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Numerical experiments with least-squares catch-at-age analysis Lawson, Timothy Adair 1980

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NUMERICAL EXPERIMENTS WITH LEAST-SQUARES CATCH-AT-AGE ANALYSIS by TIMOTHY ADAIR IAWSON E.Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACUITY OF GRADUATE STUDIES (Department of Zoology) and ( I n s t i t u t e o f Animal Resource Ecology) 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 c the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1980 © Timothy A d a i r Lawson 1S80 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 a d v a n c e d d e g r e e 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 a g r e e 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 a n d s t u d y . I f u r t h e r a g r e e 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 D e p a r t m e n t o r by h i s 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 . Timothy A. Lawson D e p a r t m e n t nf Zoology  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 2075 W e s b r o o k P l a c e V a n c o u v e r , C a n a d a V6T 1W5 D a t e A p r i l 2 9 , 1 9 8 0 DE-6 B P 75-S1 1 E ABSTRACT Three methods of analyzing age composition from the catches of exploited pcpulaticns are compared on the basis of the i r assumptions about recruitment, harvesting, and natural mortality; the manner in which data errors are treated; and how information contained i n the catch data i s u t i l i z e d in estimating population parameters. The analysis of catch curves involves the s t r i c t e s t assumptions and uses the least amount of information i n the data. Cohort analysis, while having the most relaxed assumptions, ignores the errors and r e s t r i c t s information to within cohorts. The least-sguares approach of Doutleday (1S76) takes f u l l account of data errors and information between cohorts, but assumes that the age selection c h a r a c t e r i s t i c s cf a fishery are constant over time. The least-sguares technigue i s modified to account for changes i n the r e l a t i v e variance of data errors with age and to prevent unreasonable estimates of population parameters. In i t s most general form, the method i s capable of analyzing catches-at-age, estimates of catch-at-age error variances, e f f o r t data, reprcducticn s t a t i s t i c s , and prior information about parameter values, a l l simultaneously and i n a way that guarantees consistent r e s u l t s . The method i s applied to northwestern At l a n t i c harp seal catch-at-age data. The results indicate that the seal population i i i i s more abundant than previous analyses have shown. However, projections with a guota of 180,000 animals predict a decline in the stock tc twc-tbirds i t s present abundance over the next ten years,. Monte Carlo studies were performed to investigate the general r e l a t i o n s h i p between r e l i a b i l i t y of population forecasts and the information content of the data, as determined by data quantity, data errors, and the contrast i n abundance and exp l o i t a t i o n rates during the period over which the data were taken. The c o e f f i c i e n t of variation for predictions of abundance ranged frcm 1% for high contrast, high guantity data sets to 91% for low quantity, low contrast data. Estimates of the natural mortality rate with the least-sguares technigue are precise when the data set i s large, contrast i s high, and errors are moderate. i v TABLE OF CONTENTS ABSTRACT i i TABLE CF CONTENTS .................................... i v LIST OF TABLES v i LIST OF FIGURES ........... v i i i AKNOWLEDGEMENIS . .. i x 1, INTRODUCTION ........ 1 2. CAICH-AT-AGE ANALYSIS ........... 5 2.1 C a t c h C u r v e s , C o h o r t A n a l y s i s , And The Le a s t Sguares Approach 6 2.2 F o r m u l a t i o n Cf The L e a s t - S q u a r e s E s t i m a t o r .14 2.2.1 The Model .15 2.2.2 The O b j e c t i v e F u n c t i o n 18 2.3 Seme M o d i f i c a t i o n s To The.log Transform Model ... 21 2.3.1 W e i g h t i n g And A l t e r n a t i v e E s t i m a t i o n C r i t e r i a ....................... 21 2.3.2 Unreasonable Ccnvergence And C o n s t r a i n t s .. 23 2.3.3 M o d i f y i n g The Model With A S t o c k - R e c r u i t R e l a t i o n s h i p .............................. 27 2.3.4 The L e a s t - S q u a r e s Approach For E s t i m a t i n g N a t u r a l M o r t a l i t y ................ 28 2.4 I n t e r p r e t a t i o n Of The L e a s t - S g u a r e s E s t i m a t e s ... 29 2.4.1 The S t a t e R e c o n s t r u c t i o n And F o r e c a s t i n g ...29 2.4.2 The C o v a r i a n c e M a t r i x And The R e l i a b i l i t y Of The E s t i m a t e s 32 2.4.3 The R e s i d u a l s And Lack Of F i t .36 2,.5 E x t e n s i o n s Cf The L e a s t - S g u a r e s Approach ........ 37 2.6 Summary . 39 3m CASE STUDY: NORTHWESTERN ATLANTIC HABP SEALS .42 3.1 D a t a . , , 43 3,.2 Pc r m u l a t i c n Of The Weighted Least-Squares E s t i m a t o r , N a t u r a l M o r t a l i t y Unknown .44 3,.3 E e s u l t s With The Basic Model .. , 50 3.4 I n i t i a l E e s u l t s Using A Beverton-Holt S t o c k - E e c r u i t R e l a t i o n s h i p ...................... 53 3.5 F i n a l E e s u l t s Using A Beverton-Holt S t o c k - E e c r u i t E e l a t i o n s h i p ...................... 61 3.6 P o p u l a t i o n P r o j e c t i o n s With A Quota Of 180,000 .. 69 3.7 D i s c u s s i o n 77 4. NUMERICAL EXPERIMENTS WITH LEAST^SQUARES CATCH-AT-AGE ANALYSIS: A PINNIPED FISHERY ., .80 4.1 F a c t o r s Determining The R e l i a b i l i t y Of The Estimates ....................................... 84 4.2 The Case Study Continued .86 4.3 R e s u l t s And D i s c u s s i o n .......................... 88 5. NUMERICAL EXPERIMENTS WITH LEAST'SQUARES CATCH-AT-AGE ANALYSIS: A CLUPECID FISHERY , 105 5.1 Four Cases Cf Low Data Quantity .................106 5.2 R e s u l t s And D i s c u s s i o n .......................... 115 6. CONCLUSION , 135 6.1 Summary ............137 L i t e r a t u r e C i t e d ,, 139 Appendix A. REPARAMETEEIZATICN OF THE BEVERTON-HOLT AGE STRUCTURE MODEL . . 142 v i LIST OF TAELES Table 1 . Data error c o e f f i c i e n t s cf variation for northwestern A t l a n t i c harp seal catchesTat-age..47 Table 2. Least-sguares catch-at-age analysis parameter constraints for northwestern A t l a n t i c harp seals ............................ 49 Table 3 . Results of least-sguares catch-at-age analysis for northwestern A t l a n t i c harp seals ........... 52 Table 4. Extreme values cf northwestern A t l a n t i c harp seal Eeverton-Holt stcck-recruit function parameters a and p are determined from extreme values cf N o , A, and Sn ............... 56 Table 5. Parameter constraints of least-squares catch-at-age analysis with a Beverton-Holt stcck-recruit function for northwestern A t l a n t i c harp seals ............................ 58 Table 6. The age d i s t r i b u t i o n for northwestern Atlantic harp seals i n 1952, from i n i t i a l r e s u l t s of least-sguares catch-at-age analysis with a Beverton-Hclt stcck- r e c r u i t function, and the revised d i s t r i b u t i o n ....................... 63 Table 7. Northwestern A t l a n t i c harp seal pup production and annual pup exploitation rates from the f i n a l least-squares results are compared to res u l t s ficm sequential population analysis ....68 Table 8 . Northwestern A t l a n t i c harp seal age selection factors frcm the f i n a l least-sguares results ...71 Table 9, F o r e c a s t c f n o r t h w e s t e r n A t l a n t i c harp s e a l abundance f o r 1979 from the f i n a l l e a s t -s q u a r e s r e s u l t s ............................... 73 Table 10. Monte C a r l o r e s u l t s f o r Case A 90 Table 11. Monte C a r l o r e s u l t s f o r Case B .,,...93 Table 12. Monte C a r l o r e s u l t s f o r Case C ... 96 Table 13. F o r e c a s t e r r o r s f o r Cases A-C .................99 Table 14. Data e r r c r c o e f f i c i e n t s of v a r i a t i o n f o r Cases D-G 108 Table 15. L e a s t - s q u a r e s c a t c h - a t - a g e parameter c o n s t r a i n t s f o r Cases D-G 114 Table 16. Monte C a r l o r e s u l t s f o r Case D ...............117 Table 17. Monte C a r l o r e s u l t s f o r Case E ., ,..,.,120 Table 18. Mcnte C a r l o r e s u l t s f o r Case F ..123 Table 19. Monte C a r l o r e s u l t s f o r Case G 126 Table 20. F o r e c a s t e r r o r s f o r Cases D-G ,.,.,,...,......129 LIST 0F FIGURES F i g u r e 1. A p r o j e c t i o n o f t h e sum of sguares s u r f a c e i n n-space t c two dime n s i o n s .................. 26 F i g u r e 2. A B e v e r t o n - H o l t s t o c k - r e c r u i t d i s t r i b u t i o n ....34 F i g u r e 3. The age d i s t r i b u t i o n f o r n o r t h w e s t e r n A t l a n t i c harp s e a l s i n 1952, from i n i t i a l r e s u l t s of l e a s t - s g u a r e s c a t c h - a t - a g e a n a l y s i s w i t h a B e v e r t o n - H o l t s t o c k - r e c r u i t f u n c t i o n , and the r e v i s e d d i s t r i b u t i o n .................. 60 F i g u r e 4. E s t i m a t e d t r e n d s i n n o r t h w e s t e r n A t l a n t i c harp s e a l abundance ........................... 65 s F i g u r e : 5. P o p u l a t i o n p r o j e c t i o n s f o r n o r t h w e s t e r n A t l a n t i c harp s e a l s under a quota of 180,000 ..76 F i g u r e 6. The data s e t " i n f o r m a t i o n - s p a c e " .............. 83 F i g u r e 7. "True" h i s t o r i e s of abundance f o r Cases D-G ..112 i x ACKNOWLEDGEMENTS Foremost, I wish t o thank my s u p e r v i s o r , C.J. W a l t e r s , f o r h i s s u p p o r t and encouragement d u r i n g the course o f t h i s s t u d y . I wculd a l s o l i k e t c thank N.J. W i l i m c v s k y and P.A. L a r k i n f o r t h e i r h e l p f u l comments on an e a r l i e r d r a f t , and P.F. L e t t f c r p r o v i d i n g e s s e n t i a l i n f o r m a t i o n . Programming a d v i c e from Mike P a t t e r s o n and B i l l Webb was much a p p r e c i a t e d , as was s e c r e t a r i a l a s s i s t a n c e from Joan Anderson and Sandy Masai,.1 am g r a t e f u l t o Monica G u t i e r r e z f o r drawing t h e f i g u r e s . A masters degree t a k e s an average of t h r e e y e a r s a t the I n s t i t u t e : an e x p e r i e n c e s u f f i c i e n t l y arduous t h a t g r a d u a t e s t u d e n t s are s e v e r e l y tempted t o s e n t i m e n t a l i z e i n t h e i r acknowledgements. In t h i s r e s p e c t I would p a r t i c u l a r l y l i k e t o thank my o f f i c e - m a t e s : Max L e d b e t t e r , Greg S t e e r , L a r r y S m i t h , and C h r i s Wood; "the S c u l p i n s " : E r i c Woodsworth, J i m Jo n e s , Ken Lertzman, and A r t h u r Pouchet; and my house-mates: James C u r r i e -J c h n s o n , John Lyon, K a r l N e u e n f e l d t , and Mike Moore f o r t h e i r c r u c i a l moral s u p p o r t . 1 Chapter 1..INTRODUCTION The age c o m p o s i t i o n o f h a r v e s t s has l o n g been c o n s i d e r e d a v a l u a b l e o b j e c t of study by f i s h p o p u l a t i o n a n a l y s t s . The age s t r u c t u r e o f t h e c a t c h c o n t a i n s i n f o r m a t i o n about the past abundance of the s t o c k , and t h e e x p l o i t a t i o n and n a t u r a l m o r t a l i t y r a t e s t o which i t has been s u b j e c t . R e c r u i t m e n t , n a t u r a l m o r t a l i t y , and f i s h i n g a r e the p r o c e s s e s t h a t r e g u l a t e p o p u l a t i o n numbers. Each p l a y s a r o l e i n a f f e c t i n g changes i n th e age s t r u c t u r e of the s t o c k ; t o g e t h e r they have t h e i r v i s i b l e outcome i n the age s t r u c t u r e o f the c a t c h . The e x t e n t t o which we can a c c u r a t e l y a s s e s s a s t o c k depends, e s s e n t i a l l y , cn the i n f o r m a t i o n c o n t e n t of the data from which we s t a r t . F o r a s e t o f c a t c h e s - a t - a g e , i n f o r m a t i o n c o n t e n t i s determined p r i m a r i l y by t h e q u a n t i t y of d a t a , the degree o f e r r o r w i t h which i t was measured, and t h e " c o n t r a s t " , the range i n e x p l o i t a t i o n r a t e s and abundance d u r i n g t h e p e r i o d i t was c o l l e c t e d . But the r e s u l t s are not independent of t h e e s t i m a t i o n t e c h n i q u e a p p l i e d t o the d a t a . The r e l i a b i l i t y of the r e s u l t s r e a l l y depends cn both i n f o r m a t i o n c o n t e n t and the e f f i c i e n c y w i t h which the a n a l y s i s e x t r a c t s t h e i n f o r m a t i o n . S e v e r a l methods developed t o a n a l y z e c a t c h - a t - a g e data have seen numerous a p p l i c a t i o n s . However, g e n e r a l g u a n t i t a t i v e p r i n c i p l e s r e l a t i n g the r e l i a b i l i t y c f e s t i m a t e s of p o p u l a t i o n parameters 2 (measured by t h e i r b i a s and v a r i a n c e ) t o a t t r i b u t e s of the data have teen slew t c emerge. D n t i l r e c e n t l y , methods used f o r e s t i m a t i n g e x p l o i t a t i o n r a t e s and abundance of f i s h s t o c k s belonged t o one of two major c l a s s e s . The f i r s t , which uses c a t c h and e f f o r t d a t a , has as a fundamental assumption t h a t c a t c h - p e r - u n i t - e f f o r t i s p r o p o r t i o n a l t c abundance o r , e g u a l l y , t h a t t h e e x p l o i t a t i o n r a t e i s p r o p o r t i o n a l t c f i s h i n g e f f o r t . T h i s a s s u m p t i o n has come under c o n s i d e r a b l e a t t a c k (e.g. E o t h s c h i l d 1977) w i t h c r i t i c i s m s f o c u s e d on the o b s e r v a t i o n t h a t c a t c h - p e r - u n i t - e f f o r t as an i n d e x o f abundance can be confounded by changes i n t h e g e o g r a p h i c a l d i s t r i b u t i o n o f t h e s t o c k , by uneven mi x i n g o f the age g r o u p s , and by l o c a l d e p l e t i o n of s u b - s t o c k s . A l s o , t h e r e i s e v i d e n c e t h a t t h e r e l a t i o n between e x p l o i t a t i o n r a t e and e f f o r t can be n o n l i n e a r due t o gear c o m p e t i t i o n , gear s a t u r a t i o n , and s e a r c h c o o p e r a t i o n . S t i l l f u r t h e r d i f f i c u l t i e s i n v o l v e the s t a n d a r d i z a t i o n c f e f f o r t i n m u l t i - g e a r f i s h e r i e s . The second type of method, u s u a l l y termed s e g u e n t i a l p o p u l a t i o n a n a l y s i s or c o h o r t a n a l y s i s , was d e v e l o p e d p a r t l y to a v o i d the use c f e f f o r t d a t a . I t uses th e t i m e - s e r i e s o f c a t c h e s of a y e a r - c l a s s t o r e c o n s t r u c t i t s p a s t abundance. Though e f f o r t d a t a i s n o t r e g u i r e d , t h e r e must be independent e s t i m a t e s of the " t e r m i n a l " ( o l d e s t age c l a s s ) e x p l o i t a t i o n r a t e and the r a t e of n a t u r a l m o r t a l i t y . Over the l a s t few y e a r s a t h i r d c l a s s has r e c e i v e d a t t e n t i o n , termed the l e a s t - s q u a r e s approach ( a f t e r Doubleday 1976). T h i s t e c h n i g u e a l s o a n a l y s e s the s t o c k ' s c a t c h ~ a t - a g e h i s t o r y independent c f e f f o r t d a t a , but does not r e g u i r e 3 estimates of terminal exploitation rates, nor, in certain cases, an estimate of natural mortality. In the least-squares approach a general model i s hypothesized as to how observed catches have arisen as functions cf h i s t c r i c a l recruitment, e x p l o i t a t i o n , and natural mortality parameters: these parameters are then estimated in such a way that as much information as possible i s extracted from a given data set. Taking as i t s point cf departure that catch-per-unit-effort methods are becoming inadequate with the trend towards greater power, mobility, and d i v e r s i f i c a t i o n of fis h i n g f l e e t s , t h i s paper attempts to explore seme of the theory of catch-at-rage analysis. In p a r t i c u l a r , my objectives are, f i r s t , to examine and suggest improvements to the least-sguares technique; second, to use the least-sguares estimator as a vehicle by which general quantitative relationships between the r e l i a b i l t y of assessment and properties of the data can be formulated. In chapter 2 a comparison i s made between the least-sguares approach and cohort analysis together with t h e i r forerunner, the catch curve, tc point cut the t h e o r e t i c a l s i m i l a r i t i e s and differences among these methods of analyzing age composition data. Some modifications to the least-squares approach are then developed, and in chapter 3 an application i s given with data f o r northwestern A t l a n t i c harp seals. Chapter 4 considers the hias and variance of least-sguares estimates and predictions of abundance, and how they are affected by data quantity, errors, and contrast, with a series of Monte Carlo studies using a model of the northwest At l a n t i c harp seal population. Chapter 5 repeats the error analysis, but with a model of a hypothetical 4 clupecid stcck. Chapter 6 presents a summary and conclusions. 5 Chapter 2. CATCH-AT-AGE ANALYSIS T h i s c h a p t e r i s concerned w i t h the t h e o r e t i c a l f o u n d a t i o n s of c a t c h - a t - a g e a n a l y s e s i n g e n e r a l , and w i t h t h e l e a s t - s q u a r e s approach i n p a r t i c u l a r , - A f t e r a comparison w i t h o t h e r methods of a n a l y z i n g age c o m p o s i t i o n d a t a ( s e c t i o n 2 . 1 ) , a t t e n t i o n w i l l be f o c u s s e d on the l e a s t - s q u a r e s e s t i m a t o r . . I t s d e t a i l e d f o r m u l a t i o n w i l l be p r e s e n t e d i n a h i s t o r i c a l l i g h t ( s e c t i o n 2.2), t h e n m o d i f i e d t c d e a l w i t h c e r t a i n s h o r t c o m i n g s t h a t i m p a i r i t s p r a c t i c a l i t y as a g e n e r a l t e c h n i q u e ( s e c t i o n 2.3). The use o f l e a s t - s q u a r e s e s t i m a t e s c f p o p u l a t i o n parameters i n r e c o n s t r u c t i n g the h i s t o r y of the s t o c k and i n f o r e c a s t i n g abundance i s d i s c u s s e d next ( s e c t i o n 2 . 4 ) , t o g e t h e r w i t h some comments cn e v a l u a t i n g t h e r e l i a b i l i t y o f the e s t i m a t e s and the f o r e c a s t , and a s s e s s i n g the r e l a t i v e i m p o r t a n c e of s t o c h a s t i c e n v i r o n m e n t a l v a r i a t i o n i n r e c r u i t m e n t . . The remainder of the t h e s i s i s concerned w i t h the a p p l i c a t i o n o f the t e c h n i q u e s developed below t o a stu d y o f a n o r t h w e s t e r n A t l a n t i c p i n n i p e d p o p u l a t i o n and a f i c t i t i o u s c l u p e o i d f i s h e r y . 6 2.1 Catch C u r v e s , C o h o r t A n a l y s i s , and the L e a s t Squares Approach.. C o n s i d e r t h r e e t e c h n i q u e s t o e v a l u a t e c a t c h - a t - a g e d a t a : a n a l y s i s o f c a t c h c u r v e s , c o h o r t a n a l y s i s , and the l e a s t - s g u a r e s approach. . How the methods s t a n d i n r e l a t i o n t o one a n o t h e r can b e s t be determined by e x a m i n i n g the u n d e r l y i n g models a l g e b r a i c a l l y , comparing the assumptions upon which each i s based. These assumptions c o n c e r n (1) r e c r u i t m e n t , (2) n a t u r a l m o r t a l i t y , (3) f i s h i n g m o r t a l i t y , and (4) whether o r not s a m p l i n g e r r o r s are r e c o g n i z e d or i g n o r e d i n the a n a l y s i s . . F l u c t u a t i o n s i n abundance due t o i m m i g r a t i o n and emmigration a r e assumed t o t e n e g l i g i b l e and a r e i g n o r e d i n each o f t h e models. The use c f c a t c h c u r v e s has a l o n g h i s t o r y , b e g i n n i n g i n t h e e a r l y 1900s w^.th many v a r i a t i o n s on the b a s i c theme (summarized i n E i c k e r 1S75). Coh o r t a n a l y s i s , a l s o termed s e g u e n t i a l p o p u l a t i o n a n a l y s i s , was suggested by B i c k e r (1948) and a n t i c i p a t e d by the v i r t u a l p o p u l a t i o n a n a l y s i s of F r y (1949). The method has s i n c e been developed by Jones (1961, 1968), G u l l a n d (1965), Murphy (1965), and g i v e n i t s most w i d e l y used form by Pope (1972). The l e a s t - s g u a r e s approach was f i r s t d e s c r i b e d by Agger e t a l (1971) and e x p l o r e d by Pope (1974), D c u t l e d a y (1976), and H a l t e r s (MS 1S76). I n c a t c h c u r v e a n a l y s i s , t o t a l s u r v i v a l (S) i s e s t i m a t e d from the s l o p e c f a l i n e f i t t e d t o the p o i n t s on t h e d e s c e n d i n g r i g h t - h a n d l i m b o f the c a t c h c u r v e , a p l o t of l o g - c a t c h - a t - a g e a g a i n s t age. The model i m p l i c i t i n the method i s g i v e n by: 7 where C i s the c a t c h ; ID i s the f i s h i n g m o r t a l i t y r a t e ; N i s abundance i n number of a n i m a l s ; j i s a s u b s c r i p t r e f e r r i n g t o age ( j = C,1>...,J u n l e s s noted o t h e r w i s e ) ; Nt) i s t h e abundance at the youngest age f u l l y e x p l o i t e d by t h e f i s h e r y , age t, ; and S i s the t o t a l s u r v i v a l r a t e . I n o t h e r words, the c a t c h a t age j i s t h e product of t h e number of age j f i s h and an e x p l o i t a t i o n r a t e , where t h e number o f age j f i s h are the number o f a c o h o r t at age t, t h a t have s u b s e q u e n t l y s u r v i v e d from age t, t o age j . The term " r e c r u i t m e n t " has come t o have a v a r i e t y of meanings. Here i t i s r e s e r v e d f o r the number o f a n i m a l s a t the age c f f i r s t c a p t u r e (N t o or E) ; i t s h o u l d not be confused w i t h the number of a n i m a l s a t the youngest age of complete v u l n e r a b i l i t y (N t ) ) , nor w i t h the number a t age 0 (N 0) , a l t h o u g h E =N0 when t 0 = 0. Age t 0 i s t h e youngest a t which t h e p r o p o r t i o n of the c o h o r t a v a i l a b l e t o f i s h i n g i s g r e a t e r t h a n 0.0 and age t, i s the youngest a t which t h e p r o p o r t i o n i s 1.0 . I n n a t u r a l l o g a r i t h m form, Eg. (2-1) becomes: When l n C i s r e g r e s s e d a g a i n s t age, the s l o p e i s an e s t i m a t e of l n S. E q u a t i o n (2^1) i s the s i m p l e s t p o s s i b l e model of r e c r u i t m e n t and m o r t a l i t y . The assumptions can be summarized as f o l l o w s : (1) r e c r u i t m e n t i s c o n s t a n t over y e a r s ; (2) n a t u r a l m o r t a l i t y i s c o n s t a n t ever ages and y e a r s ; (3) f i s h i n g m o r t a l i t y i s c o n s t a n t over ages and y e a r s ; and (4) e r r o r s a r e r e c o g n i z e d : 8 they must have ze r o mean f o r t h e e s t i m a t e of l n S t o he un b i a s e d and they must be u n c o r r e l a t e d and h c m o s c e d a s t i c f o r the e s t i m a t e t o be minimum v a r i a n c e . T o t a l m o r t a l i t y (S) c o n t a i n s b o t h f i s h i n g and n a t u r a l m o r t a l i t y components; they can o n l y be s e p a r a t e d by assuming one or the o t h e r i s known. Once an assumption has been made about the n a t u r a l m o r t a l i t y r a t e , t h e n m can be s e p a r a t e d from the e s t i m a t e c f S, and an e s t i m a t e of Nt> can be computed (the v i r t u a l p o p u l a t i o n e s t i m