x f o r U(0) can be o b t a i n e d by p e r m u t a t i o n t h e o r y arguments ( L a w l e s s , 1 9 8 2 ) . The a^'s can be chosen so t h a t E ( U ( 0 ) ) = 0 . L e t V be the c o v a r i a n c e m a t r i x of U ( 0 ) . Then s i n c e U(0) i s a s y m p t o t i c a l l y n o r m a l , H O : 0 = O can be t e s t e d by compar ing U ( 0 ) ' V - 1 U(0) t o X 2 ( P ) t a b l e s . T h i s t e s t can be ex tended t o accommodate c e n s o r i n g as f o l l o w s : suppose t h a t t h e r e a r e k d i s t i n c t o b s e r v e d l o g l i f e t i m e s and n-k - 13 - c e n s o r i n g t i m e s . I f z(±) i s the i n d i c a t o r v a r i a b l e a s s o c i a t e d w i t h y ( i ) , l e t S ( i ) be the sum of t h e s e v e c t o r s f o r a l l i n d i v i d u a l s c e n s o r e d i n [ y ( i ) > v ( i + l ) ) « Then the s c o r e s t a t i s t i c s u g g e s t e d by P r e n t i c e (1978) i s U(0)(p x 1) where k V 0 ) ( z ( D j a i + s ( D j a i ) } J-I.--.P- So i n d v i d u a l s whose l i f e t i m e s a r e c e n s o r e d a r e g i v e n d i f f e r e n t s c o r e s a^ . The s c o r e s may be chosen so t h a t E[U(0)] = 0. I n p a r t i c u l a r , t a k i n g g ( y ) = e x p ( y - e^) as b e f o r e , and u s i n g P r e n t i c e ' s method of o b t a i n i n g the and a^ l e a d s t o the l o g r a n k t e s t v i a the s c o r e s 1 1 1 1 a i =.E,^7 - l> a i = .\ nT' where n^ = the number at r i s k j u s t p r i o r t o t ( ^ ) = e x p ( y ( . j j . ) Then we o b t a i n as b e f o r e k Uj(0) = - 2 ^ ( d j ± - n j ± d i / n 1 ) j = l , . . . , p , w i t h n o t a t i o n as b e f o r e . The e x p e c t a t i o n of U(0) i s 0. P r e n t i c e o b t a i n s a p e r m u t a t i o n v a r i a n c e f o r U(0). E i t h e r t h i s or the v a r i a n c e o b t a i n e d b e f o r e i n the c o n t e x t of C o x ' s model can be u s e d . S i n c e the s c o r e s were m o t i v a t e d by l e t t i n g g ( y ) = e x p ( y - e ^ ) , the extreme v a l u e d i s t r i b u t i o n , the t e s t i s a s y m p t o t i c a l l y f u l l y e f f i c i e n t f o r d e t e c t i n g l o c a t i o n d i f f e r e n c e s under an extreme v a l u e model f o r l o g l i f e t i m e s or e q u i v a l e n t l y , f o r t e s t i n g e q u a l i t y o f l i f e t i m e d i s t r i b u t i o n s i n a p r o p o r t i o n a l h a z a r d s or Lehmann f a m i l y when t h e r e i s no c e n s o r i n g o r e q u a l c e n s o r i n g i n a l l s a m p l e s . T h i s t e s t was d e r i v e d under the - 14 - a s s u m p t i o n of no t i e s but may be used w i t h a s m a l l number of t i e s . b. W i l c o x o n Test The W i l c o x o n ( o r as i t i s sometimes c a l l e d , P r e n t i c e ' s g e n e r a l i s e d W i l c o x o n t e s t ) can a l s o be d e r i v e d as a l i n e a r rank t e s t of the f o rm k U . ( 0 ) = ^ ( z ( i ) . a ± + s ( i ) . a ±) J - 1 . . . . . P f o r t e s t i n g e q u a l i t y of l i f e t i m e d i s t r i b u t i o n s . I n t h i s case the scores a r e d e f i n e d as ( P r e n t i c e , 1978) : a , = 1 - 2 n ( n . - d . + l ) / ( n . + 1) j =1 a , = 1 - n ( n . - d . + l ) / ( n . + 1) j = l J J J w i t h n o t a t i o n as b e f o r e . S u b s t i t u t i n g t h e s e s c o r e s i n U j ( u ) g i v e s n . . U . ( 0 ) = - E F , ( d . . - d , ) j = l , . . . , p j i j i i i ( n . - d . + 1) where F, = II — - ^ An e s t i m a t e of the c o v a r i a n c e m a t r i x of U(0) (see P r e n t i c e and Marek ( 1 9 7 9 ) ) i s V where k d . ( n . - d . ) n . . n, . TT y „2 i i i J i ( f , h i ^ V . , = £ F. — 7 - r r — - — 16., - - — I i , h = l , . . . , p . J h i (n^ - 1) n^ j h J J ' • , v - 15 - A g a i n U ( 0 ) ' V ^ l K O ) i s compared t o X 2 ( p ) t a b l e s . The s c o r e s i n t h i s case a r e g e n e r a t e d by t a k i n g g ( y ) = e^/( l + e y ) 2 , the l o g i s t i c d e n s i t y . So t h i s t e s t i s a s y m p t o t i c a l l y f u l l y e f f i c i e n t f o r d e t e c t i n g l o c a t i o n s h i f t s when the u n d e r l y i n g d i s t r i b u t i o n s a r e l o g i s t i c . Whereas the l o g rank t e s t g i v e s e q u a l w e i g h t t o a l l terms ( d j ^ - d ^ n ^ / n ^ ) , the W i l c o x o n g i v e s more w e i g h t to e a r l i e r e v e n t s than t o l a t e r o n e s . Hence t h i s t e s t i s good a t d e t e c t i n g d i f f e r e n c e s e a r l y o n . A g a i n i t was d e r i v e d under the a s s u m p t i o n of no t i e s , but may be used w i t h a s m a l l number of t i e s . c . L o g r a n k T e s t f o r Grouped D a t a I n the d e r i v a t i o n of the l o g rank t e s t i n the c o n t e x t of the p r o p o r t i o n a l h a z a r d s model i t was assumed t h a t i f t i e s were p r e s e n t t h i s was because the d a t a a r o s e f rom a d i s c r e t e m o d e l . I t may i n s t e a d be the c a s e t h a t the d a t a a r i s e s f rom a c o n t i n u o u s model but t h a t i t i s g rouped - the a c t u a l s u r v i v a l t ime i s not r e c o r d e d , o n l y the i n t e r v a l i n t o w h i c h i t f a l l s . T h i s would g i v e r i s e t o a s l i g h t l y d i f f e r e n t l i k e l i h o o d and thus t o a s l i g h t l y d i f f e r e n t t e s t than t h a t o b t a i n e d i n a , as w i l l now be d e s c r i b e d . C e n s o r i n g i s assumed o n l y to o c c u r j u s t p r i o r t o the end of an i n t e r v a l . Assuming a p r o p o r t i o n a l h a z a r d s model f o r the c o n t i n u o u s d a t a , t h e n , i f x± r e p r e s e n t s f a i l u r e i n t h e i t h i n t e r v a l , [ a ^ - i , a ^ ) , t h e h a z a r d c o n t r i b u t i o n a t x-£ f o r c o v a r i a t e z i s 1 - (1 - X, ) e X p ( z p ) , where (1 - X . ) = exp [ -/ 1 X (u) du] and X ( t ) 1 l a . . o o - 16 - i s the b a s e - l i n e h a z a r d f u n c t i o n . L e t y± = l o g [ - l o g ( l - \ ± ) ] , 1 = 1 , . . , k . Then the l i k e l i h o o d i s k L ( y , B ) = £ ( E l o g { l - e x p [ - e x p ( Y , + z B ) ] } - £ e x p ( y , + z B ) ) i = l 1€D ± 1 16 R 1 x where i s the se t of l a b e l s a t t a c h e d t o i n d i v i d u a l s f a i l i n g a t x^ and R̂  i s the s e t of l a b e l s a t t a c h e d t o i n d i v i d u a l s c e n s o r e d a t x^ or o b s e r v e d t o s u r v i v e p a s t x^. A t e s t of Ho:8=0 i s based on u ( y ( 0 ) , 0) where U . ( y , 8 ) = -rs— l o g L and y(0) i s the maximum l i k e l i h o o d e s t i m a t e of y a t 6 = 0 . A g a i n u ( y ( 0 ) , 0) i s a s y m p t o t i c a l l y n o r m a l w i t h mean 0 and 2 c o v a r i a n c e m a t r i x e s t i m a t e d by l ( y ( 0 ) , 0) where ^JJJCYJP) = • Suppose t h a t t h e r e a re p + 1 s a m p l e s . When z i s an i n d i c a t o r v a r i a b l e f o r the samples i t f o l l o w s t h a t U J ( Y ( 0 ) , 0) = - E n±/d± l o g ( l - d . / n i ) ( d j ± - d ± n j ± / n ± ) j = l , . . . , p and the e l e m e n t s of the c o v a r i a n c e m a t r i x a r e q i V j h = E ( n 7 ) n j i ( 6 j h " n h i / n i ) j ' h = 1 P x i J n i ( n i " d i } r , „ d i ^ 2 where -l± = — a : ( l o g ( 1 - S 7 » and o t h e r n o t a t i o n i s as b e f o r e . The t e s t s t a t i s t i c i s U ( y ( 0 ) , 0 ) ' V - 1 U ( y ( 0 ) , 0) w h i c h can be compared t o X ( p ) 2 t a b l e s . n i d i I f d . / n . i s s m a l l t h e n - j — l o g ( l ) » -1 and the t e s t i s i l i i - 17 - a p p r o x i m a t e l y the same as the l o g r a n k t e s t i n a . A p p l i c a t i o n o f t h e T e s t s I n our case we have c o n t i n u o u s d a t a wh ich has been grouped - the a c t u a l s u r v i v a l t i m e s a r e not r e c o r d e d , o n l y the i n t e r v a l i n t o w h i c h they f a l l . T h i s g i v e s r i s e to a l a r g e number of t i e s . However t h e r e a r e a l s o a l a r g e number at r i s k and d^/n^ i s r a r e l y b i g g e r than 0.2 and u s u a l l y much s m a l l e r . C e n s o r i n g i s due o n l y to the f a c t t h a t some b a r n a c l e s a r e s t i l l a l i v e a t the end o f the s t u d y - t h i s i s t ype I c e n s o r i n g . So we have the same c e n s o r i n g p a t t e r n i n a l l s a m p l e s . The u n d e r l y i n g d i s t r i b u t i o n of the s u r v i v a l t i m e s i s not known so i t i s not c l e a r whether the l o g r a n k t e s t o r the W i l c o x o n t e s t w i l l be more p o w e r f u l . We do know, however , t h a t the W i l c o x o n i s more s e n s i t i v e t o d i f f e r e n c e s e a r l y on whereas the l o g r a n k i s more s e n s i t i v e t o d i f f e r e n c e s l a t e r . I f t h e l o g r a n k t e s t i s t o be used i t would be more a p p r o p r i a t e t o use the t e s t d e s c r i b e d i n c . as we have grouped d a t a . The l o g r a n k t e s t a . on t h e o t h e r hand assumes a d i s c r e t e m o d e l . A l s o the a p p r o x i m a t i o n to the l i k e l i h o o d used i n the d e r i v a t i o n of t h i s t e s t assumed s m a l l v a l u e s of d-^/n^. I t i s not c l e a r whether our v a l u e s o f d^/n^ a r e s u f f i c i e n t l y s m a l l . The W i l c o x o n was d e r i v e d under the a s s u m p t i o n of no t i e s and i s o n l y an a p p r o x i m a t i o n when, as i n our c a s e , a l a r g e number o f t i e s a r e p r e s e n t . \ - 18 - E a c h t e s t was a p p l i e d t o v a r i o u s g rous of l i n e s t o t e s t the e q u a l i t y of the s u r v i v a l d i s t r i b u t i o n s of the l i n e s , w i t h the f o l l o w i n g r e s u l t s : L o g r a n k f o r L i n e s Compared L o g r a n k W i l c o x o n grouped d a t a 01,02,03,04,05,05 2 X = 5 34 .7 2 X = 5 31 .8 2 X = 5 3 5 . 9 07,08,09.0io,0n,0i2 2 X = 5 2 8 . 9 2 X = 5 36 .6 2 X = 5 3 3 . 6 D l , D 2 2 X = 1 32 .5 2 X = 1 29 .3 2 X = 1 32 .9 Dl.D2.D3 2 X = 2 129.8 2 X = 2 164.2 2 X = 2 126.7 Ni,N 2 2 X = 1 129.8 2 X = 1 301.6 2 X = 1 196.2 D 3 , N 2 2 X = 1 5.8 2 X = 1 12.6 2 X = 1 5.1 A l l p - v a l u e s a r e < .001 e x c e p t t h a t o b t a i n e d i n the c o m p a r i s o n of D3 and N2. Here the l o g r a n k t e s t f o r grouped d a t a g i v e s p = . 0 2 5 , the l o g r a n k g i v e s p - .017 and the W i l c o x o n g i v e s p < . 0 0 1 . F o r each group of l i n e s , the h y p o t h e s i s t h a t t h e i r s u r v i v a l d i s t r i b u t i o n s a r e the same i s s t r o n g l y r e j e c t e d . The p v a l u e s a r e v e r y s m a l l p o s s i b l y due t o the l a r g e amount of d a t a . The s m a l l e s t p - v a l u e s o c c u r when l i n e s f rom d i f f e r e n t s y s t e m s , f o r example Ni and N2, a r e compared . On the o t h e r hand the s u r v i v a l d i s t r i b u t i o n s f o r D2 and N3, l i n e s of d i f f e r e n t t y p e s w i t h i n the same s y s t e m , appear t o be much more s i m i l a r j u d g i n g by the - 19 - r e l a t i v e l y s m a l l c h i - s q u a r e d v a l u e . T h i s s u g g e s t s t h a t sys tem may be a more i m p o r t a n t f a c t o r than m a t e r i a l t ype i n d e t e r m i n i n g the s u r v i v a l d i s t r i b u t i o n . In any case d i f f e r e n c e s e x i s t even between l i n e s of the same type w i t h i n a sys tem and i t i s c l e a r l y not r e a s o n a b l e to c o l l a p s e o v e r t h e s e g r o u p s . The d i s c r e p a n c y between the r e s u l t s of the two l o g r a n k t e s t s i s s u r p r i s i n g l y s m a l l e x c e p t i n the c o m p a r i s o n of N i and N2. T h i s c o u l d be b e c a u s e d-j/n^ i s a lways q u i t e s m a l l i n s p i t e o f the l a r g e number of t i e s . F o r the f i r s t t h r e e compar i sons the W i l c o x o n g i v e s a s i m i l a r r e s u l t to the l o g r a n k t e s t s but f o r the l a s t t h r e e compar i sons i t g i v e s a much s m a l l e r p - v a l u e than the o t h e r two t e s t s . T h i s c o u l d r e f l e c t the f a c t t h a t the W i l c o x o n i s b e t t e r at d e t e c t i n g d i f f e r e n c e s e a r l y o n , w h i c h i s where the b i g g e s t d i f f e r e n c e s l i e i n t h e s e c a s e s . F o r example the p e r c e n t a g e d y i n g between weeks 6 and 7 on l i n e N i i s 23.7 whereas on N 2 t h i s p e r c e n t a g e i s 1 .4 . L a t e r on the d i s c r e p a n c i e s a r e not as g r e a t . 2.2 E x p o n e n t i a l M o d e l s H a v i n g compared the s u r v i v a l d i s t r i b u t i o n s f o r d i f f e r e n t l i n e s n o n p a r a m e t r i c a l l y , i t was t h e n o f i n t e r e s t t o l o o k f o r a p a r a m e t r i c model t o f i t the s u r v i v a l c u r v e s . T h e r e were two r e a s o n s f o r d o i n g t h i s . F i r s t l y the shape of the s u r v i v a l c u r v e s was of i n t e r e s t i n i t s e l f and s e c o n d l y under a p a r a m e t r i c model the t a s k of compar ing s u r v i v a l d i s t r i b u t i o n s f o r d i f f e r e n t l i n e s would be s i m p l e r - i n s t e a d of h a v i n g to compare a l a r g e amount of d a t a , namely number of d e a t h s and - 20 - number at r i s k at each time, we would j u s t have to compare a small number of parameters, for example slope. With so much data i t was l i k e l y that any parametric model would be rejected by a goodness-of-fit t e s t . This shouldn't matter provided that the model captures the most important features of the data. Because of i t s s i m p l i c i t y the exponential was the f i r s t choice of model. The exponential was f i r s t f i t t e d to each of the 17 l i n e s s t a r t i n g at t=0. So the following model was assumed: P(T > t) = -Bt e t > 0 0 otherwise where T = l i f e t i m e of barnacles. For each of the 17 l i n e s the maximum l i k e l i h o o d estimate of 8 was obtained. Notation i s defined as follows: to=0 ti=5 t2=6 ts=14 tg=17 d^ deaths d 2 deaths d 9 deaths N-Zd^ Under t h i s model the l i k e l i h o o d i s proportional to L - (1 - e ^ f l ( e ^ l - e - e ^ .... - e'**')*9 ( e " ^ 9 Since there were no deaths p r i o r to t i = 5 weeks on any of the l i n e s , d i = 0 and the f i r s t term i s i d e n t i c a l l y one. The l a s t term i s the contribution of the censored observations: under our model the p r o b a b i l i t y that a barnacle survives at least u n t i l the end of the study - 21 - at t 9 = 17 i s e ' P ^ . The number that do survive t h i s long i s N-Edi where N i s the i n i t i a l number on the l i n e . To obtain the maximum l i k e l i h o o d estimate of B, note that -Bt L -Bt I d i ^ i - l e + t i e ) = Z ~ * ' - B t TBT t 9 ( N - Ed.) 96 i = 2 Bt j Bt x (e - e ) a 2 i o g L I . , -P<V*i-i> ^ i - i - , E d |-e }. (e - e ) 2 ^ ^ o g L i s c l e a r l y < 0 for a l l 8 since the d are >̂  0 (and c l e a r l y at oB 2 l e a s t one d i s > 0). So the turning point at the so l u t i o n to ̂ ^ ° g L = 0 1 Op i s 8, the maximum l i k e l i h o o d estimate of p. This was obtained by doing a Newton Raphson i t e r a t i o n : P J .1 - P - U p )/ UB ) Mn+1 r n r n K n and 8 L = 1/T where T = median s u r v i v a l time (not exact due to discreteness) and ' _ ologL _ o 2logL When convergence to desired accuracy i s achieved -L(B) i s the observed Fisher information and leads to an estimate of the variance of B: Var 8 = [ - U p ) ] - 1 , SE(B) = [Var p ] 1 / 2 A f t e r obtaining 6, two goodness of f i t tests were ca r r i e d out to check - 22 - the f i t of the model for each line. 1. Pearson's Goodness of F i t Test k+1 (0 - E ) 2 X = £ — i = i i=l E i where 0̂  = observed number of deaths in i t h interval and E^ = expected number of deaths in i t h interval -8t -Bt Under our model the estimate of is E^ = N(e - e ) and k + 1 2 2 is the number of time intervals. X i s compared to X (k- s) tables where s = number of parameters estimated in the model (=1). If X is too large the f i t is poor. It wasn't necessary to combine any intervals as for a l l lines the expected frequency in a l l intervals was at least 5. 2. L i k e l i h o o d Ratio Goodness of F i t Test k+1 0, 2 i 2 X = 2 S 0.1og(—) is compared to X,. . tables where i=l 1 h± Ck-s; notation i s as for 1. Results Results are given in Table 2. Comparing to chi-squared tables with eight degrees of freedom, the p-values for the goodness-of-fit tests are a l l < .001. Clearly the model doesn't f i t . The largest residuals were found to be at the beginning which is not surprising since the model gives a positive probability of death for t€ [0,5] but no deaths were T a b l e 2 - R e s u l t s of F i t t i n g E x p o n e n t i a l Mode l S t a r t i n g a t t = 0 . 8 ± SE(8) P e a r s o n ' s X 2 L i k e l i h o o d R a t i o X 2 0 i 0 .0520 + 0.0047 299.3 242 .7 0 2 0.0506 + 0.0048 221 .5 192.4 0 3 0.0672 + 0.0069 263.7 216 .8 0 .6540 + 0.0067 187.7 167 .0 05 0.0623 + 0.0055 251 .9 226 .2 o6 0.0374 + 0.0031 232.2 2 1 5 . 5 D l 0 .0500 + 0.0021 368.1 380 .4 D 2 0.0680 + 0.0028 94 .6 106.8 N i 0 .0638 + 0.0025 506.4 4 7 7 . 0 0 7 0.0444 + 0.0026 183.3 154.6 o8 0.0391 + 0.0041 500.1 508 .6 0 9 0.0455 + 0.0025 218.7 238 .8 OlO 0 .0475 + 0.0051 905.4 8 4 4 . 5 O i l 0 .0432 + 0.0022 1168.2 1008.0 0 l 2 0 .0318 + 0.0022 743 .9 402 .8 D3 0 .0402 + 0.0016 1692.0 1419 .0 N 2 0.0367 + 0.0013 1707.3 1976.0 - 24 - a c t u a l l y observed i n t h i s period. So the same model was then f i t t e d s t a r t i n g at t=5: P(T > t) - -B(t-5) t > 5 « otherwise L e t t i n g t' = t - 5, we have the following s i t u a t i o n t ' - 0 t{ - 1 t' - 9 t' - 12 di deaths d 2 deaths d 8 deaths N-Zd For t h i s model the l i k e l i h o o d i s -Bt' d -BtJ -Bt' d L = (l - e ) (e - e ' ,"2 , " B t J - B t8> d8 , - B t 8 ^ M l J [e " e J (e J where t j =1, t£ = 2 t£ = 6, t j =9, t£ = 12. Now i v i - i -at-1 i=2 so T 8 d.C-t! , e ologL _ v i - i 96 " 4_. . -Bt! -Bt;., -pt' -BtJ + t' e (e - e J Z | t T - tJCN - Ed.). (1 - e ') The new maximum l i k e l i h o o d estimates of p were obtained and the goodness-of-fit tests repeated. Results appear i n Table 3. The p-values f o r the tests are much larger t h i s time but are again a l l < .001. The f i t i s much better but s t i l l poor. The model appears to f i t better for the oyster l i n e s than for the dowling or netting l i n e s , though t h i s might be because there i s a smaller amount of data for the oyster l i n e s . It was possible that a better f i t might be obtained by s t a r t i n g at some point other than t=5. So the same model was f i t t e d again, t h i s time T a b l e 3 - R e s u l t s of F i t t i n g t h e E x p o n e n t i a l M o d e l S t a r t i n g a t t = 5 B ± SE(B) 2 P e a r s o n ' s X L i k e l i h o o d R a t i o X 2 O i 0 .0927 + 0.0084 94 .5 103.0 0 2 0.0883 + 0.0084 6 2 . 5 6 8 . 9 03 0.1290 + 0.0130 77 .2 9 4 . 3 0 4 0.1240 + 0.0130 40 .6 4 5 . 9 05 0 .1160 + 0.0100 59 .3 69 .7 o6 0.0613 + 0.0050 69 .2 7 0 . 5 0 .0867 + 0.0037 80 .8 7 9 . 8 D 2 0.1298 + 0.0054 17.9 18.3 N l 0 .1200 + 0.0047 148.1 141.9 0 7 0.0740 + 0.0043 58 .3 59 .2 o8 0.0630 + 0.0065 127.7 136.9 0 9 0.0755 + 0.0042 54 .1 53 .6 OlO 0 .0818 + 0.0087 221 .8 235 .0 O i l 0 .0713 + 0.0037 242.5 253 .5 0l2 0.0497 + 0.0034 171.8 187.8 D 3 0.0656 + 0.0026 495 .5 5 9 9 . 4 N 2 0.0588 + 0.0021 681 .4 6 3 9 . 