a t e a t age t, , c o r r e c t e d f o r l o s s e s due t c n a t u r a l m o r t a l i t y ) . I n t h e second c l a s s c f t e c h n i q u e s , the c a t c h h i s t o r y of a c o h o r t i s used t o r e c o n s t r u c t numbers-at-age and age-year-s p e c i f i c f i s h i n g m o r t a l i t y r a t e s . For s i m p l i c i t y , c o n s i d e r a f i s h e r y i n which n a t u r a l m o r t a l i t y d u r i n g the f i s h i n g season i s n e g l i g i b l e compared t o the r e s t of the year (a Type 1 f i s h e r y , a f t e r B i c k e r 1S75). Cohort a n a l y s i s i s based on Eg. ( 2 - 3 ) : Here Sn i s the n a t u r a l s u r v i v a l r a t e over the p e r i o d between h a r v e s t s and i i s a s u b s c r i p t r e f e r r i n g t o y e a r : i = 1 , 2 , . . . , I u n l e s s o t h e r w i s e noted. ( H e r e i n , we have two s u b s c r i p t s f o r N, m, and C. The f i r s t s t a n d s f o r year and t h e second s t a n d s f o r age,. Where s u b s c r i p t s i n v o l v e a r i t h m e t i c o p e r a t i o n s , they a r e s e p a r a t e d by a comma.) The uodel s i m p l y says t h a t the number i n a c o h o r t next year i s e q u a l t o the number p r e s e n t t h i s year t h a t s u r v i v e e x p l o i t a t i o n and n a t u r a l m o r t a l i t y . R e a r r a n g i n g E g . ( 2 - 3 ) 9 g i v e s an i t e r a t i v e f o r m u l a f o r numbers-at-age and e x p l o i t a t i o n r a t e : from which the past h i s t o r y of t h e c o h o r t can be computed, g i v e n an e s t i m a t e of Sn and of m f o r the e l d e s t age i n the s e r i e s . From t h i s " t e r m i n a l e x p l o i t a t i o n r a t e " and the c o r r e s p o n d i n g c a t c h datum, an e s t i m a t e c f the c o r r e s p o n d i n g abundance i s o b t a i n e d . T h i s becomes N i n Eg. ( 2 - 4 ) , from which an H I , yvl ' ' e s t i m a t e c f the abundance and e x p l o i t a t i o n r a t e f o r the n e x t - t o -l a s t age i n t h e s e r i e s are c a l c u l a t e d , and so on, back t o t h e age o f f i r s t c a p t u r e . I n t h e i t e r a t i v e scheme, numbers-at-age and e x p l o i t a t i o n r a t e s f o r younger ages depend on the e s t i m a t e s f o r c i d e r ages. The assumptions embodied i n Eg. (2-4) a r e : (1) r e c r u i t m e n t i s y e a r - s p e c i f i c ; (2) n a t u r a l m o r t a l i t y i s known and c o n s t a n t over ages and y e a r s ; (3) f i s h i n g m o r t a l i t i e s a r e age-y e a r - s p e c i f i c ; and (4) t h e s a m p l i n g e r r o r s are i g n o r e d . (However, Pope 1972, Appendix C, o f f e r s f o r m u l a s f o r t h e v a r i a n c e s o f the e s t i m a t e s g i v e n t h e s a m p l i n g e r r o r v a r i a n c e s . ) Compared wit h the use c f c a t c h c u r v e s , c o h o r t a n a l y s i s r e l a x e s a s s u m p t i o n s (1) and ( 3 ) , but i s g u a l i t a t i v e l y d i f f e r e n t i n t h a t i t i g n o r e s the data e r r o r s i n the e s t i m a t i o n procedure. I n c o n t r a s t , the t h i r d c l a s s c f t e c h n i g u e s , t h e l e a s t -s g u a r e s approach, makes f u l l use of t h e e r r o r s t r u c t u r e , but c o n s t r a i n s f i s h i n g m o r t a l i t y t o a g r e a t e r e x t e n t than does 10 c o h c r t a n a l y s i s . For a Type 1 f i s h e r y t h e u n d e r l y i n g model i s u s u a l l y w r i t t e n a s : i . e . , the e x p l o i t a t i o n r a t e i s now t h e pr o d u c t of a y e a r -s p e c i f i c t e rm, y, and an age s e l e c t i o n f a c t o r , a.. The assumptions summarized a r e : (1) r e c r u i t m e n t i s y e a r - s p e c i f i c ; (2) n a t u r a l m o r t a l i t y i s c o n s t a n t over ages and y e a r s (as i n t h e o t h e r t e c h n i g u e s ) , but i s not n e c e s s a r i l y known b e f o r e h a n d ; (3) f i s h i n g m o r t a l i t y i s expanded i n t o an a g e - s p e c i f i c term c o n s t a n t o v e r y e a r s and a y e a r - s p e c i f i c term c o n s t a n t over ages; and (*l) the e r r o r s t r u c t u r e forms t h e b a s i s o f the e s t i m a t i o n procedure. T h i s l a s t p r o p e r t y i s d i s c u s s e d i n d e t a i l i n s e c t i o n 2.2 . i l l t h e above models can be e x p r e s s e d as s p e c i a l c a s e s of the g e n e r a l f o r m u l a t i o n : Here the s u b s c r i p t of E r e f e r s t o year and the s u b s c r i p t s o f m and Sn r e f e r t c both age and y e a r . ( t 0 = 0, so t h a t E, = N J 0 .) The E's , m's, and Sn's a r e , i n p r i n c i p l e , a l l o w e d t o vary a c r o s s a l l p a s t y e a r s and ages,. However, t h i s r e s u l t s i n many more parameters t h a n t h e r e are data p o i n t s ( C ' s ) : some " c o l l a p s e 1 1 (assumed s i m i l a r i t y ) of the parameters i s n e c e s s a r y . The c a t c h c u r v e t e c h n i g u e c o l l a p s e s the m's and Sn's t o g e t h e r i n t o S, and assumes E ( o r , more c o r r e c t l y , ) i s c o n s t a n t over t i m e . Both c o h o r t a n a l y s i s and the l e a s t - s q u a r e s approach c o l l a p s e n a t u r a l m o r t a l i t y t o a s i n g l e Sn, and the l a t t e r c o l l a p s e s the m's to 11 t h e a's and y's. Other b l o c k i n g schemes l e a d t o s t a t i s t i c a l l y t r a c t a b l e s o l u t i o n s . . I n W a l t e r s 1 (MS 1976) a n a l y s i s of harp s e a l d a t a , f o r i n s t a n c e , Sn i s a l l o w e d t o vary between o l d e r and younger ages, and the annual components c f e x p l o i t a t i o n f o r 24 y e a r s are c o l l a p s e d t o 7 b l o c k s o f s i m i l a r y e a r s . I n i t i a l l y , t h e m e r i t of the l e a s t - s g u a r e s approach v e r s u s c o h o r t a n a l y s i s , a t l e a s t i n t h i s d i s c u s s i o n of u n d e r l y i n g models, appears t o concern o n l y the t r a d e - o f f between the c o n s t r a i n t cn f i s h i n g n c r t a l i t y and the i n c l u s i o n of t h e e r r o r s t r u c t u r e . I f t h i s were the c a s e , t h e l e a s t - s g u a r e s approach would be p r e f e r r e d inasmuch as i t s s t a t i s t i c a l p r o p e r t i e s a r e v a l u e d o v e r the apparent l o s s i n p r e c i s i o n of e s t i m a t i n g the e x p l o i t a t i o n r a t e s . I n f a c t , f a c t o r i z i n g the e x p l o i t a t i o n r a t e s l e n d s a c o m p l e t e l y new dimensi o n t o the e s t i m a t i o n . Cohort a n a l y s i s i s be s t d e s c r i b e d as s e q u e n t i a l : e s t i m a t e s f o r younger ages depend on those of c i d e r ages w i t h i n t h e same c o h o r t ; the i n f o r m a t i o n upon which the e s t i m a t e s f o r younger ages are made i s c o n t a i n e d o n l y i n t h e data f o r o l d e r ages w i t h i n t h e c o h o r t ; t h a t i s , c o h o r t s are a n a l y z e d i n d e p e n d e n t l y . On the o t h e r hand, l e a s t - s g u a r e s a n a l y s i s i s b e s t d e s c r i b e d as i n t e g r a l , c r c o m p a r a t i v e : e s t i m a t e s f o r any age and year depend cn those f o r every o t h e r age and y e a r ; the i n f o r m a t i o n upon which e s t i m a t e s of abundance f o r any age and year are made i s c o n t a i n e d i n a l l of t h e c a t c h d a t a , though p r i m a r i l y i n t h o s e d a t a from the same age, y e a r , and c o h o r t . I n t h i s r e s p e c t the method e n a b l e s each datum i n a c o h o r t t o r e c a l l i t s memory, so to speak, of p a s t age and year e f f e c t s o f m o r t a l i t y and to c o n t r i b u t e t h i s i n f o r m a t i o n t o the e s t i m a t i o n o f e x p l o i t a t i o n 12 rates and abundance of other cohorts* For example, the estimate of the number of age 5 f i s h i n the l a s t year of the time series depends on the estimate of the exploitation rate on age 4 f i s h i n the next to l a s t year. But information about the annual component cf exploitation i n the next to l a s t year i s not contained only i n the catch datum for age 4 f i s h that year, nor just i n the catch data for a l l ages that year. I t i s contained in the catch data for a l l ages that year and a l l ages, except the youngest, the next, by virtue of t h i s property of cohorts to "remember" past recruitment, harvesting, and natural mortality e f f e c t s . In short, the least-sguares approach views each catch as the re s u l t of an "experimental design" involving a complex of treatment e f f e c t s ; the analysis depends on recognizing contrasts among these effects,. Though a l l data contain some information relevant to the estimation of every unknown, the greater part i s i n data of the appropriate age, year, or cohort. The information relevant to the estimation of an age selection factor i s strongest i n the catch data for that age,. For an annual exploitation parameter, i t i s i n the catch data for that year. Herein, I refer to catches that are highly informative of a p a r t i c u l a r parameter as that parameter's "supporting" data. The concept w i l l be u s e f u l l i n l a t e r chapters due tc the strong relationship between the variance of certain parameter estimates and the guantity of supporting data. Catches ultimately arise as the re s u l t of the interactions between three kinds cf e f f e c t s : recruitment, harvest rates, and natural mortality. &t this point i t should te emphasized that, 13 of t i e three methods of analyzing age composition data discussed here, cnly the least-squares approach i s capable, in p r i n c i p l e , cf providing estimates of a l l three. I f prior information exists concerning cne of the three factors, the a b i l i t y of the estimator tc separate the effects of the remaining two i s enhanced.. This i s , in a sense, the s i t u a t i o n for cohort analysis, which assuaes natural mortality and the terminal exploitation rates are known, i n order to estimate the remaining harvest rates and abundance. The indeterminacy between harvest rates and abundance presents a fundamental problem in explaining any catch data. Somehow, we must diagncse whether catches were the r e s u l t of high numbers and low harvest rates, or low numbers and high harvest rates, or any cf the i n f i n i t e possible combinations in between.. The diagnosis becomes even more complex and uncertain considering the three-way indeterminacy among harvest rates, abundance, and natural mortality. Given a technigue e f f i c i e n t in extracting information from the data, the a b i l i t y to separate the three e f f e c t s that give r i s e to observed catches-at-age depends d i r e c t l y on the amount of information to be had from the data. Lew guantity data sets with large errors and poor contrast w i l l not be able to resolve the problem. High quantity, low error, high contrast data sets ought to do well. The problem i s i d e n t i c a l tc that cf catch per unit e f f o r t analyses, where the indeterminacy in explaining catch per unit e f f o r t i s between abundance and c a t c h a b i l i t y . The following relationships emerge. The analysis of catch curves invclves the most constraining set of assumptions and 14 u t i l i z e s the l e a s t amount of i n f o r m a t i o n i n t h e d a t a . C o h o r t a n a l y s i s , w h i l e h a v i n g the most r e l a x e d a ssumptions c o n c e r n i n g t h e c a t c h e q u a t i o n , i g n o r e s the e r r o r s t r u c t u r e and r e s t r i c t s i n f o r m a t i o n t o w i t h i n c o h o r t s . F i n a l l y , t h e l e a s t - s g u a r e s approach t a k e s f u l l a ccount of the data e r r o r s and uses i n f o r m a t i o n between c o h o r t s , though a t t h e c o s t o f making th e p o s s i b l y dangerous assumption o f c o n s t a n t age s e l e c t i o n . 2.2 F o r m u l a t i o n o f the L e a s t - S q u a r e s E s t i m a t o r . C o n s i d e r a m a t r i x of c a t c h - a t - a g e d a t a , i n which rows c o r r e s p o n d to y e a r s and columns t o ages,. I n the l e a s t - s q u a r e s method a model i s h y p o t h e s i z e d t h a t s i m u l a t e s , or " p r e d i c t s " , each c a t c h - a t - a g e f o r which t h e r e i s an observed datum, based on a s e t o f unknown p o p u l a t i o n p a r a m e t e r s . Given a p a r t i c u l a r v a l u e f o r every unknown, we can c o n s t r u c t a m a t r i x of p r e d i c t e d c a t c h e s - a t - a g e . The l e a s t - s q u a r e s e s t i m a t i o n c r i t e r i a s p e c i f i e s t h a t we t a k e as our e s t i m a t e s those v a l u e s of the unknowns which t o g e t h e r g i v e the minimum sum of sguared d i f f e r e n c e s between the observed and p r e d i c t e d c a t c h e s - a t - a g e . The sum of s g u a r e s i s termed t h e o b j e c t i v e f u n c t i o n , and i s t o be m i n i m i z e d w i t h r e s p e c t t o the unknown parameters. I n what f o l l o w s , I c o n s i d e r , f i r s t , t h e model, then the problem of m i n i m i z i n g t h e o b j e c t i v e f u n c t i o n . , 15 2,2,1 The model, l e t t h e f i r s t year i n the d a t a s e r i e s be d e s i g n a t e d year 1 and the f i r s t age, age 0 ( i . e . , ta= 0 ) , then t h e c a t c h e g u a t i o n . Eg. (2-5) can be w r i t t e n as a f u n c t i o n of number-at-age-(j-i+1) i n year 1, c r c f number-at-age-0, or r e c r u i t m e n t , i n year i - j , depending upon whether i i s g r e a t e r o r l e s s t h a n j : E q u a t i o n (2-7) i s i d e n t i c a l t o t h e g e n e r a l f o r m u l a t i o n , Eg. (2-6 ) , e x c e p t : t h e e x p l o i t a t i o n r a t e (m) i s f a c t o r e d i n t o an age s e l e c t i o n term (a) and an annual e x p l o i t a t i o n component ( y ) ; Sn i s c o n s t a n t over ages and y e a r s ; and, f o r those c a t c h e s whose age s u b s c r i p t i s g r e a t e r o r e q u a l t o i t s year s u b s c r i p t , t h e number p r e s e n t i n the c o h o r t i n year 1 i s the parameter, i n s t e a d of r e c r u i t m e n t . Pope (1974) suggested t a k i n g l o g a r i t h m s of Eg. (2-7) so t h a t the e r r o r v a r i a n c e s would be more h c m c s c e d a s t i c . Dcubleday (1976) and W a l t e r s (MS 1976) both p o i n t e d out the l i n e a r i z i n g e f f e c t t h e l o g t r a n s f o r m has on t h e o b j e c t i v e f u n c t i o n . . E g . (2-7) then becomes: 16 j toy. +taa) + l nN i 0 - j in^r, + Z.\r\(\-y.^ a . J , i>] Inyj - to a, - In N l # i . i n * (i-oin 5n Z \n( l - ^ Q ^ , i «-j If natural mortality i s assumed known and there are I years and J + 1 ages, then there are 2I + 2J unknowns, i,.e., the following set cf parameters: ln N„ , ln N1JL l n N,7 , ln Nlo , l n Nl0 l n N I 0 , ln y{ , l n y^  , l n y x , ln a, , ln a^ , l n a-j- • There are only J age selection terms because one of the l n y's cr In a's must be fixed due to an indeterminacy i n the model. That i s , since the ln y's and In a's add together to give the logarithms cf the exploitation rates, a constant added to a l l the l n y's and subtracted from a l l the ln a's w i l l not af f e c t the r e s u l t s . In practice one l n aj can be set to zero ( i . e . , aj = 1 ), the other terms then being scaled r e l a t i v e to the fixed term, Now consider the data errors,. These result from inaccuracies in aging, and e f f e c t s due to the aggregation of samples by gear type or s t a t i s t i c a l area. The simplest 17 assumption i s t h a t i n t i e l o g t r a n s f o r m model the e r r o r s i n l n C are n o r m a l l y d i s t r i b u t e d w i t h z e r o mean and c o n s t a n t v a r i a n c e . T h i s i s t h e same as assuming t h a t e r r o r s i n o b s e r v i n g t h e c a t c h e s - a t - a g e a r e l o g n o r m a l , and i m p l i e s t h a t t h e c a t c h e s - a t -age have c o e f f i c i e n t s c f v a r i a t i o n (cv- ) c o n s t a n t over ages and y e a r s . That i s , i f we have: then i t f e l l o w s t h a t : cv^ = Var ( C i j )''X I E l C i j l -,(e V-0 V l (z-'o) i . e . , cv i s c o n s t a n t . C- i s t h e t r u e c a t c h - a t - a g e , observed as C\y . (The n o t a t i o n e~N(/*,V) means the random v a r i a b l e e i s n o r m a l l y d i s t r i b u t e d w i t h mean y u and v a r i a n c e V; E[ e ] i s the mean of e and V a r ( e ) i s i t s v a r i a n c e . ) I f t h e e r r o r s i n C a r e a product c f a s e r i e s c f independent random v a r i a b l e s ( e r r o r f a c t o r s ) , t h e n by the C e n t r a l L i m i t Theorem the sum of the l o g a r i t h m s cf the e r r o r s converge i n d i s t r i b u t i o n t o a normal random v a r i a b l e ; t h u s , the s a m p l i n g e r r o r i n C w i l l be l c g n c r m a l . He use l n C^ ; - \ i n s t e a d o f l n €[• i n Eg. ( 2 - 9 ) i n o r d e r t h a t E £ ] = C;J , but i n Eg. ( 2 - 8 ) t h e i s i g n o r e d , w i t h t h e r e s u l t t h a t the p r e d i c t e d c a t c h e s are b i a s e d upward ( i n t h e range of 0.5SJ t o 11.8%, c o r r e s p o n d i n g t o v a l u e s of cv i n the range of 0 . 1 t c 0 , 5 ) . While i t may be r e a s o n a b l e t o assume l c g n c r m a l s a m p l i n g e r r o r s , i t i s l i k e l y u n r easonable t o suppose 18 a constant c o e f f i c i e n t of v a r i a t i o n . I f the younger ages are not well sampled and i f aging i s also inaccurate for older animals, the c o e f f i c i e n t of variation w i l l he smallest i n the intermediate ages. In the case where cv i s not constant, the data w i l l te inappropriately weighted and ordinary least-sguares methods w i l l not provide the most e f f i c i e n t estimator..A remedy for t h i s s i t u a t i o n i s discussed i n section 2.3.1 . 2,2,.2 The :ob jective function. Ignoring for the moment the dangerous assumption regarding data error heteroscedasticity, the least-sguares objective function i s given as: We take as the estimates for the unknown parameters that set of values which minimize ? . If Eg. (2-8) were l i n e a r i n the unknowns, the estimates could be obtained a n a l y t i c a l l y , as a simple or multiple l i n e a r regression. Unfortunately the nonlinear terms make the use of a nonlinear optimization algorithm e s s e n t i a l , so that 3? i s minimized numerically.. The problem of minimizing the least-sguares objective function, when the model i s nonlinear i n the parameters, i s a spec i a l case of a general class cf problems in which a nonlinear function i s tc be optimized with respect to one or more of i t s parameters. The study cf solutions tc such problems make up the subject matter of what i s variously termed mathematical p r e d i c t e d In Cn 1* U-il) 19 programming, n o n l i n e a r programming, c r n o n l i n e a r o p t i m i z a t i o n . . A comprehensive d i s c u s s i o n cn n o n l i n e a r programming as a p p l i e d t o n o n l i n e a r parameter e s t i m a t i o n , and the source o f much of the f o l l o w i n g comments, i s g i v e n by Bard (1974). i s m inimized n u m e r i c a l l y v i a an i t e r a t i v e p r o c e d u r e : s t a r t i n g w i t h an i n i t i a l guess of the unknowns, at each i t e r a t i o n the v a l u e s are updated such t h a t becomes s m a l l e r and s m a l l e r . The s e t c f r u l e s f o r i m p r o v i n g t h e parameter v a l u e s a t each i t e r a t i o n i s c a l l e d the o p t i m i z a t i o n a l g o r i t h m . L e t the v e c t o r c f parameter e s t i m a t e s a t t h e k i t e r a t i o n be c i K . From t h e i n i t i a l guess ( 60 ) , the a l g o r i t h m produces the s e r i e s Oar ®i» ®z/ which c u l m i n a t e s w i t h a v e c t o r o f parameter v a l u e s t h a t s a t i s f y some convergence c r i t e r i a , such as | © - c 9 K | b e i n g v e r y s m a l l . These we t a k e as our b e s t parameter e s t i m a t e s . The v e c t o r 6 which g i v e s the t r u e minimum of 5> must s a t i s f y t h e c o n d i t i o n t h a t the v e c t o r of f i r s t p a r t i a l d e r i v a t i v e s of §E be e q u a l t c the z e r o v e c t o r . The s i m p l e s t p r a c t i c a l o p t i m i z a t i o n a l g o r i t h m i s Newton's (or t h e Newtcn-Baphscn) method. The k i t e r a t i o n i s d e f i n e d by: -I 23> e K + , - 0^ ~ H K ^ iZ-\2.) where i s the g r a d i e n t v e c t o r of 3 (the v e c t o r o f f i r s t p a r t i a l d e r i v a t i v e s c f 5 w i t h r e s p e c t t o t h e unknowns) e v a l u a t e d at 6 K; and H~' i s the i n v e r s e of the H e s s i a n m a t r i x (the m a t r i x c f second p a r t i a l d e r i v a t i v e s ) e v a l u a t e d a t B K . Newton's method i s d e r i v e d by t a k i n g the T a y l o r s e r i e s e x p a n s i o n of 3i around 6 K, i n c l u d i n g terms up t o second o r d e r , and s o l v i n g t h e e g u a t i o n o b t a i n e d from s e t t i n g the g r a d i e n t o f t h e 20 ex p a n s i o n e g u a l t o the z e r o v e c t o r . E g u a t i o n (2-13) i l l u s t r a t e s two p r o p e r t i e s ccmmon t c the c l a s s c f a l g o r i t h m s c a l l e d g r a d i e n t methods: each i t e r a t i o n i n v o l v e s t h e c o m p u t a t i o n o f t h e g r a d i e n t o f 1> and a m a t r i x e g u a l o r , i n some f a s h i o n , r e l a t e d t o the H e s s i a n . I f 3? i s g u a d r a t i c and H i s p o s i t i v e d e f i n i t e , Newton's method g i v e s t h e minimum i n a s i n g l e i t e r a t i o n . . T h i s f o l l o w s because Eg. (2-12) d e f i n e s the v e c t o r f o r which the g r a d i e n t of a g u a d r a t i c e g u a t i o n i s t h e z e r o v e c t o r . I f IE i s not g u a d r a t i c , hut H i s always p o s i t i v e d e f i n i t e , the method s t i l l c o n v e r g e s , though n e t i n a s i n g l e i t e r a t i o n . U n f o r t u n a t e l y , H e i t h e r may not be p o s i t i v e d e f i n i t e c r may be d i f f i c u l t t o c a l c u l a t e . The e t h e r g r a d i e n t methods d i f f e r as t o how t h e y overcome these drawbacks. Gauss and v a r i a b l e m e t r i c methods c i r c u m v e n t t h e need f o r second p a r t i a l d e r i v a t i v e s , as do d i r e c t i o n a l d i s c r i m i n a t i o n and Marguardt's method, which a l s o d e a l w i t h t h e problem of p o s i t i v e d e f i n i t e n e s s . In the harp s e a l example below and i n the n u m e r i c a l e x p e r i m e n t s of c h a p t e r s 4 and 5, the D a v i d o n - F l e t c h e r -P o w e l l method was used: t h i s i s a v a r i a b l e m e t r i c a l g o r i t h m , s i m i l a r t c Newtcn's method, b u t w i t h s o p h i s t i c a t e d schemes f o r a p p r o x i m a t i n g the i n v e r s e of the H e s s i a n w i t h o u t p e r f o r m i n g the m a t r i x i n v e r s i o n . A m o d i f i c a t i o n of Eg. (2-8) was used by Pope (1974) i n which he u i n i m i z e d : 21 5 - X I o b s e r v e d In 'l i 1 ''+'.jt» ) _ p r e d i c t e d In 2 (2-13) by a variant of the steepest descent method. The use of log-catch-ratios was rejected by Doubleday on the grounds that they introduced autcccrrelated errors to the successive catch-ratios within a cohort, even though they reduced the number of unknowns to be estimated. Nevertheless, because the optimization algorithm he used was mere sensitive to i n i t i a l estimates i n Eg. (2-11) than i n Eg, (2-13), he used the l a t t e r to provide a staring point f c r the fcrmer. Even with a much more powerful algorithm, however, there are serious problems with the leg transform model that impair i t s usefulness as a general technigue. Solutions to these problems are presented next and examples of their use are given i n chapter 3. 