8 - 26 - with two unknown parameters - the slope parameter 6 and the loc a t i o n parameter a - both of which are to be determined by maximum l i k e l i h o o d . Since the l i k e l i h o o d i s d i f f e r e n t for a i n d i f f e r e n t time i n t e r v a l s , to A A f i n d a and 8 i t i s necessary to determine a - p r i o r i i n which i n t e r v a l the maximum l i k e l i h o o d estimate of a l i e s . A Fortunately i t was possible to determine a - p r i o r i that a would have to l i e i n the i n t e r v a l (5,6] assuming no deaths i n (0,5] and at least one death i n (5,6). In our case these conditions are s a t i s f i e d since no deaths are observed i n (0,5] but on every l i n e the f i r s t deaths are observed i n (5,6]. A Proof that a € (5,6] For a e (0,5] and no deaths i n (0,5], the l i k e l i h o o d i s i-1 8 -8 ( t 9 - a ) ( N - E d.) i=l 1 e 0 t i - 5 t2=6 t 8-14 t 9=17 0 deaths d^ deaths d 8 deaths N-Ed. Then 8 9 logLj = BctN - 8t 9(N - E d ) + E d . log(e i= l 1 i=2 1 - 1 -Bt i-1 - e ) = G( 8) + pctN - 27 - where G(B) i s a f u n c t i o n of 6 i n d e p e n d e n t of a . S i n c e 8 > 0 , l o g L i i s monoton i c i n c r e a s i n g i n a f o r a £ ( 0 , 5 ] . So a >_ 5. F u r t h e r m o r e a <̂  6 s i n c e a > 6 would mean t h a t t h e r e would be a z e r o p r o b a b i l i t y of d e a t h i n ( 5 , 6 ] , but we have assumed t h a t deaths a r e o b s e r v e d i n t h i s i n t e r v a l , a ^ 5 and a _< 6 t o g e t h e r i m p l y a € ( 5 , 6 ] . m. Determination of a F o r a £ ( 5 , 6 ] the l i k e l i h o o d i s 8 - 8 ( t 2 - a ) J 9 - 6 ( t , , - o ) - B ( t . - a ) d . . - B ( t 9 - a ) ( N - £ d , ) L 2 = (1 -e ) d l n (e 1 1 - e 1 ) 1 1 e 1 1 i=3 to-o t x = 5 t 2 = 6 t 8 = 1 4 t 9 = 1 7 I 4 i i i di d e a t h s dg dea ths N-Ed i S i n c e we have a l r e a d y a s c e r t a i n e d t h a t oc€ ( 5 , 6 ] , t h i s i s the a p p r o p r i a t e l i k e l i h o o d . ^i£|i2 = B (N - dV<l - e - P ( T 2 _ A ) ) ) ?,2, T Q 2 , - p ( t 2 ~ a ) and 9 l o g L 2 = "P d l e . 2 da ^ _ e - 8 ( t 2 - a ) j 2 2 S i n c e & l o g L 2 < 0 f o r a l l a and 8 > 0, the s o l u t i o n t o a i ° g L 2 = 0 a 2 ba da - 28 - y i e l d s a l o c a l maximum. = 0 can be s o l v e d e x p l i c i t l y f o r a i n terms o f 8: So i f we maximise L 2 over a and 8 s i m u l t a n e o u s l y we o b t a i n 1 ,_rN - d P where 8 i s the maximum l i k e l i h o o d e s t i m a t e of 6. S i n c e t2=6 , 8 > 0 and l o g f — < 0, we have a < 6 . Case 1 I f a e ( 5 , 6 ] , (<===> - l n ( N " d * ) 6 ( - 1 , 0 ] ) t h e n a i s the m * N m 8 maximum l i k e l i h o o d e s t i m a t e , a , o f a , s i n c e i t i s the s o l u t i o n t o k 2 . SlogL ̂  = 0 , 0 l o g L ? ^ Q a n ( j L 2 i s the a p p r o p r i a t e l i k e l i h o o d f o r a da ' - 2 oa i n t h i s i n t e r v a l . Case 2 I f a < 5 (<===> — l n ( N „ d J-) < - 1 1 , t h e n a i s not the maximum m v * N ' m P 9a not the a p p r o p r i a t e l i k e l i h o o d f or a K 5. I n t h i s case the t r u e maximum l i k e l i h o o d e s t i m a t e of a . I t i s the s o l u t i o n to & 1 ° f L 2 = 0 but L 2 i s l i k e l i h o o d e s t i m a t e i s a = 5 . - 29 - Proof For given p, ^ ^ ^ 2 . = Q n a s a u n i q u e sol u t i o n , a m , which i s assumed to l i e i n (0,5]. We have shown that the turning point i s a maximum and therefore logL>2 i s monotonic decreasing i n a to the right of a m and i n A A /\ p a r t i c u l a r for a € (5,6], So a <^5. But we know that a & (5,6], A So a = 5. Determination of 8 logL2 was i n i t i a l l y maximised over a and 8 simultaneously to obtain and 8 m. This was done v i a a Newton Raphson i t e r a t i o n as follows: Let L = 5 1 ° g L 2 L = a i °S L2. U e Z a 5a ' 8 58 » = Q 2logl2 L = a 2 l ° g L 2 L = a 2 l o g L 2 . Q a 2 » ap 5a5p ' pp b?2 I n i t i a l estimates were given for a and p. Then the i t e r a t i o n -1 (a ,p ) = (a ,8 ) - (L ,Lfl) - -n+1 Kn+1 n' rn a p L * ( a n ' P n ) L L aa ap L a p Lpp /\ A (VBn> was used u n t i l convergence to desired accuracy was achieved. Case 1 Suppose we obtain a m £ (5,6]. Then a and p are the true maximum l i k e l i h o o d estimates a and p. m m In t h i s case an estimate of the covariance matrix of (a, p) i s given by: - 30 - L L o" aa aB L a B L B B (a ,P) _1 t h e i n v e r s e of t h e o b s e r v e d F i s h e r i n f o r m a t i o n . I n our c a s e , s i n c e A 2 = 0 can be s o l v e d e x p l i c i t l y t o f i n d a = a ( B ) , we c o u l d i n s t e a d have s u b s t i t u t e d f o r a i n terms of 8 i n ^°f^^ = 0 . Then we would have op had j u s t one e q u a t i o n i n one unknown. T h i s c o u l d have been s o l v e d by a A Newton Raphson i t e r a t i o n to f i n d 8 and i t s s t a n d a r d e r r o r . Then A A A a = a(B) and the v a r i a n c e of a would have been e s t i m a t e d by ~ L 8 8 ( L a a L B 8 " L Case 2 ' 0 2 ) _ 1 | * * p (a,6) Suppose we o b t a i n a_ < 5. Then a and 8 a r e not the t r u e r m m "m maximum l i k e l i h o o d e s t i m a t e s . As a l r e a d y shown, a = 5. Knowing t h a t A a = 5, we can t h i n k o f a as f i x e d . We max imise w i t h r e s p e c t to 8 g i v e n t h a t a = 5. T h i s means t h a t when we f i t t e d the e x p o n e n t i a l s t a r t i n g a t t = 5, we a l r e a d y had the b e s t p o s s i b l e f i t as t = 5 was the b e s t p o i n t A a t w h i c h to s t a r t . So 8 i s as o b t a i n e d i n t h i s p r e v i o u s a n a l y s i s . As 2N-1 f o r case 1 t h e v a r i a n c e of 6 i s e s t i m a t e d by - L (L L„„ - L „ )" K aa aa pp ap (a ,B) The X v a l u e s a r e the same as t h o s e o b t a i n e d i n the p r e v i o u s a n a l y s i s s t a r t i n g a t t = 5. We have one degree of f reedom fewer but t h i s s h o u l d n ' t a f f e c t the c o n c l u s i o n s . - 31 - R e s u l t s A In t h o s e c a s e s f o r w h i c h a > 5 t h e f i t was s l i g h t l y improved by m a x i m i s i n g over two p a r a m e t e r s . T h i s improvement was r e f l e c t e d i n t h e s l i g h t l y s m a l l e r X v a l u e s . F o r seven of the s e v e n t e e n l i n e s i t was found t h a t t = 5 was not the i d e a l p l a c e to s t a r t . The r e s u l t s f o r t h e s e l i n e s a r e as f o l l o w s : A A A A L i k e l i h o o d R a t i o 6 ± S E ( 8 ) a ± S E ( a ) P e a r s o n ' s X 2 X 2 o 7 0 .0763 + .0047 5.24 ± .147 76 .1 7 7 . 8 o 8 0.0646 + .0071 5.21 ± .279 16 .9 17 .8 0 9 0 .0805 + .0046 5.47 ± .112 129 .2 130 .8 O i l 0 .0774 + .0041 5.61 ± .089 99 .7 114 .2 O 1 2 0 .0541 + . 0038 5.72 ± .107 3 7 . 3 36 .7 N i 0.1281 + .0054 5 .36 ± .079 454 . 2 584 .8 N 2 0 .0624 + .0019 5.81 ± .152 682 .1 580 .3 V a l u e s of a a r e g e n e r a l l y s m a l l e r f o r the system one o y s t e r l i n e s and f o r the d o w l i n g l i n e s than f o r o t h e r l i n e s . I t a p p e a r s t h a t b a r n a c l e s on t h e s e l i n e s s t a r t d y i n g s o o n e r . However the s c a l e pa ramete r 8, w h i c h r e l a t e s t o the r a t e a t w h i c h the b a r n a c l e s d i e o f f , i s of more i n t e r e s t . 2.3 Inference on the p 's A A l t h o u g h the model d o e s n ' t f i t v e r y w e l l , 8 i s a good summary s t a t i s t i c and c o n t a i n s a l o t of i n f o r m a t i o n about the l i n e s . So many - 32 - a n a l y s e s r e l a t i n g to the 8 v a l u e s were then c a r r i e d o u t . A l l of t h e s e a n a l y s e s were based on the e s t i m a t e s of 6 o b t a i n e d by m a x i m i s i n g o v e r two p a r a m e t e r s . P r e v i o u s l y we had compared the s u r v i v a l d i s t r i b u t i o n s o f c e r t a i n g roups o f l i n e s n o n p a r a m e t r i c a l l y . Now we were i n a p o s i t i o n to compare the d i s t r i b u t i o n s of the same groups of l i n e s p a r a m e t r i c a l l y by t e s t i n g whether a l l s y s t e m one o y s t e r l i n e s ( s a y ) had the same u n d e r l y i n g 8 v a l u e . T h i s was done by means of the l i k e l i h o o d r a t i o t e s t as f o l l o w s : To t e s t H 0: 81=82=83=64=65= 86 o 2[logL(j3, a) - logL(j3 0, SLO^ i s compared t o X t a b l e s and H Q i s r e j e c t e d i f t h i s s t a t i s t i c i s t o o l a r g e . Here B j d e n o t e s t h e t r u e s l o p e f o r t h e * j t h o y s t e r l i n e on sys tem one . T h i s t e s t i s v a l i d p r o v i d e d t h a t samples a r e l a r g e , w h i c h they a r e . Here q = number o f A A c o n s t r a i n t s imposed by Ho , l o g L ( j 3 , a) = v a l u e of l o g l i k e l i h o o d a t t h e A A maximum l i k e l i h o o d e s t i m a t e s and logL(j3o> jxo) = v a l u e of l o g l i k e l i h o o d a t the maximum l i k e l i h o o d e s t i m a t e s o b t a i n e d under HQ. In our case L(j? , a) i s the l i k e l i h o o d of o b t a i n i n g the o b s e r v e d numbers of dea ths on e a c h l i n e i f the j t h l i n e f o l l o w s the e x p o n e n t i a l model w i t h p a r a m e t e r s Bj and otj w h i c h may be d i f f e r e n t f o r each l i n e . A A So t o o b t a i n logL(j3, jx) we maximize f o r each l i n e s e p a r a t e l y ( w h i c h we have a l r e a d y done) and add the maximum v a l u e s of the l o g l i k e l i h o o d . L(_B_o» J*o) I s the l i k e l i h o o d of o b t a i n i n g the o b s e r v e d numbers o f d e a t h s on e a c h l i n e i f the j t h l i n e f o l l o w s the e x p o n e n t i a l model w i t h p a r a m e t e r s 8 and a * . So a l l l i n e s must have a common B but t h e i r - 33 - l o c a t i o n p a r a m e t e r s ai,...,06 may be d i f f e r e n t . T h i s t ime the s i x p a r t s cannot be maximized s e p a r a t e l y . 9 ~ B t 1 _ 1 - B t . 8 l o g L ( 8 0 , <xo) = £ d , , l o g ( e - e ) - 6 t9(N - E d . ) i=3 1 1 i = l 1 + B [ a i (N i - d u ) + a 2 ( N 2 - d i 2 ) + . . . + a 6 ( N 6 - d i e ) ] + d n l o g ( l - e - e ( t 2 _ a i ) ) + . . . + d 1 6 l o g ( l - e " B ( t 2 " a 6 ) ) where Nj = i n i t i a l number of b a r n a c l e s on l i n e j , d^j = number of d e a t h s a t t ime t^ on l i n e j , 6 d^= £ d = t o t a l number of dea ths at t ime i and N=Nĵ  + N 2 + . . + N 6 . j = l In t h i s c a s e , = 0 can be s o l v e d t o g i v e i , N . - d, . a. = t 2 + j l o g ( J N J ) j = l , 2 , . . . , 6 . 2 T h i s i s c l e a r l y a maximum s i n c e — < 0 f o r a l l a , and 8 > 0 . T h i s can be s u b s t i t u t e d i n 9 ^ ° g L t o g i v e op - B t - B t , a i L 8 9 d . , ( t , e 1 - t e 1 A ) - ^ f - = - t 9 ( N - E d ) + E i~- + t 2 ( N - d l ) . °P 1 = i 1 i = 3 8 t i _ l P t 1 (e - e ) A Newton Raphson i t e r a t i o n u s i n g < ^ S k = I d l o B 2 i=3 1 - e h-i - c i - 6 t i - i - B t i e - e - 34 - w h i c h i s c l e a r l y < 0 then y i e l d s 8. I t i s t h e n s t r a i g h t f o r w a r d t o A o b t a i n a j , j = l , 2 , . . . , 6 and t o s u b s t i t u t e the maximum l i k e l i h o o d A A e s t i m a t e s i n t o logL(J3o >ato) to o b t a i n logL(8_o, ap). R e s u l t s 1. T e s t H Q : t h a t a l l sys tem one o y s t e r l i n e s have the same 8 v a l u e . A A A A 2[ logL(£ , a) - l o g L ( £ 0 , o 0 ) ] = 2( -2190 - ( - 2 2 5 7 ) ) = 135. R e f e r t o X 2 5 t a b l e s . P « . 0 0 1 . 2. T e s t Ho: t h a t a l l sys tem two o y s t e r l i n e s have the same 8 v a l u e . A A A A 2 [ l o g L ( p , o) - l o g L ( j 5 0 , «0)1 = 2 ( - 4 7 4 0 - ( - 4 7 5 3 ) ) = 2 6 . R e f e r t o X 2 5 t a b l e s . P « . 0 0 1 . 3. T e s t H Q : p D ^ = p ^ = Pn 3> t h a t a l l the d o w l i n g l i n e s have the same P v a l u e . A A A A 2 [ l o g L ( j 3 , a) - l o g L ( p o , oo)] = 2( -5648 - ( - 5 7 5 9 ) ) = 222. R e f e r t o X 2 2 t a b l e s . P « . 0 0 1 . 4 . T e s t Ho: p n = p n , t h a t the sys tem one d o w l i n g l i n e s have t h e u l u2 same P v a l u e . S i n c e we a r e compar ing o n l y two p v a l u e s , a t - t e s t can be u s e d , p r o v i d e d t h a t the sample f rom w h i c h p,, and p_ a r e e s t i m a t e d i s u l u2 r e a s o n a b l y l a r g e . I n our case i t i s v e r y l a r g e . The use of a t - t e s t t can be j u s t i f i e d as f o l l o w s : i n g e n e r a l , i f 9 i s a maximum l i k e l i h o o d A e s t i m a t e o f 9, t h e n 9 i s a s y m p t o t i c a l l y norma l w i t h mean 9 and - 35 - 9 2 2 covariance matrix {E[ ] } - 1 which can be estimated by {—5—i£li. 0 90 2 50 2 This arises from the central l i m i t theorem applied to the sum of the A A contributions to the log l i k e l i h o o d . Since £L. and 6_ are maximum D l D2 l i k e l i h o o d estimates and since t h e i r variances are estimated, under HQ 8 " " 8 and for large samples, ^1 ^2 might reasonably be A A / ( S E ( 8 D i ) 2 + s*%y) compared to t-tables. This s t a t i s t i c works out to 6.6 and HQ i s rejected with p < .001. 5. Test Hn: 8XT = 8„ , that the two netting l i n e s have the same 8 N i N 2 value. = 11.5 i s referred to a t - d i s t r i b u t i o n . Again, A A 2 / ( S E ( 8 N i ) + S E ( 8 N 2 ) ^ p < .001 and HQ i s rejected. Differences were found i n every set of 8 values. The 6 values for the system two oyster l i n e s have a much smaller range of values than the 2 values for the system one oyster l i n e s as indicated by the smaller X value. The non-parametric tests also found the s u r v i v a l d i s t r i b u t i o n s to be d i f f e r e n t for d i f f e r e n t l i n e s but found the system one oyster l i n e s and the system two oyster l i n e s to be approximately equally homogenous i n t h e i r s u r v i v a l d i s t r i b u t i o n s . This discrepancy could a r i s e from the fact that our parametric model doesn't f i t the data very w e l l . - 36 - Other Parametric Models The Weibull would have been the natural choice of a model to f i t next. Since t h i s model has two parameters (or three i f a l o c a t i o n parameter i s included) and i s a generalization of the exponential, i t i s l i k e l y that we would obtain a somewhat better f i t . However the Weibull was not pursued for two reasons. F i r s t l y , with so much data, the improvement i n f i t would probably be s l i g h t ( i n fact any model would probably be rejected). Secondly, the subsequent i n t e r p r e t a t i o n would be d i f f i c u l t - i t would not be clear which parameter to focus on, and meaningful r e s u l t s would be hard to obtain with so many parameters and 2.4 Random E f f e c t s Model Instead of looking at these p a r t i c u l a r seventeen l i n e s , i t might be of more in t e r e s t to think of these l i n e s as a sample from an i n f i n i t e population of l i n e s . So we could think of an i n f i n i t e population of l i n e s of each type (material) within each system. For each of these subpopulations there would be a true underlying 8 value. It would be of i n t e r e s t to compare the underlying 8 values for the d i f f e r e n t "treatments" (types of material) and for the two systems, by examining A. the 8 values obtained for our sample of seventeen l i n e s . With only one netting l i n e i n each system and only one dowling l i n e i n system two i t would be d i f f i c u t to obtain an estimate of l i n e - t o - l i n e variance separately for these treatments. So i n i t i a l l y a random e f f e c t s model was developed incorporating only the oyster l i n e s . Throughout this s e c t i o n , f o r those l i n e s f o r w h i c h a = 5 , the e s t i m a t e of the v a r i a n c e A. of 8 t h a t was used was t h a t o b t a i n e d when the e x p o n e n t i a l model s t a r t i n g A at t= 5 was f i t t e d . A l t h o u g h , s t r i c t l y s p e a k i n g , the v a r i a n c e o f 8 s h o u l d have been e s t i m a t e d i n s t e a d i n the manner d e s c r i b e d i n S e c t i o n 2 . 2 , any r e s u l t i n g i n a c c u r a c i e s were l i k e l y to be n e g l i g i b l e . L e t B i and 82 be the t r u e u n d e r l y i n g p a r a m e t e r s f o r the h y p o t h e t i c a l p o p u l a t i o n of sys tem one and system two o y s t e r l i n e s r e s p e c t i v e l y . We w i s h t o t e s t H 0 : B i = 8 2 . I f HQ i s r e j e c t e d , t h i s would sugges t t h a t a s i g n i f i c a n t sys tem e f f e c t i s p r e s e n t . We have a sample of s i x l i n e s f rom each p o p u l a t i o n . L e t B^j be the t r u e 8 v a l u e A f o r the j t h l i n e i n the i t h sys tem and 8 ^ be the maximum l i k e l i h o o d 2 e s t i m a t e o f 8 ^ . = SECB^j) i s assumed t o be a known c o n s t a n t (no t u n r e a s o n a b l e c o n s i d e r i n g t h e l a r g e amount of d a t a w i t h w h i c h 8 j j was e s t i m a t e d ) . A T h e n , p r o v i d e d t h a t the model f i t s , s i n c e 8-jj i s a maximum l i k e l i h o o d e s t i m a t e based on a l a r g e s a m p l e , i t seems r e a s o n a b l e t o assume P i j | B i r N ( B l J , V i . ) . We assume f u r t h e r , t h a t the B j j a r e i n d e p e n d e n t l y d i s t r i b u t e d and t h a t 8 i ; j ~ N ( B i , a 2 ) 2 where o"j_ i s the l i n e - t o - l i n e v a r i a n c e (unknown) o f the t r u e p^j v a l u e s f o r the i n f i n i t e p o p u l a t i o n of l i n e s i n sys tem i . I t i s no t c l e a r t h a t t h i s second a s s u m p t i o n i s e n t i r e l y r e a s o n a b l e . We can o n l y p r o c e e d w h i l e b e a r i n g i n mind t h a t a s t r o n g a s s u m p t i o n i s b e i n g made. I t f o l l o w s t h a t - 38 - P i j ~ N ( p i ' ° i + V i j } N a i v e A n a l y s i s I n i t i a l l y a n a i v e a n a l y s i s was c a r r i e d out i n w h i c h two f a i r l y s t r o n g a s s u m p t i o n s were made: A 1. I t was assumed t h a t the s t a n d a r d e r r o r s of the B i j were n e g l i g i b l e 2 compared to the l i n e - t o - l i n e v a r i a n c e , i . e . , V^j « • I t seemed r e a s o n a b l e t o e x p e c t t h a t the V^j would be s m a l l s i n c e B j j was e s t i m a t e d w i t h a l a r g e amount of d a t a . By i g n o r i n g V^j we were i n e f f e c t , t r e a t i n g 8-y as known e x a c t l y . S i n c e we a r e n e g l e c t i n g the V^j an e s t i m a t e of i s p r o v i d e d by the A sample s t a n d a r d d e v i a t i o n of t h e B^-j's. The p r e v i o u s r e s u l t s y i e l d A A A A A a 2 = s 2 = 3- S ( B 1 . - 8 X ) 2 where pj = | E = 0 . 1 0 1 9 , 3 3 3 A A A A A a2 = S 2 = 5 E ( 8 2 j " P 2 ) 2 w h e r e 6 2 = 6 S P 2 j = ° ' 0 7 2 4 * We o b t a i n ô  = .026 and = . 0 1 1 . The v a l u e s of i n sys tem one range f rom .0050 t o .0130 and i n system two f rom .0034 to . 0 0 8 7 ; w h i l e 2 the V^j a r e s m a l l e r than our e s t i m a t e s of the a -p t h e y a r e not e n t i r e l y n e g l i g i b l e . 