2.3 Some Modifications to the Log Transform Model.. 2.3.1 Weighting and alternative estimation c r i t e r i a . . An objective function consisting of a simple sum of sguares for the leg transform model improperly weights the data i f either the c o e f f i c i e n t s of sampling error variation are not constant or the catch data varies over more than about three orders of magnitude, regardless of the d i s t r i b u t i o n assumed for the errors. This makes i n t u i t i v e sense. I f the data for some ages were less accurate than for others, we would want our 22 parameter e s t i m a t e s t o t e l e s s i n f l u e n c e d by t h e s e t h a n by the more r e l i a b l e d a t a . I f the c a t c h e s f o r some ages were huge i n comparison w i t h t h e r e s t , the e s t i m a t o r would s e r i o u s l y n e g l e c t the i n f o r m a t i o n c o n t a i n e d i n the l a r g e c a t c h e s because t h e l o g t r a n s f o r m reduces a l l the da t a t o w i t h i n one or two o r d e r s o f magnitude. Somehow, the da t a must be p r o p e r l y weighted. I f t h e s a m p l i n g e r r o r s were normal and u n c o r r e l a t e d , and i f the model were l i n e a r i n the unknowns, the w e i g h t s g i v i n g the minimum v a r i a n c e e s t i m a t e s would be the i n v e r s e of the data e r r o r v a r i a n c e s , Var(C) (Eard 1974, Chapter 4 ) . . I n the case of a non-normal d i s t r i b u t i o n and a n o n l i n e a r model the use of Var (C) i s o n l y a p p r o x i m a t e l y o p t i m a l , but s t i l l r e a s o n a b l e . Our knowledge about the v a r i a t i o n i n a p a r t i c u l a r c a t c h - a t - a g e datum i s r e s t r i c t e d , however, by t h e absence o f r e p l i c a t i o n s of the . o b s e r v a t i o n . N o n e t h e l e s s , w i t h some assumptions about r e g u l a r i t i e s i n the e r r o r s , e s t i m a t e s o f Var(C) can u s u a l l y be computed from the same data used t o compute the c a t c h e s - a t - a g e ( e . g . , G u l l a n d 1955). U n f o r t u n a t e l y , t h i s e x t r e m e l y v a l u a b l e i n f o r m a t i o n i s i n v a r i a b l y m i s s i n g from p u b l i s h e d d a t a s e t s . When such i n f o r m a t i o n i s u n a v a i l a b l e , some a l t e r n a t i v e means o f e s t i m a t i n g t h e data e r r o r v a r i a n c e s i s r e g u i r e d . I f t h e c o e f f i c i e n t s o f v a r i a t i o n depend on age, we have: Of c o u r s e , the C are th e m s e l v e s unknown, but a u s e f u l a p p r o x i m a t i o n f o r Var (C) can s t i l l be o b t a i n e d by s u b s t i t u t i n g C f o r C, so t h a t : -I -I (1-/*) 23 T h i s e x p r e s s i o n h o l d s w i t h o u t any assumptions r e g a r d i n g t h e d i s t r i b u t i o n c f t h e e r r o r s . A l l t h a t i s r e q u i r e d a r e independent e s t i m a t e s (cr educated guesses) of the a g e - s p e c i f i c c o e f f i c i e n t s c f v a r i a t i o n . I f l i t t l e p r i o r , i n f o r m a t i o n e x i s t s , v a r i o u s a l t e r n a t i v e a ssumptions s h o u l d be f o r m u l a t e d t o e x p l o r e t h e s e n s i t i v i t y of the a n a l y s i s t o the weights. The weighted l e a s t - s q u a r e s o b j e c t i v e f u n c t i o n uses the b a s i c model. Eg. (2-7) , t o g i v e : 1|> = ^ [ obser ved — predicted C,j j ' J j CVJ observed Cj- J (l-\b) or observed C M — predicted C-,; i ] I 'J J 1 - i V a r i e s ( 2 - l 7 ) depending on whether or net the Var (C) c a l c u l a t e d from the data are a v a i l a b l e . 2, 3,.2 Unreasonable convergence and c o n s t r a i n t s . Here we a r e concerned w i t h the problem of i n s u r i n g t h a t the v a l u e s o f the unknowns which m i n i m i z e the o b j e c t i v e f u n c t i o n a r e , i n f a c t , r e a s o n a b l e e s t i m a t e s of th o s e unknowns. Le t the number of parameters i n the . e s t i m a t i o n be n (n = 2 I + 2 J ) . The l e a s t - s q u a r e s p r o c e d u r e i m p l i e s s e a r c h i n g the s u r f a c e o f v a l u e s i n n-space. To g i v e a f e e l i n g f o r how t h e 24 sum c f s quares s u r f a c e behaves, F i g u r e 1a shows i t c o l l a p s e d from n t c two dimensions r e p r e s e n t i n g the average o f the "abundance" parameters ( r e c r u i t m e n t and numbers-at-age i n year 1) and the average c f the " e x p l o i t a t i o n " parameters (the age and year components o f the h a r v e s t r a t e s ) . The diagram i l l u s t r a t e s two p r o p e r t i e s of t h e l e a s t - s g u a r e s technigue,. F i r s t , t h e r e i s a c e r t a i n degree of i n d e t e r m i n a c y i n the model between abundance and e x p l o i t a t i o n r a t e s , d e p i c t e d i n the sum of s guares c o n t o u r s by t h e i r g e n e r a l l y i n v e r s e r e l a t i o n s h i p . T h i s s a y s t h a t i f t h e i n f o r m a t i o n c o n t e n t of the d a t a i s low, the e s t i m a t i o n w i l l be unable t o determine whether c a t c h e s were t h e p r o d u c t o f h i g h numbers-at-age and low e x p l o i t a t i o n r a t e s or low numbers-at-age and h i g h e x p l o i t a t i o n r a t e s . . S e c o n d l y , the sum of s g u a r e s s u r f a c e s can have more than one l o c a l minimum. Some v a l u e s of the unknowns a r e c l e a r l y u n r e a s o n a b l e as e s t i m a t e s f o r the model, i . e . , n e g a t i v e y's, extreme 's or N;0 ' s , e t c . D e f i n e T as t h e r e g i o n i n n-space c o n s i s t i n g of a l l p o i n t s t h a t r e p r e s e n t r e a s o n a b l e , or b e l i e v e a b l e , v a l u e s f o r t h e s e t of parameters. I f we are l u c k y , i n t h e case of two o r more l o c a l minima o n l y one w i l l l i e w i t h i n T; y e t , i f t h a t one i s not the g l o b a l minimum, we must somehow c o n f i n e t h e e s t i m a t o r ' s s e a r c h so t h a t i t does n o t wander o u t s i d e the r e g i o n i n n-space c o n t a i n i n g r e a s o n a b l e estimates,. T h i s can be done q u i t e e a s i l y by i m p o s i n g upper and lower c o n s t r a i n t s cn the p o s s i b l e parameter v a l u e s , c r e a t i n g a r e c t a n g u l a r r e g i o n T' t h a t i s an a p p r o x i m a t i o n t c T. The c o n s t r a i n t s r e p r e s e n t p r i o r i n f o r m a t i o n about parameter v a l u e s , l i g h t c o n s t r a i n t s c o r r e s p o n d t o a g r e a t e r ancunt of independent or p r i o r knowledge c o n c e r n i n g the 25 F i g u r e 1a. .A p r o j e c t i o n o f the sum of sguares s u r f a c e i n n-space to two dimensions,. The p o i n t s A and B r e p r e s e n t l o c a l minima. F i g u r e 1b. The unreasonable minimum (A) i s e x c l u d e d from the c o n s t r a i n e d r e g i o n T'. " EXPLOITATION " 27 p a r a m e t e r s , l o o s e c o n s t r a i n t s t o l i t t l e or no knowledge except the s e n s i b l e extremes. I n the o p t i m i s t i c case the two-d i m e n s i c n a l analogue would l o o k l i k e F i g u r e 1b, where the u n r e a s o n a b l e ninimum i s e x c l u d e d from 1 ' , l e a v i n g o n l y one minimum, I n t h e p a t h o l o g i c a l c a s e T* might s t i l l c o n t a i n more than cne l o c a l minimum or might not c o n t a i n any ( p l a c i n g some parameter e s t i m a t e s cn t h e i r c o n s t r a i n t b o u n d a r i e s ) . 2.3.3 M o d i f y i n g t h e model w i t h a s t o c k - r e c r u i t r e l a t i o n s h i p . . I n a p o p u l a t i o n f o r which r e c r u i t m e n t depends s t r o n g l y on abundance, such as harp s e a l s , t h e r e p r o d u c t i v e p r o c e s s can be modelled by a d e t e r m i n i s t i c s t o c k - r e c r u i t r e l a t i o n s h i p . I f the form of t h e r e l a t i o n s h i p can be p o s t u l a t e d , t h i s i n f o r m a t i o n can be i n c o r p o r a t e d i n t o the e s t i m a t o r by r e p l a c i n g r e c r u i t m e n t (N-0 ) i n Eg, (2-7) w i t h the s t o c k - r e c r u i t f u n c t i o n . T h i s e x p r e s s i o n w i l l be i n terms of some component o f t h e s t o c k and seme unknown s t c c k - r e c r u i t parameters. I n s t e a d o f the a n n u a l r e c r u i t m e n t s , the unknown parameters of the s t o c k - r e c r u i t r e l a t i o n s h i p a r e e s t i m a t e d . B e f o r m u l a t i n g Eg. (2-7) , we g e t : 28 N. ( I - y. a . ) Sn i - i ) - \ • • • 1 N: 10 and t 0 i s the age o f f i r s t c a p t u r e . Because r e c r u i t m e n t depends cn p a s t ( c r p r e s e n t ) a d u l t abundance, the o b j e c t i v e f u n c t i o n must be e v a l u a t e d r e c u r s i v e l y , cne year a t a t i m e , a c c u m u l a t i n g the sum cf s g u a r e s . U n l e s s t e = 0, the s t o c k - r e c r u i t f u n c t i o n c a n n c t p r o v i d e a v a l u e f o r r e c r u i t m e n t i n year 1, nor i n any year up t o t 0 ; t h u s , these i n i t i a l r e c r u i t m e n t s must be t r e a t e d as unknown parameters, 2,3,4 The l e a s t - s g u a r e s approach f o r e s t i m a t i n g n a t u r a l m o r t a l i t y . I f n a t u r a l m o r t a l i t y i s unknown, i n s t e a d o f g u e s s i n g a t a " r e a s o n a b l e " v a l u e , the p o s s i b i l i t y e x i s t s t o e s t i m a t e i t as a parameter i n t h e o p t i m i z a t i o n . The b a s i c c o n d i t i o n f o r t h i s t o be p o s s i b l e i s t h a t t h e r e must be enough c o n t r a s t i n the y's f o r the n a t u r a l m o r t a l i t y e f f e c t s on t o t a l s u r v i v a l t o become " v i s i b l e " . R e s u l t s u s i n g t h i s procedure are g i v e n i n c h a p t e r s 3, 4, and 5. . 29 2.4 I n t e r p r e t a t i o n c f the le a s t - S q u a r e s Estimates. 2.4,1 The s t a t e r e c o n s t r u c t i o n and f o r e c a s t i n g , The o p t i m i z a t i o n produces e s t i m a t e s f o r the unknowns i n Eg. (2-7), Eq. (2-8), or Eq. (2-18). From these, the past h i s t o r y of the stock can he r e c o n s t r u c t e d and a g e - y e a r - s p e c i f i c e x p l o i t a t i o n r a t e s c a l c u l a t e d . A f o r e c a s t of the p o p u l a t i o n f o r the year f o l l o w i n g the l a s t year i n the data s e r i e s i s obtained by s u b t r a c t i n g f i s h i n g m o r t a l i t y and n a t u r a l m o r t a l i t y from the numbers-at-age i n the l a s t year. When the parameters of a s t o c k - r e c r u i t r e l a t i o n s h i p have been e s t i n a t e d i n the c p t i m i z a t i i o n , a p r e d i c t i o n of recruitment f o l l o w s frcm the breeding component of the a p p r o p r i a t e 1+ p o p u l a t i o n : the p r e d i c t e d 1+ p o p u l a t i o n , i f t Q = 0, or the 1+ p o p u l a t i o n i n year I+1-t 0, i f t 0 > 0. When the annual r e c r u i t m e n t s themselves have been estimated, i t i s p o s s i b l e to take account of the s t o c h a s t i c nature of the re c r u i t m e n t process i n the f o r e c a s t . Instead of making a s p e c i f i c value p r e d i c t i o n , we can make a p r o b a b i l i s t i c statement about f u t u r e r e c r u i t m e n t . A methcd f o r dcing s c , t h a t i s based on the assumption t h a t the r e l a t i v e v a r i a t i o n of recru i t m e n t i s constant over a l l breeding stock s i z e s , i s dis c u s s e d i n the f o l l o w i n g paragraphs. The methcd presupposes the form of the d i s t r i b u t i o n r e l a t i n g the rec r u i t m e n t t o the breeding s t o c k , then uses the rec r u i t m e n t s and breeding stock s i z e s from the s t a t e r e c o n s t r u c t i o n to f i t the d i s t r i b u t i o n parameters. Arguments as to the form of such a d i s t r i b u t i o n u s u a l l y assume t h a t recruitment depends cn a s e r i e s of m u l t i p l i c a t i v e 30 s u r v i v a l factors, and note that, by the Central Limit Theorem, the sum cf the logarithms cf these factors i s normally d i s t r i b u t e d . This implies recruitment i s lognormal (e.g. Walters 1975, Eeterman 1978), Empirical evidence for the lognormal d i s t r i b u t i o n i s given by Allen (1973). The lognormal d i s t r i b u t i o n has two parameters, one for the mean of the normal random variate and one related to i t s r e l a t i v e variance.. We wish tc construct a series of d i s t r i b u t i o n s , one for each breeding stock size. We therefore require a function r e l a t i n g any breeding stock size to i t s mean recruitment..For t h i s purpose we must reinterpret what i s usually meant by a stock-recruit r e l a t i o n s h i p . Usually we interpret the stock-r e c r u i t function without regard to the stochastic properties of the relationship i n nature; here i t w i l l be modified to mean the average recruitment as a function of the breeding stock s i z e . Our lognormal "stcck-recruit d i s t r i b u t i o n " written i n terms of i t s probability density function i s given by: (a-n) where £(B | C) i s the lcgnormal probability density function for recruitment (E) given any breeding stock size (P) ; h (P) i s the average recruitment at P; and LTX i s a variance parameter. From Eq. (2-1S) i t fcllcws that: E£ B|P ] = h (P) and Var( B|P ) = h (P) 2 (e^ - 1) The c o e f f i c i e n t of variation of recruitment given breeding stock size i s (e - 1) ..After the parameters of the stock-recruit 31 f u n c t i o n (h) have been f i t u s i n g the r e c r u i t m e n t s and a p p r o p r i a t e component cf the s t o c k , as r e c o n s t r u c t e d from t h e l e a s t - s q u a r e s c a t c h - a t - a g e e s t i m a t e s , an e s t i m a t e of o > i s ' g i v e n crx T T T - I [in K - \n h(P*Vl (2-20) lo A K « \ where I - t 0 i s the number of d a t a p o i n t s used t o f i t h, and x i s the number of unknown parameters o f h. From h, <yx, and an e s t i m a t e of the b r e e d i n g s t o c k (P) , we can p r e d i c t r e c r u i t m e n t i n year 1+1 as an approximate 95% c o n f i d e n c e i n t e r v a l : h (P) ± 2 h(P) (e -1) ; and, second, d i s p l a y t h e p r o b a b i l i t y d i s t r i b u t i o n of r e c r u i t m e n t g r a p h i c a l l y , u s i n g Eg. (2-19), Note t h a t i f the parameters o f h a r e e s t i m a t e d i n the l e a s t - s q u a r e s c a t c h - a t - a g e o p t i m i z a t i o n , no e s t i m a t e o f Q 1 - i s p o s s i b l e and, t h e r e f o r e , no e s t i m a t e of the v a r i a n c e i n r e c r u i t m e n t . However, the n u m e r i c a l s t u d i e s i n c h a p t e r 4 i n d i c a t e t h a t when (jx i s l a r g e , b e t t e r r e s u l t s can p r o b a b l y be o b t a i n e d w i t h o u t i n c l u d i n g a s t o c k - r e c r u i t r e l a t i o n s h i p ; i . e . , t h e problem c n l y e x i s t s when s t o c h a s t i c v a r i a t i o n i n r e c r u i t m e n t i s u n i m p o r t a n t . Sometimes the parameters o f h w i l l n o t have an o b v i o u s i n t e r p r e t a t i o n . C o n s i d e r the parameters of the B e v e r t o n - H o l t c u r v e : 32 h(PV _ P C2 -2 I ) ocP + p F c r age s t r u c t u r e models, th e s i g n i f i c a n c e o f a and p i s not d i r e c t l y a p p a r e n t . F u r t h e r i n f o r m a t i o n about t h e s t o c k can be o b t a i n e d by i n t e r p r e t i n g ex and i n a more b i o l o g i c a l l y m e a n i n g f u l way. I n appendix a two such p o p u l a t i o n parameters are d e r i v e d : the e q u i l i b r i u m u n f i s h e d p o p u l a t i o n (N«>) and t h e r e c r u i t m e n t r a t e a t h a l f t h e u n f i s h e d e q u i l i b r i u m The f o r m u l a s are analagous t o the no age s t r u c t u r e case ( E i c k e r 1S75, appendix I I I ) , b u t now and A depend on Sn and J , i n a d d i t i o n t o a and p>. An example of a E e v e r t o n - H o l t s t o c k -r e c r u i t d i s t r i b u t i o n u s i n g Eg. (2-19) and Eq, (2-21) i s shown i n F i g u r e 2. . 2.4.2 The c c v a r i a n c e m a t r i x and r e l i a b i l i t y of the e s t i m a t e s . . C o n s i d e r a h y p o t h e t i c a l s i t u a t i o n i n which i t would be p o s s i b l e t c go back i n time and resample t h e f i s h e r y , measuring the same v a r i a b l e s but c r e a t i n g a d a t a s e r i e s s l i g h t l y d i f f e r e n t frcm the o r i g i n a l . I f the new data s e t were then used i n the l e a s t - s g u a r e s c a t c h - a t - a g e a n a l y s i s , t h e r e s u l t i n g e s t i m a t e s would be d i f f e r e n t . I n o t h e r words, th e e s t i m a t e o f an unknown i s i t s e l f a randem v a r i a b l e c h a r a c t e r i z e d by a p r o b a b i l i t y d i s t r i b u t i o n , w i t h t h e v a r i a n c e o f the e s t i m a t e o f the unknown r e l a t e d t c t h e v a r i a n c e of the s a m p l i n g e r r o r s . Each t i m e we went back i n t i m e , r e s a m p l i n g the f i s h e r y and computing new e s t i m a t e s , we would have a n o t h e r r e a l i z a t i o n o f our random v a r i a b l e and e v e n t u a l l y we c o u l d o b t a i n a good i d e a o f i t s 33 F i g u r e 2. A E e v e r t o n - H c l t s t o c k - r e c r u i t d i s t r i b u t i o n . The d i s t r i b u t i o n parameters are OL= 6.67 x 10"*, (* = 0.33, and or = 0.29 ( c o r r e s p o n d i n g t o N«> = 1000, A= 1.5, and a c o e f f i c i e n t of v a r i a t i o n f o r r e c r u i t m e n t of 0.30, f o r a model w i t h o u t age s t r u c t u r e ) . 34 35 variance. Of course, the s i t u a t i o n i s only hypothetical, yet we need tc knew the variances cf our estimates i n order to assess the i r r e l i a b i l i t y . . A trustworthy estimate i s one with a small variance and l i t t l e bias. Bard (1974,p.177,Eq. (7-5-13)) gives a formula for approximating the parameter covariance matrix from the data covariance matrix and the gradient cf $ . The r e s u l t s , he warns, are themselves random variables and, unfortunately, should be regarded as no more than a rough estimate, correct to within an crder of magnitude. A second p o s s i b i l i t y i s to estimate the parameter covariance matrix using a Monte Carle approach,. The idea here i s to simulate the resampling process described above on a computer, creating sets of fake data from a model of the f i s h e r y , mimicking observation errors with a pseudo-random number generator. Each observed C has as i t s mean the "true" A catch (C) , and a r e l a t i v e variance, s p e c i f i e d beforehand, of cv.. Each set of fake data i s the r e a l i z a t i o n of a d i f f e r e n t seguence of random numbers; each data set gives r i s e to a unique set of parameter estimates. After a number of r e p l i c a t i o n s of the data generation and data analysis process, estimates of bias and variance can be calculated for each estimated parameter and the population forecast. Explorations with the Monte Carlo approach w i l l be the subject cf chapters 4 and 5. 36 2.4.3 The r e s i d u a l s and l a c k o f f i t . F o r the l o g t r a n s f o r m model the r e s i d u a l s a re the d i f f e r e n c e s between observed l n C and p r e d i c t e d l n C computed w i t h the s o l u t i o n t o the o p t i m i z a t i o n . . (For the weighted l e a s t -s g uares model, the r e s i d u a l s a r e the d i f f e r e n c e s between observed C and p r e d i c t e d C.) I f the model were e x a c t l y c o r r e c t the r e s i d u a l s would be r e a l i z a t i o n s c f the e r r o r s i n o b s e r v i n g l n C (or C ) . I f t h e r e were no such e r r o r s , the r e s i d u a l s would be e x a c t l y z e r o . The i m p l i c a t i o n i s t h a t t o determine how t h e d i s t r i b u t i o n o f the r e s i d u a l s d e p a r t s from t h a t o f the e r r o r s i s t o determine how w e l l the model s t r u c t u r e f i t s t h e d a t a . For the l o g t r a n s f o r m model we p o s t u l a t e d t h a t the e r r o r term was n o r m a l l y d i s t r i b u t e d w i t h z e r o mean and v a r i a n c e V, and t h a t the e r r o r s were i n d e p e n d e n t . To a s c e r t a i n whether t h e r e s i d u a l s are c o n s i s t e n t w i t h t h e assumption of n o r m a l i t y , a c u m u l a t i v e f r e g u e n c y p l o t can be made on normal p r o b a b i l i t y paper (e . g . , Dcubleday 1976). I n the weighted l e a s t - s g u a r e s case the r e s i d u a l s d i v i d e d by Var(C) w i l l have a r e s i d u a l mean square (S z) w i t h a v a l u e c l o s e t o 1.0 i f the Var(C) a r e a c c u r a t e and i f the model f i t s e x a c t l y . I n f a c t , i f the Var (C) a re "good", the p r o p o r t i o n of the r e s i d u a l mean square g r e a t e r t h a n 1.0, s*. , must be a t t r i b u t e d t o e r r o r s i n the model. T h i s r e l a t i o n s h i p i l l u s t r a t e s a b a s i c advantage c f l e a s t - s g u a r e s t e c h n i q u e s as compared t o c c h c r t a n a l y s i s : knowledge of t h e magnitude of t h e e r r o r s i n the d a t a i m p l i e s knowledge of goodness o f f i t , and v i c e v e r s a . P l o t s of the r e s i d u a l s a g a i n s t age, y e a r , and c o h o r t r e v e a l any non-independence. I f p a t t e r n s i n the r e s i d u a l s e x i s t , o r i f 37 they a re c f an u n a c c e p t a b l e magnitude, o r i t i s u n r e a s o n a b l e t o i g n o r e o u t l i e r s , the model i s c o n s i d e r e d t o be somehow d e f i c i e n t . Lack o f f i t can be reduced e i t h e r by r e p l a c i n g a p p a r e n t l y e r r o n e o u s d a t a w i t h ones more c o n s i s t e n t , o r by r e f o r m u l a t i n g the model w i t h d i f f e r e n t a s sumptions about age-s e l e c t i o n c r n a t u r a l m o r t a l i t y . The c l a s s i c t r e a t m e n t of r e s i d u a l s i s g i v e n by Draper and Smith (1966).. 