2. The second a s s u m p t i o n was t h a t the l i n e - t o - l i n e v a r i a n c e was the same f o r b o t h s y s t e m s , i . e . a\ = o^. S i n c e we have assumed t h a t the - 39 - Pij's are normally distributed we can test the hypothesis c± = 0 2 by 2 2 comparing Ŝ  /S 2 to F 5 5 tables. o 2 •51- = 5.67 ===> p = .05. c 2  s2 So the assumption o"i = o 2 appears to be a f a i r l y strong one. Under assumptions 1. and 2. we have a sample of six 8 values from each of two normal populations with equal variance. So H Q : 8i = 82 can be tested by a two sample t-test: 6 1 " 6 2 = 2.57 ===> p = 0.015 S 4 + 7T p 6 6 where S = / 5 S ^ * 5 S ^ = 0.020. p 10 This seems to suggest that there is a difference between the underlying B values for the two systems. However this analysis relied on two fa i r l y strong assumptions. The second assumption, a\ - 02 w a s tested by means of an F-test, which is particularly sensitive to our assumption that the 8 i j ' s &re normal. A more careful analysis is called for. Second A n a l y s i s A This time the standard error of the P ij's was not neglected. This analysis was done both assuming 01 = a 2 and also without this assumption. - 40 - Case 1: Assume Q\ = OY - O If we assume that the line-to-line variance is the same in both systems then there are three parameters - Bi, B2 and a to be estimated. A A 2 since 8^ NC^, a + V^)> the likelihood of obtaining the twelve B's that we obtained is L = 2 n 6 n i=l j=l /2* / o 2 + exp -1/2 Our estimate of the variance of B-jj was made with a large amount of data, so i t can be assumed to be close to the true variance V J J . So we replace V ^ j with this estimate, thus treating V ^ j as a known constant. logL Z E i j 2 ( P i i " hy log(o 2 + V ) + — i J i - i J ( a 2 + V ) + constant ologL 5 8 i = E (Pi, - Pi) j (a + Vx ) o 2logL o 6 i 2 = -E « 0) j (a + Vj ) ologL ap2 » E Cp2j " P2) j (<r + v 2 j ) 5 2logL ap2 2 = -E « 0) j (a + V 2 j) »i2Bt- - I E E i J ( P i 1 - Pl>' (o2+V±.) (a 2 + V ) 2 - 41 - To f i n d the maximum l i k e l i h o o d estimates of 61, 82 and a we need to solve simultaneously = = ̂ 2Sk = 0 . ° P i o82 S o2 Let w. . = - i = 1,2. 1 J a2 + V, . I j A T h e n ^ k , „ „ . > h . i — - 1.1 . 2 . 1 • l j 3 Substituting the above values (81* and 82* s a y ) i n S l o g L , 9 a 2 2 w i i 8 i i — « 0 ===> Z Z[w - w (6. . =— ) ] = 0 and 9a 2 i j i j 1 3 1 J 2 W i j a 2 l o g L ( 8 i * , 8?*, a 2) = I s E w2 A A E w. . 8., * E w. . 8. . j ^ ^ . j ^ * J i j J J . i j J J j i j A A E w2. p±. £ w ± j - E w ± j p ± j E w2. + (J J 1 )} . (E w ) 2 3 aiogL - 2 — - — = 0 can be solved by a Newton Raphson i t e r a t i o n to obtain cr . A numerical check can be done to v e r i f y that dlogL = 0 does i n fact aa 2 2 2 y i e l d a maximum. Furthermore since ^ l°gk a m j 0 l°gL a r e both < 0, 2 2 a p i a p 2 - 42 - d ^ g L =0 and Qjjj° g L = 0 w i l l c e r t a i n l y y i e l d maxima. H a v i n g o b t a i n e d a 2 A A i t i s s t r a i g h t f o r w a r d to o b t a i n 6i and 62 as we have e x p l i c i t A 2 e x p r e s s i o n s f o r them i n terms of a . To t e s t whether the u n d e r l y i n g p a r a m e t e r s 81 and 82 a re the same, we need to know the v a r i a n c e of our e s t i m a t e s . An e s t i m a t e of the A A A 2 c o v a r i a n c e m a t r i x o f (81, P2> o ) i s p r o v i d e d by the i n v e r s e of the n e g a t i v e of the m a t r i x of second d e r i v a t i v e s of l o g L e v a l u a t e d at t h e maximum l i k e l i h o o d e s t i m a t e s . A l t e r n a t i v e l y we can p r o c e e d as f o l l o w s : A A Z w. . 6, . i j i j A 1 8 = .3 t where w . = i = 1 ,2. i ^ 12 Suppose the w^j were known. Then we would have A E w , . B. . , i j K i j * l g = and v a r 8 = •= 0 i L w. . i L w. . i = 1 ,2. i j i j A S u b s t i t u t i n g our e s t i m a t e w „ of w .̂. we o b t a i n the f o l l o w i n g rough a p p r o x i m a t i o n t o the v a r i a n c e o f p^: v a r 8 ± = — j i = 1 ,2. - 43 - R e s u l t s /v A . B i = 0 . 0 9 8 3 , 8 2 = 0 . 0 7 2 2 , a = 0 . 0 1 5 8 . A A E s t i m a t e d S E ( p i ) = 0 . 0 0 7 6 , e s t i m a t e d S E ( B 2 ) - 0 . 0 0 6 8 . These l e a d t o A A PI - P2 2 > 5 6 A A A A / ( S E ( B i ) 2 + S E ( 8 2 ) 2 ) w h i c h i s a l m o s t i d e n t i c a l t o the v a l u e of the s t a t i s t i c o b t a i n e d i n the n a i v e a n a l y s i s . Compar ing the v a l u e 2.56 to normal t a b l e s would a g a i n s u g g e s t t h a t t h e r e i s a s i g n i f i c a n t d i f f e r e n c e between B i and 6 2. However t h i s t e s t i s not c o m p l e t e l y l e g i t i m a t e f o r two r e a s o n s : 2 F i r s t l y 8̂  and 82 a r e not i n d e p e n d e n t s i n c e they bo th i n v o l v e a and s e c o n d l y i t would seem more r e a s o n a b l e t o compare the v a l u e of the A A s t a t i s t i c t o t - t a b l e s as the v a r i a n c e s of p± and p 2 a r e o n l y e s t i m a t e s . The p r o b l e m i s t h a t i t i s not c l e a r how many degrees of f reedom a r e a p p r o p r i a t e . I f the square of the denomina to r of our s t a t i s t i c were a l i n e a r c o m b i n a t i o n of i n d e p e n d e n t c h i - s q u a r e d v a r i a b l e s , S a t t e r t h w a i t e ' s a p p r o x i m a t i o n ( S a t t e r t h w a i t e , 1946) c o u l d be used to e s t i m a t e the degrees of f r e e d o m . A A However our e s t i m a t e s of the v a r i a n c e s of 81 and 82 a r e o f a more c o m p l i c a t e d form and S a t t e r t h w a i t e ' s a p p r o x i m a t i o n i s not a p p l i c a b l e h e r e . A l s o , t h e whole a n a l y s i s was based on our a s s u m p t i o n of n o r m a l i t y f o r the B-LJ 'S. Because of t h e s e prob lems an e x a c t p - v a l u e cannot be g i v e n . However the v a l u e of our t e s t s t a t i s t i c i s s u f f i c i e n t l y l a r g e t o - 44 - suggest a s i g n i f i c a n t difference between 61 and 82> even allowing for s l i g h t inaccuracies. C a s e 2: o~i and a? n o t assumed e q u a l If a\ and o"2 are not assumed to be equal then we need estimates of four parameters - 6 1 , 6 2 , cri, 0 2 . For the system one oyster l i n e s A A A the l i k e l i h o o d under our model of obtaining 6 n , 612,.. . ,P16 i s L = n j exp ^2TC , 2 _,_ „ • a + V i (Pi, " Pi)' -1/2 ^ (o^ + VU) 1 J and logL = - Y E , (Pi, - Pi)' In ( c r + V j . ) + J - (<T + V i . ) 1 3 + constant. If we l e t w, , then £ "1.1 "'.1 E wj . As before we can substitute i n d l o g l Then - = 0 can be solved by bai' boi' Newton Raphson to obtain a\ . Then i t i s easy to f i n d 81• Analogously A to the previous case, an estimate of the variance of 6 l Is given by A var Bj * j m j o i 2 + V L - 45 - S i m i l a r l y the l i k e l i h o o d of obtaining 6 2 1 , . . ' 8 2 6 i s 6 L = n exp i=l /2n /cr + Vo. 2 Z3 ( 6 2 , " 8 2 ) ' - 1/2 2 (cr2 + V 2.) 2 J and o"2 , 82 as well as an estimate of the variance of 82 can be obtained i n the same way. Results A A Bi = 0.0997, p 2 = 0.0719. A A ai = 0.0219, a2 = 0.0089. A A Estimated SECp^ = 0.0098, estimated SE(p 2) = 0.0043. Pi " P2 A A A A ,2 , o r , , „ N 2 - = 2.59 / ( S E ( p i )  + S E ( p 2 ) 2 ) The same problems a r i s e as for the case where we assumed ai = 02, except that t h i s time Pi and P2 aren't correlated. Again an exact p-value cannot be quoted, but again a s i g n i f i c a n t difference between Pi and P2 i s strongly suggested. Our test s t a t i s t i c w i l l be found s i g n i f i c a n t at the 5% l e v e l i f i t i s compared to t f tables for any f >_ 3. It seems to make very l i t t l e d ifference whether or not we assume o"i = o"2; the r e s u l t i s almost i d e n t i c a l for both cases. So the fact that A A for case 1, Pi and P2 are correlated probably doesn't a f f e c t the ana l y s i s too much. However the fact remains that our analysis depends on our assumption of normality for the P i j ' s an^ 0 1 3 7 he se n s i t i v e to t h i s assumption. A summary of the r e s u l t s i s given below: - 46 - E s t i m a t e ± s t a n d a r d e r r o r Second A n a l y s i s Second A n a l y s i s w i t h o u t assuming P a r a m e t e r N a i v e A n a l y s i s assuming o"i = a 2 al ~ 02 Bi 0.1019 ± .0041 0.0983 ± .0076 0.0997 ± .0098 62 0.0724 ± .0024 0.0722 ± .0068 0.0719 ± .0043 Our a n a l y s e s s u g g e s t t h a t 81 and 82 a r e s i g n i f i c a n t l y d i f f e r e n t w h i c h can be i n t e r p r e t e d as meaning t h a t sys tem i s an i m p o r t a n t f a c t o r i n d e t e r m i n i n g the s u r v i v a l d i s t r i b u t i o n , a t l e a s t of the o y s t e r l i n e s . S i g n i f i c a n t sys tem e f f e c t i s c e r t a i n l y not s u r p r i s i n g as when the n o n p a r a m e t r i c t e s t s were c a r r i e d out t h e b i g g e s t d i f f e r e n c e s o c c u r r e d when l i n e s f rom d i f f e r e n t systems were compared . 2.5 Incorporating Netting and Dowling Lines i n t o the Model I t i s not o n l y the sys tem e f f e c t w h i c h i s of i n t e r e s t . I t i s a l s o of i n t e r e s t t o know whether the t y p e of m a t e r i a l p l a y s a r o l e i n d e t e r m i n i n g the s u r v i v a l d i s t r i b u t i o n . W i t h o n l y one n e t t i n g l i n e i n e a c h sys tem and one d o w l i n g l i n e i n sys tem two i t w i l l not be p o s s i b l e to o b t a i n an e s t i m a t e of l i n e - t o - l i n e v a r i a n c e f o r each type of m a t e r i a l - 47 - within each system. It w i l l have to be assumed that l i n e - t o - l i n e 2 2 variance i s independent of type of material. As before cr̂  and o"2 are the l i n e - t o - l i n e variances for systems one and two respectively (regardless of material type). The previous model i s extended to incororate netting and dowling l i n e s as follows: Let P^jk be the true 8 value for the kth l i n e of the j t h type i n system i . Let B^jk be our maximum l i k e l i h o o d estimate of B-jjk and l e t V\£jk = variance of S ^ j ^ . assumed to be a known constant. Assume ^kKjk ^i jk 'W and 8 ± j k N(8 + f± + t., o 2) where f = system e f f e c t i=l,2, and £ f = 0, i t. = type e f f e c t j=l,2,3, and E t = 0. 1 j j So t i = e f f e c t due to type 1 ( o y s t e r ) , t 2 = e f f e c t due to type 2 (dowling), t3 = e f f e c t due to type 3 ( n e t t i n g ) . We are assuming no system-type i n t e r a c t i o n . The variance of 2 2 P i j k * s °i instead of a±^ as we don't have s u f f i c i e n t r e p l i c a t i o n to allow the variance to depend on type. As before the assumption of normality f o r the 8 i j k ' s * s c e r t a i n l y reasonable, but the assumption of normality for the 8 i j k ' s ^ s P u r e ± y a n assumption and cannot be v e r i f i e d . It follows that Bijk N(8 + f i + t . , o 2 + V i j k ) . In order to compare the slopes of the s u r v i v a l curves for the two systems and for the three types of material, we need estimates of 8, f i , - 48 - 2 2 f 2 , t j , t 2 , t 3 , o i and o 2 and e s t i m a t e s of the v a r i a n c e s and c o v a r i a n c e s of f j , f 2 , t i , t 2 , t 3 . The l i k e l i h o o d under our model o f A o b t a i n i n g the s e v e n t e e n P i j k ' s t h a t we o b t a i n e d i s L = n n n • exp i j k VZ% / a 2 + l i j k S u b s t i t u t i n g f 2 = -fy and t 3 = - t i - t 2 , the maximum l i k e l i h o o d 2. 2 e s t i m a t e s (01 , o 2 , f^, t^, t 2 , P) a r e o b t a i n e d by Newton Raphson . 2 2 D u r i n g the i t e r a t i o n , 0̂  , and o 2 a re c o n s t r a i n e d to be n o n n e g a t i v e . R e s u l t s A 8 = 0 . 0 8 6 2 . A f x = 0 .0176 ===> f 2 = - 0 . 0 1 7 6 . A A A t i = 0 . 0 0 2 4 5 , t 2 = 0 . 0 0 1 0 5 , t 3 » - . 0 0 3 5 0 . a i = . 0 2 3 4 , 0 2 = . 0 0 7 2 . A A A A The P^jk = p + f^ + t j a r e g i v e n be low: Type System O y s t e r N e t t i n g D o w l i n g 1 0 .1063 0 .1003 0 .1049 2 0 .0711 0.0651 0.0697 Note t h a t the e s t i m a t e s of the s l o p e f o r r e s p e c t i v e l y the system one and - 49 - t h e system two o y s t e r l i n e s (0 .1063 and 0 .0711) a r e s i m i l a r but n o t i d e n t i c a l t o the e s t i m a t e s o b t a i n e d b e f o r e under t h e model f o r t h e o y s t e r l i n e s a l o n e ( r e s p e c t i v e l y 0.0997 and 0 . 0 7 1 9 ) . E s t i m a t e s o f the v a r i a n c e s o f t h e s e e s t i m a t e s a r e o b t a i n e d as u s u a l u s i n g the m a t r i x of second d e r i v a t i v e s e v a l u a t e d a t the maximum l i k e l i h o o d e s t i m a t e s , wh ich was a l r e a d y c a l c u l a t e d d u r i n g the Newton Raphson p r o c e s s . A A SE 8 = . 0 0 3 5 . A A A A SE f i = SE f 2 = . 0 0 3 1 . A A A A SE ti = . 0 0 3 0 , SE t 2 = . 0 0 3 8 , SE t 3 = . 0 0 3 9 . I t was t h e n o f i n t e r e s t to t e s t f o r d i f f e r e n c e s between f i and f 2 and between t^, t 2 and t 3 : f i ~ f; = 5.62 / v a r ( f i - f 2 ) A A t l " t 2 = 0 .25 A A A / v a r ( t i - t 2 ) A A 1 : 1 " t 3 = 1.02 A A A / v a r ( t i - t 3 ) S i n c e t i + t 2 + t 3 = 0 i t i s u n n e c e s s a r y a l s o to compare t 2 to t 3 . As b e f o r e , i t seems r e a s o n a b l e to compare t h e s e q u a n t i t i e s t o t - t a b l e s s i n c e the v a r i a n c e s a r e e s t i m a t e d . A g a i n i t i s not c l e a r how many - 50 - d e g r e e s of f reedom s h o u l d be u s e d . S i n c e we a r e d o i n g t h r e e c o m p a r i s o n s the s i g n i f i c a n c e l e v e l s h o u l d be a d j u s t e d a c c o r d i n g l y : i f the o v e r a l l s i g n i f i c a n c e l e v e l i s t o be 5%, t h e n s i g n i f i c a n c e l e v e l of e a c h c o m p a r i s o n c o u l d be 5/3% = 1.66%. F o r the above c o m p a r i s o n s , whatever the d e g r e e s of f reedom we r e a c h the s a m e c o n c l u s i o n , namely t h a t t h e r e i s a s i g n i f i c a n t d i f f e r e n c e between t h e 8 v a l u e s f o r the two sys tems (as was found p r e v i o u s l y ) , but no s i g n i f i c a n t d i f f e r e n c e between the 8 v a l u e s f o r the t h r e e t y p e s of m a t e r i a l . T h i s s u g g e s t s t h a t r e g a r d i n g the r a t e a t w h i c h the b a r n a c l e s d i e o f f , t h e r e i s a s i g n i f i c a n t d i f f e r e n c e between the two sys tems but not between the t h r e e t y p e s o f m a t e r i a l . Discussion I n s p e c t i n g t h e 8 v a l u e s , i t i s seen t h a t the v a l u e s f o r s y s t e m one a r e g e n e r a l l y l a r g e r than t h o s e f o r sys tem two, whereas p v a l u e s f o r any p a r t i c u l a r type o f m a t e r i a l show no t e n d e n c y to be b i g g e r or s m a l l e r t h a n t h o s e f o r any o t h e r t y p e . These o b s e r v a t i o n s a r e bo rne out by t h e above a n a l y s e s w h i c h s u g g e s t a s i g n i f i c a n t sys tem e f f e c t but n o t a s i g n i f i c a n t t y p e e f f e c t . I t i s seen f rom a\ and a 2 t h a t 8 v a l u e s a r e more v a r i a b l e i n sys tem one t h a n i n sys tem two. A. S t a t i s t i c a l a n a l y s e s r e l a t i n g to the a v a l u e s were not c a r r i e d b u t an i n s p e c t i o n of the v a l u e s s u g g e s t s t h a t t h e r e i s b o t h a sys tem and t y p e e f f e c t . The a v a l u e s f o r sys tem one a r e g e n e r a l l y s m a l l e r than v a l u e s f o r sys tem two and v a l u e s f o r d o w l i n g l i n e s a r e g e n e r a l l y s m a l l e r t h a n v a l u e s f o r the o t h e r t y pe s of l i n e . The p h y s i c a l i n t e r p r e t a t i o n of - 51 - these observations i s that barnacles on dowling and system one l i n e s tend to st a r t dying e a r l i e r than other barnacles. Barnacles i n system one tend to die at a f a s t e r rate than those i n system two. Furthermore the rate at which the system one barnacles die shows more v a r i a b i l i t y from l i n e to l i n e . The f i n a l observation relates to the i n i t i a l number of barnacles on the l i n e s . There are more barnacles i n i t i a l l y on a l i n e of a given type i n system two than on a l i n e of the same type i n system one. Also A A we found the 8 values to be generally smaller and the a values generally larger f o r system two l i n e s than for system one l i n e s . Formal s t a t i s t i c a l tests are not appropriate and even the simple c o r r e l a t i o n c o e f f i c i e n t i s not very meaningful with such a small number of l i n e s of a given type. However i t i s worth noting that f o r the system one oyster l i n e s , the c o r r e l a t i o n c o e f f i c i e n t between the i n i t i a l number of barnacles and 6 i s - .93. This c o r r e l a t i o n suggests that on l i n e s with fewer barnacles i n i t i a l l y , barnacles tend to st a r t dying e a r l i e r and at a fas t e r rate. However the c o r r e l a t i o n may also be due to a t h i r d variable related to both 6 and i n i t i a l number. This w i l l be discussed again l a t e r i n r e l a t i o n to the growth data. A l l i n a l l i t appears that system i s more important than type as regards s u r v i v a l d i s t r i b u t i o n . However t h i s conclusion i s somewhat tentative since i t i s based on the analysis of parameters p and a of a model which doesn't adequately f i t the data. Furthermore the subsequent analyses required a strong normality assumption. - 52 - 3 . EXAMINATION OF THE GROWTH DATA 3 .1 Growth D a t a Growth d a t a was c o l l e c t e d at t = 1, 2 , . . . , 9 , 10, 11 , 14, 17 weeks . A t each of t h e s e t i m e s samples were taken as f o l l o w s : f rom each s h e l l on each o y s t e r l i n e and f rom e a c h p i e c e of wood on each d o w l i n g l i n e a h a p h a z a r d sample of 5 l i v e b a r n a c l e s was s e l e c t e d . From each p i e c e o f r u b b e r on each n e t t i n g l i n e a h a p h a z a r d sample of 10 l i v e b a r n a c l e s was s e l e c t e d . I t w i l l be assumed t h a t on each i t e m e v e r y l i v e b a r n a c l e was e q u a l l y l i k e l y t o be s a m p l e d , so f o r example the p r o b a b i l i t y of b e i n g sampled was no t r e l a t e d t o s i z e . F o r each sampled b a r n a c l e s h e l l l e n g t h and neck l e n g t h were m e a s u r e d . Measurements were r e c o r d e d i n cm. The sampled b a r n a c l e s were not removed f rom the i t e m s t o w h i c h they were a t t a c h e d . I n t o t a l t h e r e was an enormous amount of growth d a t a : the t o t a l number of b a r n a c l e s measured a t each t ime was: From o y s t e r l i n e s : 2 x 6 x 1 0 x 5 = 6 0 0 , From d o w l i n g l i n e s : 3 x 20 x 5 = 300, From n e t t i n g l i n e s : 2 x 10 x 10 = 200 . The d a t a was i n c o m p l e t e i n s e v e r a l ways: a t t = 1 week, o b s e r v a t i o n s were t a k e n o n l y on b a r n a c l e s f rom 3 o y s t e r l i n e s and 1 d o w l i n g l i n e i n sys tem 2. A l s o a t t = 10 and 11 weeks o b s e r v a t i o n s were t a k e n on b a r n a c l e s f rom some of the l i n e s but not a l l of them. I t somet imes happened t h a t fewer than f i v e ( o r t e n ) b a r n a c l e s were sampled f rom a p a r t i c u l a r i t e m . T h i s was because the p o p u l a t i o n f rom wh ich the - 53 - b a r n a c l e s were sampled was c o n t i n u a l l y g e t t i n g s m a l l e r due t o b a r n a c l e s d y i n g . Somet imes , towards the end of the s tudy ( p a r t i c u l a r l y f o r the s y s t e m one o y s t e r l i n e s where i n i t i a l sample s i z e s were s m a l l ) fewer than f i v e ( o r t e n ) l i v i n g b a r n a c l e s rema ined on a p a r t i c u l a r i t e m . I n t h i s case a l l r e m a i n i n g b a r n a c l e s were s a m p l e d . The a n a l y s e s w h i c h were used assumed t h a t measurements were t a k e n a t the same t ime p o i n t s f o r a l l i n d i v i d u a l s . So o n l y the d a t a c o l l e c t e d a t t = 2 , 3 , 4 . . . , 9 , 14, 17 weeks was u s e d . The a n a l y s e s c o u l d have been e x t e n d e d t o accommodate the m i s s i n g d a t a and methods of h a n d l i n g m i s s i n g d a t a w i l l be d i s c u s s e d l a t e r . However t h i s would have made the c o m p u t a t i o n much more l a b o r i o u s and s i n c e t h e amount of d a t a n e g l e c t e d i s r e l a t i v e l y s m a l l i t w a s n ' t c o n s i d e r e d w o r t h w h i l e . A l s o i t ems were n o t i n c l u d e d i f a t any p o i n t i n t ime fewer than f i v e ( o r t e n ) b a r n a c l e s were sampled f rom them. A g a i n not much i n f o r m a t i o n was l o s t by d o i n g t h i s . O n l y t h e d a t a r e l a t i n g t o s h e l l l e n g t h was a n a l y z e d . The d a t a r e l a t i n g to neck l e n g t h c o u l d have been t r e a t e d i n an i d e n t i c a l way. I n summary: on e a c h o f 17 l i n e s t h e r e a r e a number of i t e m s ( o y s t e r s h e l l s , p i e c e s of wood or r u b b e r ) . At each p o i n t i n t i m e , measurements a r e made on a random sample of b a r n a c l e s t a k e n f rom each i t e m . The p o p u l a t i o n f rom w h i c h t h e s e b a r n a c l e s a r e sampled i s c o n t i n u a l l y d i m i n i s h i n g because b a r n a c l e s a r e d y i n g . The p r o p o r t i o n of b a r n a c l e s sampled on a p a r t i c u l a r i t e m i s a p p r o x i m a t e l y 10% i n i t i a l l y and 20-30% towards the end of the s t u d y e x c e p t f o r i t ems on the sys tem one o y s t e r l i n e s where the p r o p o r t i o n s a r e somewhat l a r g e r - a p p r o x i m a t e l y 25% i n i t i a l l y and c l o s e t o 100% by - 54 - the end of the study. These proportions are only estimates as we don't a c t u a l l y have the s u r v i v a l data for each item, only for each l i n e . I ndividual barnacles cannot be i d e n t i f i e d . We would l i k e to f i t growth curves for the s h e l l length of the barnacles and to compare these curves for the two systems and for the three types of material. 3.2 Growth Curves f o r Items Sophisticated growth curve models have been developed f o r l o n g i t u d i n a l data - for the s i t u a t i o n where an observation i s obtained at each time point on each of a number of i n d i v i d u a l s . In our case we don't have l o n g i t u d i n a l data for the barnacles: at each point we have measurements on a sample of u n i d e n t i f i e d barnacles - each barnacle may or may not have been measured at the previous time point. The items, however, can be i d e n t i f i e d and i f we were to average over the sample obtained from each item at each time point, we could think of our data as l o n g i t u d i n a l data for the items: on each l i n e we would have a number of items and we would have a measurement representing each item at each time. This measurement would a c t u a l l y be the average s h e l l length over a sample of barnacles picked from the remaining l i v i n g barnacles on that item at that time. Since each barnacle has equal p r o b a b i l i t y of being sampled, regardless of s i z e , and since the proportion of barnacles sampled i s quite large, i t should be reasonable to allow this average to represent a l l the remaining l i v i n g barnacles on that items at that time. So - 55 - i n s t e a d of f o l l o w i n g i n d i v i d u a l b a r n a c l e s we would be f o l l o w i n g i n d i v i d u a l i t e m s . Suppose we would l i k e t o f i t p o l y n o m i a l growth c u r v e s . W i t h e f f e c t i v e l y o n l y one o b s e r v a t i o n per i t e m we w i l l have to assume t h a t a l l i t e m s w i t h i n a l i n e have t h e same p a r a m e t e r s . So c o m p a r i s o n of i t e m s w i l l be i m p o s s i b l e . But i n any case the main i n t e r e s t l i e s i n c o m p a r i s o n o f the 17 l i n e s . We may a l l o w e a c h of the 17 l i n e s t o have d i f f e r e n t p a r a m e t e r s and we may t h i n k o f the i t ems on each l i n e as a sample f rom a c o n c e p t u a l p o p u l a t i o n o f i t ems on t h a t l i n e . We may t h e n f i t the a v e r a g e growth c u r v e f o r i t e m s on each l i n e . What w i l l t h i s c u r v e r e p r e s e n t ? C o n s i d e r the growth c u r v e f o r the f i r s t o y s t e r l i n e . T h i s w i l l show how the a v e r a g e s h e l l l e n g t h of the r e m a i n i n g l i v i n g b a r n a c l e s on a t y p i c a l i t e m f rom t h i s l i n e changes w i t h t i m e . (We hope t h a t the a v e r a g e over a l l b a r n a c l e s on a p a r t i c u l a r i t e m a t a p a r t i c u l a r t ime w i l l be w e l l - r e p r e s e n t e d by the a v e r a g e over a sample of b a r n a c l e s t a k e n f rom t h a t i t e m a t t h a t t i m e ) . T h i s a v e r a g e s i z e changes f o r two r e a s o n s - f i r s t l y because the b a r n a c l e s grow and s e c o n d l y because the p o p u l a t i o n o f l i v i n g b a r n a c l e s changes due t o b a r n a c l e s d y i n g . So the growth c u r v e r e p r e s e n t s the n a t u r a l p o p u l a t i o n r a t h e r than i n d i v i d u a l b a r n a c l e s . We would e x p e c t the growth c u r v e s f o r the n a t u r a l p o p u l a t i o n t o be s i m i l a r t o the growth c u r v e s f o r i n d i v i d u a l b a r n a c l e s , p r o v i d e d t h a t the p r o b a b i l i t y of d y i n g i s u n r e l a t e d t o s i z e . I f the b a r n a c l e s t h a t d i e i n a p a r t i c u l a r t i m e - i n t e r v a l a r e n e i t h e r p a r t i c u l a r l y b i g n o r p a r t i c u l a r l y s m a l l , t h e n the change i n p o p u l a t i o n w i l l no t a f f e c t t h e a v e r a g e s i z e o f - 56 - t h e l i v i n g b a r n a c l e s . The changes i n t h i s a v e r a g e s i z e w i l l then be e n t i r e l y due to the growth of the b a r n a c l e s . I f , on the o t h e r h a n d , t h e r e i s a t e n d e n c y f o r the l a r g e r b a r n a c l e s , s a y , t o d i e f i r s t , t h e n w h i l e each i n d i v i d u a l b a r n a c l e grows b i g g e r , the average s i z e of the l i v i n g b a r n a c l e s might a c t u a l l y get s m a l l e r . I f t h i s were the c a s e , the growth c h a r a c t e r i s t i c s of the n a t u r a l p o p u l a t i o n would not r e f l e c t t h e growth c h a r a c t e r i s t i c s of i n d i v i d u a l b a r n a c l e s . W i t h o u t b e i n g a b l e to i d e n t i f y i n d i v i d u a l b a r n a c l e s i t i s d i f f i c u l t t o a s c e r t a i n whether the p r o b a b i l i t y of d y i n g i s r e l a t e d to s i z e . T h i s q u e s t i o n w i l l be a d d r e s s e d l a t e r a l o n g w i t h a d i s c u s s i o n of the l e g i t i m a c y of t h e a p p r o a c h and of the a s s u m p t i o n s t h a t a r e i m p l i c i t l y b e i n g made. So u s i n g the " l o n g i t u d i n a l " d a t a f o r i t e m s , p o l y n o m i a l g rowth c u r v e s were f i t t e d f o r each of the 17 l i n e s . Two growth c u r v e models were employed - the f i r s t was due t o P o t h o f f and Roy (1964) and the second t o Rao (1965 , 1 9 6 6 ) . The c u r v e s o b t a i n e d f o r the two d i f f e r e n t systems and f o r the t h r e e t y p e s of m a t e r i a l were t h e n compared . 3.3 The Growth Curve Models Pothoff and Roy's Model I t i s assumed t h a t l o n g i t u d i n a l d a t a i s a v a i l a b l e - s u c c e s s i v e measurements on a number of i n d i v i d u a l s . The measurements f o r any one i n d i v i d u a l a r e t h e n c l e a r l y c o r r e l a t e d . T y p i c a l l y the i n d i v i d u a l s a r e d i v i d e d i n t o groups w h i c h a r e to be compared . I t i s assumed t h a t each i n d i v i d u a l has measurements t a k e n a t the - 57 - same q points i n time. The set of q measurements for any one i n d i v i d u a l form one row of the data matrix Yo. The model i s as follows: E[Y 0] = ' A I P nxq nxm mxp pxq where n = number of i n d i v i d u a l s , m = number of groups, q = number of time points, p = number of parameters f i t t e d within each group (p _< q), and A i s the design matrix across i n d i v i d u a l s , £ i s a matrix of parameters to be estimated, P i s a matrix of known constants related to time (assumed to be of f u l l rank, p). It i s assumed that the rows of YQ are mutually independent (measurements on d i f f e r e n t i n d i v i d u a l s are independent) and that the q elements i n any one row follow the multivariate normal d i s t r i b u t i o n with unknown covariance matrix T.Q (q x q), the same for every i n d i v i d u a l . So i s not diagonal since successive measurements on the same i n d i v i d u a l are correlated. It would be possible to extend the model to allow the covariance matrices for d i f f e r e n t i n d i v i d u a l s to be multiples of each other and also to allow d i f f e r e n t i n d i v i d u a l s to have measurements taken at d i f f e r e n t times. These extensions are appropriate i n our s i t u a t i o n and w i l l be considered l a t e r . To i l l u s t r a t e the model consider the following s i t u a t i o n : we have m groups of i n d i v i d u a l s with nj i n the j t h group. The growth curves for i n d i v i d u a l s i n a l l groups are from the same family, for example - 58 - p o l y n o m i a l s of degree p-1. But the p a r a m e t e r s may be d i f f e r e n t f o r the d i f f e r e n t g r o u p s . So the growth c u r v e a s s o c i a t e d w i t h the j t h group i s Then A = { n { n { m • 0 0 0 1 0 1 '10 '1 p-1 m̂0 ^m p-1 P = .P-1 .P-1 A l t e r n a t i v e l y i f the t ime p o i n t s a r e e q u a l l y s p a c e d , o r t h o g o n a l c o e f f i c i e n t s may be used f o r P. T h i s can be g e n e r a l i s e d t o the s i t u a t i o n where t h e r e a r e two or more f a c t o r s each w i t h a number of l e v e l s or t o the m u l t i - r e s p o n s e - 59 - s i t u a t i o n where more t h a n one growth c h a r a c t e r i s t i c i s measured . Any of the u s u a l f a c t o r i a l or b l o c k d e s i g n s can be accommodated by t h i s m o d e l . We want to e s t i m a t e the p a r a m e t e r s E,^ and to t e s t v a r i o u s h y p o t h e s e s of the form C I V = 0. s x m m x p p x u F o r example , i f the h y p o t h e s i s i s t h a t a l l m growth c u r v e s a re the same, i . e . t h a t a l l p a r a m e t e r s f o r a l l g roups a r e e q u a l , t h e n C - ( I -1 ) and V = I . ( m - l ) x ( m - l ) ( m - l ) x l pxp I f the h y p o t h e s i s i s t h a t a l l c u r v e s a r e of degree p-2 or l e s s t h e n C = I and V mxm ( p x l ) S o l u t i o n P o t h o f f and Roy s u g g e s t a t r a n s f o r m a t i o n w h i c h r e d u c e s the model t o a more s t a n d a r d model w h i c h has been t r e a t e d e x t e n s i v e l y i n the l i t e r a t u r e . The t r a n s f o r m a t i o n i s X = Y 0 G - 1 P 1 (P G - 1 P ' ) - 1 nxp nxq qxq qxp pxp where G i s an a r b i t r a r y symmetr i c p o s i t i v e d e f i n i t e m a t r i x such t h a t P G - 1 P» i s of f u l l r a n k . Then X i s such t h a t : ( a ) i t s rows a r e m u t u a l l y i n d e p e n d e n t , (b) t h e p e l e m e n t s i n any one row f o l l o w the 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 w i t h unknown p o s i t i v e d e f i n i t e c o v a r i a n c e m a t r i x - 60 - E = [P(G')- 1 P ' ] " 1 P ( G ' ) - 1 E 0 G"1 P' (P G"1 P » ) _ 1 , pxp (c) E[X] = A?. This i s the usual multivariate analysis of variance (MANOVA) model. Under t h i s model a test of C£V = 0 i s based on S^(u x u) and S e ( u x u) where S h = (C I V)' [CjCA* A j ) - ^ ' ] - 1 (C I V) = V X' A J ( A J A ^ - 1 C J I C J C A J A j ) - 1 C | ] _ 1 CjCA' A ^ - 1 A^ X and S g = V X'[I - A^Aj^ A ^ - 1 AJ] X V, where £ i s the le a s t squares estimate of £, A^ = f i r s t r columns of A, = f i r s t r columns of C, and rank A = r ( _< m, < n) , rank C = s ( <m, <_ r) , rank V = u (< p). Several possible tests are a v a i l a b l e : (1) Roy's test i s based on the largest c h a r a c t e r i s t i c root of S, S - 1 n e (2) Hot e l l i n g ' s test i s based on the trace of S, S - 1 , h e | Se (3) Wilk's test i s based on the r a t i o of determinants S h + S e In order to test a hypothesis C£V = 0 under the o r i g i n a l model, X = Y 0 G - 1 p' (p G - 1 P')" 1 i s substituted i n t o the expressions for S h and S e. In a l l analyses which follow, Roy's test was used. This proceeds as follows: - 61 - Let X = largest c h a r a c t e r i s t i c root of S h S "~ . Compare ^ + to Heck tables (e.g. Morrison, 1976) with parameters s* = min (s, u), m* = 1/2 (|s-u|-l), n* = 1/2 (n-r-u-1). Reject the n u l l hypothesis If . ^ .. i s too large. 1 T A. (n* + 1 ) If s* = 1, the s t a t i s t i c X can be compared to F tables with (m* + 1) (2m* + 2) and (2n* + 2) degrees of freedom. The least squares estimator of £ i s given by I = (AJ A x ) _ 1 A* X = (AJ A j ) " 1 A^ Y q G _ 1 P•(P G _ 1 P ' ) _ 1 . The best l i n e a r unbiased estimator of the function C£V i s then C£V. Confidence bounds can also be found for these functions. Choice of G The analysis i s v a l i d for any G s a t i s f y i n g the given conditions. However the choice of G a f f e c t s the power of the tests and the variance of the estimators. The minimum variance unbiased estimator of £ i s (A! A . ) - 1 A! Y E - 1 P' (P S - 1 P ' ) - 1 . Comparing t h i s with the l e a s t l l l o o o squares estimator of £ given above, would suggest that the optimal choice of G i s G = E Q . Pothoff and Roy suggest that the more G d i f f e r s from E Q the worse the power of the test w i l l be and the greater the variance of the estimators. However E 0 i s unknown and an estimate of E 0 obtained from Y Q may not be used. If the - 62 - e x p e r i m e n t e r has no i d e a about the form of E Q the s i m p l e s t p r o c e d u r e i s t o use G=I. I f an e s t i m a t e of E 0 can be o b t a i n e d f rom a s i m i l a r b u t i n d e p e n d e n t exper iment i t would be p r e f e r a b l e t o use t h i s e s t i m a t e . A l t e r n a t i v e l y , i f a guess of E Q can be made b e f o r e the e x p e r i m e n t i s r u n , t h i s may be u s e d . F o r examp le , i t might be assumed t h a t the c o r r e l a t i o n between any two o b s e r v a t i o n s d p e r i o d s a p a r t i s and t h a t the v a r i a n c e i s c o n s t a n t w i t h r e s p e c t t o t i m e . Then E Q i s p r o p o r t i o n a l t o ~ i 2 3 i p p p . . . . p 1 p p . . . . 2 , p p 1 p . . . . 3 2 , p p p I . . . . . . . . . . . . U n f o r t u n a t e l y , however , the c h o i c e of p i s a r b i t r a r y u n l e s s an e s t i m a t e of i t can be o b t a i n e d f rom an i n d e p e n d e n t e x p e r i m e n t . R a o ' s M o d e l The p r o b l e m w i t h P o t h o f f and R o y ' s model i s t h a t G must be chosen a r b i t r a r i l y . Rao p o i n t s out t h a t t h e i r model d o e s n ' t u t i l i z e a l l t h e a v a i l a b l e i n f o r m a t i o n u n l e s s G = E Q and t h e y do not a l l o w an e s t i m a t e o f E Q o b t a i n e d f rom the d a t a t o be u s e d . So Rao s u g g e s t s an a l t e r n a t i v e way of r e d u c i n g the growth model t o the u s u a l MANOVA m o d e l : c o n s t r u c t a q x q n o n s i n g u l a r m a t r i x H = (H. H„) s u c h t h a t the columns - 63 - o f B-i f o rm a b a s i s f o r the v e c t o r space spanned by the rows of P and PH} = I , PH2 = 0. When rank P = p we can choose H x = G " 1 P ' ( P G - 1 P * ) - 1 , H 2 = I - P where G i s an a r b i t r a r y p o s i t i v e d e f i n i t e m a t r i x . The c h o i c e of G does no t a f f e c t e s t i m a t e s or t e s t s . L e t X = Y H = Y G " 1 P' ( P G - 1 P * ) - 1 and Z = Y H„ . o 1 o , , N o 2 nxp n x ( q - p ) T h e n , as f o r the P o t h o f f and R o y ' s m o d e l , E [X] = A£, but i n a d d i t i o n E [ Z ] =0. Hence E[x|z] = A£ + ZT where Z i s a m a t r i x of q-p c o v a r i a b l e s and T i s a m a t r i x o f unknown r e g r e s s i o n c o e f f i c i e n t s . ( q - p ) x p Rao c l a i m s t h a t the e s t i m a t e o f £ o b t a i n e d under t h i s model i s more e f f i c i e n t than t h a t o b t a i n e d under P o t h o f f and R o y ' s model as i t uses i n f o r m a t i o n c o n t a i n e d i n t h e c o v a r i a b l e s Z , w h i c h i s n e g l e c t e d under P o t h o f f and R o y ' s m o d e l . However t h i s i s not t r u e . The b e s t l i n e a r u n b i a s e d e s t i m a t o r of E, under R a o ' s model i s I = (A{ A ^ - 1 A J Y q S - 1 P' (P S " 1 P ' ) " 1 where S i s an e s t i m a t e o f E 0 o b t a i n e d f rom the d a t a : s = Y ; [ I - A i ( A - V 1 A'] Y O T h i s i s p r e c i s e l y what we would o b t a i n under P o t h o f f and R o y ' s model i f we s e t G = S, thus a l l o w i n g an e s t i m a t e of E Q o b t a i n e d f rom the d a t a as Rao d o e s . To t e s t h y p o t h e s e s of the fo rm C£V = 0, m a t r i c e s S h and S g a r e a g a i n f o u n d . T h i s t ime t h e y have a s l i g h t l y d i f f e r e n t f o r m : L e t B = [ S _ 1 - S - 1 P ' ( P S - 1 P ' ) " 1 P S - 1 ] . - 64 - Then S h - V X' A1 (Aj A ^ - 1 C ^ C j R C p - 1 (A^ A ^ - 1 A{ X V S = V (P S " 1 P ' ) - 1 V e where R = (A.'A. ) - 1 + ( A . ' A . ) - 1 A.' Y B Y ' A. (A.' A . ) - 1 1 1 1 1 l o o l l l and X = Y q S " 1 P ' ( P S " 1 P ' ) " 1 . The e x p r e s s i o n f o r S e i s i d e n t i c a l t o the e x p r e s s i o n f o r S g i n P o t h o f f and R o y ' s model w i t h G s e t e q u a l t o S. However i s not the same. The t e s t s based on and S e a r e the same as f o r P o t h o f f and R o y ' s m o d e l . However when compar ing . ^ .. t o Heck t a b l e s the I i A. paramete r n* i s now s l i g h t l y d i f f e r e n t , n * = n - s - u - l - ( q - p ) . Sometimes as d i s c u s s e d by G r i z z l e and A l l e n (1969) i t i s p r e f e r a b l e t o use fewer than q-p c o v a r i a b l e s . 