2.5 E x t e n s i o n s of the L e a s t - S g u a r e s Approach. I n f a c t , t h e l e a s t - s g u a r e s approach i s c a p a b l e of i n c o r p o r a t i n g a v a r i e t y of data o t h e r t h a n c a t c h e s - a t - a g e . Annual e f f o r t d ata ( f ( ) can be i n c l u d e d by s u b s t i t u t i n g the e x p r e s s i o n : Yi * %{ $i ( 1-12) i n t c Eg. (2-7) , Eg. (2-8) , or Eg. (2-18) . Here t h e g's, the annual c a t c h a b i l i t y c o e f f i c i e n t s , are e s t i m a t e d i n s t e a d of the y's. Assumptions about r e g u l a r i t i e s i n t h e g's l e a d t o a r e d u c t i o n i n the number of parameters. E f f o r t d ata can o n l y be used i n c o h o r t a n a l y s i s i f independent e s t i m a t e s of c a t c h a b i l i t y are a v a i l a b l e . F e r t i l i t y d a t a can a l s o be i n c l u d e d i n the l e a s t - s g u a r e s e s t i m a t o r . For example, i f a g e - s p e c i f i c annual f e c u n d i t i e s and sex r a t i o s are m o n i t o r e d , t h e g e n e r a l e x p r e s s i o n f c r r e c r u i t m e n t i n Eg, (2-18) becomes: 38 Ni0 = h(a ip,... )P i_ +J (z-23) where b (j i s the f e c u n d i t y - a t - a g e c o r r e c t e d f o r the sex r a t i o . I n the s i m p l e s t c a s e , N i o i s e g u a l t c P,-^  . I f s t r o n g v a r i a b l i t y i n r e c r u i t m e n t does not a l l o w t h e use o f Eg. ( 2 - 1 8 ) , Eg. (2-23) can s t i l l t e used t o f i t the s t o c k - r e c r u i t 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 from the s t a t e r e c o n s t r u c t i o n . The approach can be a p p l i e d even i f t h e age c o m p o s i t i o n i s u n a v a i l a b l e . A l l e n (1977) d e s c r i b e s a method of a n a l y z i n g whale data u s i n g a mcdel f o r the t o t a l a n n u a l c a t c h , i n s t e a d of c a t c h e s - a t - a g e , g i v e n by: N, , i - i N H (\-^-fj_() Sn + rN,-., , z * i f e l (2-2.4-) where r , r e p r e s e n t i n g the r e c r u i t m e n t r a t e , i s e s t i m a t e d a l o n g w i t h K, , Sn, and g. . Though the l e a s t - s q u a r e s t e c h n i g u e was d e v e l o p e d s t r i c t l y t o a n a l y z e c a t c h e s - a t - a g e , the approach s e r v e s as a paradigm f o r t h e a n a l y s i s of a v a r i e t y of o t h e r f i s h e r i e s s t a t i s t i c s . I t i s c a p a b l e c f e x t r a c t i n g i n f o r m a t i o n from enormous amounts o f d a t a : the m a t r i x of c a t c h e s , the m a t r i x of c a t c h e r r c r v a r i a n c e s , the s e r i e s of e f f o r t d a t a , p l u s t h e m a t r i c e s of age-year s p e c i f i c f e c u n d i t i e s and sex r a t i o s . I t n o t o n l y 39 provides a framework for the simultaneous analysis of (1) the age structure cf the catch, (2) e f f o r t s t a t i s t i c s , and (3) reproduction data, but guarantees the r e s u l t i n g assessment i s consistent with each, In the past, these three species of fishery s t a t i s t i c s have usually been investigated independently. The least-sguares technigue allows a l l the information to be c o l l e c t i v e l y represented ty a much smaller set of s t a t i s t i c s , i . e . , the estimated parameters (and t h e i r covariance matrix) of our model eguations. The method can also be used to estimate c a t c h a b i l i t y and natural mortality frcm tag recovery data. I f the number of tagged fished that are released at one point i n time i s N0 , then a model f o r subseguent recoveries (C) i s given by: where the time units can be on the order of months, rather than years. It i s possible tc use Eg.(2-25) to estimate separate natural mortality rates f o r a series of time i n t e r v a l s , and/or estimate c a t c h a b i l i t y c o e f f i c i e n t s for d i s t i n c t s t a t i s t i c a l areas and gear types through which the f i s h migrate. Aside frcm guesticns cf bias and e f f i c i e n c y , a comparison of catch curves, cohort analysis, and the least-sguares approach can be made cn the basis of the assumptions underlying each model and the manner i n which the information i n the data i s (Z-25) 2.6 Summary. 40 u t i l i z e d . C atch c u r v e s i n v o l v e the s t r i c t e s t a ssumptions and u t i l i z e the l e a s t i n f o r m a t i o n . C o h o r t a n a l y s i s has the most l a x as s u m p t i o n s , but i g n o r e s d a t a e r r o r s and r e s t r i c t s i n f o r m a t i o n t o w i t h i n c o h o r t s . The l e a s t - s g u a r e s approach e x t r a c t s t h e most i n f o r m a t i o n , b u t assumes the age s e l e c t i o n c h a r a c t e r i s t i c s o f th e f i s h e r y a re c o n s t a n t ever t i m e . The fundamental problem t o be r e s o l v e d i n e x p l a i n i n g c a t c h -at-age d a t a i s the three-way i n d e t e r m i n a c y between abundance, e x p l o i t a t i o n , and n a t u r a l m o r t a l i t y . I n the l e a s t - s g u a r e s method o f Pope (1974), Boubleday (1976), and W a l t e r s (MS 1976), the l o g a r i t h m of t h e c a t c h e q u a t i o n i s used and t h e sum of squares i s m i n i m i z e d n u m e r i c a l l y . Problems o f w e i g h t i n g and u n r e a s o n a b l e convergence have been d i s c u s s e d and i t i s suggested t h a t some m o d i f i c a t i o n s be made t c the l o g t r a n s f o r m model. These i n c l u d e u s i n g t h e weighted l e a s t - s g u a r e s c r i t e r i a w i t h t h e i n v e r s e of the da t a e r r o r v a r i a n c e s as weights and c o n s t r a i n i n g the p o s s i b l e parameter v a l u e s w i t h upper and lower bounds,. M o d i f i c a t i o n s f o r s p e c i a l c a s e s i n c l u d e m o d e l l i n g r e c r u i t m e n t w i t h a s t o c k - r e c r u i t f u n c t i o n , when s t o c h a s t i c e f f e c t s a re s m a l l , and t r e a t i n g n a t u r a l m o r t a l i t y as an unknown, when the d a t a i s h i g h l y i n f o r m a t i v e . The r e s u l t s of the a n a l y s i s a l l o w a h i s t o r y of s t o c k abundance t o be r e c o n s t r u c t e d , from which a p r e d i c t i o n can be made. When annual r e c r u i t m e n t s are e s t i m a t e d as parameters i n the o p t i m i z a t i o n , i t i s p o s s i b l e t o f i t a 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 r e l a t i n g r e c r u i t m e n t t o b r e e d i n g s t o c k s i z e . . a t e c h n i g u e i s suggested f o r f i t t i n g a l o g n o r m a l d i s t r i b u t i o n which assumes the r e l a t i v e v a r i a n c e c f r e c r u i t m e n t i s c o n s t a n t . 41 The parameter covariance matrix obtained from the data i s unreliable and can be estimated using a Mcnte Carlo approach. Examination of the residuals reveals lack of f i t and may point to better s t r u c t u r a l assumptions. 42 Chapter 3. CASE STDDY: NOBTHWESTEBN ATLANTIC HABP SEALS. The a n a l y s i s o f harp s e a l d a t a p r e s e n t e d i n t h i s c h a p t e r s e r v e s two purposes i n r e l a t i o n t o o v e r a l l o b j e c t i v e s of the s t u d y : f i r s t , i t i l l u s t r a t e s t h e methodology developed i n c h a p t e r 2, and, s e c c n d , the r e s u l t s w i l l p r o v i d e a model t o be used i n t h e n u m e r i c a l e x p e r i m e n t s o f c h a p t e r 4.. The harp s e a l s were chosen t o i l l u s t r a t e the l e a s t - s q u a r e s approach p r i m a r i l y f c r the f o l l o w i n g r e a s o n s . They are marine mammals, thus t h e i r p a t t e r n of r e c r u i t m e n t s h o u l d not show such v i o l e n t s t o c h a s t i c f l u c t u a t i o n s as a r e c h a r a c t e r i s t i c o f most f i s h s p e c i e s . T h i s s u g g e s t s t h e harp s e a l s are a good t e s t case f o r i n c l u d i n g a s t o c k - r e c r u i t r e l a t i o n s h i p i n the e s t i m a t i o n scheme. What i s more i m p o r t a n t , the d a t a s e t i s l a r g e , w i t h c a t c h e s - a t - a g e observed over 26 ages and 27 y e a r s . I t i s s u s p e c t e d t h a t , i n the p e r i o d over which t h e d a t a were c o l l e c t e d (1952-1S78), t h e abundance of s e a l s d e c l i n e d d r a m a t i c a l l y as a r e s u l t o f the i n t e n s e h a r v e s t i n g p r e s s u r e f o l l o w i n g World War I I , then i n c r e a s e d a f t e r the i n t r o d u c t i o n of g u o t a s i n the e a r l y 1970s ( L e t t and Benjaminsen 1977). These two f a c t o r s ( l a r g e d a t a s e t and s t r o n g c o n t r a s t i n both abundance and e x p l o i t a t i o n r a t e s ) i m p l y t h a t the i n f o r m a t i o n c o n t e n t o f the d a t a s e t i s l i k e l y h i g h . The harp s e a l s s h o u l d , t h e r e f o r e , be a s u i t a b l e t e s t case f o r e s t i m a t i n g n a t u r a l m o r t a l i t y . L a s t l y , t h e d a t a 13 have p r e v i o u s l y been a n a l y z e d w i t h a v a r i a n t o f s e q u e n t i a l p o p u l a t i o n a n a l y s i s ( L e t t e t a l 1978, Mohn e t a l 1978) and so a b a s i s f o r comparison of the r e s u l t s i.s p r o v i d e d . Here, the data were a n a l y z e d f i r s t u s i n g t h e l e a s t - s g u a r e s approach w i t h o u t i n c l u d i n g a s t c c k - r e c r u i t r e l a t i o n s h i p . E s t i m a t e s o f the d a t a e r r o r v a r i a n c e s were u n a v a i l a b l e , so the a n a l y s i s was r e p e a t e d under v a r i o u s assumptions about the magnitude c f t h e e r r o r s ( s e c t i o n s 3.2 and 3,3),. As each s e t of a ssumptions l e a d s t o an a l t e r n a t i v e w e i g h t i n g scheme, the g u e s t i o n i s posed: How s e n s i t i v e i s the a n a l y s i s t o the w e i g h t s ? Then, u s i n g the w e i g h t i n g scheme t h a t g i v e s t h e " b e s t " r e s i d u a l s ( t h a t g i v i n g reduced t r e n d s w i t h a g e ) , the a n a l y s i s i s performed a g a i n , but t h i s time i n c l u d i n g a s t o c k - r e c r u i t r e l a t i o n s h i p ( s e c t i o n s 3.4 and 3.5), These r e s u l t s are s u b s e q u e n t l y used t o a s s e s s the e f f e c t on the s e a l p o p u l a t i o n o f the c u r r e n t a n n u a l quota of 180,000 a n i m a l s , T h i s i s done by making s t o c h a s t i c p r o j e c t i o n s over t e n years from th e p r e d i c t i o n f o r 1979 ( s e c t i o n 3.6). 3.1 Data. The harp s e a l f i s h e r y c o n s i s t s o f f o u r c a t e g o r i e s of h u n t e r s ^ landsmen, l a r g e v e s s e l s , s m a l l v e s s e l s , and A r c t i c n a t i v e s . The landsmen a r e those h u n t e r s who work from s h o r e , h i k i n g o r snowmobiling over t h e i c e , or u s i n g s m a l l b o a t s , t o t a k e t h e s e a l s by c l u b b i n g them, s h o c t i n g them, or drowning them i n n e t s . The l a r g e v e s s e l s are s h i p s g r e a t e r t h a n 65 f e e t w i t h 44 crews of 25-30 men who s t r i k e out from the s h i p , f a r from shore i n the r a f t i n g i c e c f f the coast of Newfoundland or the G u l f of St, Lawrence, to h a r v e s t the s e a l s i n t h e i r pupping and moulting patches. The s m a l l v e s s e l s , or l o n g l i n e r s , are c r a f t 35-65 f e e t operated by s m a l l groups cf men who, l i k e the l a r g e v e s s e l hunters, work from the s h i p , sometimes using s m a l l boats to make t h e i r way amcng the i c e f l o e s . F i n a l l y , there are the Canadian A r c t i c and west Greenland n a t i v e s . a s e r i o u s problem i s to d e r i v e age f r e g u e n c i e s f o r the t o t a l c a t c h t h a t represent the combined e f f e c t s of the i n d i v i d u a l f i s h e r i e s ; the data used here are those presented i n L e t t and Benjaminsen (1977), L e t t e t a l (1S78), and Mohn et a l (1978) f o r ages 0 to 25 and years 1952 to 1976. The sampling h i s t o r y and the method used to weight i n d i v i d u a l samples are d i s c u s s e d i n L e t t and Benjaminsen (1977). 3.2 Formulation of the Weighted Least-Squares E s t i m a t o r , N a t u r a l M o r t a l i t y Unknown, The model given by Eg.(2-7) was used with the o b j e c t i v e f u n c t i o n Eq. (2-*16) and weights from Eq. (2-15). Eg. (2-7) i s the model wherein rec r u i t m e n t i s estimated as a separate parameter f o r each year, r e f e r r e d to below as the " b a s i c " model, Given the mcdel and the o b j e c t i v e f u n c t i o n , the f o r m u l a t i o n i s complete when the a g e - s p e c i f i c c o e f f i c i e n t s of v a r i a t i o n of the data e r r o r s ( c v ) , the c o n s t r a i n t s , the i n i t i a l parameter e s t i m a t e s , and the o p t i m i z a t i o n algorithm are s p e c i f i e d . . The cv's were i n i t i a l l y d e r i v e d i n d i s c u s s i o n with P.F.Lett 45 as very c r u d e a p p r o x i m a t i o n s , though based cn L e t t ^ s f a m i l i a r i t y w i t h the d a t a . I n the course of t h e a n a l y s i s they were m o d i f i e d t w i c e t c c o r r e c t f o r t r e n d s i n the r e s i d u a l s , p r o d u c i n g a t o t a l o f t h r e e w e i g h t i n g schemes (Table 1 ) . In each, t h e c v ' s e x h i b i t the same p a t t e r n : t h o s e f o r pup c a t c h e s are s m a l l e s t , b e i n g the l a r g e s t p r o p o r t i o n of the sample and e a s i e s t t o age; t h o s e f o r j u v e n i l e s are s m a l l e r than those f o r a d u l t s ; and t h o s e f o r o l d e r ages a r e l a r g e s t , as both numbers o f a n i m a l s and a c c u r a c y of age d e t e r m i n a t i o n decrease w i t h age. The c o n s t r a i n t s are p r e s e n t e d i n Table 2. On t h e whole, they are very wide, i m p l y i n g l i t t l e o r no p r i o r i n f o r m a t i o n c o n c e r n i n g the parameter v a l u e s . . A case c o u l d be made f o r t i g h t e r c o n s t r a i n t s cn the b a s i s o f p r e v i o u s a n a l y s e s , but i t was thought more o b j e c t i v e r e s u l t s c c u l d be a c h i e v e d o t h e r w i s e . There i s u s u a l l y no g u a r a n t e e t h a t an a l g o r i t h m w i l l converge t o a unigue s o l u t i o n from a r b i t r a r i l y d i f f e r e n t s t a r t i n g p o i n t s , so i t i s c r u c i a l t c demonstrate t h a t the s o l u t i o n o b t a i n e d i s independent of the i n i t i a l e s t i m a t e . L u c k i l y , i n a n a l y z i n g c a t c h - a t - a g e d a t a i t i s p o s s i b l e t o e x p l o i t the s t r u c t u r e shewn i n F i g u r e 1b: g i v e n t h e c o n s t r a i n t s d e f i n i n g T', i f the s o l u t i o n o b t a i n e d from s t a r t i n g near the high-abundances- a n d - l o w - e x p l o i t a t i c n - r a t e s extreme and t h a t of i t s o p p o s i t e a r e i d e n t i c a l , we have r e a s o n to be c o n f i d e n t the s o l u t i o n i s unigue. I n t h i s s t u d y a v a r i a n t o f the Davidon-F l e t c h e r - P o w e l l a l g o r i t h m produced s o l u t i o n s s i m i l a r t o a t l e a s t s i x s i g n i f i c a n t f i g u r e s f o r a l l parameters ( i n one i n s t a n c e numbering more than a hundred) a t a c o n s i s t e n t l y a c c e p t a b l e r a t e of convergence. More p r e c i s e l y , the low-abundance-high-46 Table 1. Data errcr c o e f f i c i e n t s cf variation for northwestern Atlantic harp seal catches-at-age. Age 0 1 - 2 3 - 7 8 - 2 0 2 1 - 2 5 W e i g h t i n g #1 . 0 2 5 . 0 5 . 0 5 . 1 5 . 3 5 W e i g h t i n g #2 . 1 0 . 2 0 . 1 5 . 2 0 . 3 5 W e i g h t i n g #3 . 2 0 . 2 5 . 2 0 . 2 5 . 4 0 4 8 Table 2. Least-squares catch-at-age analysis parameter constraints for northwestern A t l a n t i c harp seals. A l l numbers-at-age i n year 1: [ 1, i o 6 ] A l l annual components of exploitation: [.001, 1.0 ] A l l age selection factors: [.001, 1.0 ] A l l annual recruitments: [ 1, 5 x 10 6 ] Survival rate through natural mortality [ 0.7, 1.0 ] 50 e x p l o i t a t i o n i n i t i a l p o i n t s were t a k e n as 101X of the l o w e r c o n s t r a i n t s cn the N,: 's and t h e N 1 0 ' s , and 995? o f t h e upper c o n s t r a i n t s cn t h e a ' s , the y ' s , and Sn. The o p p o s i t e p o i n t s were t a k e n t c t e the r e v e r s e extremes. 3.3 R e s u l t s With t h e B a s i c Model. The i n i t i a l w e i g h t i n g scheme l e d to a q u i t e h i g h r e s i d u a l mean sguare and a t r e n d o f i n c r e a s i n g r e s i d u a l s a t younger ages; t h e r e f o r e a second w e i g h t i n g was t e s t e d . W e i g h t i n g #2, w i t h much h i g h e r c o e f f i c i e n t s of v a r i a t i o n f o r younger ages than W e i g h t i n g #1, e f f e c t i v e l y reduced t r e n d s i n t h e r e s i d u a l s . A t h i r d w e i g h t i n g was t r i e d i n o r d e r t c a s s e s s the e f f e c t of i n c r e a s i n g the c o e f f i c i e n t s o f v a r i a t i o n s t i l l f u r t h e r , a l t h o u g h t h e s e v a l u e s a r e c o n s i d e r e d t o be t o o h i g h f o r harp s e a l d a t a . Together the t h r e e w e i g h t i n g s r e p r e s e n t rough e s t i m a t e s of v a r i a t i o n s p a n n i n g a r e l a t i v e l y wide range o f v a l u e s . Some s t a t i s t i c s from the r e s u l t s of each o p t i m i z a t i o n are shown i n Table 3 . I t i s c l e a r t h a t t h e t e c h n i g u e i s not r o b u s t t o l a r g e changes i n the w e i g h t s , a t l e a s t f o r the harp s e a l data._ Each w e i g h t i n g g i v e s a unigue c o m b i n a t i o n of n a t u r a l m o r t a l i t y , f i s h i n g m o r t a l i t y , and abundance; each i m p l i e s a r a d i c a l l y d i f f e r e n t e x p l a n a t i o n o f the c a t c h h i s t o r y . Whereas t h i s r e s u l t i s u n f o r t u n a t e i t i s net unexpected. There w i l l always be a s t r c n g c o v a r i a n c e among n a t u r a l m o r t a l i t y , e x p l o i t a t i o n r a t e , and abundance parameters. The lower the i n f o r m a t i o n c o n t e n t of t h e d a t a , the more s e n s i t i v e t h e e s t i m a t o r w i l l be t o the 51 Tafcle 3 . R e s u l t s of l e a s t - s q u a r e s c a t c h - a t - a g e a n a l y s i s f o r n o r t h w e s t e r n A t l a n t i c harp s e a l s . The l e a s t - s g u a r e s r e s u l t s a r e compared t c r e s u l t s frcm s e g u e n t i a l p o p u l a t i o n a n a l y s i s . , 52 mean mean S .2 Sn ^ i 0 N i 0 W i t h t h e b a s i c m o d e l , w e i g h t i n g #1 2 3 . 97 . 9 4 . 4 7 384 W i t h t h e b a s i c m o d e l , w e i g h t i n g #2 3 . 98 . 8 7 .17 1080 W i t h t h e b a s i c m o d e l , w e i g h t i n g #3 2 . 23 . 9 2 . 1 2 1386 W i t h a B e v e r t o n - H o l t s t o c k - r e c r u i t f u n c t i o n , w e i g h t i n g #2 4 . 51 . 9 0 . 3 3 509 S e q u e n t i a l p o p u l a t i o n a n a l y s i s . 9 0 . 4 7 423 53 w e i g h t s . I f the i n f o r m a t i o n c o n t e n t i s h i g h , i . e . , h i g h q u a l i t y and h i g h g u a n t i t y data over a wide range o f s t o c k s i z e s , the c o v a r i a n c e among the t h r e e k i n d s of e f f e c t s might be reduced enough t c g i v e r e l i a b l e r e s u l t s , i n c l u d i n g an e s t i m a t e of n a t u r a l m o r t a l i t y . The a b i l i t y t o e s t i m a t e n a t u r a l m o r t a l i t y would be enhanced, f i r s t , by u s i n g a c c u r a t e e s t i m a t e s of the d a t a e r r o r v a r i a n c e s as weights ( i n f o r m a t i o n t h a t i s g e n e r a l l y a v a i l a b l e but not used) and, second, by i n c o r p o r a t i n g e x t r a i n f o r m a t i o n (assumptions) i n t o the e s t i m a t i o n i n t h e form of m o d i f i c a t i o n s to the model and t h e c o n s t r a i n t s . An example of t h e l a t t e r i s g i v e n n e x t . 4 I n i t i a l R e s u l t s Using a B e v e r t o n - H o l t S t o c k - R e c r u i t R e l a t i o n s h i p . Mechanisms f o r d e n s i t y - d e p e n d e n t r e g u l a t i o n of harp s e a l abundance have been proposed t o o p e r a t e i n t h r e e ways: th r o u g h t h e m o r t a l i t y r a t e c f pups, the mean age of m a t u r i t y , and the pregnancy r a t e ( L e t t e t a l 1978). T h i s s u g g e s t s a number of p o s s i b i l i t i e s f o r the s t o c k - r e c r u i t f u n c t i o n (h) i n Eg.(2-18). A l l t h r e e mechanisms c o u l d be i n c l u d e d e x p l i c i t l y , but t h i s would l e a d t o a t l e a s t 6 r e c r u i t m e n t parameters and i t i s d o u b t f u l t h a t r e l i a b l e e s t i m a t e s c o u l d be o b t a i n e d f o r each. Seme s i m p l i f i c a t i c n i s r e q u i r e d , t h e r e f o r e , somewhat a r b i t r a r i l y , I chose t c use a B e v e r t c n - H o l t c u r v e . The b r e e d i n g s t o c k was d e f i n e d as the sum o f a l l a d u l t a n i m a l s , m u l t i p l i e d by 1,06 ( f o l l o w i n g L e t t and Benjaminsen, who assumed t h a t 6% of the 54 b r e e d i n g p o p u l a t i o n was ever 25). The model i m p l i c i t l y a c c o u n t s f o r changes i n the pregnancy r a t e and pup m o r t a l i t y , but n e g l e c t s changes i n the age o f m a t u r i t y , which was assumed c o n s t a n t a t age 5. We have: Pi » 1.06 Z N J ; (3-1) where a. and p a r e t h e unknowns. The B e v e r t o n - H o l t c u r v e s h o u l d be c o n s i d e r e d as an a p p r o x i m a t i o n t o the t r u e r e l a t i o n s h i p , which L e t t e t a l suggest i s s l i g h t l y s i n u s o i d a l . As b e f o r e , i t remains t o s p e c i f y the c o n s t r a i n t s , the w e i g h t s , t h e o p t i m i z a t i o n a l g o r i t h m , and the i n i t i a l parameter e s t i m a t e s . C o n s t r a i n t s on a and p were determined by d e r i v i n g an e x p r e s s i o n f o r them i n terms o f the e g u i l i b r i u m u n f i s h e d p o p u l a t i o n s i z e (Noo ) , t h e r e c r u i t m e n t r a t e at h a l f the u n f i s h e d e g u i l i b r i u m (A) , and Sn, analagous t c Eg. (A-7) and Eg. (A-8) but u s i n g Eg. ( 3 - 1 ) ; then c a l c u l a t i n g extreme v a l u e s of cx and p» frcm extreme v a l u e s c f Hgo , A , and Sn (Table 4 ) . The c o n s t r a i n t s a r e shown i n T a b l e 5,. W e i g h t i n g #2 was used, w i t h the D a v i d o n - F l e t c h e r - E o w e l l a l g o r i t h m and i n i t i a l e s t i m a t e s determined as i n s e c t i o n 3.2 . The i n i t i a l r e s u l t s u s i n g t h e s t o c k - r e c r u i t c u r v e were c o n s i d e r e d u n a c c e p t a b l e due t o an ancmaly i n the e s t i m a t e s of numbers-at-age i n t h e f i r s t y e a r . The age f r e g u e n c i e s f o r 1952 f o r the B e v e r t o n - H o l t model a r e shown i n F i g u r e 3 .. The e s t i m a t e s f o r ages 22 t o 25 are u n r e a s o n a b l e , w i t h l a r g e v a r i a n c e s due t c low g u a l i t y and g u a n t i t y of s u p p o r t i n g 55 Table 4. Extreme values of northwestern Atlantic harp seal Beverton-flolt stock-recruit parameters CL and (b are determined frcm extreme values of N. , * , and Sn, Sn X N CO a 3 .80 .10 106 -3, .833 x IO"5 18, .279 .80 .10 107 -3. .833 x IO - 6 18. .279 .925 .50 106 1. .570 x -5 10 3 -3. .752 .925 .50 IO7 1. .570 x 10"6 -3. .752 57 Table 5. Parameter constraints of least-squares catch-at-age analysis with a Beverton-Holt stock-r e c r u i t function, f o r northwestern A t l a n t i c harp seals. A l l n u m b e r s - a t - a g e i n y e a r 1 : [ 1 , i o 6 ] A l l a n n u a l c o m p o n e n t s o f e x p l o i t a t i o n : [ . 0 0 1 , 1 ] A l l a g e s e l e c t i o n f a c t o r s : [ . 0 0 1 , 1 ] S t o c k - r e c r u i t p a r a m e t e r a: [ 0 , 1 . 5 7 0 x 1 0 ~ 5 ] S t o c k - r e c r u i t p a r a m e t e r 3: [ 0 , 1 8 . 2 7 9 ] S u r v i v a l r a t e t h r o u g h n a t u r a l m o r t a l i t y : [ 0 . 7 , 1 . 0 ] 59 F i g u r e 3. The age d i s t r i b u t i o n f o r n o r t h w e s t e r n A t l a n t i c harp s e a l s i n 1952, from (A) i n i t i a l r e s u l t s c f l e a s t - s q u a r e s c a t c h - a t -age a n a l y s i s w i t h a B e v e r t o n - H c l t s t o c k -r e c r u i t f u n c t i o n and (E) t h e r e v i s e d d i s t r i b u t i c n . A G E D ISTRIBUTION IN 1952 ( P E R C E N T A G E OF T H E I + P O P U L A T I O N ) 09 61 d a t a . Cne s o l u t i o n t o t h i s problem i s t o modify the i n i t i a l r e s u l t s by p o s t u l a t i n g r e a s o n a b l e v a l u e s f o r ages 22 t o 25 r e l a t i v e t o a d j a c e n t ages, then n o r m a l i z i n g t h e r e s u l t i n g age d i s t r i b u t i o n (Table 6, F i g u r e 3 ) . The e s t i m a t i o n was r e p e a t e d u s i n g t h i s p r e - e s t i m a t e d d i s t r i b u t i o n i n a model i d e n t i c a l t o Eg, (3-1), e x c e p t : N>j m p^ j , I ~ j * T (3-2) where pr- i s the p r o p o r t i o n o f the t o t a l p o p u l a t i o n a t age j . The t o t a l p o p u l a t i o n s i z e , N 1 + , i s the unknown i n s t e a d o f each N|j . With t h e p r e - e s t i m a t e d age d i s t r i b u t i o n , t h e c o n s t r a i n t s were as b e f o r e , e x c e p t N l + was c o n s t r a i n e d t o ( 106 ,10 7 ], based r o u g h l y on L e t t e t a l ' s s i m u l a t i o n r e s u l t s of maximum p o p u l a t i o n s i z e . The same weights and a l g o r i t h m were used, w i t h i n i t i a l p o i n t s d e r i v e d frcm the c o n s t r a i n t s . 3.5 F i n a l E e s u l t s u s i n g t h e B e v e r t c n - H o l t S t o c k - R e c r u i t R e l a t i o n s h i p . Some s t a t i s t i c s from the f i n a l r e s u l t s are p r e s e n t e d i n T able 3 and the s t a t e r e c o n s t r u c t i o n i s compared t o t h e s e g u e n t i a l p o p u l a t i o n a n a l y s i s o f L e t t e t a l and Mohn e t a l i n F i g u r e 4 . I n s h o r t , the e s t i m a t e f o r s u r v i v a l t h r ough n a t u r a l m o r t a l i t y i s the same i n b o t h , Sn = 0.9, but the l e a s t - s g u a r e s method shews the p o p u l a t i o n has been more abundant w i t h l o w e r e x p l o i t a t i o n r a t e s . L e t t e t a l assumed a v a l u e of Sn = 0.9 based on the a v a i l a b l e evidence from 7 s o u r c e s (summarized t h e r e i n ) , 62 Table 6, The age d i s t r i b u t i o n for northwestern A t l a n t i c harp seals i n 1952, from i n i t i a l r e s u l t s of least-sguares catch-at-age analysis with a Beverton-Holt stock-recruit function, and the revised d i s t r i b u t i o n . Age I n i t i a l r e s u l t s (%) R e v i s e d [%) 1 5.78 6.70 2 6.19 7.17 3 5.05 5.85 4 5.03 5.83 5 5.22 6.05 6 3.97 4.60 7 4.33 5.02 8 . 1.61 1.97 9 3.33 3.86 10 3.49 4.04 11 3.25 3.77 12 3.01 3.49 13 2.94 3.41 14 3.72 4.31 15 3.59 4.16 16 1.88 2.18 17 3.02 3.50 18 2.69 3.12 19 2.83 3.28 20 2.87 3.33 21 3.01 3.49 22 0.30 3.19 23 4.56 2.90 24 8.82 2.60 25 9.53 2.32 64 F i g u r e 4. E s t i m a t e d t r e n d s i n n o r t h w e s t e r n A t l a n t i c harp s e a l abundance, 1952-1979. F i n a l r e s u l t s c f l e a s t - s q u a r e s c a t c h - a t - a g e a n a l y s i s w i t h a E e v e r t o n - H o l t s t o c k - r e c r u i t f u n c t i o n a r e compared t o r e s u l t s from s e q u e n t i a l p o p u l a t i o n a n a l y s i s . 66 each depending, i n seme manner, on the a n a l y s i s of s u r v i v o r s h i p i n d i c e s . Pup p r o d u c t i o n and pup e x p l o i t a t i o n r a t e s are compared by year i n Table 7 . The g r e a t e s t d i f f e r e n c e s i n pup p r o d u c t i o n occur i n the years f o r which the l e a s t - s g u a r e s c a t c h p r e d i c t i o n e r r o r s a r e l a r g e : 1956, 1958, 1961, 1971, and 1977. The l a c k of f i t f o r these years might be a t t r i b u t a b l e t o the e f f e c t of changes i n age s e l e c t i o n over time, although other years f i t g u i t e w e l l , with most e r r o r s l e s s than 35,000 pups. The trend i n 1+ p o p u l a t i o n i s s l i g h t l y d i f f e r e n t from the s e q u e n t i a l p o p u l a t i o n a n a l y s i s estimates, which decrease s t e a d i l y frcm 1952 to the mid-1970s and then i n c r e a s e . The l e a s t - s g u a r e s r e s u l t s have the p o p u l a t i o n d e c r e a s i n g u n t i l the mid-1970s, but l e v e l i n g out and not i n c r e a s i n g t h e r e a f t e r . . The 1+ p o p u l a t i o n s i z e i n 1952 (N | + ) i s 8.86 m i l l i o n , compared to 2.53 m i l l i o n f o r s e g u e n t i a l p o p u l a t i o n a n a l y s i s , a d i f f e r e n c e c f over 300%. While the l e a s t - s g u a r e s f i g u r e may appear i n c r e d u l o u s i n view of previous a n a l y s e s , s t i l l , the r e s u l t i s c o n s i s t e n t with the catch-at-age data. The e q u i l i b r i u m u n e x p l o i t e d 1+ p o p u l a t i o n (N w) was estimated as 4.3 m i l l i c n , compared with L e t t e t a l ' s value using the Lett-Benjaminsen model under f u l l density-dependence of 4.2 m i l l i c n . The g r e a t e s t i n c o n s i s t e n c y i n the l e a s t - s g u a r e s a n a l y s i s i s that the 1+ p o p u l a t i o n i n 1952 was twice the estimated u n e x p l o i t e d e g u i l i b r i u m l e v e l . T h i s suggests t h a t e i t h e r cr both of the estimates are u n r e l i a b l e . The e r r o r i s probably with the estimate c f Noo which i s very s e n s i t i v e to s m a l l changes i n a . The recruitment r a t e at h a l f the u n f i s h e d 67 Table 7. Northwestern A t l a n t i c harp seal pup production and annual pup exploitation rates frcm the f i n a l least-squares r e s u l t s are compared to re s u l t s from sequential population analysis. Values for pup production are in thousands. final least-squares results sequential population analysis Year N i 0 Vo N i 0 • ' i 0 1952 692 .27 559 .35 1953 680 .21 549 .36 1954 668 .23 558 .33 1955 656 .28 560 .46 1956 642 .21 566 .61 1957 625 .31 567 .30 1958 609 .42 555 .27 1959 593 .37 518 .47 1960 578 .44 498 .33 1961 563 .10- 468 .37 1962 552 .39 433 .49 1963 533 .60 403 .71 1964 514 .53 381 .71 1965 497 .34 362 .52 1966 491 .54 364 .62 1967 475 .51 358 .78 1968 454 .35 362 .44 1969 437 .44 368 .64 1970 430 .45 361 .61 1971 413 .20 347 .61 1972 400 .21 329 .36 1973 391 .30 329 .31 1974 380 .32 317 .37 1975 370 .28 308 .46 1976 368 .23 312 .42 1977 366 .12 327 .40 1978 363 .25 349 .34 69 e q u i l i b r i u m p o p u l a t i o n (A) was e s t i m a t e d as 0 . 2 4 . a and p> were 1 . 0 8 2 x 10~fe and 2 . 5 4 6 , r e s p e c t i v e l y . The age s e l e c t i o n f a c t o r s a r e shown i n Table 8. They show a d e f i n i t e t r e n d w i t h age, i m p l y i n g d e c r e a s e d v u l n e r a b i l i t y as the ani m a l s g e t c i d e r . T h i s c c u l d be r e a s o n a b l e ( e . g . , i t i s not i n c o n c e i v a b l e t h a t the a n i m a l s l e a r n t o a v o i d s e a l h u n t e r s ) , or i t c o u l d s i m p l y be an a r t i f a c t o f assuming n a t u r a l m o r t a l i t y i s c o n s t a n t ever a l l ages. The s t a t e p r e d i c t i o n f c r 1979 i s g i v e n i n Table 9 . The l e a s t - s g u a r e s method p r e d i c t s a 1+ p o p u l a t i o n and pup p r o d u c t i o n c f 2 . 2 8 3 a i l l i c n and 3 5 8 , 0 0 0 , compared t o 1 . 3 9 7 m i l l i o n and 3 5 8 , 0 0 0 f o r s e q u e n t i a l p o p u l a t i o n a n a l y s i s combined w i t h t h e L e t t - E e n j a m i n s e n model (Mchn e t a l 1 9 7 8 ) . The l e a s t - s q u a r e s p r e d i c t i o n c f pup p r o d u c t i o n i s c o n s i s t e n t w i t h r e c e n t e s t i m a t e s f r c m e t h e r t e c h n i q u e s , e.g., t a g g i n g s t u d i e s and th e DeLury method, summarized i n L e t t et a l and Mohn e t a l . The f i n a l model e x p l a i n e d 85% c f the v a r i a t i o n i n weighted c a t c h e s . However, i t s h o u l d be noted t h a t t h e number c f parameters r e l a t i v e t c the amount o f data i s l a r g e . 3 . 6 P o p u l a t i o n P r o j e c t i o n s With a Quota o f 1 8 0 , 0 0 0 . The c u r r e n t t o t a l a l l o w a b l e c a t c h of harp s e a l s i s 1 8 0 , 0 0 0 . . P r o j e c t i o n s c f p o p u l a t i o n s i z e over the next t e n y e a r s w i t h t h i s guota were made by Mohn e t a l ( 1 9 7 8 , T a b l e 3) u s i n g t h e L e t t -Ben jamihsen model. Under t h r e e d i f f e r e n t assumptions of d e n s i t y -dependence, i n each case they found the s t o c k t o i n c r e a s e s l o w l y 70 Table 8. Northwestern At l a n t i c harp seal age selection factors from the f i n a l least-sguares r e s u l t s . Age 71 0 1 2 3 4 5 6 7 8 9 1.0000 .0427 .0521 .0445 .0431 .0436 .0386 .0368 .0366 .0309 10 11 12 13 14 15 16 17 18 19 .0321 .0266 .0279 .0243 .0246 .0265 .0231 .0193 .0184 .0149 20 21 22 23 24 25 .0139 .0076 .0019 .0053 .0053 .0017 72 Table 9. Forecast of northwestern Atlantic harp seal abundance for 1979 from the f i n a l l e a s t -sguares r e s u l t s . 73 Age A b u n d a n c e 0 358258 1 246069 2 259367 3 203080 4 170089 5 147326 6 138262 7 142983 8 132969 9 85896 10 78033 11 83390 12 59253 13 50710 14 64342 15 42126 16 32875 17 46693 18 62517 19 36142 20 37540 21 31307 22 34532 23 36583 24 30907 25 30298 1+ 2283289 74 but s t e a d i l y . For the p r o j e c t i o n s given here the i n i t i a l p o p u l a t i o n was the p r e d i c t i o n f o r 1979, s u r v i v a l through n a t u r a l m o r t a l i t y was 0.9, and the quota was broken down over ages using a d i s t r i b u t o n c a l c u l a t e d ty n o r m a l i z i n g the estimated age s e l e c t c n f a c t o r s . Recruitment was modelled using Eq. (3-2) , with the l e a s t - s q u a r e s estimates of o( and p . The r e s u l t s i n F i g u r e 5 show the mean p l u s or minus two standard d e v i a t i o n s c f 100 s t o c h a s t i c runs. & lognormal d i s t r i b u t i o n mimicked random f l u c t u a t i o n s i n r e c r u i t m e n t , while catches were assumed normally d i s t r i b u t e d . Two d i f f e r e n t assumptions about v a r i a b i l i t y were used. The high v a r i a b i l i t y assumption had a value c f 0. 2 f o r the c o e f f i c i e n t of v a r i a t i o n of r e c r u i t m e n t , 0.4 f o r the 1+ catches, and 0.1 f o r pup c a t c h e s . The lew v a r i a b i l i t y assumption had values of 0.1, 0.2, and C.05 . I t gave a p r o j e c t i o n almost i d e n t i c a l t o the high v a r i a b i l t y assuitption, t u t with s m a l l e r standard d e v i a t i o n s , and so i s net shewn i n F i g u r e 5. In c o n t r a s t to Mchn e t a l ' s p r o j e c t i o n s , those given here show a c o n t i n u a l decrease i n the stock cf about 3.6% per year f o r both cases. T h i s r e p r e s e n t s a d e c l i n e of 700,000 animals over ten years, or o n e - t h i r d of the present p o p u l a t i o n . The question i s : Why do the two methods give such d i f f e r e n t f o r e c a s t s ? Two e x p l a n a t i o n s come to mind. F i r s t , the r e p r o d u c t i v e p o t e n t i a l embodied i n the Lett-Eenjaminsen model co u l d be g r e a t e r than t h a t embodied i n the Beverton-Holt model. The density-dependent f u n c t i o n s i n the L-B model were f i t with f e r t i l i t y r a t e s sampled s i n c e the e a r l y 1950s and estimates of 75 figure 5. Population projections for northwestern Atl a n t i c harp seals under a guota of 180,000.. 77 abundance from a s e q u e n t i a l p o p u l a t i o n a n a l y s i s . I f t h e s e s t o c k s i z e s were to o lew, as i s suggested h e r e , then t h e r e p r o d u c t i v e p c t e n t i a l i n the model would be b i a s e d . However, s i n c e the p r o p o r t i o n of t h e s t c c k r e c r u i t e d d e c r e a s e s w i t h i n c r e a s i n g abundance, i f a n y t h i n g t h e L-B model u n d e r e s t i m a t e s p r o d u c t i v i t y . The second e x p l a n a t i o n i s t h a t t h e r e p r o d u c t i v e p o t e n t i a l s are s i m i l a r , but t h a t the decrease i n r e c r u i t m e n t r a t e a t t h e h i g h p o p u l a t i o n e s t i m a t e d by l e a s t - " s q u a r e s , r e l a t i v e t o the p o p u l a t c n from s e q u e n t i a l p o p u l a t i o n a n a l y s i s , i s g r e a t e r t h a n the c o r r e s p o n d i n g decrease i n e x p l o i t a t i o n r a t e . T h i s must be the c a s e , w i t h t h e d i f f e r e n c e b e i n g l a r g e enough t o produce the p r e d i c t e d d e c l i n e . 3, 7 D i s c u s s i o n . The harp s e a l a n a l y s i s i s d e f i c i e n t i n s e v e r a l r e s p e c t s . F i r s t , t h e c c e f f i c i e n t s of v a r i a t i o n f o r t h e d a t a e r r o r s were a s s i g n e d somewhat a r b i t r a r i l y . . B e t t e r e s t i m a t e s can be c a l c u l a t e d from the h i s t o r i c a l age samples. A l a r g e s a m p l i n g experiment wculd a l s o be w o r t h w h i l e . T h i s w i l l e n a b l e t h e c o e f f i c i e n t s o f v a r i a t i o n t o be f i x e d a c c u r a t e l y and w i l l r e v e a l how they change w i t h age. F e c u n d i t y and sex r a t i o d a t a have been c o l l e c t e d a n n u a l l y f o r harp s e a l s s i n c e t h e e a r l y 1950s, but a r e n e t i n c l u d e d i n t h i s a n a l y s i s . I f e s t i m a t e s of both a g e - s p e c i f i c sex r a t i o s and p r o p o r t i o n s - a t - a g e o f females whelping can be s u p p l i e d f o r each y e a r i n t h e c a t c h - a t - a g e data s e r i e s , t h e n , u s i n g Eg. (2-23), the 78 l e a s t - s g u a r e s s t a t e r e c o n s t r u c t i o n w i l l be c o n s i s t e n t w i t h h i s t o r i c a l r e p r o d u c t i o n s t a t i s t i c s . I e x p e c t t h a t doing so w i l l b r i n g the r e s u l t s more i n l i n e w i t h L e t t et a l ' s s e q u e n t i a l p o p u l a t i o n a n a l y s i s , which a l r e a d y i n c o r p o r a t e s some o f t h i s i n f c r m a t i c n . Some of the pup c a t c h p r e d i c t i o n e r r o r s are u n r e a s o n a b l y l a r g e , p r o b a b l y as r e s u l t of v a r i a t i o n i n t h e age s e l e c t i o n f a c t o r s f o r pups. The a n a l y s i s can be m o d i f i e d such t h a t i n t h o s e y e a r s w i t h l a r g e pup c a t c h p r e d i c t i o n e r r o r s , t h e pup s e l e c t i o n f a c t o r s a r e not f i x e d a t 1.0, but a r e e s t i m a t e d i n t h e o p t i m i z a t i o n . T h i s would have t h e same a f f e c t as the extreme assumption t h a t the data e r r o r s f o r t h o s e c a t c h e s a r e n e g l i g i b l e . Another s o u r c e o f s t r u c t u r a l weakness i n t h e f o r m u l a t i o n e x i s t s i f n a t u r a l m o r t a l i t y d e c r e a s e s w i t h age, though how t h i s wculd a f f e c t t h e e s t i m a t e s i s not c l e a r . A r e l a t e d p o s s i b i l i t y i s t h a t t h e age s e l e c t i o n f a c t o r s are not i n f a c t v a r i a b l e w i t h age f o r o l d e r a n i m a l s . The e s t i m a t o r has succeeded i n p r o d u c i n g a r e a s o n a b l e e s t i m a t e of n a t u r a l m o r t a l i t y . A p p a r e n t l y , i n i t s a b i l i t y t o e x t r a c t i n f c r m a t i c n , the l e a s t - s g u a r e s approach c o u l d prove t o be a p o w e r f u l t o o l f o r a n a l y z i n g c a t c h - a t - a g e d a t a . However, whether t h i s t u r n s out to be t h e case w i l l b e s t be determined not by p r o v i d i n g mere case s t u d i e s i n which r e a s o n a b l e r e s u l t s a re o b t a i n e d . The problem i s t o determine whether the r e s u l t s are r e l i a b l e , and t h i s i m p l i e s a knowledge of the v a r i a n c e s and c o v a r i a n c e s o f the e s t i m a t e s . S i m u l a t i o n s t u d i e s w i l l be u s e f u l i n t h i s r e s p e c t , net o n l y p r o v i d i n g e s t i m a t e s o f t h e parameter 79 v a r i a n c e s , but the v a r i a n c e of the s t a t e p r e d i c t i o n as w e l l . T h i s p o i n t i s the focus c f chapters 4 and 5 . . 80 Chapter 4. NUMEBIC AL EXPERIMENTS WITH LEAST-SQUABES CATCH-AT-AGE ANALYSIS: A PINNIPED FISHEEY Chapter 2 was concerned w i t h d e v e l o p i n g a s t o c k assessment methodology, and c h a p t e r 3 with a p p l y i n g i t t o r e a l d a t a . I n t h i s c h a p t e r and the n e x t , an attempt i s made t o g u a n t i f y the r e l a t i o n s h i p between the l e a s t - s q u a r e s e s t i m a t e s and some a t t r i b u t e s o f data which a f f e c t t h e i r r e l i a b i l i t y . Two t y p e s of f i s h e r y a r e examined: a p i n n i p e d s t o c k and a c l u p e o i d s t o c k . Each t y p e c f f i s h e r y d i f f e r s i n t h e e s t i m a t o r t h a t i s used. For the p i n n i p e d s t o c k , the method i s the same used t o o b t a i n the f i n a l harp s e a l r e s u l t s i n c h a p t e r 3 : the parameters of a s t o c k -r e c r u i t f u n c t i o n a r e e s t i m a t e d i n t h e o p t i m i z a t i o n , and a p r e -e s t i m a t e d age d i s t r i b u t i o n f o r y e a r 1 i s i n c l u d e d . F o r the c l u p e c i d s t c c k , the annual r e c r u i t m e n t s are e s t i m a t e d , i n s t e a d o f t h e parameters of a stock-* r e c r u i t f u n c t i o n ; t h e n t h e parameters o f a s t o c k - r e c r u i t d i s t r i b u t i o n a r e e s t i m a t e d u s i n g the s t a t e r e c o n s t r u c t i o n . The numbers-at-age i n y e a r 1 are a l s o e s t i m a t e d , i n s t e a d of i n c l u d i n g a p r e - e s t i m a t e d age d i s t r i b t u t i c n . The i n f o r m a t i o n c o n t e n t of a g i v e n d a t a s e t i s c h a r a c t e r i z e d by i t s s i z e , i t s degree o f e r r o r , and i t s l e v e l of c o n t r a s t ( i n abundance and e x p l o i t a t i o n r a t e s ) . One c o u l d d e s c r i b e , f c r example, the P a c i f i c H a l i b u t (Area 2 ) d a t a s e t as 81 huge (18 ages sampled s i n c e 1932), w i t h enormous e r r o r s ( u n t i l r e c e n t l y ) , and p r o b a b l y moderate c o n t r a s t (Hoag and McNaughton MS 1978, Southward MS 1S76). T h i s s i m p l e c l a s s i f i c a t i o n h e l p s t o g i v e form to a s e a r c h f o r g u a n t i t a t i v e g e n e r a l i t i e s about r e l i a b i l i t y i n assessments o f c a t c h - a t - a g e d a t a . The t h r e e c o n t i n u a can be thought of as d e f i n i n g d i m e n s i o n s i n space ( F i g u r e 6 ) . The problem i s to determine the magnitude o f the parameter v a r i a n c e s and hew t h e y change a t d i f f e r e n t p o i n t s i n t h i s "space". We expect they w i l l decrease as data q u a n t i t y i n c r e a s e s , i n c r e a s e when e r r o r s are i n c r e a s e d , and so on. By q u a n t i f y i n g the v a r i a n c e s of t h e p a r t i c u l a r parameter e s t i m a t e s and the f o r e c a s t of abundance, u s i n g models g e n e r a t i n g d a t a w i t h a t t r i b u t e s c o r r e s p o n d i n g t o v a r i o u s p o i n t s i n " i n f o r m a t i o n -s p a c e " , we s h o u l d be a b l e t c a n t i c i p a t e the r e l i a b i l i t y o f the e s t i m a t e s we might o b t a i n i n some u n i n v e s t i g a t e d c i r c u m s t a n c e . . The Mcnte C a r l e approach i s a means t o e v a l u a t e t h e r e l i a b i l i t y c f b o t h the parameter e s t i m a t e s and f o r e c a s t e d abundance, as measured by b i a s e s and v a r i a n c e s . . T h e b a s i c i d e a i s q u i t e s i m p l e , The a t t r i b u t e s o f the d a t a d e t e r m i n i n g i n f o r m a t i o n c o n t e n t ( i . e . , d a t a e r r o r s , c o n t r a s t , data g u a n t i t y ) a r e mimicked by a computer model r e p r e s e n t i n g not j u s t the p o p u l a t i o n dynamics w i t h f i s h i n g , but a l s o the d a t a c o l l e c t i o n p r o c e s s . R e p l i c a t i o n s c f f a k e data are g e n e r a t e d , then a re a n a l y s e d by the l e a s t - s q u a r e s e s t i m a t o r . The c o n t r i b u t i o n of v a r i o u s f a c t o r s t o t h e b i a s and v a r i a n c e , o f t h e e s t i m a t e s , i s a s s e s s e d t y comparing the r e s u l t s of the e s t i m a t o r w i t h t h e " t r u e " or known v a l u e s i n the s i m u l a t i o n model. The approach has th e advantage t h a t the c o n t r i b u t i o n t o b i a s and v a r i a n c e o f the 82 F i g u r e 6 . The d a t a s e t " i n f o r m a t i o n - s p a c e " . . C a t c h - a t -age data s e t s can be c l a s s i f i e d by q u a n t i t y c f d a t a , degree of e r r o r s , and l e v e l of c o n t r a s t i n abundance and e x p l o i t a t i o n r a t e s . The p o s i t i o n s c f t h e boxes i n d i c a t e the c o n d i t i o n s f o r t h e Monte C a r l o Cases A-G. 84 e s t i m a t e s can be e v a l u a t e d f o r each f a c t o r i n d e p e n d e n t l y . The o v e r a l l v a r i a n c e can be determined by m o d e l l i n g a l l the f a c t o r s c o l l e c t i v e l y . However, a drawback i s t h a t the r a t i o n a l e i s somewhat c i r c u l a r . We assume we knew the t r u e v a l u e s i n n a t u r e of the parameters used i n the model g e n e r a t i n g the data r e p l i c a t e s , i n c r d e r t c a s s e s s how l i t t l e we know about them. I f t h e a n a l y s i s i s r e l i a b l e t h i s i s not an i m p o r t a n t problem, but i f i t i s u n r e l i a b l e i t i s p o s s i b l e t h a t t h e model g e n e r a t i n g the f a k e c a t c h e s - a t - a g e w i l l produce i n f o r m a t i v e d a t a , when t h e r e a l d a t a are a c t u a l l y u n i n f o r m a t i v e . I n t h i s c a s e , the Monte C a r l o e s t i m a t e s of parameter and f o r e c a s t v a r i a n c e s w i l l be i n a p p r o p r i a t e l y o p t i m i s t i c , 4.1 F a c t o r s D e t e r m i n i n g t h e E e l i a b l i t y of the E s t i m a t e s , F a c t o r s a f f e c t i n g r e l i a b i l i t y c f e s t i m a t e s f a l l i n t o two c l a s s e s : t h o s e t h a t reduce t h e i n f o r m a t i o n c o n t e n t of t h e d a t a and those t h a t i n h i b i t i n f o r m a t i o n e x t r a c t i o n by t h e e s t i m a t o r . I n f o r m a t i o n c o n t e n t i s d e c r e a s e d due t o ( 1 ) low d a t a g u a n t i t y , (2) h i g h d a t a e r r o r v a r i a t i o n , (3) lew c o n t r a s t i n s t o c k s i z e s and e x p l c i t a t i c r r a t e s d u r i n g the p e r i o d over which t h e data were t a k e n , and (4) e r r o r s i n t h e e s t i m a t e s o f data e r r o r v a r i a n c e s . I n f o r m a t i o n e x t r a c t i o n i s i m p a i r e d by (5) n a t u r a l m o r t a l i t y unknown, (6) s t r u c t u r a l l a c k o f f i t due t o changes i n n a t u r a l m o r t a l i t y o ver ages c r y e a r s or b o t h , and (7) s t r u c t u r a l l a c k o f f i t due t o changing age s e l e c t i o n . When s t o c k - r e c r u i t parameters are e s t i m a t e d i n the o p t i m i z a t i o n , a d d i t i o n a l f a c t o r s 85 are (8) l a c k o f f i t due t o an i n c o r r e c t form o f t h e s t c c k -r e c r u i t f u n c t i o n / (9) l a c k o f f i t due t o n a t u r a l s t o c h a s t i c v a r i a t i o n i n r e c r u i t m e n t , and (10) l a c k o f f i t due t o changes i n the s t c c k - r e c r u i t parameters over t i m e . I n t h e remainder c f t h i s c h a p t e r , the r e s u l t s of the harp s e a l a n a l y s i s from c h a p t e r 3 a r e used t o p a r a m e t e r i z e a s i m u l a t i o n model t h a t g e n e r a t e s l a r g e d a t a s e t s , w i t h moderate e r r o r s and tremendous c o n t r a s t . Each of the t h r e e c a s e s examined below c o r r e s p o n d s t o t h i s s i n g l e p o i n t i n " i n f o r m a t i o n - s p a c e " . S p e c i f i c a l l y , we are i n t e r e s t e d i n q u a n t i f y i n g t h e e f f e c t s of (A) data e r r o r s , i n the absence o f any o t h e r f a c t o r s ; (B) d a t a e r r o r s p l u s n a t u r a l m o r t a l i t y unknown; and (C) these p l u s a c o l l e c t i o n c f f a c t o r s c a u s i n g s t r u c t u r a l l a c k of f i t . The r e s u l t s p r o v i d e a range frcm o p t i m a l e s t i m a t i o n c o n d i t i o n s ( w i t h n a t u r a l m o r t a l i t y known, p e r f e c t w e i g h t i n g of the d a t a , and no l a c k of f i t ) t o mere r e a l i s t i c c o n d i t i o n s . With n a t u r a l m o r t a l i t y known, p e r f e c t w e i g h t i n g , and no s t r u c t u r a l l a c k of f i t , assessment r e l i a b i l i t y w i l l he a t a minimum, and the b i a s and v a r i a n c e c f parameter e s t i m a t e s s h o u l d r e p r e s e n t an approximate l o w e r bound. 