3.4 A p p l i c a t i o n o f t h e M o d e l s The two growth c u r v e models were a p p l i e d t o the l o n g i t u d i n a l d a t a f o r i t e m s . E a c h row of the d a t a m a t r i x c o n t a i n e d n i n e e n t r i e s . The i t h e n t r y was the a v e r a g e s h e l l l e n g t h over a random sample of l i v i n g b a r n a c l e s t a k e n a t t = f rom the i t e m c o r r e s p o n d i n g to t h a t row. The i t e m s were d i v i d e d i n t o g roups a c c o r d i n g t o the l i n e t h a t they were o n . I n t h e i r p r e s e n t form the models r e q u i r e the f o l l o w i n g a s s u m p t i o n s : (1) The rows of the d a t a m a t r i x a r e m u t u a l l y i n d e p e n d e n t . (2) The s e t of e n t r i e s i n any row f o l l o w s the m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n w i t h the same c o v a r i a n c e m a t r i x E n f o r e a c h row. - 65 - Provided that we are prepared to assume that observations on barnacles from d i f f e r e n t items are independent, then the rows of our data matrix w i l l c e r t a i n l y be mutually indepndent since each row of our data matrix corresponds to a d i f f e r e n t item and no two items have any barnacles i n common. Furthermore, i f we are prepared to assume that the set of observations on an i n d i v i d u a l barnacle i s multivariate normal then the f i r s t part of assumption 2 would also be s a t i s f i e d since each entry i n our data matrix would then be the average of a number of variables assumed to be normally d i s t r i b u t e d . Whether or not i t i s reasonable to assume that the covariance matrix i s the same for every row of our data matrix w i l l be discussed l a t e r . Subsequently another analysis was c a r r i e d out i n which the weaker and more reasonable assumption was made that a set of observations corresponding to items from oyster or dowling l i n e s had covariance matrix -jr E and a set of observations corresponding to items from netting l i n e s had covariance matrix jfi ^ f ° r some f i x e d , unknown £. In order to do t h i s the model had to be modified s l i g h t l y as described i n Section 3.8. It i s not clear that even this weaker assumption i s reasonable. This too w i l l be discussed l a t e r . Only those items for which every observation was an average over f i v e (ten) barnacles were included i n the a n a l y s i s . In t o t a l 141 items were used i n the analyses. The number of items used from each l i n e was as follows: - 66 - 01 02 03 04 05 06 5 8 5 6 6 8 D l D2 N l 07 08 09 11 11 8 9 8 8 010 O i l 012 D3 N2 6 10 9 13 10 The main o b j e c t i v e was to f i t p o l y n o m i a l growth c u r v e s t o each o f the l i n e s and to d e t e r m i n e whether d i f f e r e n c e s e x i s t e d between t h e s e c u r v e s f o r the t h r e e d i f f e r e n t t y p e s o f m a t e r i a l s and f o r the two s y s t e m s . As a f i r s t s t e p i t was of i n t e r e s t t o d e t e r m i n e whether l i n e s w i t h i n e a c h s y s t e m - t y p e c o m b i n a t i o n c o u l d be i g n o r e d . I f t h i s were the case i t would be r e a s o n a b l e to c o l l a p s e over l i n e s of the same type w i t h i n t h e same sys tem and we would have a two f a c t o r s i t u a t i o n w i t h the f o l l o w i n g number of i t e m s . System 1 System 2 Type O y s t e r 38 50 D o w l i n g 22 13 N e t t i n g 8 10 I n o r d e r to d e t e r m i n e whether d i f f e r e n c e s e x i s t e d between the growth c u r v e s f o r the s i x o y s t e r l i n e s i n sys tem one , p o l y n o m i a l g rowth c u r v e s were f i t t e d t o the s i x l i n e s and the p a r a m e t e r s were compared . A s i m i l a r a n a l y s i s was c a r r i e d out f o r the s i x o y s t e r l i n e s i n s y s t e m two. - 67 - 3 . 5 System One Oyster Lines The a n a l y s e s were c a r r i e d out u s i n g : (a) R a o ' s m o d e l , (b) P o t h o f f and R o y ' s model w i t h G = I , ( c ) P o t h o f f and R o y ' s model w i t h G = an i n d e p e n d e n t e s t i m a t e of £ Q namely an e s t i m a t e o f E Q o b t a i n e d f rom the system 2 o y s t e r l i n e s , G - Y ' 2 [ I - AjCA} A p " 1 A;] Y Q 2 = H where Y Q 2 and A a r e r e s p e c t i v e l y the d a t a m a t r i x and t h e d e s i g n m a t r i x f o r the sys tem two o y s t e r l i n e s . F o r a l l models Y Q (38x9) = d a t a m a t r i x and ( j , k ) t h e lement of P = ( t f c - t ) J _ 1 where t = Z t = 58/9 ( s i n c e the t ime p o i n t s a r e not e q u a l l y spaced we cannot use o r t h o g o n a l c o e f f i c i e n t s ) . So P = 1 1 1 1 1 1 1 1 1 -40 -31 -22 -13 -4 5 14 23 68 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9 • 9 • • • • • • • • • • * • • • • • • • • ,-40xp-l f 68 .p - l - 68 - where p i s the number p a r a m e t e r s f i t t e d . S i n c e we have t a k e n the ( j , k ) t h e lement of P to be ( t ^ - t ) 3 * r a t h e r than t^ the growth c u r v e f o r the j t h group w i l l be £ Q j + ( t - t ) +... + V i j ( t " ° p-1 The i t ems a r e d i v i d e d i n t o s i x groups w i t h the f o l l o w i n g number i n e a c h g r o u p : 5, 8 , 5, 6, 6, 8. So A (38x6) = 5{ 8{ 8{ 0 . . . . 0 0 1 . . . 0 . . . 0 1 . . . . 0 0 . . . 0 1 0 0 . . . 0 1 n = 38 (number of i n d i v i d u a l s ) , m = 6 (number of g r o u p s ) , q = 9 , r = 6. I n S e c t i o n 3 . 3 , i t was d e s c r i b e d how t o t e s t h y p o t h e s e s of the fo rm C£V = 0 under each m o d e l . By c h o o s i n g C and V a p p r o p r i a t e l y the f o l l o w i n g a n a l y s e s were c a r r i e d out under each m o d e l : 1. I t was d e t e r m i n e d what degree of p o l y n o m i a l a d e q u a t e l y f i t t e d t h e d a t a . T h i s was done i n the f o l l o w i n g way: i n i t i a l l y c u b i c s were f i t t e d and the h y p o t h e s i s t h a t a l l c u r v e s were of degree 2 or l e s s was t e s t e d . - 69 - T h i s r e q u i r e d a t e s t of £13 = Z23 ~ ••• ~ Zs3 = 0 w h i c h can be w r i t t e n C£V = 0 w i t h C = I ( 6 x 6 ) and V - 0 0 0 1 s = 6, u = 1. I f q u a d r a t i c s were found to be a d e q u a t e , q u a d r a t i c s were then f i t t e d i n o r d e r to d e t e r m i n e whether l i n e a r c u r v e s would a l s o be a d e q u a t e . I f q u a d r a t i c s were no t found t o be a d e q u a t e , q u a r t i c s were then f i t t e d t o d e t e r m i n e whether c u b i c s would be a d e q u a t e . T h i s p r o c e s s c o n t i n u e d u n t i l the a p p r o p r i a t e p o l y n o m i a l was d e t e r m i n e d . 2. The h y p o t h e s i s t h a t a l l s i x growth c u r v e s were i d e n t i c a l was t e s t e d . T h i s h y p o t h e s i s can be w r i t t e n ?10 = £20 = ••• = ?60 £ll = £21 = ••• = ^61 ^1 p-1 " ^2 p-1 " *•* " ^6 p-1 or C£V = 0 where C (5x6) 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 - 70 - and V = I ( p x p ) , s = 5, u = p, where p-1 i s the degree of the p o l y n o m i a l found to be a p p r o p r i a t e i n 1. 3 . The h y p o t h e s i s t h a t the c u r v e s were i d e n t i c a l e x c e p t p o s s i b l y f o r an a d d i t i v e c o n s t a n t ( i . e . p a r a l l e l ) was t e s t e d . T h i s h y p o t h e s i s can be w r i t t e n C£V = 0 where C i s as i n 2. and V = 0 ( p - 1 ) x ( p - 1 ) , s = 5, u = p - 1 . R e s u l t s 1. ( i ) C u b i c s were f i t t e d and the h y p o t h e s i s t h a t a l l c u r v e s were o f degree 2 or l e s s was t e s t e d . a . P o t h o f f and R o y ' s model w i t h G = Z o S h S _ 1 = . 1 1 5 , r e f e r 1.04 t o F 6 j 5 1 + , p > .1 b . R a o ' s mode l S h S e - 1 = . 1 3 8 , r e f e r 1.12 t o F 6 > l t 9 , p > .1 A c c o r d i n g to b o t h m o d e l s , q u a d r a t i c s a r e a d e q u a t e . I t i s p o s s i b l e t h a t l i n e a r c u r v e s a r e a l s o a d e q u a t e . ( i i ) Q u a d r a t i c s were f i t t e d and the h y p o t h e s i s t h a t a l l c u r v e s were l i n e a r was t e s t e d . a . P o t h o f f and R o y ' s w i t h G = I -1 S h S ~"1 = 1 .84 , r e f e r 16.6 t o T6,5h> p < .001 - 71 - b. P o t h o f f and R o y ' s model w i t h G = E J o S h S e - 1 = 1 .41 , r e f e r 12.7 t o F 6 > 5 4 , P < .001 c . R a o ' s model S h S _ 1 = 1 .86, r e f e r 14.3 t o F 6 > t | 6 , p < .001 C l e a r l y l i n e a r c u r v e s a re not adequate and q u a d r a t i c s a r e a p p r o p r i a t e . So f o r each l i n e we have e s t i m a t e s of t h r e e p a r a m e t e r s , w h i c h a r e as f o l l o w s : a . P o t h o f f and Roy , G = I 2.00 .113 - .0043 1.95 .119 - .0050 1.93 .127 - .0063 1.99 .117 - .0054 2.05 .106 - .0069 2 .00 .112 - .0049 b. P o t h o f f and Roy , G = E Q 2.00 .114 - .0042 1.95 .116 - .0047 1.91 .123 - .0057 1.98 .120 - .0051 2 .09 .102 - .0066 2.01 .107 - .0044 c . Rao 1.97 .112 - .0043 1.92 .119 - .0046 1.94 .132 - .0074 2.02 .126 - .0070 2.02 .104 - .0060 1.99 .114 - .0045 - 72 - A l l t h r e e models g i v e v e r y s i m i l a r e s t i m a t e s of £. In p a r t i c u l a r , the c h o i c e of G i n P o t h o f f and R o y ' s model a f f e c t s the e s t i m a t e v e r y l i t t l e . So , l e t t i n g y ( t ) denote s h e l l l e n g t h at t ime t , the growth c u r v e s t h a t we o b t a i n u n d e r , s a y , R a o ' s model f o r 0 1 , 0 2 , . . . , 06 a r e r e s p e c t i v e l y : y ( t ) = - .0043 t 2 + .167 t + 1.07 y ( t ) = - .0046 t 2 + .178 t + .96 y ( t ) = - .0074 t 2 + .227 t + .78 y ( t ) - - .0070 t 2 + .216 t + .92 y ( t ) = - .0060 t 2 + .181 t + 1.10 y ( t ) = - .0045 t 2 + .172 t + 1.07 A c c o r d i n g t o R a o ' s m o d e l , e s t i m a t e s of the a v e r a g e s h e l l l e n g t h over a sample of l i v i n g b a r n a c l e s t a k e n f rom an i t e m on 01 a t t = 2, 3 , 4 , 5, 6, 7, 8 , 9 , 14 weeks a r e r e s p e c t i v e l y : 1 .39 , 1 .53 , 1 .67 , 1 .80 , 1 .92 , 2 . 0 3 , 2 . 1 3 , 2 . 2 3 , 2.57 cm. P i c t u r e s of the s i x c u r v e s o b t a i n e d under R a o ' s model a r e g i v e n i n F i g u r e 1. The s i x c u r v e s a r e not t o o f a r f rom b e i n g l i n e a r and a r e v e r y s i m i l a r t o each o t h e r - a lmost p a r a l l e l , a l t h o u g h no t q u i t e i d e n t i c a l . The h y p o t h e s e s t h a t the c u r v e s were i d e n t i c a l and t h a t they were i d e n t i c a l e x c e p t f o r an a d d i t i v e c o n s t a n t were then t e s t e d w i t h the f o l l o w i n g r e s u l t s : 2. The h y p o t h e s i s t h a t a l l s i x c u r v e s were i d e n t i c a l was t e s t e d , a . P o t h o f f and R o y ' s m o d e l , G = I Figure 1. Growth Curves for the System 1 Oyster Lines 1 1 1 1 1 1 1 1 1 r — i 1 1 r .0 2.0 4 .0 6 .0 8.0 10.0 12.0 T I M E I V E E K 5 ) Figure 2. Growth Curves for the System 2 Oyster Lines - 74 - L a r g e s t r o o t o f S e _ 1 = .419 R e f e r = .29 to Heck t a b l e s w i t h s* = 3 , m* = 1/2, n* = 2 5 . P * , .04 b. P o t h o f f and R o y ' s m o d e l , G = E o L a r g e s t r o o t of S - 1 = .613 h e 613 R e f e r ,* _ = .38 to Heck t a b l e s 1 • D l J w i t h s* = 3, m* = 1/2, n* » 2 5 . p = .'01 c . R a o ' s model L a r g e s t r o o t o f S, S - 1 = .452 h e 452 R e f e r ^*. c o - = .31 to Heck t a b l e s 1.452 w i t h s* = 3 , m* = 1/2, n* = 2 5 . p « .05 We have a s u f f i c i e n t amount of d a t a to r e j e c t the h y p o t h e s i s t h a t the s i x growth c u r v e s a r e i d e n t i c a l , a l t h o u g h t h e y appear v e r y s i m i l a r i n F i g u r e 1. T h e r e i s a c o n s i d e r a b l e d i s c r e p a n c y between the p - v a l u e s o b t a i n e d under P o t h o f f and R o y ' s model w i t h G = I , and w i t h G = T.Q. I t seems t h a t the c h o i c e of G g r e a t l y a f f e c t s the v a r i a n c e of the e s t i m a t o r s and hence the p - v a l u e s . - 75 - 3 . The h y p o t h e s i s t h a t t h e c u r v e s were i d e n t i c a l e x c e p t f o r an a d d i t i v e c o n s t a n t was t e s t e d , a . P o t h o f f and R o y ' s m o d e l , G = I L a r g e s t r o o t of S e _ 1 = .164 .164 R e f e r = .14 t o Heck t a b l e s 1.164 p » .05 b. P o t h o f f and R o y ' s m o d e l , G = ZQ L a r g e s t r o o t of S h S e _ 1 = .224 .224 R e f e r ^ * ~ * ^ t o H e c ^ t a b l e s p » .05 c . R a o ' s model L a r g e s t r o o t of S, S - 1 = .212 n e .212 R e f e r ^ * 212 = t o ^ e c ^ t a ° l e s p » .05 Not s u r p r i s i n g l y the c u r v e s a r e found to be p a r a l l e l . T h i s c o n c l u s i o n i s c l e a r i n s p i t e of t h e c o n s i d e r a b l e d i s c r e p a n c y w h i c h i s a g a i n o b s e r v e d between the p - v a l u e s o b t a i n e d under P o t h o f f and R o y ' s model w i t h G = S q and w i t h G = I . B e f o r e d i s c u s s i n g the r e s u l t s f u r t h e r , s i m i l a r r e s u l t s w i l l now be o b t a i n e d f o r the sys tem two o y s t e r l i n e s . - 76 - 3.6 System Two Oyster Lines The same analyses were repeated for the system two oyster l i n e s . We had a new data matrix Y(50 x 9) and a new design matrix A (50x6) - 9{ 9{ 1 0 . . . . 0 . . . 0 . . . 0 1 0 . . . . 0 1 while P was unchanged. Again the analyses were carried out using Rao's model and using Pothoff and Roy's model both with G = I and with G = an estimate of E Q obtained from the system one oyster l i n e s , E = Y ' [ I - A.(A! A . ) - 1 AJ]Y o o i l l l o where Y q and A are respectively the data matrix and the design matrix for the system one oyster l i n e s . Results 1. ( i ) Cubics were f i t t e d and the hypothesis that a l l curves were of degree 2 or less was tested. a. Pothoff and Roy's model with G = E J o S h S e _ 1 = .066, re f e r .59 to F 6 > 5 1 t , p > .1 b. Rao's model S h S g 1 = .196, refer 1.6 to F g ^ g , p > .1 - 77 - C l e a r l y q u a d r a t i c s a r e a d e q u a t e . The c o n c l u s i o n i s c l e a r d e s p i t e the d i s c r e p a n c y between the two v a l u e s of the t e s t s t a t i s t i c . ( i i ) Q u a d r a t i c s were t h e n f i t t e d and the hypothe s i s t h a t a l l c u r v e s were l i n e a r was t e s t e d . a . P o t h o f f and Roy , G = I S h S e - 1 e 8 . 3 7 , r e f e r 7 5 . 3 t o F 6 > 5 1 | , P « .001 A b. P o t h o f f and Roy G = E o S h S _ 1 = 4 . 5 1 , r e f e r 4 0 . 5 t o F 6 ) 5 1 t , p « .001 c . R a o ' s model S h S e - 1 = 8 . 2 6 , r e f e r 6 2 . 9 t o F 6 > l t 6 , p « .001 The h y p o t h e s i s t h a t l i n e a r c u r v e s a r e adequate i s v e r y s t r o n g l y r e j e c t e d . So q u a d r a t i c s a r e a g a i n a p p r o p r i a t e . The e s t i m a t e s of % a r e : a . P o t h o f f and R o y , G = I 2 .23 .094 - .0052 2 .15 .104 - .0070 2 .28 .076 - .0053 2.17 .125 - .0099 2 .28 .097 - .0063 2.16 .106 - .0082 b. P o t h o f f and Roy , G = £ 2.21 .091 - .0042 2.16 .101 - .0069 2.29 .077 - .0059 2.18 .128 - .0105 2.27 .095 - .0063 2.14 .103 - .0080 - 78 - c . Rao 2.24 .094 - .0054 2.16 .107 - .0074 2.25 .081 - .0055 2.17 .122 - .0096 2.28 .099 - .0064 2.15 .107 - .0084 A g a i n , the e s t i m a t e s o b t a i n e d from a l l t h r e e methods a re a l m o s t i d e n t i c a l . The g rowth c u r v e s t h a t we o b t a i n under R a o ' s model f o r 0 7 , 08 012 a r e r e s p e c t i v e l y : y ( t ) = - .0054 t 2 + .164 t + 1.41 y ( t ) = - .0074 t 2 + .202 t + 1.16 y ( t ) = - .0055 t 2 + .152 t + 1.50 y ( t ) = - .0096 t 2 + .246 t + 0.98 y ( t ) = - .0064 t 2 + .181 t + 1.38 y ( t ) = - .0084 t 2 + .215 t + 1.11 A c c o r d i n g to R a o ' s m o d e l , e s t i m a t e s of the a v e r a g e s h e l l l e n g t h over a sample o f l i v i n g b a r n a c l e s t a k e n f rom an i t e m on 07 a t t = 2, 3 , 4 , 5, 6, 7, 8, 9 , 14 weeks a r e r e s p e c t i v e l y : 1 .72, 1 .85 , 1 .98, 2 . 1 0 , 2 . 2 0 , 2 . 2 9 , 2 . 3 8 , 2 . 4 5 , 2.65 cm. P i c t u r e s of the s i x c u r v e s o b t a i n e d under R a o ' s model a r e g i v e n i n F i g u r e 2. The c u r v e s a r e l e s s l i n e a r than t h o s e o b t a i n e d f o r the sys tem one o y s t e r l i n e s ( t h e h y p o t h e s i s o f l i n e a r i t y was r e j e c t e d w i t h a much s m a l l e r p - v a l u e f o r the sys tem two o y s t e r l i n e s than f o r the sys tem one o y s t e r l i n e s ) . They are a l s o l e s s homogeneous than the c u r v e s o b t a i n e d f o r the sys tem one o y s t e r l i n e s - whereas the c u r v e s f o r the sys tem one o y s t e r l i n e s were found to be p a r a l l e l , t h e s e c u r v e s a re f a r f rom b e i n g p a r a l l e l . T h i s o b s e r v a t i o n was c o n f i r m e d by the r e s u l t s o f the f o l l o w i n g t e s t s : 2. The h y p o t h e s i s t h a t a l l s i x c u r v e s were i d e n t i c a l was t e s t e d . a . P o t h o f f and R o y ' s m o d e l , G = I L a r g e s t r o o t of S L S _ 1 = .801 h e 801 R e f e r ^ 'gQ^ = «44 t o Heck t a b l e s 1% c r i t i c a l v a l u e = . 3 2 , p < .01 At b. P o t h o f f and R o y ' s m o d e l , G = E o L a r g e s t r o o t of S , S - 1 = .894 h e R e f e r = .47 t o Heck t a b l e s , 1% c r i t i c a l v a l u e = . 3 2 , 1% c r i t i c a l v a l u e = . 3 2 , p < .01 c . R a o ' s model L a r g e s t r o o t of S, S - 1 = .660 ° h e R e f e r ^ * = .40 to Heck t a b l e s 1% c r i t i c a l v a l u e = . 3 6 , p < .01 As e x p e c t e d the c u r v e s a r e no t found to be i d e n t i c a l . 3 . The h y p o t h e s i s t h a t the c u r v e s were p a r a l l e l was then t e s t e d . a . P o t h o f f and R o y ' s m o d e l , G = I L a r g e s t r o o t of S, S - 1 = .648 h e - 80 - R e f e r = .39 to Heck t a b l e s 1% c r i t i c a l v a l u e = . 3 0 , p < .01 b. P o t h o f f and R o y ' s m o d e l , G = E L a r g e s t r o o t of S L S _ 1 = .678 n e 678 R e f e r . *,-. Q = .40 to Heck t a b l e s , 1 . o / o 1% c r i t i c a l v a l u e = . 3 0 , p < .01 c . R a o ' s model L a r g e s t r o o t of S, S - 1 = . 4 9 0 h e .49 R e f e r ~ .33 t o Heck t a b l e s , 1% c r i t i c a l v a l u e = . 3 3 , p = .01 The p - v a l u e s a re somewhat l a r g e r than i n 2 . , bu t n e v e r t h e l e s s , even the h y p o t h e s i s t h a t the c u r v e s a r e p a r a l l e l must be r e j e c t e d . Discussion As a l r e a d y o b s e r v e d , the growth c u r v e s f o r the sys tem one o y s t e r l i n e s a r e more homogenous than the c u r v e s f o r the sys tem two o y s t e r l i n e s . But even f o r the sys tem one l i n e s d i f f e r e n c e s were f o u n d , so i t won ' t be r e a s o n a b l e t o i g n o r e l i n e s w i t h i n a p a r t i c u l a r s y s t e m - t y p e c o m b i n a t i o n . The c o n s t a n t term i s g e n e r a l l y l a r g e r f o r the sys tem two c u r v e s than f o r the sys tem one c u r v e s w h i c h s u g g e s t s t h a t b a r n a c l e s on s y s t e m two a r e g e n e r a l l y l a r g e r than t h o s e on sys tem one . The sys tem - 81 - one c u r v e s a r e more l i n e a r than the sys tem two c u r v e s . The a v e r a g e s i z e of b a r n a c l e s on i t ems i n sys tem two a p p e a r s to i n c r e a s e f a i r l y q u i c k l y t o s t a r t w i t h and much l e s s r a p i d l y l a t e r on , whereas the a v e r a g e s i z e of b a r n a c l e s on i t e m s i n sys tem one i n c r e a s e s at a more c o n s t a n t r a t e . We cannot c o n c l u d e however , t h a t b a r n a c l e s on system two t e n d t o grow more q u i c k l y at f i r s t than b a r n a c l e s on sys tem one and l e s s q u i c k l y l a t e r on , as the c u r v e s r e p r e s e n t the n a t u r a l p o p u l a t i o n , no t i n d i v i d u a l b a r n a c l e s . I t i s p o s s i b l e t h a t the d i f f e r e n c e does r e f l e c t d i f f e r e n t g rowth c h a r a c t e r i s t i c s f o r the two s y s t e m s , but i f the p r o b a b i l i t y of d y i n g i s r e l a t e d to s i z e , i t may a l s o r e f l e c t the d i f f e r e n t s u r v i v a l p a t t e r n s f o r the two s y s t e m s . Suppose , f o r examp le , t h a t t h e r e i s a t e n d e n c y f o r the b i g g e r b a r n a c l e s t o d i e f i r s t , then a l a r g e number o f d e a t h s i n a p a r t i c u l a r i n t e r v a l would cause the i n c r e a s e i n average s i z e i n t h a t i n t e r v a l t o be s m a l l e r than would be e x p e c t e d f o r the i n c r e a s e i n s i z e of any i n d i v i d u a l . On system one o y s t e r l i n e s b a r n a c l e s d i e o f f v e r y q u i c k l y between t = 5 and t = 8 weeks , but a t a much s l o w e r r a t e a f t e r t h a t , whereas b a r n a c l e s on system two o y s t e r l i n e s d i e o f f a t a more c o n s t a n t r a t e between t = 5 and t = 17 weeks . T h i s c o u l d a c c o u n t f o r the more l i n e a r c u r v e s f o r sys tem one l i n e s . The M o d e l s The e s t i m a t e s of J; a r e v e r y s i m i l a r f o r a l l t h r e e methods . As a l r e a d y p o i n t e d o u t , the c h o i c e o f G i n P o t h o f f and R o y ' s model d o e s n ' t g r e a t l y a f f e c t the e s t i m a t e o f £, but i t does a f f e c t i t s v a r i a n c e and hence the p - v a l u e s of the t e s t s . The p - v a l u e o b t a i n e d u s i n g G = I - 82 - was usually considerably d i f f e r e n t from that obtained using G = E Q . The p-value obtained with G = I tended to agree more cl o s e l y with the A p-value obtained under Rao's model. So the p-value obtained with G = E r o may be suspect, p a r t i c u l a r l y since the estimate of the covariance matrix obtained from the system one oyster l i n e s turned out to be very d i f f e r e n t from that obtained from the system two oyster l i n e s . The estimates of the vector of variances obtained respectively from the system one oyster l i n e s and the system two oyster l i n e s were: [.015, .018, .016, .014, .0096, .026, .035, .051, .040] and [.025, .024, .041, .020, .012, .0094, .018, .014, .024]. The estimates of the c o r r e l a t i o n matrix were respectively: 1 .62 1 .48 .61 1 .44 .57 .55 1 .23 .32 .29 .43 1 .13 .27 .47 .41 .40 1 .20 .25 .40 .37 .23 .77 1 .19 .23 .19 .36 .07 .62 .69 1 .03 .09 .17 .28 .09 .19 .19 .26 and - 83 - 1 .72 1 .74 .73 1 .52 .59 .61 1 .54 .61 .58 .68 1 .47 .54 .49 .55 .63 1 .24 .22 .13 .30 .27 .31 1 .35 .35 .35 .50 .57 .61 .47 1 .25 .22 .15 .29 .32 .43 .45 .59 O b s e r v a t i o n s f o r the sys tem one o y s t e r l i n e s a r e more v a r i a b l e towards the end of the s tudy than at the b e g i n n i n g , whereas measurements f o r the sys tem two o y s t e r l i n e s a r e more v a r i a b l e a t t h e b e g i n n i n g o f the s t u d y than l a t e r o n . The two c o r r e l a t i o n m a t r i c e s have a r o u g h l y s i m i l a r p a t t e r n a l t h o u g h the c o r r e l a t i o n s between o b s e r v a t i o n s at t = 7 and t = 8 weeks and between o b s e r v a t i o n s at t = 8 and t = 9 weeks a r e p a r t i c u l a r l y low f o r sys tem two and p a r t i c u l a r l y h i g h f o r sys tem one . In v iew of t h e s e d i f f e r e n c e s i t may be t h a t u s i n g an e s t i m a t e o f £ 0 o b t a i n e d f rom one system as the c h o i c e of G i n the model f o r the o t h e r s y s t e m , l e d to d u b i o u s p - v a l u e s . 3.7 A p p l i c a t i o n of the Model to A l l Seventeen Lines The next s t e p was to i n c o r p o r a t e t h e n e t t i n g and d o w l i n g i n t o the mode l and to t e s t f o r d i f f e r e n c e s not o n l y between the two systems but - 84 - also between the three types of material. Since differences had been found between the system one oyster l i n e s and between the system two oyster l i n e s , each of the seventeen l i n e s was allowed d i f f e r e n t parameters. After f i t t i n g growth curves for each of the seventeen l i n e s , contrasts were examined. This time only Rao's model was used. Since no estimate of E 0 was a v a i l a b l e from a s i m i l a r but independent experiment i n t h i s case, the only reasonable choice of G i n Pothoff and Roy's model would be I, which could d i f f e r s u b s t a n t i a l l y from £ 0 . Y Q (141 x 9) i s the data matrix for a l l 17 l i n e s . The data for system one appears f i r s t , then the data for system two. P i s unchanged. A (141x17) = 5{ 8{ 10f 1 0 0 . . 1 . . 1 . . 0 . 1 0 1 . 0 . 0 . 0 . 0 . 0 . 0 . 0 0 q = . . . . o 1 141 (number of i n d i v i d u a l s ) , m 9 (number of time points), r 17 (number of groups), 17 (rank A). - 85 - At t h i s stage i t was assumed that each set of nine measurements had the same covariance matrix, whether measurements were averages over f i v e as for oyster and dowling, or over ten as for netting. Suppose we wanted an o v e r a l l test for differences between the growth curves for the two systems. This could be done by t e s t i n g the hypothesis: 8Ll + 8 l 2 + ••• + 8 l 9 - 9 I l O " ... ~ 9 J L i 7 = 0 where l± = [?1Q 5±1 . . . l± ^1 i s the set of p parameters for group i . Since the f i r s t nine groups correspond to system one l i n e s , and the other eight to system two l i n e s , t h i s i s equivalent to t e s t i n g that the average of the s over the system one l i n e s i s the same as the average of the C j / s over the system two l i n e s . A r e j e c t i o n of th i s hypothesis would indicate a difference between the two systems i n the average value of at least one of the parameters. The hypothesis can be written C£V = 0 where C = (8 8... 8 -9 -9 ... -9) and V = I (p x p). (1x17) Other contrasts were examined i n order to determine whether differences existed between the three types of material. F i r s t of a l l , i t was determined what degree of polynomial was needed to f i t the growth curves f o r a l l 17 l i n e s adequately. This was done i n the same way as for the oyster l i n e s previously. I t was found that quartics were needed. The f i t t e d growth curves obtained for Dl, D2, D3 were respectively: - 86 - y(t ) = - .0002 tH + .0067 t 3 - .086 t 2 + .578 t + .44 y(t ) = .0005 tk - .0143 t 3 + .126 t 2 - .224 t + 1.06 y ( t ) - .00001 t 4 + .0009 t 3 - .030 t 2 + .387 t + .51 and for Nl and N2: y ( t ) = .0001 tH - .0023 t 3 + .014 t 2 + .100 t + 1.24 y(t) - .0002 t 1* - .0059 t 3 + .058 t 2 - .113 t + 1.62 Pictures of the curves f o r the dowling l i n e s appear i n Figure 3 and for the netting l i n e s i n Figure 4. The curves for the netting l i n e s are almost l i n e a r . A test of the hypothesis of l i n e a r i t y f o r these curves alone yielded a p-value as large as 7%. We have already ascertained that quadratics are adequate for the oyster l i n e s . C l e a r l y i t i s because of the dowling l i n e s , and because of D2 i n p a r t i c u l a r , that I. quartics are needed. The c o e f f i c i e n t of t i s much larger for D2 than for any other l i n e and the growth curve for D2 looks remarkably d i f f e r e n t from any of the other growth curves (see Figure 2). Before discussing t h i s , r e s u l t s of the tests comparing the curves for the two systems and for the three material types w i l l be presented: 1. The growth curves f o r the two systems were compared by t e s t i n g CIV = 0 where C = ( 8 8 8 -9 -9 ... -9) and V = I (5x5), (1x17) Largest root of S,S ~ 1 = .732, h e Refer 16.9 to F 5 1 1 6 , p < .001. - 87 - Figure 3. Growth Curves for the Dowling Lines Figure 4. Growth Curves for the Netting Lines ^ i I 1 1 I 1 1 1 1 1 1 1 1 1 1 0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 TIME (WEEKS) - 88 - 2. The growth curves for the oyster and dowling l i n e s were compared by t e s t i n g C£V = 0 where C = (1 1 1 1 1 1 -4 -4 0 1 1 1 1 1 1 -4 0) and V = I- (5x5). Largest root of S,S - 1 = 1.121-, h e Refer 25.9 to F 5 ) 1 1 6 , p < .001. 3. The growth curves for the oyster and netting l i n e s were compared by te s t i n g C^V = 0 where C = (1 1 1 1 1 1 0 0 - 6 1 1 1 1 1 1 0 - 6 ) and V = I (5x5). Largest root of S.S - 1 = .426, h e ' Refer 9.9 to F 5 > 1 1 6 , p < .001. 4. The growth curves for the dowling and netting l i n e s were compared by t e s t i n g C£V = 0 where C = ( 0 0 0 0 0 0 2 2 - 3 0 0 0 0 0 0 2 - 3 ) and V = I (5x5). Largest root of S h S e _ 1 = .524. Refer 12 to F 5 } n 6 , p < .001. The same analyses were then repeated t h i s time allowing the curves to d i f f e r by an additive constant. So this time £ Q was not included; f o r each test C remained unchanged, but V was replaced by 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 - 89 - Results 1. The two systems L a r g e s t r o o t of S, S - 1 = .412 h e R e f e r 12 .2 t o F 4 ) 1 1 9 p < .001 . 2. O y s t e r and d o w l i n g L a r g e s t r o o t of S , S - 1 = . 7 3 0 h e R e f e r 21 .7 t o F 4 > 1 1 9 p < .001 . 3 . O y s t e r and n e t t i n g L a r g e s t r o o t of S - 1 = .35 7 R e f e r 10 .7 t o Yh 1 1 9 p < .001 . 4 . D o w l i n g and n e t t i n g L a r g e s t r o o t of S . S - 1 = .471 h e R e f e r 13 .9 t o Fi+ p < .001 . Discussion D i f f e r e n c e s a r e found b o t h between the two systems and between the t h r e e t y p e s of m a t e r i a l . T h i s i s i n c o n t r a s t to what was found f o r t h e - 90 - s u r v i v a l d i s t r i b u t i o n s , namely that only system was an important f a c t o r . With such a large amount of data i t i s not su r p r i s i n g that d i f f e r e n c e s are found everywhere. The r e l a t i v e size of the test s t a t i s t i c s indicates that the biggest differences l i e between dowling and the other two types of material. This was to be expected i n view of the unusual growth curve for D2. For the comparisons of the two systems and of oyster and dowling l i n e s the p-values were much larger when the curves were allowed to d i f f e r by an additive constant. On the other hand for the other two comparisons the p-values did not change s u b s t a n t i a l l y when the curves were allowed to d i f f e r by a constant. So the difference between netting and dowling and between netting and oyster i s (almost) s o l e l y a t t r i b u t a b l e to the d i f f e r e n t shapes of the growth curves. The difference between the two systems and between oyster and dowling i s due pa r t l y to the d i f f e r e n t shapes of the curves and p a r t l y to d i f f e r e n t l o c a t i o n s . We conclude that barnacles on system two are generally bigger than those on system one and that barnacles on oyster l i n e s are generally bigger than those on dowling l i n e s . The smoothness of the growth curves for the netting l i n e s may r e f l e c t the averaging over ten instead of f i v e , rather than a growth c h a r a c t e r i s t i c . The average size of barnacles on netting l i n e s appears to increase at an almost constant rate except towards the end of the study when i t increases a l i t t l e f a s t e r , whereas the average size of barnacles on dowling l i n e s i s p a r t i c u l a r l y small i n i t i a l l y , increases rapidly to - 91 - s t a r t w i t h and l e s s r a p i d l y l a t e r on . The b e h a v i o r o f the a v e r a g e s i z e of b a r n a c l e s on o y s t e r l i n e s l i e s somewhere between t h e s e two e x t r e m e s . The a v e r a g e s i z e o f b a r n a c l e s on D2 i s e x c e p t i o n a l l y s m a l l a t t = 2 weeks , i n c r e a s e s f a i r l y r a p i d l y u n t i l t = 8 weeks , appears t o r e m a i n the same u n t i l t = 12 weeks and t h e n t o i n c r e a s e e x t r e m e l y r a p i d l y . T h i s c o u l d be a growth c h a r a c t e r i s t i c , but s i n c e t h i s b e h a v i o r i s so d i f f e r e n t , i t may a l s o be t h a t p a r t i c u l a r l y s m a l l b a r n a c l e s were sampled at t = 8 and 9 weeks and p a r t i c u l a r l y l a r g e b a r n a c l e s were sampled at t = 14 weeks . 3.8 Extension of the Model to Allow D i f f e r e n t Covariance Matrices f o r D i f f e r e n t Items An e s t i m a t e of E Q was o b t a i n e d s e p a r a t e l y f rom the o y s t e r l i n e s , t h e d o w l i n g l i n e s and the n e t t i n g l i n e s . Not s u r p r i s i n g l y the e s t i m a t e o b t a i n e d f rom the n e t t i n g l i n e s was c o n s i d e r a b l y s m a l l e r than the o t h e r e s t i m a t e s s i n c e measurements f rom i t e m s on n e t t i n g l i n e s a r e a v e r a g e s over t e n i n s t e a d of f i v e . I t would seem more r e a s o n a b l e t o assume t h a t t h e c o v a r i a n c e m a t r i x f o r i t e m s on o y s t e r or d o w l i n g l i n e s i s (1/5)E f o r some E, and (1/10)E f o r i t e m s on n e t t i n g l i n e s . The growth c u r v e mode ls can be e x t e n d e d to the case where the c o v a r i a n c e m a t r i c e s f o r d i f f e r e n t i n d i v i d u a l s a r e assumed t o be known m u l t i p l e s of each o t h e r ( I t o , 1 9 6 8 ) . Suppose t h a t t h e r e a r e N i n d i v i d u a l s who a r e assumed to have c o v a r i a n c e m a t r i c e s G l E, e 2 E, e 3 E,..., where the 9^ a r e s c a l a r s and E i s f i x e d but unknown. Suppose the - 92 - i n d i v i d u a l s are divided into groups with nj i n the j t h group. Individuals i n the same group are assumed to have the same covariance matrix, so 0i = 02 = .... = Qn^ and s i m i l a r l y for the other groups. Define a diagonal matrix of weights W (NxN) 1/0! i / e 2 i / e N Instead of choosing £ to minimise (X - A£)' (X - A£) i t i s now chosen to minimise (X - A £ ) f W (X - Ac;) (weighted least squares). So more weight i s given to i n d i v i d u a l s with smaller covariance matrix. It follows that £ = (A^ W A T 1 A{ W X. Tests of C£V = 0 are based on S = ( q I V)« ( C i R C i ) ~ l ( d I V) where R = (A^ W A j ) " 1 + (AJ W A j ) " 1 AJ W X B X' W A j(A{ W ^ ) -1 and S = V'(P S - 1 P * ) _ 1 V. I f W = I ( t h i s means equal weight i s given to a l l observations) we obtain the same expression for R and £ and hence for S^ as previously. The estimate of E i s given by S, where S=-g Sx+g- s2 + 9" S 3 + .... 1 n^+1 ni+n2 +l and S t i s the estimate of the covariance matrix for group t, namely Ŝ  = Y'[I - A,(A» A , ) " 1 A.1] Y where Y i s the data matrix for group t t o l l l l o O a r - 93 - and A, the design matrix for group t, i s a ( n t x 1) vector of l ' s . Individuals i n group 2 have covariance matrix 9 n i + ^ £> those i n group 3 have covariance matrix 0 , ,,£ and so on. If 0i = 0o = ... = 9„ = 1 n]+n 2+l 1 N ( a l l i n d i v i d u a l s have the same covariance matrix) then S w i l l be as for the standard model, namely Y'[I - A.(A! A . ) - 1 A'] Y where Y and A o i l l 1 o o are the data and design matrices for the complete set of data. The test s based on and S e are then as before. Quartics were again f i t t e d and a l l the tests of the previous section were repeated with the adjusted and S e. In our case S = y [covariance matrix for oysters] + •^•[covariance matrix for dowling] + [covariance matrix for netting] and 0^ = 2, i = 1,...,60 and 69,...,131 (items on oyster and dowling l i n e s ) , 0^ = 1, i = 61,...,68 and 132,...,141 (items on netting l i n e s ) . A, Y Q and P are unchanged. Results The same tests as i n the previous section, to compare the growth curves for various pairs of subgroups were carried out: 1. The two systems Largest root of S h S - 1 = .595 Refer 13.9 to F 5 > 1 1 6 , p < .001 - 94 - 2. O y s t e r and D o w l i n g L a r g e s t r o o t of S , S _ 1 = .902 h e R e f e r 20 .9 t o F 5 , 1 1 6 , p < .001 3 . O y s t e r and N e t t i n g L a r g e s t r o o t of S, S - 1 = .610 h e R e f e r 14.2 t o F 5 , i 1 6 , p < .001 4 . N e t t i n g and D o w l i n g L a r g e s t r o o t of S, S - 1 = .640 n e R e f e r 14 .8 t o F 5 > 1 1 6 , p < .001 Comments The e s t i m a t e of £ i s u n a f f e c t e d by i n c l u d i n g W i n the a n a l y s i s , s i n c e w i t h i n each group ( l i n e ) the same we igh t i s g i v e n to a l l i t ems ( t h e same c o v a r i a n c e m a t r i x i s assmed f o r i n d i v i d u a l s i n t h e same g r o u p ) . However the p - v a l u e s of the t e s t s a r e a f f e c t e d . In compar i sons i n v o l v i n g n e t t i n g , the p - v a l u e o b t a i n e d i n the o r i g i n a l unwe ighted a n a l y s i s was l a r g e r . T h i s I s because the e s t i m a t e of the c o v a r i a n c e was t o o b i g because i t was o b t a i n e d f rom a l l i t e m s , whether f rom o y s t e r , n e t t i n g or d o w l i n g l i n e s , and t h e s e were a l l assumed to have the same £ Q . F o r c o m p a r i s o n s not i n v o l v i n g n e t t i n g the o r i g i n a l p v a l u e was - 95 - s m a l l e r . T h i s i s because the e s t i m a t e of the c o v a r i a n c e m a t r i x was t o o s m a l l because i t was o b t a i n e d no t o n l y f rom i t e m s on o y s t e r and d o w l i n g l i n e s but a l s o f rom i t e m s on n e t t i n g l i n e s . The p - v a l u e s o b t a i n e d i n the w e i g h t e d a n a l y s i s s h o u l d be e x p e c t e d t o be more r e l i a b l e . The main c o n c l u s i o n s rema in the same however , namely t h a t d i f f e r e n c e s e x i s t b o t h between the two systems and between the t h r e e t y p e s of m a t e r i a l s w i t h the b i g g e s t d i f f e r e n c e b e i n g between the o y s t e r and d o w l i n g l i n e s . 3 .9 Collapse Over Lines Within Each System-Type Combination A l t h o u g h d i f f e r e n c e s had been found between l i n e s of the same t y p e w i t h i n a s y s t e m , ( p a r t i c u l a r l y w i t h i n sys tem t w o ) , a n o t h e r a n a l y s i s was then c a r r i e d out i n wh ich t h e s e d i f f e r e n c e s were i g n o r e d . I g n o r i n g l i n e s we get a t w o - f a c t o r s i t u a t i o n as p r e v i o u s l y d e s c r i b e d . The number of i t e m s i n each c a t e g o r y i s g i v e n be low: Type System 1 System 2 O y s t e r 38 50 N e t t i n g 22 13 D o w l i n g 8 10 The a n a l y s i s was unwe ighted - t h e same c o v a r i a n c e m a t r i x was assumed f o r a l l i n d i v i d u a l s ( i t e m s ) . I t was assumed t h a t the growth c u r v e f o r an i t e m on the j t h type o f l i n e i n the i t h sys tem was - 96 - + ( ^ o j + ^ l j fc + ••* + V l j t P _ 1 ) So E[Y Q] = A I P, where A (141 x 5) = 38 22 50 13 10 0 0 £(5xp) = HOI H l l • H02 M-12 • HO 3 Hi 3 • £oi ^11 * ^02 S i 2 • and Y Q and P are as before. H - 1 i i p-1 1 V-i 2 p-1 3 'p-1 1 V l 2 Quartics were fitted and the usual hypotheses were tested, namely that the growth curves for various pairs of subgroups were identical: - 97 - 1) and V = 1 ( 5 x 5 ) . 0) and V = 1 ( 5 x 5 ) . 3. O y s t e r and N e t t i n g T e s t C I V = 0 where C = (1 0 -1 0 0) and V = 1 ( 5 x 5 ) . L a r g e s t r o o t of S, S - 1 = .874 h e R e f e r 22 .4 t o F 5 > 1 2 8 , p < .001 4 . N e t t i n g and D o w l i n g T e s t C H = 0 where C = (0 1 -1 0 0) and V = 1 ( 5 x 5 ) . L a r g e s t r o o t of S, S - 1 = 5 .022 n e R e f e r 128.5 t o F 5 ) 1 2 8 , P « .001 A g a i n d i f f e r e n c e s a r e found everywhere w i t h the b i g g e s t d i f f e r e n c e s b e i n g between d o w l i n g and each of the o t h e r two t y p e s . The s m a l l e s t d i f f e r e n c e i s between the two s y s t e m s . However the p - v a l u e s f o r the two c o m p a r i s o n s i n v o l v i n g d o w l i n g a r e s u s p i c i o u s l y s m a l l and t h i s method o f c o l l a p s i n g over l i n e s i s s u s p e c t . 1. The two systems T e s t C I V = 0 where C = (0 0 0 1 - L a r g e s t r o o t of S, S - 1 = . 