86 4,2 The Case Study C o n t i n u e d . For the harp s e a l a n a l y s i s , some of t h e f a c t o r s ennumerated i n s e c t i o n 4 , 1 a r e p r o b a b l y u n i m p o r t a n t . There i s a r e l a t i v e l y l a r g e s e t of d a t a , and i t has a p p a r e n t l y been c o l l e c t e d over a wide range c f s t o c k s i z e s and e x p l o i t a t i o n r a t e s . Lack of f i t due to c h a n g i n g n a t u r a l m o r t a l i t y c o u l d be a problem, but more i m p o r t a n t a re d a t a e r r o r s , w e i g h t i n g e r r o r s , unknown n a t u r a l m o r t a l i t y , changing age s e l e c t i o n , and s t o c h a s t i c v a r i a t i o n i n r e c r u i t m e n t . B e f o r e these s o u r c e s of u n c e r t a i n t y are a s s e s s e d c o l l e c t i v e l y , twc more s p e c i f i c c a s e s w i l l be e x p l o r e d . Case A addresses a g u e s t i o n about d a t a e r r o r s : are d a t a e r r o r s of t h e magnitude c o r r e s p o n d i n g t o W e i g h t i n g #2 s u f f i c i e n t t o r e s u l t i n u n r e l i a b l e e s t i m a t e s ? By e l i m i n a t i n g a l l o t h e r s o u r c e s o f u n c e r t a i n t y i n t h e model, g u a r a n t e e i n g a p e r f e c t f i t and p e r f e c t w e i g h t i n g of t h e data with n a t u r a l m o r t a l i t y known e x a c t l y , the e f f e c t o f data e r r c r s can be t e s t e d i n d e p e n d e n t l y . I f t h e s e r e s u l t s a r e u n a c c e p t a b l e , the i n c l u s i o n of a d d i t i o n a l f a c t o r s w i l l be even more s c . The second. Case B, a s k s : does v a r i a t i o n i n the d a t a p l u s n a t u r a l m o r t a l i t y unknown l e a d t o u n r e l i a b l e p r e d i c t i o n s ? The o b j e c t i s t o f i n d out i f e s t i m a t i n g n a t u r a l m o r t a l i t y i s f e a s i b l e i n the f a c e c f r e a l i s t i c data e r r o r s , g i v e n t h a t a l l e t h e r c o n d i t i o n s a re o p t i m a l . I n t h e t h i r d s t u d y , Case C, a l l i m p o r t a n t f a c t o r s are a s s e s s e d c o l l e c t i v e l y . . T h e model used i n each experiment i s d e s c r i b e d n e x t . E s s e n t i a l l y the models r e p r e s e n t the f i n a l r e s u l t s from the a n a l y s i s i n c h a p t e r 3. I n each r e p l i c a t i o n i n each c a s e , d a t a were g e n e r a t e d f o r ages 0 t o 25 and f o r 25 y e a r s . R e c r u i t m e n t 87 was modelled by a Beverton-Holt s t o c k - r e c r u i t r e l a t i o n s h i p , as i n Eg. (3-1). The i n i t i a l s t a t e , age s e l e c t i o n f a c t o r s , n a t u r a l m o r t a l i t y , , and p> were a l l taken from the r e s u l t s presented i n s e c t i o n 3.5; the annual components c f e x p l o i t a t i o n correspond to those f o r 1952 to 1S76. The e s t i m a t o r was the same: the weighted l e a s t - s g u a r e s v e r s i o n modified to i n c l u d e a s t o c k -r e c r u i t curve and a pre-estimated age d i s t r i b u t i o n f o r year 1. A l l c o n s t r a i n t s are as i n Table 5 except « , p , and Nl+ which are c o n s t r a i n e d to [ 1 0 ~ 7 , 1 0 ~ 5 ] , [ 0 , 5 ] , and [ 1 0 6 ,10^ ], r e s p e c t i v e l y . The age d i s t r i b u t i o n f o r year 1 was assumed known without e r r o r . In a l l ca s e s , observed data were modelled as a lognormal random v a r i a b l e with mean e q u a l to the " t r u e " value (C) and c o e f f i c i e n t s of v a r i a t i o n (cv) as i n Weighting #2, i . e . : c i j ^ , e ~ N ( I n C j j - \ , V : l (4-.) where In Case A only the data e r r o r s were t e s t e d , while n a t u r a l m o r t a l i t y was assumed known without e r r o r . Weights were the i n v e r s e o f the data e r r o r v a r i a n c e s , asumed to be known e x a c t l y . Becruitment was modelled without s t o c h a s t i c v a r i a t i o n . Case B was the same as Case A, except n a t u r a l m o r t a l i t y was estimated and c o n s t r a i n e d to be w i t h i n [0.7,1.0]. In both A and B, the " t r u e " s t o c k h i s t o r y i s the same as the l e a s t - s g u a r e s r e c o n s t r u c t i o n f o r years 1952-1S76, shown i n F i g u r e 4. For case C, e r r o r s i n the weights were modelled using the approximation Eg. (2-9), but the cv were assumed to be known e x a c t l y . Lack cf f i t due to age s e l e c t i o n changing with time was 88 simulated ty making the age selection factors for ages 1 to 4 ncrnal randcm variables vith a c o e f f i c i e n t of variation of 10%. Age s e l e c t i o n i s believed to vary greatest i n immature animals, l i k e l y in a random manner, cr possibly following trends in the market value of the skins,. Recruitment i s mimicked by a lcgnormal random variable with a c o e f f i c i e n t of variation of 15%, a value intermediate between the high and low v a r i a b l i t y assumptions. In a l l r e p l i c a t i o n s the same sequence of random numbers for the age selection and recruitment random variables was used so that the "true" histcry of abundance remained constant. . The "true" histcry cf 1+ abundance i s almost exactly the same as i n Cases & and B and i s , therefore, not shown.. For each case the data generation and analysis process was replicated 25 times, then the results were summarized by computing the means and variances for each of the estimated parameters and the population forecast. These re s u l t s were compared to the "true" values..Deviations in mean estimates from the true values w i l l be presented as "biases", though with 25 r e p l i c a t i o n s the deviations may i n fact be due to random sampling e f f e c t s , rather than true biases. 4.3 Results and Discussion. The results are summarized i n Tables 9 to 12 , For Cases A and B they are very encouraging. The parameter estimates i n both cases have acceptably small biases and almost a l l c o e f f i c i e n t s of variation are less than 10%. Those that exceed 10% are, as 89 Table 1 0 . Mcnte Carlo results for alpha, except variatio n , are scaled in the biases are due for Case A, A l l values the c o e f f i c i e n t of by 10 6 . Slight errors tc rounding. 90 TOTAL ABUNDANCE IN YEAB 1: MEAN TRUE VALUE VALUE "BIAS" 8710778. 8654287. -143509. STD ERROR COEF VAE 528432.. .061 ANNUAL COMPONENTS Of FISHING MORTALITY: MEAN TEUE STD COEF YEAB VALUE VALUE "BIAS" EEBOB VAE 1 .271 .267 .004 ,0138 ,051 2 .209 .208 .001 .0104 .050 3 .232 ,230 .002 . 0106 .046 4 .281 .278 .003 .01 55 .055 5 .210 . 210 . 000 .0127 .060 6 .313 .310 . 003 .0150 .048 7 .427 ,424 . 003 .0211 .050 8 .373 .366 .007 .0224 .060 9 .440 ,43 9 .001 .0204 .046 10 .096 .097 -.001 .0050 .052 1 1 .389 ,386 .003 .0143 .037 1 2 .607 .605 .002 .0180 .030 13 ,.533 ,535 -.002 .0229 ,043 14 .344 .344 .000 .0165 .048 15 .547 .538 .009 .0234 .043 16 .511 .506 .005 .0229 .045 17 .356 . 352 .004 .0270 .076 18 .447 .443 .004 .0270 .060 19 .448 .447 .001 .0293 .066 20 .199 . 201 -.002 .0137 .069 21 .211 .207 . 004 .0120 .057 22 .303 .303 . 000 .0 234 .077 23 ,324 ,321 . 003 .0206 .064 24 .283 .283 .000 .0194 .069 25 .233 . 232 .001 .0187 .080 91 AGE SELECTION FACTORS: MEAN TEUE AGE VALUE VALUE 1 ,042 .043 2 .0 53 .052 3 .045 ,045 4 .044 .043 5 .043 ,044 6 .039 .039 7 .038 .037 8 .037 .037 9 .031 .031 1 0 .032 . 032 1 1 .027 . 027 12 .028 .028 13 ,025 .024 1 4 .025 .025 15 .027 .027 16 .023 .023 17 .020 .019 1 8 .019 .018 19 .015 .015 20 .014 .014 21 .008 ,008 22 .002 .002 23 .005 .005 24 • .006 .005 25 .002 . 002 STOCK-BECEUI1 PAEAMETEBS: MEAN IBUE VALUE VALUE ALPHA 1. 106 1.082 BETA 2.484 2.545 STD COEF "BIAS" EEBOB VAR -. 000 .0021 .050 .001 ,0034 .064 .000 .0018 .041 .001 .0016 .036 -.000 .0025 .057 .000 .0018 .045 .001 .0016 .042 .000 .0021 .057 .000 .0018 .058 .000 .00 19 .058 . 000 .0016 .058 .000 .0016 .058 .000 .0016 .065 .000 .0013 .054 .000 .0015 ,055 .000 .0012 .051 .000 .0013 .064 . 000 .0014 .075 .000 .0009 .058 . 000 .0010 .070 . 000 .0009 .114 . 000 .0002 .078 -.000 .0004 .078 .000 .0005 .094 -.000 .0001 .083 STD COEF "BIAS" ERROR VAR .025 .0743 .067 1.0619 .1996 .080 92 Table 11, Monte Carlo results f o r alpha, except va r i a t i o n , are sealed i n the biases are due for Case B. a l l values the c o e f f i c i e n t of by 106 . Slight errors to rounding. 93 TOTAL ABUNDANCE IN YEAE 1: MEAN TEUE STD COEF VALUE VALUE "BIAS" EEBOE VAE 8780185. 8854287. -74102. 722905. ,082 ANNUAL COMIONENTS CF FISHING MQETAIITY: MEAN TEUE STD COEF YEAB VALUE VALUE "BIAS" EBEOB VAE 1 .269 .267 .002 .02 54 .094 2 .208 ,208 .000 .0188 .090 3 .230 .230 .000 .0177 .077 4 .280 .278 .002 .0226 .081 • 5 .210 .210 -.000 .0168 .080 6 .312 .310 .002 .0181 ,058 7 .426 ,424 .002 .0 182 .043 8 .373 ,366 .007 ,02 37 ,064 S .439 .439 . 000 .0200 .046 1 0 .096 .097 -.001 .0044 .046 11 .389 .386 .003 .0140 .036 12 .607 ,60 5 . 002 .0172 ,028 13 .534 .535 -.001 .0221 .041 14 ,345 .344 .001 .0159 .046 15 .549 .538 .011 .0227 .041 16 .513 , 506 .007 .0230 .045 1 7 .358 .352 .006 .0272 .076 18 .450 .443 .007 .0313 .070 19 .452 .447 . 005 .0353 .078 20 .202 .201 . 001 .0166 .082 2 1 .214 ,207 .007 .0180 .084 22 .3 08 . 303 . 005 .0315 .102 23 .329 .321 .008 .0328 . 100 24 .288 . 283 .005 .0291 . 101 25 .238 .232 .006 .0313 . 132 r 94 AGE SELECTION FACTORS: MEAN TEUE STD COEF AGE VALUE VALUE "BIAS" ERROR VAE 1 .042 .043 -. 000 .0022 .051 2 .053 .052 .001 .0034 .063 3 .045 .045 .000 .0018 .041 4 .044 .043 .001 .0016 .037 .044 .044 -.000 .0025 .057 6 .039 , 039 .001 .0018 .046 7 .038 .037 .001 .0016 .042 8 .037 .037 . 000 .0021 .055 9 .031 .031 .000 .0018 .058 10 .032 . 032 . 000 .0019 .058 1 1 .027 .027 .001 .0016 .058 12 ,028 .028 . 000 .0017 ,059 13 .025 .024 .000 .0016 .066 1 4 .025 .025 .000 .0014 .057 1 5 .027 .027 .001 .0015 .057 1 6 .024 ,023 . 000 .0012 .053 1 7 .020 .019 . 000 .0013 .065 1 8 ,019 ,018 . 001 .0015 .078 1 9 .015 .015 .000 .0009 .061 20 .014 .014 .000 .0010 .072 2 1 .008 .008 . 000 .0009 .119 22 .002 . 002 .000 .0002 .079 23 .005 .005 -.000 .0004 .083 24 .006 .005 .000 .0006 .100 25 .002 .002 -.000 .0002 .091 STCCK-EECEUIT PABAMEIEES: MEAN TEUE VALUE VALUE ALPHA 1. 104 1. 082 BETA 2.479 2. 546 STD COEF "BIAS" ERROR VAR .023 .0998 .090 -0.0665 .2424 .098 SURVIVAL THROUGH KATURAL MORTALITY: MEAN TEUE STD COEF VALUE VALUE "BIAS" EEEOE VAE .900 .901 -,001 .0073 .008 95 Table 12. Monte Carle results for alpha, except variatio n , are scaled i n the biases are due for Case C, A l l values the c o e f f i c i e n t of by 10 . Slight errors to rounding. 96 TOTAL ABUNDANCE IN YEAR 1: MEAN TRUE STD COEF VALUE VALUE "BIAS" EBBOE VAR 8758028, 8854287, -96259. 1150673, .131 ANNUAL COMPONENTS OF IISHING MORTALITY: MEAN TRUE STD COEF YEAR VALUE VALUE "BIAS" ERBOR VAR 1 .266 .267 -.001 .0342 .128 2 ,201 . 208 -.007 .0290 .144 3 .226 .230 -.004 .0257 . 113 4 .283 . 278 .005 .0386 .136 5 .204 .210 -.006 .0241 . 118 6 .287 .310 -. 023 .0241 .084 7 .408 .424 -.016 .0282 .069 8 .376 , 366 .010 ,0327 .087 9 ,413 .439 -.026 .0226 .055 10 .091 .097 -.006 .0066 .073 1 1 .362 .386 -.024 .0185 .051 12 .593 .605 -.012 .02 30 .039 1 3 .545 .535 .010 .0314 .058 14 .324 .344 -. 020 .0178 .055 15 .554 .538 .016 .0194 .035 16 .496 .506 -.010 .0276 .056 17 .341 .352 -.011 .0268 .079 18 .432 .443 -.011 .0329 .076 19 .450 .447 .003 .0345 .077 20 .194 , 201 -. 007 .0175 ,091 21 .208 .207 .001 .0189 .091 22 .299 .303 ^.004 .0315 .105 23 .318 .321 -.003 ,0355 .111 24 .268 .283 -.015 .0334 .125 25 .221 .232 -.011 .0313 .142 97 AGE SELECTION FACTOES: MEAN TEUE AGE VALUE VALUE 1 .041 .043 2 .048 . 052 3 .044 .045 4 .043 ,04 3 c .043 .044 6 .039 .039 7 .038 .037 8 .036 .037 S .030 .031 1 0 .031 .032 11 .026 ,027 1 2 ,027 .028 13 .024 .024 14 .024 .025 1 5 .026 .027 1 6 .023 .023 1 7 .019 .019 1 8 .018 .018 1 9 .015 .015 20 .014 .014 2 1 .006 .008 22 .002 .002 23 .004 .005 24 .005 . 005 2 5 .001 .002 STOCK-EECEUIT PAEAMETEES: MEAN TEUE VALUE VALUE ALPHA 1.367 1.082 BETA 1,708 2. 546 STD COEF "BIAS" EEBOE VAR -.001 .0028 .068 -.004 .0029 .060 -.001 .0028 .065 . 000 .0022 .052 -.000 .0025 .058 .000 .0023 .059 .001 .0021 .057 - . 000 .0025 .070 -.000 .0022 .073 -.001 .0021 .067 -.000 .0017 .064 - . 001 .0021 .077 -.001 .0017 .071 -.001 .0018 .073 -.001 .0017 .066 -.000 .0016 .069 -.000 .0014 .074 -.000 ,0014 .078 -.000 .0010 .068 - . 000 .0011 ,083 -,001 .0008 . 126 - . 000 .0002 .102 -.001 .0006 . 132 -.001 .0006 .127 - . 000 .0002 .115 STD COEF "BIAS" EEEOB VAR . 285 .2029 .148 I.8377 .43 26 . 253 SURVIVAL THROUGH KATURAL MORTALITY: MEAN TRUE STD COEF VALUE VALUE "BIAS" ERROR VAR .900 .901 -,001 .0102 .011 9 8 Table 13. Forecast errors for Cases A-C. fill values are given as a percentage of the "true" value. CASE A CASE B CASE C COEF COEF COEF AGE " B I A S " VAR AGE " B I A S " VAR AGE " B I A S " VAR 0 - 3 . 9 6 . 0 - 5 . 0 9 . 0 1 1 . 9 1 2 . 1 - 0 . 1 8 . 1 - 1 . 6 1 3 . 1 1 7 . 0 17 . 2 - 0 . 1 8 . 2 - 1 . 7 14. 2 2 2 . 6 1 9 . 3 - 0 . 5 8 . 3 - 2 . 1 1 5 . 3 - 1 . 7 1 6 . 4 - 0 . 3 8 . 4 - 1 . 7 1 4 . 4 0 . 8 1 6 . 5 - 0 . 9 6 . 5 - 2 . 2 1 2 . 5 1 4 . 4 17. 6 - 0 . 2 6 . 6 - 1 . 5 1 2 . 6 - 1 . 4 1 4 . 7 - 0 . 4 1 0 . 7 - 1 . 9 1 6 . 7 7 . 6 1 9 . 8 - 1 . 0 9 . 8 - 2 . 4 1 6 . 8 3 . 1 1 9 . 9 - 1 . 2 8 . 9 - 2 . 5 1 3 . 9 - 1 9 . 7 1 3 . 10 - 1 . 7 9 . 10 - 3 . 1 1 4 . 10 - 7 . 4 1 6 . 11 - 2 . 7 9 . 11 - 4 . 2 1 3 . 11 3 . 5 1 6 . 12 - 0 . 9 7 . 12 - 2 . 2 1 3 . 12 - 3 1 . 4 1 1 . 13 - 0 . 3 1 0 . 13 - 1 . 6 1 5 . 13 1 0 . 9 2 0 . 14 - 1 . 3 9 . 14 - 2 . 6 1 4 . 14 - 2 . 3 17 . 15 - 1 . 5 7 . 15 - 2 . 8 1 2 . 15 - 2 5 . 3 1 2 . 16 - 1 . 1 6 . 16 - 2 . 2 1 3 . 16 2 1 . 9 2 0 . 17 - 1 . 2 8 . 17 - 2 . 5 1 3 . 17 - 9 . 5 1 5 . 18 - 2 . 3 8 . 18 - 3 . 6 1 3 . 18 3 . 8 17 . 19 - 1 . 7 8 . 19 - 2 . 8 1 4 . 19 - 7 . 6 1 5 . 20 - 1 . 8 7 . 20 - 3 . 1 1 3 . 20 - 1 3 . 7 1 5 . 21 - 1 . 5 7 . 21 - 2 . 8 1 4 . 21 - 9 . 9 1 6 . 22 - 1 . 8 7 . 22 - 3 . 2 1 3 . 22 1 7 . 5 1 9 . 23 - 1 . 7 7 . 23 - 3 . 0 1 4 . 23 - 5 . 4 17. 24 - 1 . 7 7 . 24 - 3 . 0 1 4 . 24 - 0 . 4 1 8 . 25 - 2 . 0 7 . 25 - 3 . 4 14. 25 - 3 . 0 1 8 . 1+ - 0 . 8 7 . 1+ - 2 . 2 1 3 . 1+ 1 . 3 1 6 . 100 would te e x p e c t e d , those w i t h t h e l e a s t s u p p o r t i n g d a t a : th e a n n u a l components o f e x p l o i t a t i o n f o r l a t e r y e a r s and the age s e l e c t i o n f a c t o r s f o r c i d e r ages. For the p o p u l a t i o n f o r e c a s t s , h oth c a s e s show a c o n s i s t e n t , hut s m a l l , n e g a t i v e b i a s . Most remarkable i s t h a t the v a r i a t i o n i n e s t i m a t e s of n a t u r a l m o r t a l i t y i s l e s s t h a n 1%. However, the e f f e c t of not knowing n a t u r a l m o r t a l i t y i s t o a lmost double v a r i a t i o n i n t h e p r e d i c t i o n o f 1+ abundance, from 7% to 13%. I n o t h e r words, we have an a p p a r e n t paradcx: even though n a t u r a l m o r t a l i t y i s e s t i m a t e d almost w i t h c e r t a i n t y , i n c l u d i n g i t as an unknown l e a d s t o t w i c e the u n c e r t a i n t y about p r e d i c t i o n s . A p p a r e n t l y , even very s m a l l e r r o r s i n n a t u r a l m o r t a l i t y have a l a r g e e f f e c t on the p r e d i c t i o n , when the p r e d i c t i o n i s o t h e r w i s e good.. I n c h a p t e r 5, i t w i l l be seen a g a i n t h a t not knowing Sn r e s u l t s i n an i n c r e a s e i n the r e l a t i v e v a r i a n c e of the 1+ p r e d i c t i o n (over the o p t i m a l case) c f about 5% f o r low q u a n t i t y , h i g h c o n t r a s t d a t a s e t s . Eut t h e r e the i n c r e a s e i s from 43% t c 48%. F o r Case C, the mcst r e a l i s t i c t e s t , t h e r e s u l t s a r e e n c o u r a g i n g but must be i n t e r p r e t e d w i t h c a u t i o n . Except f o r ov and (b , t h e v a r i a n c e s of the parameter e s t i m a t e s are o n l y s l i g h t l y l a r g e r than i n Case B. A g a i n , th e c o e f f i c i e n t of v a r i a t i o n f o r n a t u r a l m o r t a l i t y i s j u s t 1%. But f o r the s t o c k -r e c r u i t p a r a m e t e r s , i n c l u d i n g more s o u r c e s of u n c e r t a i n t y l e a d s t o a l a r g e decrease i n r e l i a b l i t y . For p , t h e p r o d u c t i v i t y parameter i n the B e v e r t c n - H o l t c u r v e , 95% c o n f i d e n c e l i m i t s are ± 50% o f the e s t i m a t e , Presumably a f u r t h e r i n c r e a s e i n s t o c h a s t i c v a r i a t i o n t o l e v e l s c h a r a c t e r i s t i c of most f i s h p o p u l a t i o n s would i n c r e a s e u n c e r t a i n t y i n a, and p to the p o i n t 101 where estimating stcck- r e c r u i t parameters i n the optimization would be guite useless. With regard to the forecasts, Case C exhibits some g u a l i t a t i v e l y d i f f e r e n t behavior. Whereas the bias for the 1+ population, -1.351, i s even less than i n Case B, the biases in the age structure are both much larger and more inconsistent, ranging frcm -31% to +23%. This r e s u l t has important bearing cn the r e l a t i o n between forecasts of age structure and forecasts of t o t a l abundance. While we cannot place much confidence on estimates cf age structure, the estimate for the t o t a l population i s probably unbiased. This apparently strange r e s u l t can be explained guite simply:the variance of a random variable i s related tc the variance of the mean cf n of the same, i n the proportion n,. On the whole, these r e s u l t s are what we might have expected. By f i s h e r i e s standards, a value for the c o e f f i c i e n t of variation cf a forecast i n the range cf 7% to 16% i s r e l a t i v e l y good, even suspiciously small. From a large data set with high contrast and moderate errors, we cught to get r e l a t i v e l y good re s u l t s . Frcm a small data set, with peer contrast and bad errors, the results should be r e l a t i v e l y dismal. But what i s a suitable value corresponding t c "dismal"? A r e l a t i v e variance of 16% means an approximate S5% confidence i n t e r v a l of ± 32%. I f a forecast with a confidence i n t e r v a l greater than ± 50% i s bordering on i n u t i l i t y , then a r e l a t i v e variance i s dismal i f i t i s greater than 25%. At any rate, the ho r r i f y i n g implication, though, i n essence, the wisdom of old f i s h e r i e s biometricians, i s that the forecasts based on most data sets, insofar as they 102 are s m a l l e r , c r w i t h poorer c o n t r a s t or worse e r r o r s , t h a n t h a t of harp s e a l s , w i l l be e x c e e d i n g l y d i s m a l . One wishes t h a t i t c o u l d be o t h e r w i s e . N e v e r t h e l e s s , improvement i s p o s s i b l e f r c m t h r e e s o u r c e s . F i r s t , w i t h the passage c f t i m e the g u a n t i t y of data w i l l i n c r e a s e . , The o n l y a a n i p u l a b l e element i n t h i s p r o c e s s i s t h e a n a l y s t ' s p a t i e n c e . Second, t h e d a t a e r r o r s can be reduced w i t h b i g g e r samples and b e t t e r s a m p l i n g d e s i g n s . The r e s o u r c e ' s management depends here on the management's r e s o u r c e s . T h i r d , more c o n t r a s t can be i n t r o d u c e d by m a n i p u l a t i n g h a r v e s t r a t e s . D e s i g n i n g h a r v e s t p o l i c i e s w i t h a view t o p r o d u c i n g i n f o r m a t i v e c a t c h d a t a , as an a i d i n o p t i m i z i n g y i e l d , i s termed a c t i v e a d a p t i v e c o n t r o l (see w a i t e r s 1975; H a l t e r s and H i l b o r n 1976, 1S78; S i l v e r t 1S78; Smith 1S79). How do t h e s e r e s u l t s bear on the c o n t r o v e r s y about the r e l i a b i l i t y of the l e a s t - s g u a r e s harp s e a l p r e d i c t i o n s v e r s u s t h o s e frcm s e q u e n t i a l p o p u l a t i o n a n a l y s i s ? On t h e one hand we have not i n c l u d e d t i e p o s s i b i l i t y o f t r e n d s i n n a t u r a l m o r t a l i t y and have assumed t h a t Weighting #2 a c c u r a t e l y r e f l e c t s e r r o r s i n the c a t c h - a t - a g e d a t a . A l s o , t h e v a r i a b l i t y used i n m o d e l l i n g r e c r u i t m e n t and changing age s e l e c t i v i t y c o u l d be u n d e r e s t i m a t e d . I f the r e a l p o p u l a t i o n has not v a r i e d i n abundance as much as the r e s u l t s from the a n a l y s i s i n d i c a t e , the i n f o r m a t i o n c o n t e n t o f the data w i l l be overestimated,. A l l these c o n s i d e r a t i o n s l e a d to a d e c rease i n r e l i a b i l i t y o f the p r e d i c t i o n c f t h e 1+ p o p u l a t i o n , On t h e o t h e r hand, n a t u r a l m o r t a l i t y was assumed unknown i n Case . C, even though th e e s t i m a t e i s i d e n t i c a l to L e t t e t a l ' s . I f we assume Sn i s known, 103 t h e r e l i a b i l i t y i n t h e 1+ p r e d i c t i o n i s i n c r e a s e d s u b s t a n t i a l l y . These c o m p l i c a t i o n s are i m p o s s i b l e t o q u a n t i f y . I g n o r i n g the d e f i c i e n c i e s i n the a n a l y s i s d i s c u s s e d i n s e c t i o n 3.7, the r e s u l t s here i n d i c a t e the p r e s e n t p o p u l a t i o n i s p r o b a b l y w i t h i n 1.6 m i l l i o n t o 3.0 m i l l i o n and t h a t i t i s most l i k e l y 2.283 m i l l i o n . S i m i l a r n u m e r i c a l e x p e r i m e n t s have not been performed w i t h s e q u e n t i a l p o p u l a t i o n a n a l y s i s . L e t t e t a l ' s method i n v c l v e s r e c o n s t r u c t i n g t e r m i n a l e x p l o i t a t i o n r a t e s from f e r t i l i t y d a ta sampled s i n c e t h e e a r l y 1950s and pup p r o d u c t i o n e s t i m a t e d from s u r v i v a l i n d i c e s . I n v e s t i g a t i o n i n t o t h e e r r o r s i n t h i s p r o c e s s , p l u s the assumption of no e r r o r s i n the c a t c h -at-age d a t a , wculd be very i n f o r m a t i v e . Mohn e t a l ' s f o r e c a s t s are made u s i n g t h e r e s u l t s of s e q u e n t i a l p o p u l a t i o n a n a l y s i s combined w i t h a more c o m p l i c a t e d mcdel c f s e a l and s e a l i n g dynamics. An e x a m i n a t i o n of the model r e v e a l s t h a t n a t u r a l m o r t a l i t y , the d e r i v a t i o n of t h e d e n s i t y -dependent pregnancy r a t e and mean-age-at-whelping f u n c t i o n s , and, f i n a l l y , a d j u s t m e n t s i n the r e s u l t i n g s t a t e r e c o n s t r u c t i o n a r e a l l c r i t i c a l l y dependent on pup p r o d u c t i o n s from the s u r v i v a l i n d e x methcd, though r e l a t e d i n such a way as t o guarantee o v e r a l l c o n s i s t e n c y . . T h e p o i n t i s t h a t u n c e r t a i n t i e s i n the model t c a g r e a t e x t e n t r e f l e c t u n c e r t a i n t i e s i n h e r e n t i n t h e s u r v i v a l i n d e x method. Y e t , a l t h o u g h the t e c h n i q u e i s much d i s c u s s e d ( e . g . . B i c k e r MS 1971, S ergeant MS 1971), t h e r e e x i s t s nc a n a l y s i s c f the a c c u r a c y of the method, nor any c r i t i c a l e v a l u a t i o n c f i t s u n d e r l y i n g a s s u m p t i o n s . For example, i t assumes t h a t pup p r o d u c t i o n i s c o n s t a n t frcm one year t o the n e x t and then c o n c l u d e s i t i s c h a n g i n g i n the l o n g r u n . I t a l s o 104 assumes that f i s h i n g intensity i s constant over time, although t h i s i s not the case comparing the periods before and aft e r the introduction of e f f e c t i v e quotas i n 1972. The implications of these contradictions to the accuracy of the assessment are not e n t i r e l y obvious. Ike situation with respect to Mohn et al's predictions can best be described by saying that i t i s the uncertainties themselves that are uncertain. 