7 5 2 . n e R e f e r 19.2 t o F 5 > 1 2 8 , P < '001 2. O y s t e r and D o w l i n g T e s t C I V = 0 where C = (1 -1 0 0 L a r g e s t r o o t o f S h S e - 1 = 4 .561 R e f e r 116.7 t o F 5 1 2 8 , P « .001 - 98 - 3.10 Missing Data A n o t h e r g e n e r a l i s a t i o n of the model c o u l d a l s o have been u s e d , namely the g e n e r a l i s a t i o n s u g g e s t e d by K l e i n b a u m (1973) t o a l l o w m i s s i n g d a t a . In t h i s way the a s s u m p t i o n t h a t o b s e r v a t i o n s a re t a k e n a t the same t imes f o r a l l i n d i v i d u a l s ( I tems) c o u l d have been r e l a x e d . Under t h i s model i t i s assumed t h a t d a t a i s m i s s i n g at random - whether or n o t an o b s e r v a t i o n i s m i s s i n g i s u n r e l a t e d to the v a l u e i t wou ld have t a k e n . K l e i n b a u m ' s g e n e r a l i s e d model p r o c e e d s as f o l l o w s : The n i n d i v i d u a l s a r e d i v i d e d i n t o u d i s j o i n t groups S i , S 2 , . . . , S u such t h a t i n d i v i d u a l s i n the same group Sj have measurements t a k e n a t the same qj t ime p o i n t s . The number of i n d i v i d u a l s i n Sj i s n j . I n d i v i d u a l s i n d i f f e r e n t g roups may no t have measurements t a k e n a t the same t ime p o i n t s but may have measurements t a k e n at the same number o f t i m e p o i n t s (q j = 9 j ' ) * L e t q be the t o t a l number of t ime p o i n t s . Then f o r each group Sj we have a d e s i g n m a t r i x Aj (n j x m) ( s i n c e w i t h i n each group S j , i n d i v i d u a l s a r e s u b d i v i d e d a c c o r d i n g t o w h i c h of the m " t r e a t m e n t " g r o u p s t h e y b e l o n g t o ) . A l s o f o r e a c h group Sj we have an i n c i d e n c e m a t r i x Bj (q x q j ) of O 's and l ' s i n d i c a t i n g the p o s i t i o n s of m i s s i n g o b s e r v a t i o n s f o r i n d i v i d u a l s i n S j • I f ^ j ( n j x q j ) i s the d a t a m a t r i x f o r the i n d i v i d u a l s i n S j and £ Q i s the c o v a r i a n c e m a t r i x f o r a comp le te s e t of q o b s e r v a t i o n s on any i n d i v i d u a l (assumed the same f o r e v e r y i n d i v i d u a l ) , t h e n f o r each j = 1, 2 , . . . , u we h a v e : - 99 - E [ Y . ] = A . I P B. 3 3 J and f o r each row of Y . t h e c o v a r i a n c e m a t r i x i s B . ' S B . , where E and J 3 ° 3 P a r e as f o r t h e growth c u r v e models f o r comp le te d a t a d e s c r i b e d i n S e c t i o n 3 . 3 . As b e f o r e the rows o f Y j a r e assumed to be m u t u a l l y i n d e p e n d e n t and each row i s assumed t o have a m u l t i v a r i a t e norma l d i s t r i b u t i o n . Under t h i s model i t i s not p o s s i b l e to f i n d the maximum l i k e l i h o o d e s t i m a t e s of £ and Z i n c l o s e d f o r m , but K l e i n b a u m f i n d s some b e s t a s y m p t o t i c a l l y normal (BAN) e s t i m a t o r s w h i c h have the same l a r g e sample p r o p e r t i e s as the maximum l i k e l i h o o d e s t i m a t e s . L e t Z* be the column v e c t o r formed by p u t t i n g the columns o f E, u n d e r n e a t h each o t h e r and l e t y j be the v e c t o r formed by p u t t i n g the columns of Y j u n d e r n e a t h e a c h o t h e r . Then a BAN e s t i m a t o r o f ^* ( w h i c h K l e i n b a u m f i n d s by w r i t i n g the model i n the form of a g e n e r a l u n i v a r i a t e w e i g h t e d l e a s t s q u a r e s model) t u r n s out to be: A u A u A £*= [ Z P B . ( B ! Z B . ) - 1 B. P' ® A ! A . ] " Z [P B , ( B ! Z B . ) _ 1 ® A ! ] y . j = 1 3 3 0 3 j 3 3 j J o j 3 3 where M i n d i c a t e s the g e n e r a l i z e d i n v e r s e of the m a t r i x M and Z Q, an u n b i a s e d and c o n s i s t e n t e s t i m a t o r of Z Q, i s f ound as f o l l o w s : The A ( r , s ) e lement a r g o f £ q i s the u s u a l p o o l e d e s t i m a t e u s i n g o n l y t h o s e i n d i v i d u a l s f o r w h i c h measurements were o b t a i n e d at b o t h t r and t g . In o r d e r to t e s t a h y p o t h e s i s of the form H'F,* = 0 , where H (mp x w) i s of f u l l rank w and H'£* i s e s t i m a b l e ( w h i c h means t h a t i t has an u n b i a s e d e s t i m a t o r l i n e a r i n y where y i s the v e c t o r formed by p u t t i n g - 100 - t h e columns of Y u n d e r n e a t h each o t h e r ) , the t e s t s t a t i s t i c i s g i v e n by W n = ( H ' £*)'[H'[ I P B . ( B ' E B . ) _ 1 B, P» ® A ' A , ] - H ] - 1 ( H ' £*) where i s any BAN e s t i m a t o r o f £* and E q i s any c o n s i s t e n t p o s i t i v e d e f i n i t e e s t i m a t o r of £ 0 . I n p a r t i c u l a r t h e e s t i m a t o r s o b t a i n e d by K l e i n b a u m may be u s e d . Under H Q , W n i s a s y m p t o t i c a l l y d i s t r i b u t e d as X 2 . W i s the a n a l o g u e of S S - 1 i n P o t h o f f and R o y ' s model w n h e 3 d e s c r i b e d i n S e c t i o n 3 . 3 . An a l t e r n a t i v e s t a t i s t i c W n * may be used w h i c h i s the ana logue of S, S - 1 i n R a o ' s m o d e l . W * i s o b t a i n e d by h e n J r e p l a c i n g ( A j ' A j ) i n W n by a more c o m p l i c a t e d m a t r i x Qj whose g e n e r a l i z e d i n v e r s e i s a n a l o g o u s t o R o f R a o ' s m o d e l . Application of the Model to our Data T h i s g e n e r a l i s a t i o n might have been a p p r o p r i a t e f o r our d a t a s e t , because some d a t a was n e g l e c t e d i n o r d e r t o s a t i s f y the a s s u m p t i o n t h a t o b s e r v a t i o n s on a l l i n d i v i d u a l s were t a k e n a t the same t i m e s . Namely the d a t a t a k e n a t t = 1, 10 and 11 weeks was i g n o r e d . Under K l e i n b a u m ' s model t h i s d a t a c o u l d have been i n c l u d e d : The i t e m s would have been d i v i d e d i n t o g roups f i r s t a c c o r d i n g to the t i m e s a t w h i c h d a t a was t a k e n on them. F o r example i t e m s f o r w h i c h d a t a was a v a i l a b l e a t 2, 3 , . . . , 9 , 10 , 14, 17 would fo rm one g r o u p , t h o s e f o r w h i c h d a t a was a v a i l a b l e a t 2, 3 , . . . , 9 , 11 , 14, 17 would form a n o t h e r g roup and so o n . W i t h i n each group i t e m s would be d i v i d e d a c c o r d i n g t o l i n e . Thus the A^ 's would be d e t e r m i n e d . Then p o l y n o m i a l s would have - 101 - been f i t t e d as b e f o r e and the p a r a m e t e r s f o r d i f f e r e n t l i n e s compared. However s i n c e the amount of d a t a n e g l e c t e d was so s m a l l compared t o t h e t o t a l amount of d a t a , and s i n c e a p p l i c a t i o n o f the model would have been q u i t e l a b o r i o u s i t was not c o n s i d e r e d w o r t h w h i l e t o p u r s u e t h i s m o d e l . 3.11 Growth Curves f o r Individual Barnacles I n a l l the p r e v i o u s a n a l y s e s r e l a t i n g t o the growth d a t a , a f t e r a v e r a g i n g over the sample of f i v e ( t e n ) , t h e r e was o n l y one o b s e r v a t i o n at each t ime f o r each i t e m . F o r t h i s r e a s o n i t was not p o s s i b l e to f i t a d i f f e r e n t g rowth c u r v e f o r e a c h i t e m . So our c o v a r i a n c e m a t r i x c o n t a i n e d two components of v a r i a b i l i t y - b a r n a c l e - t o - b a r n a c l e v a r i a b i l i t y w i t h i n i t ems and i t e m t o i t e m v a r i a b i l i t y . I t would be n i c e to f i t the a v e r a g e growth cu rve f o r i n d i v i d u a l s on each i t e m and then to compare the growth c u r v e s f o r the d i f f e r e n t i t e m s w i t h i n a l i n e . Whether t h i s i s p o s s i b l e w i t h the l i m i t e d d a t a t h a t i s a v a i l a b l e w i l l be i n v e s t i g a t e d i n t h i s s e c t i o n . C o n s i d e r j u s t t h e f i r s t o y s t e r l i n e . I d e a l l y a d a t a m a t r i x Y^ wou ld be a v a i l a b l e , e a c h row of w h i c h would c o r r e s p o n d t o one b a r n a c l e . T h e r e would be f i v e rows c o r r e s p o n d i n g to e a c h i t e m on the l i n e . I n f a c t a l l t h a t i s a v a i l a b l e i s a d a t a m a t r i x Y 2 . In Y 2 , f i v e rows c o r r e s p o n d t o e a c h i t e m , but w i t h i n each s e t of f i v e rows , s u c c e s s i v e o b s e r v a t i o n s i n a row may o r may not c o r r e s p o n d to the same b a r n a c l e . F u r t h e r m o r e o b s e r v a t i o n s at d i f f e r e n t t ime p o i n t s i n d i f f e r e n t rows may - 102 - correspond to the same barnacle, so the rows are dependent. However, the f i v e observations at any one time are independent. But E[Y]J = E[Y 2] = A£P. So we can use our data matrix Y 2 j u s t as we would have used Y^ to obtain a least squares estimate of £. This estimate involves only the average of the observations at each time and i t doesn't matter whether or not the observations at successive times correspond to the same barnacles - our estimate i s s t i l l v a l i d . Of course i f we were to assume that a l l items have the same growth curve, we would obtain the same growth curve for the oyster l i n e as previously when we f i r s t averaged within each item. But by choosing a d i f f e r e n t design matrix A, we can now obtain a d i f f e r e n t growth curve for each item. Testing for differences between the growth curves i s not as straightforward as f i t t i n g the growth curves: i n order to test hypotheses r e l a t i n g to the parameters we need to f i n d matrices S^ and S~. These involve S and S, an estimate of the covariance matrix S c o for the set of observations on an i n d i v i d u a l barnacle. We have already obtained £, but S i s more d i f f i c u l t . Since each row of Y 2 does ot necessarily correspond to the same barnacle, Y 2 ' [ I - A^CA^' A ^ ) - 1 A]/] Y 2 does not provide an estimate of E Q. However i f a strong assumption i s made about the form of E Q , i t i s possible to obtain an estimate of i t . - 103 - Obtaining an Estimate of E n I f we assume t h a t v a r i a n c e i s c o n s t a n t over t ime and t h a t the c o r r e l a t i o n between o b s e r v a t i o n s d weeks a p a r t on an i n d i v i d u a l i s t h e n E Q i s p r o p o r t i o n a l t o : P 1 P P 1 12 P P 1 P P 1 P P 1 P P 1 P P 1 P 1 Under t h i s s t r o n g a s s u m p t i o n a l l t h a t i s needed i s an e s t i m a t e o f p . T h i s i s o b t a i n e d as f o l l o w s : A number of i t e m s were e x c l u d e d f rom the p r e v i o u s a n a l y s e s because towards the end of the s t u d y , fewer than f i v e b a r n a c l e s were s t i l l a l i v e on them. I t was assumed t h a t t h e s e b a r n a c l e s were r e p r e s e n t a t i v e o f a l l the b a r n a c l e s and an e s t i m a t e of p was o b t a i n e d u s i n g o n l y t h e s e b a r n a c l e s . Suppose t h a t on a p a r t i c u l a r i t e m , f o u r b a r n a c l e s rema ined f o r the l a s t few t ime p o i n t s . Then c l e a r l y we would be s a m p l i n g the same f o u r b a r n a c l e s a t each of t h e s e t i m e s . F u r t h e r m o r e i t was u s u a l l y c l e a r w h i c h o b s e r v a t i o n s at s u c c e s s i v e t i m e s c o r r e s p o n d e d t o t h e same b a r n a c l e s ( s i n c e b a r n a c l e s cannot get s m a l l e r ) . I f t h i s w a s n ' t o b v i o u s - 104 - i t was assumed t h a t the s m a l l e s t o b s e r v a t i o n at the f i r s t t ime c o r r e s p o n d e d to the s m a l l e s t o b s e r v a t i o n at the next t i m e , and so o n . Items w i t h o n l y one b a r n a c l e l e f t c o u l d not be u s e d , as r e p l i c a t i o n was n e e d e d . E l e v e n i t ems ( a l l f rom the sys tem one o y s t e r l i n e s ) were u s e d . L o n g i t u d i n a l d a t a was a v a i l a b l e a t t = 11, 14 and 17 weeks on 2, 3 or 4 b a r n a c l e s on each of t h e s e i t e m s ; the t o t a l number of b a r n a c l e s i n v o l v e d was 34. U s i n g t h i s d a t a , the maximum l i k e l i h o o d e s t i m a t e of p was o b t a i n e d . Koopmans (1942) d e s c r i b e s how t o f i n d the maximum l i k e l i h o o d e s t i m a t e o f p under our model of s e r i a l c o r r e l a t i o n . S i n c e our l o n g i t u d i n a l d a t a was t a k e n at t h r e e week i n t e r v a l s we l e t p' = p . Our a s s u m p t i o n about E Q i m p l i e s t h a t (y t - n t) = P'iy^ - ) + z t where the Z t a r e i n d e p e n d e n t l y d i s t r i b u t e d as N(0, a2), y t and y t _ l a r e o b s e r v a t i o n s on t h e same i n d i v i d u a l r e s p e c t i v e l y at t and t-1, and u t = E [ y t ] , I t f o l l o w s t h a t the v a r i a n c e of the y ' s i s a 2 / ( l - p ' 2 ) . The l i k e l i h o o d of o b t a i n i n g the y ' s i s , (1 " P ' 2 ) N / 2 2 2 L = 2 NT/2 e x P [ " ( A " 2 p ' B + ( 1 + P' )C)/2a ] (2itcf ) where T i s the t o t a l number of t ime p o i n t s , N i s the t o t a l number of i n d i v i d u a l s and A = E £ [ ( y ^ - H^) 2 + ( y * j - i4>2], i j B = E S [ ( y « - uJ) ( y * J - u*) + . . . + ( y j ^ - ^ ) ( y ^ - p £ ) ] , i j C = E E [ ( y ^ j - \L\)2 + . . . ( y j ^ - \ ^ _ ] ) 2 ] , i j - 105 - where y t J i s the o b s e r v a t i o n at t ime t on the j t h b a r n a c l e on the i t h i t e m and p.* = E [ y ^ ] . M a x i m i s i n g t h i s l i k e l i h o o d over the p^ a n d P 1 l e a d s t o an e s t i m a t e o f p ' as the r o o t o f a c u b i c e q u a t i o n (Koopmans, 1 9 4 2 ) . R e s u l t s The e s t i m a t e o f p ' t u r n e d out to be . 8 9 9 . So p = . 9 6 5 . The c o r r e l a t i o n between o b s e r v a t i o n s on the same i n d i v i d u a l a t s u c c e s s i v e t i m e s i s seen t o be v e r y h i g h . I t i s p o s s i b l e t o o b t a i n e s t i m a t e s o f the b a r n a c l e - t o - b a r n a c l e v a r i a n c e w i t h i n i t e m s a t e a c h t ime and thus check our a s s u m p t i o n of c o n s t a n t v a r i a n c e over t i m e . The v a r i a n c e a t t ime t i s e s t i m a t e d by ( y * j - vb2 ? E N - 1 i J where 1 i s the number of i t e m s . The e s t i m a t e s t u r n e d out t o be . 0 5 4 , . 0 6 9 , . 0 5 6 , . 0 5 4 , . 0 4 8 , . 0 4 9 , . 0 6 0 , .043 and .053 s u g g e s t i n g t h a t our a s s u m p t i o n o f c o n s t a n t v a r i a n c e over t i m e was a f a i r l y weak one . Combin ing t h e s e e s t i m a t e s would then l e a d t o an e s t i m a t e of a and thus t o an e s t i m a t e o f Z 0 of the assumed s e r i a l c o r r e l a t i o n f o r m . A l t h o u g h t h i s i s p r o b a b l y the b e s t e s t i m a t e o f S 0 t h a t can r e a l i s t i c a l l y be o b t a i n e d f rom our l i m i t e d d a t a i t i s u n l i k e l y t o be a good e s t i m a t e f o r s e v e r a l r e a s o n s : f i r s t l y a s t r o n g a s s u m p t i o n was made about the form of F,0 and s e c o n d l y the e s t i m a t e was o b t a i n e d u s i n g o n l y a s m a l l number of b a r n a c l e s and u s i n g o b s e r v a t i o n s o n l y a t the l a s t t h r e e t ime p o i n t s . - 106 - Testing Hypotheses of the Form CgV = 0 A H a v i n g o b t a i n e d an e s t i m a t e E q of E q , i s i t p o s s i b l e t o t e s t A A h y p o t h e s e s of the fo rm C£V = 0 by s u b s t i t u t i n g E q f o r S and £ o b t a i n e d A f rom Y 2 f o r £ i n the e x p r e s s i o n s f o r S, and S ? T h i s i s not c l e a r . h e The t e s t s of h y p o t h e s e s w h i c h have been d i s c u s s e d r e l y on the f a c t A t h a t i f n'v-NCu, CE) where c i s c o n s t a n t , and D ~ w p ( k , £)> t n e W i s h a r t d i s t r i b u t i o n ' w i t h k degrees of f r e e d o m , t h e n the d i s t r i b u t i o n o f — 1 2 k 1 u' D u i s known (namely H o t e l l i n g ' s T = — u D _ u). Suppose an e s t i m a t e of £ i s o b t a i n e d f rom the d a t a m a t r i x Y i ( t h e t r u e l o n g i t u d i n a l d a t a ) , namely A t = ( A J A i ) - 1 A J Yl P*(P P ' ) " 1 where we have t a k e n G = I i n P o t h o f f and R o y ' s m o d e l . Then the A c o v a r i a n c e m a t r i x o f £ i s (P P ' ) " 1 P ( A ! A . ) - 1 E P ' ( P P ' ) " 1 = (A! A , ) - 1 E 1 1 o 1 1 where £ i s the c o v a r i a n c e m a t r i x f o r one row of X = Y P ' (P P ' ) - 1 . F u r t h e r A £ ~ N(£, ( A J A j ) " 1 E ) (1) and X ' [ I - A X ( A ^ A j ) - 1 A J ] X ~ W(k, E ) . ( 2 ) T o g e t h e r (1) and (2) i m p l y t h a t the d i s t r i b u t i o n of S^1 i s known and t h i s forms the b a s i s f o r the t e s t s of C£V = 0 . Now suppose t h a t the e s t i m a t e of £ i s o b t a i n e d i n s t e a d f rom our A d a t a m a t r i x Y2 ( l e t t h i s e s t i m a t e be denoted £1) and t h a t - 107 - S = Y ' [ I - A . ( A ! A . ) A ' ] Y i s r e p l a c e d by our e s t i m a t e E o f E . o i l l l o O O Then t h e r e a r e two r e a s o n s why the same d i s t r i b u t i o n t h e o r y no l o n g e r a p p l i e s : A, 1. A l t h o u g h Ci s t i l l has a m u l t i v a r i a t e n o r m a l d i s t r i b u t i o n w i t h mean 5, i t s c o v a r i a n c e m a t r i x i s not the same as t h a t of E, and i n p a r t i c u l a r i s not p r o p o r t i o n a l t o E . 2. E Q does no t have a W i s h a r t d i s t r i b u t i o n as S does because of t h e s p e c i a l form t h a t was assumed f o r i t . So a l t h o u g h i t i s p o s s i b l e to o b t a i n growth c u r v e s f o r i n d i v i d u a l b a r n a c l e s w i t h o u t l o n g i t u d i n a l d a t a , i t does not seem p o s s i b l e to t e s t f o r d i f f e r e n c e s . 3.12 D i s c u s s i o n The growth c u r v e models t h a t were a p p l i e d the p r e v i o u s s e c t i o n s to our l o n g i t u d i n a l d a t a f o r i t ems r e q u i r e d a number of a s s u m p t i o n s . As m e n t i o n e d i n S e c t i o n 3.4 t h e o n l y a s s u m p t i o n t h a t was l i k e l y t o cause a p r o b l e m was the a s s u m p t i o n of e q u a l c o v a r i a n c e m a t r i c e s f o r e v e r y i n d i v i d u a l ( i t e m ) . T h e r e were two r e a s o n s why t h i s might be a s t r o n g a s s u m p t i o n even i f the c o v a r a n c e m a t r i c e s were assumed to be the same f o r e v e r y i n d i v i d u a l b a r n a c l e : 1. Our o b s e r v a t i o n s were a v e r a g e s o v e r f i v e b a r n a c l e s i n some c a s e s and over t e n b a r n a c l e s i n o t h e r c a s e s . 2. At each t ime the p o p u l a t i o n f rom w h i c h the b a r n a c l e s were sampled was of a d i f f e r e n t s i z e f o r each d i f f e r e n t i t e m . - 108 - I n S e c t i o n 3 .8 t h e a n a l y s i s was m o d i f i e d t o a l l o w f o r the f a c t t h a t o b s e r v a t i o n s were a v e r a g e s over sometimes f i v e and sometimes t e n b a r n a c l e s . New p - v a l u e s were o b t a i n e d w h i c h were s l i g h t l y d i f f e r e n t f rom t h o s e o b t a i n e d p r e v i o u s l y . However the b road c o n c l u s i o n s rema ined t h e same. I t i s not so s t r a i g h t f o r w a r d to accommodate the d i f f e r e n t p o p u l a t i o n s i z e s , e s p e c i a l l y s i n c e the number of b a r n a c l e s a l i v e a t each t ime on each i t e m i s no t known (we know o n l y the t o t a l number on e a c h l i n e ) . In S e c t i o n 3.6 e s t i m a t e s of the c o v a r i a n c e m a t r i c e s o b t a i n e d r e s p e c t i v e l y f rom the sys tem one o y s t e r l i n e s and f rom the sys tem two o y s t e r l i n e s "were p r e s e n t e d . The two e s t i m a t e s d i d i n d e e d show somewhat d i f f e r e n t p a t t e r n s w h i c h may be p a r t l y due t o the d i f f e r e n t p o p u l a t i o n s i z e s . I n v iew of t h i s d i f f i c u l t y , the p - v a l u e s o b t a i n e d i n S e c t i o n s 3 . 