105 Chapter 5. , NUMERICAL EXPERIMENTS WITH LEAST-SOUABES CATCH-AT-AGE ANALYSIS: A CLUEEOID FISHEBY I n t h i s c h a p t e r the s e r i e s of n u m e r i c a l e x p e r i m e n t s i n i t i a t e d i n c h a p t e r 4 are c o n c l u d e d . Four more c a s e s a r e examined, w i t h data g e nerated f o r each by a s i m u l a t i o n model of a h y p o t h e t i c a l c l u p e o i d s t o c k . The f i r s t two of t h e s e , Cases D and E, r e p r e s e n t a s i n g l e p o i n t i n " i n f o r m a t i o n - s p a c e " , i d e n t i c a l t c t h a t examined examined i n Cases A-C, except th e g u a n t i t y of data i s reduced by about 75%. E r r o r s a r e , a g a i n , moderate and c c n t r a s t i s h i g h . I n t h e f i n a l two c a s e s , F and G, a n o t h e r p o i n t i s i n v e s t i g a t e d , s i m i l a r t o D and E i n data g u a n t i t y and e r r o r s , but w i t h very lew c o n t r a s t . The c o n d i t i o n s f o r Cases D and E are a nalagous t o Cases A and B. Those f o r Case D are o p t i m a l : n a t u r a l m o r t a l i t y i s known e x a c t l y ; the weights are t h e i n v e r s e o f the data e r r o r v a r i a n c e s , a l s c known e x a c t l y ; and s t r u c t u r a l f i t i s p e r f e c t . I n Case E the c o n d i t i o n s are o p t i m a l , e x cept Sn i s unknown. Cases F and G, w i t h low c o n t r a s t , a l s o c o r r e s p o n d t o A and B: the c o n d i t i o n s f c r F a r e o p t i m a l , whereas f o r G, Sn i s unknown. A comparison of the r e s u l t s from Cases A, D, and F w i l l i n d i c a t e how the l ower bound on b i a s and v a r i a n c e changes from a h i g h g u a n t i t y , h i g h c c n t r a s t s t a t e t c a low g u a n t i t y , h i g h c o n t r a s t s t a t e to a low g u a n t i t y , low c o n t r a s t state..We have seen how 106 n a t u r a l m o r t a l i t y unknown i n c r e a s e s the r e l a t i v e v a r i a n c e o f the 1* p r e d i c t e d abundance c f Case A t y about 691 ( i n Case B ) . S i m i l a r c o m p a r i s o n s , E w i t h D and G w i t h F, ought t o f u r t h e r our u n d e r s t a n d i n g of the e f f e c t s o f n a t u r a l m o r t a l i t y unknown on p r e d i c t i o n a c c u r a c y . 5.1 Four Cases c f lew Data Q u a n t i t y . C l u p e c i d s p e c i e s , such as s a r d i n e s o r h e r r i n g , are h i g h l y f e c u n d , e x p e r i e n c e e r r a t i c r e c r u i t m e n t , and s u f f e r r e l a t i v e l y h i g h r a t e s cf n a t u r a l m o r t a l i t y . These f e a t u r e s a r e embodied i n t h e data s i m u l a t i o n model o u t l i n e d below. The model s t o c k i s comprised of 16 age groups, ages 0-15, though f i s h o l d e r than about f i v e y e a r s a r e r a r e under heavy e x p l o i t a t i o n . Observed. The o n l y elements d i f f e r i n g between the h i g h c o n r a s t and lew c o n t r a s t c a s e s a r e the a n n u a l components of e x p l o i t a t i o n . The t r u e c a t c h - a t - a g e and p o p u l a t i c n - a t - a g e a r e g i v e n by Eg.(2-18), e x c e p t r e c r u i t m e n t i s s t r c n g l y s t o c h a s t i c . The observed c a t c h e s - a t - a g e are l c g n o r m a l randem v a r i a b l e s w i t h means e g u a l t o the t r u e c a t c h and r e l a t i v e v a r i a n c e s (cv) as shown i n Table 14. To c o m p l e t e l y s p e c i f y how the t r u e c a t c h e s are g e n e r a t e d , we must g u a n t i f y : the i n i t i a l s t a t e , or the numbers-at-age i n year 1; the r a t e c f s u r v i v a l t h rough n a t u r a l m o r t a l i t y ; t h e age s e l e c t i o n f a c t o r s and the a n n u a l components of e x p l o i t a t i o n ; p l u s t h e parameters of the s t o c k - r e c r u i t d i s t r i b u t i o n . E e c r u i t m e n t i s l o g n o r m a l , w i t h mean r e c r u i t m e n t , as a f u n c t i o n of the b r e e d i n g s t o c k , g i v e n by the B i c k e r s t o c k -107 Table 14. Data error c o e f f i c i e n t s of variation for Cases D?G. 108 AGE 0 1 2 - 8 9 - 1 1 1 2 - 1 3 14 15 COEF VAR . 4 . 3 . 2 5 . 2 . 2 5 . 3 . . 4 109 r e c r u i t model. The b r e e d i n g s t o c k i s d e f i n e d as t h e 2+ p o p u l a t i o n . I n e t h e r words, we have: -pp. . h(P^ = « P , - e where c* and p> are parameters of the B i c k e r f u n c t i o n and O7" i s t h e v a r i a n c e parameter. (Here N-,0 i s t o be thought of as N;+l , /Sn.) a and a r e f i x e d i n terms of the u n f i s h e d e g u i l i b r i u m p o p u l a t i o n (Noo) and the r e c r u i t m e n t r a t e a t h a l f the u n f i s h e d e g u i l i b r i u m (A), u s i n g e g u a t i o n s s i m i l a r t o Eg. (A—11) and E g . ( A - 1 2 ) , but d e r i v e d w i t h t h e b r e e d i n g s t o c k d e f i n e d as the 2* p o p u l a t i o n , Nro i s s e t a t 10 6 and r\ a t 1.0, g i v i n g v a l u e s - 6 f o r cx and £> c f 1.358 and 0.914 x 10 , r e s p e c t i v e l y . With ^= 1.0, the r e p r o d u c t i v e p o t e n t i a l c f t h e s t o c k i s h i g h . The 2+ p o p u l a t i o n i s c a p a b l e of more than d u p l i c a t i n g i t s numbers a t l e v e l s l e s s than about Nco/2. The r e l a t i v e v a r i a n c e c f r e c r u i t m e n t i s 0.4 (or O* = 0. 385). T h i s i m p l i e s t h a t , g i v e n the b r e e d i n g s t o c k (P-, ) , t h e number of r e c r u i t s i n a p a r t i c u l a r year (N;Q ) w i l l u s u a l l y f a l l w i t h i n ± 80% o f t h e mean f o r t h a t b r e e d i n g s t c c k (h (P-, ) ) . T h i s mimics th e s t r o n g i n f l u e n c e of e n v i r o n m e n t a l v a r i a b i l t y . The same sequence of random e f f e c t s f o r r e c r u i t m e n t was used i n each r e p l i c a t i o n , so t h a t t h e " t r u e " h i s t o r y c f t h e s t c c k remained c o n s t a n t . The s u r v i v a l r a t e through n a t u r a l m o r t a l i t y i s low, 0.67 (or M = - l n Sn = 0.4), and i s assumed c o n s t a n t over ages and y e a r s . 1 1 0 The numbers-at-age i n year 1 r e p r e s e n t " e q u i l i b r i u m " u n f i s h e d c o n d i t i o n s . At e q u i l i b r i u m , the numbers one year would be the same as the n e x t , i n the t o t a l p o p u l a t i o n and f o r each age group, i f s t c c h a s t i c v a r i a t i o n i n r e c r u i t m e n t were e x c l u d e d . Then r e c r u i t m e n t would e x a c t l y b a l a n c e l o s s e s due t o n a t u r a l m o r t a l i t y and the t r u n c a t i o n o f the model a t age 1 5 . H a r v e s t i n g b e g i n s i n year 1 and c o n t i n u e s f o r 1 0 y e a r s . A c o h o r t i s p a r t i a l l y e x p l o i t e d by the f i s h e r y a t age 1 and i s f u l l y e x p l o i t e d by age 3 , . I n t h e e s t i m a t i o n , the age s e l e c t i o n f a c t o r f o r age 1 5 i s f i x e d a t 1 . 0 . C a t c h d a t a a re g e n e r a t e d f o r a l l 1 6 age groups, g i v i n g 1 6 0 o b s e r v a t i o n s , compared w i t h 6 5 0 f o r Cases A-C. In Cases D and E, w i t h h i g h c o n t r a s t , the a n n u a l components of e x p l o i t a t i o n ranged from 0 . 1 t o 0 . 9 , c a u s i n g t h e 1+ abundance t o d e c l i n e frcm t o about i N w . I n Cases F and G , w i t h low c c n t r a s t , they remained c o n s t a n t a t 0 . 1 . The 1+ p o p u l a t i o n d i d j not drop much below ^ N o o . Beth " t r u e " h i s t o r i e s of 1+ abundance are shown i n F i g u r e 7 , The e s t i m a t e s were c o n s t r a i n e d t c the i n t e r v a l s g i v e n i n Tab l e 1 5 . .Annual r e c r u i t m e n t s were e s t i m a t e d as parameters. The s t a t e r e c o n s t r u c t i o n was the n used t o e s t i m a t e a and (b o f t h e s t o c k - r e c r u i t d i s t r i b u t i o n v i a a l i n e a r r e g r e s s i o n of l n ( N ; 0 /P(- ) on E ; , i . e . , t h e B i c k e r model l o g a r i t h m i c a l l y t r a n s f o r m e d . An e s t i m a t e c f 0 - v was then c a l c u l a t e d from Eg. ( 2 - 2 0 ) w i t h 1 = 1 0 , t o = 0 , and x = 2 . I n each c a s e , 2 5 r e p l i c a t i o n s were made and the r e s u l t i n g parameter e s t i m a t e s were summarised as b e f o r e . The " t r u e " v a l u e s of the numbers-at-age i n year 1 , t h e age s e l e c t i o n f a c t o r s , the 111 F i g u r e 7 . "True" h i s t o r i e s of abundance f o r Monte C a r l o Cases E-G. 113 Table 15. Least-sguares catch-at-age analysis parameter constraints for Cases D-G., A l l numbers-at-age i n year 1: [ 1, 750,000 ] A l l annual ccmponents of exploitation: [ .00001, 1.0 ] A l l age selection factors: [ .00001, 1.0 ] A l l annual recruitments: [ 1, 750,000 ] Rate of s u r v i v a l through natural mortality [ 0.5, 0.9 ] 115 annual components o f e x p l o i t a t i o n , and r e c r u i t m e n t a re g i v e n w i t h t h e Mccte C a r l o r e s u l t s i n T a b l e s 16-20. 5.2 B e s u l t s and D i s c u s s i o n . T a b l e 16 shows the r e s u l t s f o r the low g u a n t i t y , h i g h c o n t r a s t case under o p t i m a l e s t i m a t i o n c o n d i t i o n s . I t e x h i b i t s some p r o p e r t i e s common t o a l l f o u r e a s e l , p r o p e r t i e s due bo t h to the e s t i m a t o r , and to the r e d u c t i o n i n d a t a . F i r s t , I w i l l d i s c u s s t h e s e cemmon f e a t u r e s of t h e r e s u l t s , t h e n some p a r t i c u l a r s . I n T a b l e 16, t h e c o e f f i c i e n t s of v a r i a t i o n of t h e i n i t i a l abundance parameters show c l e a r l y hew a decrease i n s u p p o r t i n g d a t a d e c r e a s e s t h e r e l i a b i l i t y c f the e s t i m a t e . Those f o r youngest ages are s m a l l e s t , t h o s e f o r c i d e r ages a re l a r g e r , w i t h t h a t f o r age 15 bei n g enormous compared t o t h e o t h e r s . The v a r i a n c e o f N. . i s l a r g e because the e s t i m a t e depends e n l y cn the o b s e r v a t i o n c f C, l 5 and the e s t i m a t e o f the e x p l o i t a t i o n r a t e (y ( a < s ) . I f t h e e s t i m a t e s of y( o r a ) 5 are poor, or the e r r o r i n o b s e r v i n g CVi,5 i s l a r g e , the e s t i m a t e o f N, 1 5 w i l l be p c o r . S i n c e t h e r e l a t i v e v a r i a n c e c f y( i s s m a l l and a,3 i s f i x e d , the l a r g e v a r i a n c e f o r N ) i ( 5 must be due t o the l a r g e o b s e r v a t i o n e r r o r s . When the e s t i m a t e of an abundance parameter depends on a s i n g l e c a t c h - a t - a g e datum, the c o r r e s p o n d i n g r e s i d u a l w i l l be z e r o , u n l e s s t h e parameter i s bounded by a c o n s t r a i n t . The e s t i m a t e s f o r the a n n u a l components o f f i s h i n g 116 Table 16. Monte Carlo results for Case D. A l l values for beta, except the ^ c o e f f i c i e n t of variation, are scaled by 10 . Slight errors in the biases are due to rounding. 117 ABUNDANCE IN YEAB 1: MEAN TEUE STD COEF AGE VALUE VALUE "BIAS" EBEOB VAR 1 3 2 9 5 7 0 . 3 3 0 5 0 0 , - 9 3 0 . 2 1 4 6 0 . . 0 6 5 2 2 1 2 6 4 0 . 22154 1. - 8 9 0 1 . 1 7 0 0 6 . . 0 8 0 3 14.6326, 1 4 8 5 0 3 . - 2 1 7 7 . 1 1 0 6 8 . . 0 7 6 a 9 7 9 8 8 . . 9 9 5 4 5 . - 1 5 5 7 . 5 6 9 5 . . 0 5 8 c 6 6 6 7 0 . 6 6 7 2 7 , - 5 7 . 3 8 2 8 . . 0 5 7 6 4 4 5 4 1 . 4 4 7 2 8 . - 1 8 7 . 3 8 5 3 , . 0 8 7 7 2 9 6 8 2 . 2 9 9 8 2 , - 3 0 0 . 23 8 0 , . 0 8 0 8 2 0 3 6 7 . 2 0 0 9 8 . 2 6 9 . 1 6 7 1 . . 0 8 2 9 13115. 13472. - 3 5 7 . 1 2 2 0 . . 0 9 3 10 9 0 7 6 . 9 0 3 0 . 4 6 . 1 0 2 0 . . . 1 1 2 1 1 6 1 2 9 . 6 0 5 3 , 7 6 . 7 6 6 . . 1 2 5 12 4 0 2 8 . 4 0 5 8 . - 3 0 . . 5 8 3 . . 145 13 2 8 3 6 . 2 7 2 0 . 116. 6 5 6 , . 2 3 1 14 1 7 6 4 . 1 8 2 3 . - 5 9 . 3 1 9 . . 1 8 1 15 1 3 3 3 . 1 2 2 2 . 111. 4 7 3 . , 3 5 5 ANNUAL COMPONENTS OF FISHING ! MGETALITY MEAN TEUE STD COEF YEAB VALUE VALUE "BIAS" ERROR VAE 1 . 1 0 7 . 100 . 0 0 7 . 0 0 6 0 . 0 5 6 2 . 2 6 2 , 2 5 0 . 0 1 2 . 0 1 4 1 . 0 5 4 3 . 5 1 7 . 5 0 0 . 0 1 7 . 0 2 2 0 . 0 4 3 4 , 5 1 6 , 5 0 0 . 0 1 6 . 0 2 3 9 . 0 4 6 5 . 8 3 6 . 8 0 0 . 0 3 6 . 0 3 6 9 . 0 4 4 6 . 9 38 . 9 0 0 . 0 3 8 . 0 3 1 6 . 0 3 4 7 . 9 3 8 . 9 0 0 . 0 3 8 . 0 3 4 1 . 0 3 6 8 . 9 3 1 , 9 0 0 . 0 3 1 . 0 4 2 1 . 0 4 5 9 . 1 0 9 . 100 . 009 . 0 3 8 4 . 3 5 4 10 . 1 0 9 . 100 . 009 . 0 4 3 0 , 3 9 5 118 AGE SELECTION FACTORS: MEAN TEUE AGE VALUE VALUE 0 .000 .000 1 .245 . 250 2 .721 .750 3 .957 1.000 4 .964 1.000 5 .957 1,000 6 .962 1.000 7 .957 1, 000 8 .958 1.000 9 .960 r.000 10 .959 1.000 11 .961 1,000 1 2 .959 1.000 1 3 = .963 1.000 1 4 .S51 1.000 iNNOAL RECRUITMENT: MEAN TEUE YEAB VALUE VALUE 1 422717. 408800. 2 318743. 319807. 3 378108. 376651. 4 308122.. 305835. 5 449874. 442427. 6 231065. 226550. 7 295564. 294356. 8 159433. . 157440, 9 177756. 181090. 10 121872. 101033. STOC^-BECEUIT DISTRIBUTION MEAN TEUE VALUE VALUE ALEHA .794 1. 358 BETA .621' .914 SIGMA .376 .385 STD COEF "BIAS" EBEOR VAR ^.000 .0000 . 179 -.005 ,0258 , 105 -.029 .0398 .055 -. 043 .0343 .036 -.036 ,0356 ,037 -.043 .0330 .034 -. 038 ,0325 .034 -.043 .0379 .040 -.042 .0317 .033 -.040 ,0352 .037 -.041 .03 07 .032 -. 039 .0337 ,035 -. 041 .0368 .038 -. 037 .0375 .039 -. 049 .0349 .037 STD COEF "BIAS" EEROR VAR 13917. 33436. .079 -1064, 28674. .090 1457. 43033. , .114 2287. 47113. .153 7448. 58069. . ,129 4515. 45160. . .195 1209. 69068, .234 1993. 54695. .343 -3334. 72680. .409 20839. 88760. . .728 'AR AMETEBS: STD COEF "BIAS" ERROR VAE I, 5637 .2375 .299 0. 293 .3999 .644 1.0091 .1047 .278 119 T a b l e 17. Monte C a r l o r e s u l t s f o r b e t a , e x c e p t v a r i a t i o n , a r e s c a l e d i n the b i a s e s are due f o r Case E . . A l l v a l u e s the c o e f f i c i e n t of by 10 . S l i g h t e r r o r s t o r o u n d i n g . 120 AEUNEANCE IN YEAE 1: MEAN TBUE STD COEF AGE VALUE VALUE "BIAS" EBBOfi VAB 1 404084. 330500. 73584. 144427. .357 2 255742. . 22154 1. 34201. 85139. .333 3 174100. 148503. 25597. 54525. .313 4 1 16833. . 99545. 17288. 36431. ,312 5 79196. 66727. 12469. 24033. .303 6 52601. 44728. 7873. 15848. .301 7 35096. 29982. 5114, 10246. .292 8 24239. 20098. . 4141. 7731. .319 9 15690. 13472. 2218. 5329. ,340 10 10838. . 9030. 1808. 3668.. .338 11 7267. 605 3, . 1214. 2392. .329 12 4744. 4058.. 686. 1461. .308 13 3360, 2720, 640. 1314. .391 14 2133. 1823. 310. 832. .390 15 1581. 1222. 359. 750. .475 ANNUAL COMPONENTS OE FISHING MOETA1ITY MEAN TBUE STD COEF YEAB VALUE VALUE "BIAS" EEROB VAB 1 .101 . 100 .001 .0373 .369 2 .248 .250 -.002 .0618 .249 3 .497 .500 -.003 .0651 . 131 4 .502 ,500 .002 .0452 .090 5 .826 .800 .026 .0477 .058 6 .934 . 900 ,034 .0353 .038 7 .934 .900 .034 .0375 .040 8 .929 .900 . 029 .0460 .050 9 .110 .100 .010 .0397 .361 10 .118 . 100 .018 .0573 .485 121 AGE SELECTION FACTORS: MEAN TEUE STD COEF AGE VALUE VALUE "EIAS" ERBOfi VAB 0 .000 .000 -.000 .0000 .279 1 .237 ,250 -.013 .0322 .136 2 .710 .750 -.040 .0564 .079 3 .954 1.000 -. 046 .0374 .039 4 . 962 1.000 -.038 .0378 .039 5 .955 1.000 045 .0353 .037 6 .960 1.000 -.040 .0351 .037 7 .955 1.000 -.045 .04 06 .042 8 .956 1.000 -.044 .0340 ,036 9 .957 1.000 -.043 .0372 .039 1 0 .956 1.000 -.044 .0338 .035 1 1 .959 1.000 -.041 .0363 .038 1 2 = . S56 1.000 -.044 .0400 .042 1 3 .961 1.000 -.039 .0401 .042 1 4 .949 1.000 -.051 .0355 .037 ANNUAL EECEUITMENT: MEAN TBUE STD COEF YEAB VALUE VALUE "BIAS" EBBOB VAB 1 543478. 408800. 134678. 222330. .409 2 387781. 31S807. 67974. 134655. .347 3 446736. 376651. 70085. 150659. .337 4 357574. 305835. 51 738. 115859. .324 5 515970. 442427. 73543. 151655. .294 6 265138. 226550. 38587. . 92661. .349 7 333056. 294356. 38700. 126663. .380 8 174440. . 157440. 17000. 75350. .432 9 185786. 181090. 4695. 85838. .462 10 122572. 101033. 21540. 89406. .729 IUBVIVAL THBOUGH KATURAL MOBTALITY: MEAN TBUE STD COEF VALUE VALUE "BIAS" EEROB VAB .648 .670 -.022 .1131 .174 STOCK-EECBDIT DISTEIEUTION MEAN TEUE VALUE VALUE ALPHA .818 1.358 EETA .605 .914 SIGMA .384 .385 PABAMETEBS: STD COEF "BIAS" EEBOB VAB -0.5403 .2352 .288 -0.309 .5030 .831 -0.0012 .1114 .290 122 Table 18. Monte Carlo results for beta, except variatio n , are scaled in the biases are due for Case F, A l l values the c o e f f i c i e n t of by 10fe ,. Slight errors tc rounding,. 123 ABUNEANCE IN YEAR 1: MEAN TRUE STD COEF AGE VALUE VALUE "BIAS" ERROR VAR 1 276684. 330500. . -53816. 51058. . 185 2 178859. 221541. -42682. 32296. .181 3 122235. 148503.. -26268. 22261.. . 182 4 79901. 99545. -19644. 14819. .185 53231. 66727. -13496. 106 57. .200 6 34594. 44728, -10134. 7069. .204 7 22590. . 29.982, -7392. . 6145. .272 8 15072. 20098. -5026. 4229. .281 9 9318. 13472. -4154. 2586. .278 10 6176. 9030. -2854. 1867. .302 1 1 4034. 6053. -2019. 1523.. .377 12 2578,. 4058. -1480. 1264. .490 13 1659. 2720. -1061. 873. .526 14 999. 1823. x824. 627. .628 15 706. 122 2. -516. 576. .816 ANNUAL COMPONENTS OF FISHING MORTALITY MEAN TRUE STD COEF YEAR VALUE VALUE "BIAS" ERROR VAR 1 .440 . 100 . 340 .3656 .830 2 .433 . 100 .333 .3588 .828 3 .429 ,100 .329 .3465 .808 4 .428 .100 .328 .3407 .795 5 .439 . 100 . 339 .3374 .769 6 .440 .100 .340 .32 84 .746 7 .441 . 100 . 341 .3157 .716 8 .448 .100 . 348 ,3037 .678 9 .477 . 100 .377 .3081 .646 1 0 .535 . 100 . 435 .3608 .675 124 AGE SELECTION FACTOES: MEAN TBUE STD COEF AGE VALUE VALUE "BIAS" EBBOB VAB 0 .000 .000 - . 0 0 0 .0000 .739 1 . 150 .250 - . 100 . 1033 .687 2 .418 ,750 - . 3 3 2 .2771 .663 3 .553 1.000 - . 447 .3606 .653 4 .577 1.000 - . 4 2 3 .3741 .648 c .576 1.000 - . 424 .3592 .623 6 .575 1.000 - . 425 .3482 i605 7 .581 1.000 - . 4 1 9 .3473 .597 8 .589 1.000 - .411 .3455 .586 S .614 1.000 - . 386 .3480 .567 10 .622 1.000 - . 378 .3244 .521 1 1 .651 1.000 ^.349 .3224 .495 1 2 .670 1.000 - . 330 .2914 .435 13 .709 1.000 - . 291 .2589 .365 14 .773 1.000 - . 2 2 7 .1874 .242 LNNUAL RECEOITMENT: MEAN TBUE STD COEF YEAB VALUE VALUE "BIAS" EEEOB VAB 1 349699, 408800. -59101. 82460. . .236 2 259920. 319807. - 59886 . 69567. .268 327023 . . 401284. -74261 . 95346. .292 4 328345. 409325. - 8 0 9 7 9 . 102937. .314 5 532837. 655778. . -122941. 171455. .322 6 391374. 485016. - 93641 . 168574. .431 7 506258. . 647282. -141024, . 229666. .454 8 402413. 513472, -111059 . 232939. .579 9 416430. 585546. . -169116. 231404. .556 10 237291. 278280, - 40989 . 217360. .916 STOCK-EECBOil DISTRIBUTION PABAMETEBS MEAN TEUE STD COEF VALUE VALUE "BIAS" EBBOB VAB ALPHA BETA SIGMA 1.584 1.080 .373 1.358 . 914 .385 .2264 . 167 -0 .0120 1. 1798 1. 1444 i0972 .745 1.059 .261 125 T a J 3 l € 19. .Monte Carlo results for beta, except var i a t i o n , are scaled in the biases are due for Case G, a l l values the ^ c o e f f i c i e n t of by 10 . Slight errors tc rounding. ABUNIANCE IN YEAB 1: MEAN TEUE STD COEF AGE VALOE VALUE "EIAS" EEECE VAE 1 232315. 330500. -98185. 213940. .921 2 150994. 22154 1. -70547. 120054. .795 3 105384. . 148503. -43119. 81918. .777 4 70374. 99545, -29171. 55151, ,7 84 5 46895. 66727. -19832. 35533. .758 6 31664. 44728. -13064. 25478. .805 7 20557. . 29982. -9425. 15858. .771 8 13903. 20098. -6195. 10605. .763 9 8966. 13472. -4506. 7024. .783 10 5963. 9030. -3067. 4357. .731 1 1 3910. . 6053. -2143. 2834. .725 12 2707. 4058. -1351. 2438. .900 13 1843. . 2720. . -877. 1705. .925 14 1181. 1823. -642. 1227. 1.038 15 851. 1222. -371. 933. 1.097 ANNUAL COMPONENTS OF FISHING MCETALITY MEAN TEUE STD COEF YEAB VALUE VALUE "BIAS" EEEOR VAE 1 .378 . 100 . 278 .2896 .766 2 .365 . 100 . 265 .27 66 .758 3 .353 . 100 . 253 ,2649 .749 4 .344 .100 . 244 .2648 ,769 5 .338 . 100 . 238 .2637 .781 6 .316 . 100 .216 .2358 .746 7 .301 , 100 . 201 .2198 .731 8 .296 . 100 . 196 .2275 .770 9 .313 . 100 .213 .2645 .846 10 .366 . 100 . 266 .3663 1.000 127 AGE SELECTION FACTORS: MEAN TEUE STD COEF AGE VALUE VALUE "BIAS" ERROR VAR 0 .000 .000 -.000 ,0000 .379 1 .203 .250 -.047 .0644 .318 2 .546 .750 -. 204 .1787 .327 3 .710 1. 000 -.290 .2371 .334 4 .726 1.000 -.274 .2394 .330 5 .735 1.000 -.265 .2375 ,323 6 .733 1. 000 -.267 .2333 .318 7 .733 1.000 -. 267 .2330 .318 8 .730 1. 000 -. 270 .2240 ,307 9 .752 1.000 -. 248 .22 18 .295 1 0 .758 1.000 -. 242 .2057 .272 1 1 .781 1.000 -.219 .2122 .272 12 .792 1.000 -. 208 . 1931 .244 13 .817 1.00 0 -. 183 .1722 .211 14 . 847 1.000 -. 153 .1074 .127 INNUAL RECRUITMENT: MEAN TBUE STD COEF YEA E VALUE VALUE "BIAS" ERROR VAR 1 274074. 408800.. -134726. 262558. .958 2 193475. 319807. -126332. 168006. .868 229449. 401284. -171835. 175226. .764 4 227219. 409325. -182106. 146881. .646 5 370028. 655778. -285750. 209296. .566 6 272615. 485016. -212401. 126184. .463 7 367546. 647282. -279736. 172338.. .469 8 337029. 513472. -176443. 224300. .666 9 377097. 585546. . -208449. 243872. .647 10 245441. 278280. -32839. 228454. .931 SURVIVAL THROUGH NATURAL MORTALITY: MEAN TRUE STD COEF VALUE VALUE "BIAS" ERROR VAR .783 ,670 . 113 .. 1455 . 186 STOCK-EECRUIT DISTRIBUTION PARAMETERS: MEAN TRUE STD COEF VALUE VALUE "BIAS" ERROR VAR ALPHA .577 1.358 -0.7811 .2362 .410 BETA .499 .914 -0.415 1.6357 3.280 SIGMA .405 .385 .0197 .0967 .239 128 Table 20. Forecast errors for Cases D-G. A l l values are given as a percentage of the "true" value. CASE D CASE E COEF COEF AGE "BIAS" VAR AGE "BIAS" VAR 0 -37.4 32. 0 -44.9 26. 1 20.6 88. 1 18.3 90. 2 -1.7 41. 2 -6.1 41. 3 1.9 37. 3 -3.6 40. 4 1.2 29. 4 -3.1 44. 5 4.5 31. 5 -1.1 46. 6 10.1 37. 6 5.1 54. 7 4.9 37. 7 0.0 50. 8 7.7 43. 8 1.8 51. 9 4.8 40. 9 -0.9 49. 10 12.1 40. 10 5.3 50. 11 10.6 40. 11 5.9 55. 12 7.1 37. 12 1.2 48. 13 9.5 37. 13 3.5 50. 14 6.6 37. 14 1.3 51. 15 7.8 41. 15 0.8 46. 1+ 6.1 43. 1+ 2.0 48. CASE F CASE G COEF COEF AGE "BIAS" VAR AGE "BIAS" VAR 0 -35.5 40. 0 -44.9 44. 1 -14.7 78. 1 8.9 108. 2 -29.4 40. 2 -0.9 80. 3 -23.7 50. 3 18.3 105. 4 -26.5 43. 4 21.2 104. 5 -26.5 47. 5 28.6 112. 6 -29.5 41. 6 27.1 109. 7 -33.0 42. 7 25.4 112. 8 -34.1 42. 8 26.5 112. 9 -36.9 41. 9 27.5 117. 10 -37.3 42. 10 32.2 120. 11 -43.0 39. 11 24.3 114. 12 -48.3 39. 12 18.9 109. 13 -50.8 41. 13 17.1 109. 14 -55.3 41. 14 13.6 108. 15 -58.9 43. 15 12.2 111. 1+ -24.8 46. 1+ 11.8 91. 130 mortality i n years 9 and 10 are also affected by a lack of supporting data: the r e l a t i v e variances are almost ten times as great as f c r the other y*s. The conseguence of a poor estimate of the annual component of exploitation i n the l a s t year (y^) i s d r a s t i c , Eecruitment i n the l a s t year (N x o ) depends on a single datum in the same manner as the number-^at-age in year 1 for the oldest age (N(T ) . If the r e l a t i v e variance of y i s large, that of B ' w i l l also be large, with the r e s u l t that the estimate of recruitment i n the l a s t year w i l l often be nonsense or bounded only by i t s constraints, I t follows that the forecast for age 1 w i l l alsc be very poor. For example, in Case D the r e l a t i v e variance cf Nlo o and the prediction of age 1 (N ( M ) are twice as great as for the other recruitment parameters and predictions. S i m i l a r l y , the estimate f c r recruitment in the next to l a s t year (N V ( 0) w i l l have a large variance, i f that of the annual component of exploitation (y ) i s large, except, here, the estimate cf recruitment i s supported by more data and the e f f e c t w i l l net be as large. In Cases ft-C, recruitment i s determnined frcm a s t c c k - r e c r u i t function, embodying enough information to counteract the effect of unreliable estimates of y x . The selection factors for older ages are not as unreliable as the annual components cf fi s h i n g mortality f o r l a t e r years. This i s because they are t i g h t l y bound by t h e i r upper constraint. Though the amount of supporting data i s low, the upper constraint (representing valuable prior information) helps to f i x the estimate accurately. The a's are scaled r e l a t i v e to a 1 5 and are constrained to be no greater than a v 5 . . I f age selection decreases at cider ages (e.g., by older f i s h migrating 131 out o f the f i s h e r y , by avo i d a n c e b e h a v i o r ) , i t i s i m p o r t a n t t o e i t h e r i n c r e a s e the upper c o n s t r a i n t from 1.0 o r lower t h e age of t h e f i x e d term. I n each o f the f o u r c a s e s , the b i a s e s f o r a l l t h e y's a r e of the same s i g n , as are t i e b i a s e s f o r a l l t h e a's. I n Case D, those f o r the y's a r e p o s i t i v e , w h i l e t h o s e f o r the a's a r e n e g a t i v e . T h i s c o n s i s t e n c y r e f l e c t s the i n d e t e r m i n a c y between the y's and the a ' s , f o r which reason i t was n e c e s s a r y t o f i x one c f t h e para m e t e r s . The r e s u l t f o r Cases D and E i s t h a t t h e e x p l o i t a t i o n r a t e s (y. a. ) are r e l a t i v e l y u n b i a s e d . . In Cases F J and G, the same c o n s i s t e n c y i n b i a s i s seen, but the r e s u l t i n g e x p l c i t a t i c n r a t e s a r e o v e r e s t i m a t e d . T h i s r e f l e c t s the " h i g h e r o r d e r " i n d e t e r m i n a c y between e x p l o i t a t i o n and abundance f o r Case F, and among e x p l o i t a t i o n , abundance, and n a t u r a l m o r t a l i t y f o r Case G.. For the s t o c k - r e c r u i t d i s t r i b u t i o n p a r a m e t e r s , f i t u s i n g t h e s t a t e r e c c n s t r u c t i c n , t h e r e seems t o be no s y s t e m a t i c b i a s , a l t h o u g h t h i s c c u l d w e l l be an a r t i f a c t o f t h e t e s t p r o c e d u r e . For each r e p l i c a t i o n i n each c a s e , the same sequence o f 10 random numbers were used t o s i m u l a t e s t o c h a s t i c v a r i a t i o n i n r e c r u i t m e n t . F o r seme r e a s o n , the e s t i m a t e o f (b always has a r e l a t i v e v a r i a n c e much l a r g e r than cx , r e g a r d l e s s o f whether t h e r e i s h i g h c r low c o n t r a s t i n abundance. The b i a s e s i n <x and p> a r e u s u a l l y l a r g e , whereas the b i a s f o r cr i s s m a l l . Good e s t i m a t e s of 0* are o b t a i n e d i n the s e s t u d i e s because t h e r e l a t i v e v a r i a n c e i n r e c r u i t m e n t i s c o n s t a n t . I n n a t u r e t h i s may c r may not be the c a s e . Now c o n s i d e r seme p a r t i c u l a r r e s u l t s . I n Case D, w i t h low 132 q u a n t i t y and h i g h c o n t r a s t d a t a , t h e c o e f f i c i e n t s of v a r i a t i o n o f t h e parameter e s t i m a t e s a r e not much g r e a t e r t h a n i n Case ft, w i t h h i g h g u a n t i t y and h i g h c o n t r a s t d a t a , except f o r the s t o c k -r e c r u i t p a r a m e t e r s , which i s t o be expected s i n c e Case A d i d not i n c l u d e s t o c h a s t i c v a r i a t i o n . The abundance and e x p l o i t a t i o n parameters a r e w e l l s e p a r a t e d . A l t h o u g h t h e r e i s a c o n s i s t e n t b i a s i n t h e y's and a's, the o v e r a l l b i a s f o r the e x p l o i t a t i o n r a t e s (y. a. ) i s s m a l l . The most s i g n i f i c a n t d i f f e r e n c e between A and D c o n c e r n s the f o r e c a s t . I n Case A the b i a s and r e l a t i v e v a r i a n c e of p r e d i c t e d r e c r u i t m e n t a r e - 3 % and 6%, r e p e c t i v e l y . I n Case D, where mean r e c r u i t m e n t i s p r e d i c t e d from t h e s t o c k - r e c r u i t d i s t r i b u t i o n and compared to t h e " t r u e " mean, they are -37% and 32%. Presumably, w i t h more ye a r s o f d a t a , e r r o r s i n t h e e s t i m a t e s of <x and p i n Case D would d e c r e a s e . . More i m p o r t a n t l y , t h e d i f f e r e n c e s i n the r e s u l t s f o r t h e p r e d i c t e d 1+ p o p u l a t i o n a re huge: f o r Case A, t h e b i a s and c o e f f i c i e n t of v a r i a t i o n are - 1 % and 7%, f o r Case D, 6% and 43%. The f o r e c a s t i n D i s much l e s s r e l i a b l e , though r e l a t i v e l y u n b i a s e d . I n Case E, where Sn i s unknown, the r e l a t i v e v a r i a n c e o f the p r e d i c t e d 1+ p o p u l a t i o n i s 48%, an i n c r e a s e o v er Case D of 5%, and the b i a s i s 2%, a decrease of 4%. T h i s i s about the same magnitude of change as f r c m Case A t o Case B, which seems t o i n d i c a t e t h a t f o r h i g h c o n t r a s t data s e t s , n a t u r a l m o r t a l i t y unknown does not a p p r e c i a b l y a l t e r the r e l i a b i l i t y o f the 1+ f o r e c a s t (though the i n c r e a s e appears d r a m a t i c i n Case B s i n c e t h e p r e d i c t i o n i n Case A i s very good) . In Case E, t h e e f f e c t s of the i n d e t e r m i n a c y between 133 abundance, e x p l o i t a t i o n , and n a t u r a l m o r t a l i t y are v i s i b l e . Net o n l y has n a t u r a l m o r t a l i t y unknown r e s u l t e d i n s u b s t a n t i a l i n c r e a s e s i n most o f the parameter v a r i a n c e s , compared t o Case D, but t h e r e i s a d e f i n i t e p o s i t i v e b i a s i n the i n i t i a l numbers-at-age and r e c r u i t m e n t p a r a m e t e r s , and a n e g a t i v e b i a s i n s u r v i v a l . I n Cases B and C, the r a t e o f s u r v i v a l t h r o u g h n a t u r a l m o r t a l i t y was e s t i m a t e d almost w i t h c e r t a i n t y . . I n Cases E and G, the r e s u l t s are much l e s s e n c o u r a g i n g . The b i a s f o r Sn i n Case E i s s m a l l , but the s t a n d a r d e r r o r i s 0.11, g i v i n g an approximate 95% c o n f i d e n c e . i n t e r v a l of [ 0 . 5 4 , 0.76]. I n Case G, w i t h low c o n t r a s t , the e s t i m a t e i s a l s o b i a s e d . T h i s i n d i c a t e s an even g r e a t e r i n a b i l i t y t o r e s c l v e t h e i n d e t e r m i n a c y among abundance, e x p l o i t a t i o n , and n a t u r a l s u r v i v a l than i n Case E. For both o f the low g u a n t i t y , low c o n t r a s t c a s e s , the i n i t i a l abundance and r e c r u i t m e n t parameters show a s t r o n g n e g a t i v e b i a s , w h i l e the e x p l o i t a t i o n r a t e s (y.( a. ) show an e q u a l l y s t r o n g p o s i t i v e b i a s . I n Case G, Sn a l s o shows a p o s i t i v e b i a s o f about 17%. I n t h e low c o n t r a s t c a s e s , i t i s c o m p l e t e l y i m p o s s i b l e to s e p a r a t e the e f f e c t s o f abundance and m o r t a l i t y . A l s c , the v a r i a n c e s o f t h e parameter e s t i m a t e s a r e much l a r g e r than i n Cases D and E, o f t e n t e n t i m e s as g r e a t . The b i a s e s and r e l a t i v e v a r i a n c e o f the 1+ f o r e c a s t f o r Case F a r e -25% and 46%. While the b i a s i s much g r e a t e r t h a n i n Cases D and E, w i t h h i g h c o n t r a s t , s t r a n g e l y enough the r e l a t i v e v a r i a n c e i s about the same. When Sn i s known, the r e d u c t i o n i n c o n t r a s t does not seem to have such a d r a m a t i c e f f e c t on the p r e d i c t i o n e r r o r s as the r e d u c t i o n i n data q u a n t i t y . I t c o u l d be t h a t f l u c t u a t i o n s 134 i n recruitment cause enough v a r i a b i l i t y in the age structure to offset the e f f e c t s of constant e x p l o i t a t i o n , and that contrast in recruitment, rather than i n 1+ abundance, i s a better measure of information content for clupeoid f i s h e r i e s , At any rate, i n Case G where Sn i s unknown, the r e l a t i v e variance of the 1+ prediction i s 91%. The effect of low contrast and data quantity, when Sn i s unknown, i s to render any prediction t o t a l l y unreliable. 135 Chapter 6. CONCLUSION The p r i m a r y concern of t h i s t h e s i s has been t o i n v e s t i g a t e how our u n d e r s t a n d i n g of f i s h p o p u l a t i o n dynamics i s a f f e c t e d by t h e q u a l i t y c f c a t c h - a t - a g e d a t a , o r , more p r e c i s e l y , by t h e " i n f o r m a t i o n s t a t e " , as r e f l e c t e d i n a few s i m p l e a t t r i b u t e s of the data. But u n d e r s t a n d i n g i s not the u l t i m a t e o b j e c t i v e of the management p r o c e s s . The u l t i m a t e o b j e c t i v e i s t h e m a x i m i z a t i o n of seme y i e l d , such as biomass o r net revenue, i n l i e u of c e r t a i n e c o l o g i c a l , economic, or s o c i a l c o n s t r a i n t s . A f u l l a c c ount o f the management p r o c e s s would i n v e s t i g a t e how the i n f o r m a t i o n s t a t e i s r e l a t e d t o our a b i l i t y t o a c h i e v e t h e s e g o a l s , and hew management d e c i s i o n s c o n s e q u e n t l y a f f e c t t h e i n f o r m a t i o n s t a t e , H i l b o r n (1979) i n g e n i o u s l y c o n t r i v e d t o s i m u l a t e t h e f u l l p r o c e s s on a computer, by c o u p l i n g a h a r v e s t c o n t r o l procedure and a d a t a a n a l y s i s r o u t i n e t o a s t o c h a s t i c model of an age-<-s t r u c t u r e d p o p u l a t i o n . Each year i n h i s s i m u l a t i o n s , he f i r s t e s t i m a t e d the parameters of t h e S c h a e f e r p r o d u c t i o n model frcm p a s t c a t c h e s and e f f o r t s , then c a l c u l a t e d an o p t i m a l e f f o r t c o n t r o l . He then " f i s h e d " the p o p u l a t i o n , t h e r e b y g e n e r a t i n g c a t c h and e f f o r t data to be used t h e f o l l o w i n g y e a r . I n t h i s way, he c o u l d compare v a r i o u s h a r v e s t p o l i c i e s and e s t i m a t i o n p r o c e d u r e s , u s i n g as an i n d e x of performance the t o t a l c a t c h 1 3 6 over t i e s i m u l a t e d d u r a t i o n . Presumably, the same t h i n g c o u l d be dene w i t h e s t i m a t o r s t h a t use c a t c h e s - a t - a g e or c a t c h e s - a t - a g e p l u s e f f o r t d a t a . H i l b c r n p c i n t e d out t h a t , w i t h the S c h a e f e r model, management o f t e n f a i l e d because o f poor parameter e s t i m a t e s due t o i n s u f f i c i e n t c o n t r a s t i n abundance and/or e x p l o i t a t i o n r a t e s , r e s u l t s a n a l a g o u s to those o f Cases F and G above. F i s h p c p u l a t i c n a n a l y s t s r e a l l y f a c e two problems. The f i r s t i s s c i e n t i f i c , t h a t of u n d e r s t a n d i n g t h e s t o c k , o f g u a n t i f y i n g demographic p r o c e s s e s . The second i s p r a g m a t i c , t h a t of d e s i g n i n g c c n t r c l s t o a c h i e v e o p t i m a l y i e l d s . Whereas i n t h e p a s t the two have been thought t o be synonomous, t h e r e i s r e c e n t e v i d e n c e t o suppose t h a t t h i s may not be the case. H i l b o r n ( p e r s o n a l ccmmunicaticn) has been a b l e t o show t h a t the i n a b i l i t y of an e s t i m a t o r t o a c c u r a t e l y i d e n t i f y t h e t h r e e parameters of the S c h a e f e r model ( i . e . , the i n t r i n s i c r a t e o f i n c r e a s e , t h e c a r r y i n g c a p a c i t y , and t h e c a t c h a b i l i t y c o e f f i c i e n t ) does not n e c e s s a r i l y i m p a i r i t s c a p a c i t y t o t o i d e n t i f y the o p t i m a l e q u i l i b r i u m l e v e l o f f i s h i n g e f f o r t . T h i s i s because the o p t i m a l e f f o r t depends o n l y cn the r a t i o o f two o f t h e p a r a m e t e r s . While t h e r e may not be enough i n f o r m a t i o n t o e s t i m a t e each of the two a c c u r a t e l y (or each o f t h e t h r e e ) , i n c e r t a i n c a s e s t h e r e may be enough t o e s t i m a t e t h e i r r a t i o . Perhaps a s i m i l a r r e s u l t w i l l be found f o r c a t c h - a t - a g e a n a l y s e s : w h i l e t h e r e may not be enough i n f o r m a t i o n to a c c u r a t e l y s e p a r a t e the e f f e c t s of abundance from the m o r t a l i t y p a r a m e t e r s , t h i s may not be c r i t i c a l i n i d e n t i f y i n g the o p t i m a l c a t c h q u c t a or e f f o r t r e g u l a t i o n . In view o f our o f t e n d i s m a l 1 3 7 a t t e m p t s a t assessment, t h i s i s a welcome b a s i s f o r optim i s m . . 6.1 Summary. (1) The a n a l y s i s of c a t c h - a t - a g e d a t a i n v o l v e s r e s o l v i n g the i n h e r e n t i n d e t e r m i n a c y among abundance, n a t u r a l m o r t a l i t y , and e x p l o i t a t i o n r a t e . (2) C o h o r t a n a l y s i s r e g u i r e s an independent e s t i m a t e of n a t u r a l m o r t a l i t y and l i m i t s i n f o r m a t i o n t o w i t h i n c o h o r t s . The l e a s t - s g u a r e s approach c an, i n p r i n c i p l e , e s t i m a t e n a t u r a l m o r t a l i t y , and e x p l o i t s comparisons between c c h c r t s . . (3) The l e a s t - s g u a r e s method u t i l i z e s d a ta o t h e r than c a t c h e s - a t - a g e . E f f o r t d a t a , r e p r o d u c t i o n s t a t i s t i c s , and c a t c h - a t - a g e e r r o r v a r i a n c e s can be a n a l y s e d s i m u l t a n e o u s l y i n a s t a t i s t i c a l l y c o n s i s t e n t manner. . a l s o , t a g g i n g data can be used t o e s t i m a t e c a t c h a b i l i t y and n a t u r a l m o r t a l i t y by t h e l e a s t - s g u a r e s t e c h n i g u e . E r i o r i n f o r m a t i o n about p o p u l a t i o n parameter v a l u e s can be i n c l u d e d as c o n s t r a i n t s . (4) The r e l i a b i l t y of assessment i s measured by t h e b i a s , v a r i a n c e , and c o v a r i a n c e o f the parameter e s t i m a t e s and f o r e c a s t . I t depends on t h e i n f o r m a t i o n c o n t e n t of t h e d a t a and s t r u c t u r a l f i t o f the model, and can be e v a l u a t e d by Monte C a r l o methods. (5) Data i n f o r m a t i o n c o n t e n t i s c h a r a c t e r i z e d by d a t a g u a n t i t y , the magnitude o f da t a e r r o r s , and t h e l e v e l of 138 contrast i n abundance and exploitation rates from which the data were taken. (6) Natural mortality can be estimated with precision when data guantity i s high, contrast i s high, catch-at-age errors are moderate (relat i v e variances on the order of 25%), and st r u c t u r a l f i t of the model i s good. (7) The c o e f f i c i e n t of variation for forecasts of abundance ranged from 7% i n a high contrast, moderate error, high data guantity case i n which natural mortality was known, to 9 1% i n a low contrast, moderate error, low guantity case in which' natural mortality was unknown. (8) To accurately determine optimal harvest p o l i c i e s , i t may not be necessary to f u l l y resclve the indeterminacy among abundance, exploitation, and natural mortality. 139 LITEEATURE CITED. Agger,P., I . B o e t i u s , and H.Lassen. MS 1971. On e r r o r s i n t h e v i r t u a l p o p u l a t i o n a n a l y s i s . ICES C.M. 1971..Doc. No. H:16 A l l e n , K . R , 1973. The i n f l u e n c e of random f l u c t u a t i o n s i n the s t o c k - r e c r u i t m e n t r e l a t i o n s h i p on t h e economic r e t u r n from salracn f i s h e r i e s . Cons. I n t . E x p l o r . Mer Rapp. 164: 350-359 A l l e n , K , R . 1977. Whales, i n F i s h P o p u l a t i o n Dynamics, 335-358. J . A . G u l l a n d ( e d . ) , John W i l e y and Sons, New York, 372p. Bard,Y. 1974. N o n l i n e a r Parameter E s t i m a t i o n . Academic P r e s s , New Y c r k , 341p. Doubleday,W.G. 1S76. A l e a s t - s g u a r e s approach t o a n a l y z i n g c a t c h a t age d a t a . I n t . Comm, Northwest A t l . F i s h . . Res. . B u l l . . 12: 6 9-8 1 Draper,N.R. and H.Smith,Jr. 1966. A p p l i e d R e g r e s s i o n A n a l y s i s . John Wiley and Sons, New York, 407p. F r y , F . E . J . 1949, S t a t i s t i c s o f a l a k e t r o u t f i s h e r y , B i o m e t r i c s 5: 27-67 G u l l a n d , J . A . 1955.. E s t i m a t i o n o f growth and m o r t a l i t y i n c o m m e r c i a l f i s h p o p u l a t i o n s . F i s h . I n v e s t , , Lond. (2) , 1 8 ( 9 ) . G u l l a n d , J . A . MS 1965. E s t i m a t i o n of m o r t a l i t y r a t e s . Annex t o Rep.. A r c t i c F i s h . Working Grcup, I n t . Ccunc. E x p l o r . Sea CM. 1S65 (3) : 9p. H i l h o r n , Bay. 1979. Comparison of f i s h e r y c o n t r o l systems t h a t u t i l i z e c a t c h and e f f o r t d a t a . J . F i s h . Res. Board Can., 36 (1 2 ) : 1477-148S Hcag,S.H.,and E.J.McNaughton. MS 1978. Abundance and f i s h i n g m o r t a l i t y c f P a c i f i c h a l i b u t , c o h o r t a n a l y s i s , 1935-1976. I n t e r n a t i o n a l P a c i f i c H a l i b u t Commission, S c i e n t i f i c E e p ort No.65 H i l b c r n , E . . 1 9 8 0 . A comparison of f i s h e r i e s c o n t r o l systems t h a t u t i l i z e c a t c h and e f f o r t d a t a . Canadian J o u r n a l o f F i s h e r i e s and A g u a t i c S c i e n c e s , (submitted) Jones,R. 1961. The assessment o f l o n g - t e r m e f f e c t s of changes i n gear s e l e c t i v i t y and f i s h i n g e f f o r t . Mar. Bes. ( S c o t l a n d ) 1961 ( 2 ) : 1-19 Jones,R. 1S68. Appendix t o t h e r e p o r t o f the North-West Working 140 Group, I n t . Counc. E x p l o r . Sea 2p. Lett,P.F. and T.Benjaminsen. 1977. A s t o c h a s t i c model f o r the management of the northwestern A t l a n t i c harp s e a l (Pagophilus grcenlandicus) p o p u l a t i o n . J , F i s h . Ees.. Board Can., 34: 1155-1 187 L e t t , P . F . , S.K.Mohn, and E.F.Gray. MS 1978. Density-dependent processes and management s t r a t e g y f o r the northwest A t l a n t i c harp s e a l p o p u l a t i o n . ICNAF Bes. Doc. 78/XI/84, S e r i a l No. 5299 Mohn,E.K., P.F.Lett, and E.Eeck. . MS 1978. Some new a n a l y s i s r e l e v a n t to the 197S assessment of harp s e a l s . ICNAF Working Paper 78/XI/65 Murphy,G.I. 1965. A s o l u t i o n t o the c a t c h eguation. J . F i s h . Ees. Beard Can., 22: 191-202 Peterman,E.M. 1978. T e s t i n g f o r density-dependent marine s u r v i v a l i n P a c i f i c salmonids. . J . F i s h . Ees. Board Can., 35 (1 1): 1434-1450 1 Pope,J.G. 1972. An i n v e s t i g a t i o n c f the accuracy of v i r t u a l p o p u l a t i o n a n a l y s i s using cohort a n a l y s i s . I n t . . Comm. Northwest A t l . F i s h , Ees, B u l l . 9: 65-74 Pope,J.G.. MS 1974. A p o s s i b l e a l t e r n a t i v e method t o v i r t u a l p o p u l a t i o n a n a l y s i s f o r the c a l c u l a t i o n of f i s h i n g m o r t a l i t y frcm c a t c h a t age data. ICNAF Bes. Doc. 74/20, S e r i a l No. 3166 Bicker,W,E. 1948. Methods of e s t i m a t i n g v i t a l s t a t i s t i c s of f i s h p o p u l a t i o n s . . Indiana Univ. Publ. S c i , Ser. 15: 101p. Bicker,W.E. MS 1971. Comments on the West A t l a n t i c harp s e a l herd and proposals f c r the 1972 h a r v e s t . Presented t i ICNAF Meeting c f Panel A e x p e r t s , C h a r l o t t e n l u n d , 23-24 Sept. 1S71 (unpublished manuscript) Bicker,W. E. 1975. Computation and i n t e r p r e t a t o n of b i o l o g i c a l s t a t i s t i c s c f f i s h p o p u l a t i o n s . Fish..Ees.,Bd. Canada B u l l . 191. . 382pp. B o t h s c h i l d , E . J . 1977, F i s h i n g e f f o r t . In F i s h P o p u l a t i o n Dynamics, 96-115. J.A.Gulland (ed.) , John Wiley and Sons, New York, 372p. Sergeant,D.E. MS 1971, C a l c u l a t i o n of production of harp s e a l s i n the Western Ncrth A t l a n t i c . . I C N A F Ees. Doc 71/7 S i l v e r t , W . 1978. The p r i c e of knowledge: f i s h e r i e s management as a r e s e a r c h t o o l . J, F i s h , Bes. Eoard Can,, 35 (2): 208-212 Smith,A.D.M. 1979. Adaptive Management of Benewable Besources With Unce r t a i n Dynamics. . Ph.D. T h e s i s , I n s t i t u t e of Animal 141 Resource E c o l o g y , U n i v e r s i t y o f B r i t i s h Columbia Southward,G.H. . HS 1976. Sampling l a n d i n g s of h a l i b u t f o r age c o m p o s i t i o n . I n t e r n a t i o n a l P a c i f i c H a l i b u t Commission, S c i e n t i f i c R eport No.58 W a l t e r s , C . J . 1S75. O p t i m a l h a r v e s t s t r a t e g i e s f o r salmon i n r e l a t i o n t o e n v i r o n m e n t a l v a r i a b i l i t y and u n c e r t a i n p r o d u c t i o n parameters. J . F i s h . Res. Board Can., 3 2 ( 1 0 ) : 1777-1784 W a l t e r s , C . J . BS 1S76. An a l t e r n a t i v e a n a l y s i s o f s t o c k changes i n t h e n o r t h w e s t e r n A t l a n t i c harp s e a l . I n s t i t u t e of Animal Resource E c o l o g y , U n i v e r s i t y of B r i t i s h Columbia ( u n p u b l i s h e d manuscript) W a l t e r s , C . J . and Ray H i l b o r n . 1976. A d a p t i v e c o n t r o l of f i s h i n g systems. J . F i s h . Res. Board Can., 33 (1) : 145-159 W a l t e r s , C . J . and Ray H i l b o r n . 1S78. E c o l o g i c a l o p t i m i z a t i o n and a d a p t i v e management. Ann. Rev. E c o l . S y s t . 9:157-188 142 APPENDIX A R e p a r a m e t e r i z a t i o n f o r t h e B e v e r t o n - H o l t A g e - S t r u c t u r e M o d e l The p r o b l e m i s t o f i n d t h e u n f i s h e d e q u i l i b r i u m s t o c k s i z e ( N j and t h e p r o p o r t i o n o f t h e s t o c k r e c r u i t e d a t h a l f t h e u n f i s h e d e q u i l i b r i u m (A) i n t e r m s o f a a n d 8 . I n t h i s w a y , we c a n i n t e r p r e t t h e p a r a m e t e r s o f a n a g e - s t r u c t u r e d s t o c k - r e c r u i t model i n a more b i o l o g i c a l l y m e a n i n g f u l f a s h i o n . Assume r e c r u i t m e n t (R) o c c u r s a t a g e 0 , and t h a t t h e b r e e d i n g s t o c k ( P ) i s t h e 1+ p o p u l a t i o n . A l s o , a s s u m e t h a t s t o c h a s t i c v a r i a t i o n i n r e c r u i t m e n t i s n e g l i g i b l e . We h a v e : R = P ( A - l ) aP + 6 J p = I N . . j= i J A t t h e u n f i s h e d e q u i l i b r i u m : R = V N CO OO ( OO ( A - 2 ) a n d T h e r e f o r e : 1 ( A - 3 ) i s t h e p r o p o r t i o n o f t h e 1+ p o p u l a t i o n d i s a p p e a r i n g e a c h y e a r d u e t o n a t u r a l m o r t a l i t y and t h e t r u n c a t i o n o f t h e m o d e l a t a g e J . S u b s t i t u t i n g E q . ( A - 2 ) i n t o E q . ( A - l ) , we g e t : 143 V. N «> a N + 3 f l ° ° a (A-4) The h a l f e q u i l i b r i u m r a t e o f r e c r u i t m e n t i s d e f i n e d a s : 0 0 a -FT + T h e r e f o r e : X = a ~Y + 6 S u b s t i t u t i n g E q . ( A - 4 ) i n t o E q . ( A - 5 ) , we o b t a i n : (A-5) a 2 1 a 2V a n d s o . \ = ( A-6) T h u s , c o m b i n i n g E q . ( A - 4 ) a n d E q . ( A - 6 ) w i t h E q . ( A - 3 ) g i v e s N a n d X i n t e r m s o f a , 3 , S n , a n d J . F o r a and 3 i n t e r m s o f and A , we h a v e : a JL I V " A 3 = 2 JL A " V (A-7) (A - 8 ) F o r t h e R i c k e r s t o c k - r e c r u i t f u n c t i o n , d e f i n e d b y : R = a P e -BP we h a v e : I n a a n d a r v 2 , in I n (A-9) (A-10) (A-ll) (A-12) 

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