5 , 3.6 and 3.7 s h o u l d not be c o n s i d e r e d c o m p l e t e l y r e l i a b l e . However t h e p - v a l u e s t h a t we o b t a i n e d were so s m a l l , t h a t even i f the e r r o r i n v o l v e d were q u i t e l a r g e , we would s t i l l r e a c h the same c o n c l u s i o n s . I n S e c t i o n s 3 . 5 , 3.6 and 3 . 7 , growth c u r v e s f o r i t e m s were f i t t e d . I n t h e s e s e c t i o n s each d a t a p o i n t was an a v e r a g e o v e r a number o f b a r n a c l e s a l i v e a t t h a t t i m e . So our g rowth c u r v e s r e p r e s e n t e d the n a t u r a l p o p u l a t i o n . I n S e c t i o n 3.11 we d i s c u s s e d f i t t i n g growth c u r v e s f o r i n d i v i d u a l b a r n a c l e s . But a g a i n our e s t i m a t e of £ was o b t a i n e d f rom d a t a on the b a r n a c l e s w h i c h happened to be a l i v e at any p a r t i c u l a r t i m e . So a g a i n the c u r v e s r e p r e s e n t e d the n a t u r a l p o p u l a t i o n . I d e a l l y we would l i k e to f i t growth c u r v e s t h a t would r e p r e s e n t the g rowth o f a t y p i c a l i n d i v i d u a l b a r n a c l e between t = 2 and 17 weeks . I t - 109 - i s not c l e a r t h a t t h i s i s p o s s i b l e w i t h the l i m i t e d d a t a a v a i l a b l e . I d e a l l y the f o l l o w i n g i n f o r m a t i o n would be a v a i l a b l e f o r e a c h b a r n a c l e : ( a ) Growth d a t a at each t ime p o i n t u n t i l t = 17 weeks or u n t i l i t s d e a t h w h i c h e v e r i s the e a r l i e r . ( b ) Time of d e a t h i f t h i s i s b e f o r e t = 17 weeks or knowledge t h a t i t s u r v i v e d p a s t t = 17. ( c ) Cause of d e a t h - f o o d s h o r t a g e or p r e d a t o r . (d) Number of the i t e m and l i n e t o w h i c h i t i s a t t a c h e d . I f t h i s were the case s e v e r a l a p p r o a c h e s would be p o s s i b l e d e p e n d i n g on the o b j e c t i v e s : 1. Suppose we wanted t o f i t growth c u r v e s f o r i n d i v i d u a l b a r n a c l e s w h i l e a c c o u n t i n g f o r the c e n s o r i n g . The d e a t h s would a f f e c t the a n a l y s i s o n l y i n one way, namely the growth d a t a f o r each i n d i v i d u a l would be t r u n c a t e d a t i t s d e a t h t ime i f t h i s i s b e f o r e t = 17 weeks; we would have growth d a t a w i t h m i s s i n g o b s e r v a t i o n s . K l e i n b a u m ' s model i s s u f f i c i e n t l y g e n e r a l t o a l l o w t h i s p a t t e r n of m i s s i n g d a t a , so t h i s model c o u l d be used to f i t growth c u r v e s and to t e s t f o r d i f f e r e n c e s between v a r i o u s g r o u p s . The growth c u r v e s t h a t we would o b t a i n would r e p r e s e n t not the n a t u r a l p o p u l a t i o n , but the growth e x p e r i e n c e of a t y p i c a l b a r n a c l e between t = 2 and 17 weeks . 2. I f we were i n t e r e s t e d i n s t e a d i n whether s u r v i v a l was r e l a t e d t o g rowth we c o u l d t h i n k of our d a t a as s u r v i v a l d a t a w i t h a t ime dependent c o v a r i a t e . We would have t y p e I c e n s o r i n g i n t h e s u r v i v a l d a t a ; f o r b a r n a c l e s w h i c h s u r v i v e p a s t t = 17 weeks , the e x a c t l i f e t i m e - 110 - would not be known. The time dependent covariate could be s h e l l length. Time dependent covariates can be incorporated into Cox's proportional hazards model, which i s described i n Section 2.1. The hazard function would then be \(t; z ( t ) l = X (t) e 2 ^ ^ where i n our o case: \ Q ( t ) i s an unknown baseline hazard function, B i s an unknown parameter, z(t) Is our time dependent covariate - s h e l l length at time t. In t h i s context a test of 8 = 0 would be a test that growth c h a r a c t e r i s t i c s and s u r v i v a l time are not r e l a t e d . The method of t e s t i n g H Q: 8 = 0 i s described i n Section 2.1. A r e j e c t i o n of t h i s hypothesis would suggest some dependency between growth c h a r a c t e r i s t i c s and s u r v i v a l time, for example a tendency for larger barnacles to l i v e longer. 3 . There are two possible causes of death for the barnacles, namely they may be eaten by predators or they may die due to food shortage. Instead of t e s t i n g whether growth i s related to s u r v i v a l time, i t may be of i n t e r e s t to test separately whether each cause of death i s related to growth. Then we would be able to address questions such as "do the bigger barnacles tend to be eaten f i r s t by the predators?" and " i s i t the smaller barnacles which tend to die of food shortage?" This can be done i n the context of Cox's proportional hazards model by f i n d i n g the hazard function s p e c i f i c to each cause of death. Growth - I l l - i s a g a i n i n c l u d e d as a t ime dependent c o v a r i a t e but t h i s t ime 8 may v a r y o v e r the two t y p e s of f a i l u r e . So e s t i m a t e s of 8̂  and P 2 would be o b t a i n e d and a t e s t of B^ = 0 would be a t e s t t h a t the i t h cause of d e a t h i s u n r e l a t e d to growth . To o b t a i n an e s t i m a t e o f 8^, a l l f a i l u r e s o t h e r than t h o s e of t ype i a r e t r e a t e d as c e n s o r e d o b s e r v a t i o n s , and the u s u a l maximum l i k e l i h o o d methods a r e u s e d . U n f o r t u n a t e l y we do not have a l l the i n f o r m a t i o n t h a t we would l i k e : a t each t ime p o i n t we do not have growth d a t a on a l l b a r n a c l e s , o n l y on f i v e f rom e a c h i t e m . F u r t h e r m o r e i n d i v i d u a l b a r n a c l e s cannot be i d e n t i f i e d so i t i s not known whether or n o t s u c c e s s i v e o b s e r v a t i o n s c o r r e s p o n d t o the same b a r n a c l e s , nor w h i c h dea ths c o r r e s p o n d t o w h i c h growth measurements . The cause of d e a t h i s not known. S u r v i v a l d a t a i s a v a i l a b l e o n l y f o r the l i n e s not f o r the i t e m s w i t h i n the l i n e s . F o r t h e s e r e a s o n s we cannot pursue any of the a p p r o a c h e s d e s c r i b e d a b o v e . However , the growth e x p e r i e n c e of a t y p i c a l b a r n a c l e between t = 2 and 17 weeks on a p a r t i c u l a r l i n e w i l l be w e l l r e p r e s e n t e d by the growth c u r v e t h a t was f i t t e d f o r t h a t l i n e i n S e c t i o n 3 . 7 , p r o v i d e d t h a t the p r o b a b i l i t y of a b a r n a c l e d y i n g i s u n r e l a t e d t o i t s s i z e . T h i s can be e x p l a i n e d as f o l l o w s : i f the b a r n a c l e s t h a t d i e between two t ime p o i n t s a r e n e i t h e r p a r t i c u l a r l y b i g or p a r t i c u l a r l y s m a l l , then the change i n a v e r a g e s i z e o f l i v i n g b a r n a c l e s over t h i s t ime i n t e r v a l w i l l be due s o l e l y t o b a r n a c l e s g r o w i n g , not to the c h a n g i n g p o p u l a t i o n . W i t h our l i m i t e d d a t a i t i s not easy to a s c e r t a i n whether the p r o b a b i l i t y of d y i n g i s r e l a t e d t o s i z e : s i n c e s u r v i v a l d a t a i s a v a i l a b l e o n l y f o r the l i n e s , growth d a t a was a v e r a g e d over i t ems w i t h i n - 112 - e a c h l i n e . So f o r e a c h of the 17 l i n e s a v e r a g e s h e l l l e n g t h and the p e r c e n t a g e of d e a t h s were a v a i l a b l e at each t i m e . U s i n g a l l 17 l i n e s a s i g n i f i c a n t c o r r e l a t i o n was found between s i z e a t week 5 and the p e r c e n t a g e d y i n g between weeks 5 and 6 . On t h o s e l i n e s where a v e r a g e s i z e a t week 5 was l a r g e , a s m a l l e r p e r c e n t a g e tended t o d i e between weeks 5 and 6 . However we cannot c o n c l u d e f rom t h i s t h a t w i t h i n a p a r t i c u l a r l i n e the s m a l l e r b a r n a c l e s tend t o d i e f i r s t . The c o r r e l a t i o n may be due to a t h i r d u n d e r l y i n g f a c t o r r e l a t e d t o b o t h s u r v i v a l and g r o w t h . I t may r e f l e c t , f o r example , a sys tem or t r e a t m e n t ( t y p e of m a t e r i a l ) e f f e c t - b a r n a c l e s on system two tend t o s t a r t d y i n g l a t e r and f u r t h e r m o r e they a r e g e n e r a l l y b i g g e r than b a r n a c l e s on sys tem one . To e l i m i n a t e a p o s s i b l e s y s t e m or t r e a t m e n t e f f e c t , we s h o u l d u s e , f o r example , j u s t the sys tem one o y s t e r l i n e s . However t h e r e a r e o n l y s i x of t h e s e so a c o r r e l a t i o n o b t a i n e d j u s t f r o m t h e s e l i n e s w o u l d n ' t be v e r y m e a n i n g f u l . I f s u r v i v a l d a t a were a v a i l a b l e f o r e v e r y i t e m , we would have a v e r a g e s i z e and p e r c e n t a g e d y i n g at each t ime f o r 60 i t e m s f rom system one o y s t e r l i n e s . In t h i s case i t would even be p o s s i b l e to e l i m i n a t e a p o s s i b l e l i n e e f f e c t by u s i n g j u s t the i t ems f rom one l i n e . Then a s i g n i f i c a n t c o r r e l a t i o n at a p a r t i c u l a r t ime between average s i z e and p e r c e n t a g e d y i n g would c e r t a i n l y s u g g e s t dependency between the p r o b a b i l i t y of d y i n g and s i z e . From a b i o l o g i c a l p o i n t of v iew the e x p e r i m e n t e r e x p e c t e d t h a t s i z e and l i f e t i m e would be u n r e l a t e d . W i t h o u t d e t a i l e d s u r v i v a l d a t a we can o n l y assume t h a t t h i s i s t r u e . One o t h e r p o i n t i s o f i n t e r e s t : i n S e c t i o n 2 .5 i t was n o t e d t h a t t h e r e was a s t r o n g c o r r e l a t i o n between the i n i t i a l number on a l i n e and - 113 - the e s t i m a t e s of 8 and a f o r t h a t l i n e . The c o r r e l a t i o n s u g g e s t e d t h a t on l i n e s w i t h fewer b a r n a c l e s i n i t i a l l y ( e . g . sys tem one o y s t e r l i n e s ) , b a r n a c l e s tended to s t a r t d y i n g sooner and a t a f a s t e r r a t e . We now have the a d d i t i o n a l o b s e r v a t i o n t h a t b a r n a c l e s on sys tem one a r e g e n e r a l l y s m a l l e r i n i t i a l l y than t h o s e on system two. We c o u l d s p e c u l a t e f rom t h e s e o b s e r v a t i o n s t h a t sys tem two i s a more i d e a l s i t e f o r the b a r n a c l e s w h i c h i s why more b a r n a c l e s become a t t a c h e d t o sys tem two. But maybe t h e r e i s n ' t room f o r a l l the b a r n a c l e s on sys tem 2 and t h e y have t o compete f o r a p l a c e . The b i g g e r h e a l t h i e r b a r n a c l e s may have a b e t t e r chance of f i n d i n g a p l a c e on sys tem 2, wh ich would e x p l a i n why the b a r n a c l e s on sys tem two are i n i t i a l l y b i g g e r . The b a r n a c l e s on sys tem one tend to s t a r t d y i n g e a r l i e r and a t a f a s t e r r a t e . A l s o b a r n a c l e s on the d o w l i n g l i n e s a r e p a r t i c u l a r l y s m a l l i n i t i a l l y and s t a r t d y i n g e a r l y and at a f a s t r a t e . The f a c t t h a t the p o p u l a t i o n o f b a r n a c l e s on sys tem one and on d o w l i n g l i n e s d i m i n i s h e s q u i c k l y may be due to the f a c t t h a t the b a r n a c l e s he re a r e s m a l l t o s t a r t w i t h or may be a t t r i b u t a b l e t o a p o o r e r env i ronment - i t i s d i f f i c u l t t o say w h i c h . S i m i l a r l y i t i s d i f f i c u l t t o know whether t o a t t r i b u t e d i f f e r e n c e s i n the growth c u r v e s to the d i f f e r e n t e n v i r o n m e n t s or t o the i n i t i a l d i f f e r e n c e s i n the b a r n a c l e s . - 114 - CONCLUSION A d a t a s e t p r o v i d e d by Mr . H. G o l d b e r g r e l a t i n g t o the s u r v i v a l and g r o w t h o f b a r n a c l e s was examined . N o n p a r a m e t r i c t e s t s , namely the l o g r a n k and the W i l c o x o n , i n d i c a t e d t h a t d i f f e r e n c e s e x i s t e d between the s u r v i v a l d i s t r i b u t i o n s even of l i n e s of the same m a t e r i a l t ype w i t h i n the same s y s t e m . The t e s t s s u g g e s t e d t h a t sys tem had more i n f l u e n c e t h a n m a t e r i a l t y p e on the s u r v i v a l d i s t r i b u t i o n . The e x p o n e n t i a l model was f i t t e d f o r the s u r v i v a l d i s t r i b u t i o n s o f each o f the 17 l i n e s and the e s t i m a t e d s l o p e s , 8 were compared . A random e f f e c t s model was d e v e l o p e d f o r the 8 v a l u e s , i n i t i a l l y i n c o r p o r a t i n g o n l y the o y s t e r l i n e s but s u b s e q u e n t l y i n c l u d i n g a l l 17 l i n e s . Maximum l i k e l i h o o d e s t i m a t e s were o b t a i n e d o f the t r u e u n d e r l y i n g 8 v a l u e f o r l i n e s of each t y p e w i t h i n each s y s t e m . A c o m p a r i s o n of t h e s e e s t i m a t e s s u g g e s t e d t h a t s l o p e was dependent on sys tem but not on m a t e r i a l t y p e . A g a i n sys tem was found t o be the more i m p o r t a n t f a c t o r . The growth d a t a was l e s s s t r a i g h t f o r w a r d to examine as we d i d no t have l o n g i t u d i n a l d a t a . We a v e r a g e d over the s e t of measurements o b t a i n e d at each t ime f rom each i t e m and t r e a t e d the r e s u l t i n g d a t a s e t as l o n g i t u d i n a l d a t a f o r the i t e m s . U s i n g t h i s s e t o f a v e r a g e s , p o l y n o m i a l growth c u r v e s f o r i t ems were f i t t e d t o each l i n e u s i n g b o t h P o t h o f f and R o y ' s model and R a o ' s m o d e l . Q u a d r a t i c s were adequate e x c e p t i n the case of the second d o w l i n g l i n e . A c o m p a r i s o n of t h e p a r a m e t e r s i n d i c a t e d t h a t d i f f e r e n c e s e x i s t e d b o t h between systems and - 115 - between m a t e r i a l t y p e s , w i t h the b i g g e s t d i f f e r e n c e s b e i n g between d o w l i n g and the o t h e r two t y p e s of m a t e r i a l . The p - v a l u e s may be somewhat u n r e l i a b l e as the s t r o n g a s s u m p t i o n of e q u a l c o v a r i a n c e m a t r i c e s f o r e v e r y i n d i v i d u a l ( i t e m ) had to be made. A l t h o u g h , we d i d n ' t have l o n g i t u d i n a l d a t a f o r b a r n a c l e s i t was found t o be p o s s i b l e t o f i t the a v e r a g e growth c u r v e f o r i n d i v i d u a l b a r n a c l e s on each i t e m but not t o t e s t f o r d i f f e r e n c e s between the c u r v e s o b t a i n e d . I n c o n c l u s i o n , a l t h o u g h we had a l a r g e amount of growth d a t a , a s m a l l e r amount of l o n g i t u d i n a l d a t a would have been more u s e f u l . - 116 - BIBLIOGRAPHY B r e s l o w , N . E . ( 1 9 7 4 ) . C o v a r i a n c e a n a l y s i s o f c e n s o r e d s u r v i v a l d a t a . B i o m e t r i c s 30 , 8 9 - 9 9 . Cox , D.R. ( 1 9 7 2 ) . R e g r e s s i o n models and l i f e t a b l e s ( w i t h d i s c u s s i o n ) . J o u r n a l o f the R o y a l S t a t i s t i c a l S o c i e t y B, 34 , 187-202 . G r i z z l e , J . and A l l e n , D.M. ( 1 9 6 9 ) . A n a l y s i s of growth and dose r e s p o n s e c u r v e s . B i o m e t r i c s 25 , 3 5 7 - 3 8 1 . I t o , K. ( 1 9 6 8 ) . On the e f f e c t o f h e t e r o s c e d a s t i c i t y and n o n n o r m a l i t y upon some m u l t i v a r i a t e t e s t p r o c e d u r e s . I n : K r i s h n a i a h , P .R. e d . ; M u l t i v a r i a t e A n a l y s i s I I . New Y o r k : Academic P r e s s , 8 7 - 1 2 0 . K a l b f l e i s c h , J . G . and P r e n t i c e , R . L . ( 1 9 8 0 ) . The s t a t i s t i c a l a n a l y s i s of f a i l u r e t ime d a t a . New Y o r k : W i l e y . K h a t r i , C . G . ( 1 9 6 6 ) . A n o t e on a manova model a p p l i e d t o p rob lems i n g rowth c u r v e . A n n a l s of the I n s t i t u t e of S t a t i s t i c a l M a t h e m a t i c s 18, 7 5 - 8 6 . K l e i n b a u m , D .G . ( 1 9 7 3 ) . A g e n e r a l i z a t i o n of the growth c u r v e model w h i c h a l l o w s m i s s i n g d a t a . J o u r n a l of M u l t i v a r i a t e A n a l y s i s 3, 117-124 . Koopmans, T . ( 1 9 4 2 ) . S e r i a l c o r r e l a t i o n and q u a d r a t i c forms i n n o r m a l v a r i a b l e s . A n n a l s of M a t h e m a t i c a l S t a t i s t i c s 13, 1 4 - 3 3 . L a i r d , N.M. and Ware, J . H . ( 1 9 8 2 ) . Random e f f e c t s models f o r l o n g i t u d i n a l d a t a . B i o m e t r i c s 38 , 963-974 . L a w l e s s , J . F . ( 1 9 8 2 ) . S t a t i s t i c a l models f o r l i f e t i m e d a t a . New Y o r k : W i l e y . L e e , Y . K . ( 1 9 7 4 ) . A n o t e on R a o ' s r e d u c t i o n of P o t h o f f and R o y ' s g e n e r a l i z e d l i n e a r m o d e l . B i o m e t r i k a 6 1 , 3 4 9 - 3 5 1 . Lehmann, E . L . ( 1 9 1 7 ) . N o n p a r a m e t r i c s : s t a t i s t i c a l methods based on r a n k s . H o l d e n Day , San F r a n c i s c o . Machen , D. ( 1 9 7 5 ) . On a d e s i g n p r o b l e m i n growth s t u d i e s . B i o m e t r i k a 3 1 , 7 4 9 - 7 5 3 . M o r r i s o n , D . F . ( 1 9 7 6 ) . M u l t i v a r i a t e s t a t i s t i c a l m e t h o d s . New Y o r k : M c G r a w - H i l l . P e t o , R. ( 1 9 7 2 ) . C o n t r i b u t i o n t o the d i s c u s s i o n o f p a p e r by D.R. Cox. J o u r n a l of the R o y a l S t a t i s t i c a l S o c i e t y B, 34 , 2 0 5 - 2 0 7 . I - 117 - P o t h o f f , R . F . and Roy , S . N . ( 1 9 6 4 ) . A g e n e r a l i z e d 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 u s e f u l e s p e c i a l l y f o r g rowth c u r v e p r o b l e m s . B i o m e t r i k a 5 1 , 3 1 3 - 3 2 6 . P r e n t i c e , R . L . ( 1 9 7 8 ) . L i n e a r rank t e s t s w i t h r i g h t - c e n s o r e d d a t a . B i o m e t r i k a 6 5 , 167-179 . P r e n t i c e , R . L . and M a r e k , P. ( 1 9 7 9 ) . A q u a l i t a t i v e d i s c r e p a n c y between c e n s o r e d d a t a rank t e s t s . B i o m e t r i c s 35 , 8 6 1 - 8 8 6 . R a o , C R . ( 1 9 6 5 ) . The t h e o r y of l e a s t s q u a r e s when the p a r a m e t e r s a r e s t o c h a s t i c and i t s a p p l i c a t i o n to the a n a l y s i s of growth c u r v e s . B i o m e t r i k a 52 , 447-458 . Rao , C R . ( 1 9 6 6 ) . C o v a r i a n c e a d j u s t m e n t and r e l a t e d prob lems i n m u l t i v a r i a t e a n a l y s i s . I n : K r i s h n a i a h , P .R . e d . ; M u l t i v a r i a t e A n a l y s i s . New Y o r k : Academic P r e s s , 8 7 - 1 0 3 . S a t t e r t h w a i t e , F . E . ( 1 9 4 6 ) . An a p p r o x i m a t e d i s t r i b u t i o n of e s t i m a t e s o f v a r i a n c e components . B i o m e t r i c s B u l l e t i n 2, 110-114. Timm, N.H. ( 1 9 8 0 ) . M u l t i v a r i a t e a n a l y s i s o f r e p e a t e d measurements . I n : K r i s h n a i a h , P .R . e d . ; A n a l y s i s of V a r i a n c e . New Y o r k : N o r t h H o l l a n d , 4 1 - 8 7 .

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