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Field evaluation of the dichotomous population estimation technique Wood, Frederick Ernest Allen 1966

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A FIELD EVALUATION OF THE DICHOTOMOUS POPULATION ESTIMATION TECHNIQUE by F.; E. ALLEN WOOD B.Sc, University of British Columbia, 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE In the Department of Zoology We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1966 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study„ I further agree that permission., for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives„ I t i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada , A B S T R A C T A f i e l d evaluation of the dichotomous technique was made using 1965 data from the Rivers Inlet sockeye salmon population. An estimate of pre-exploitation population size was made with the compositions of the pre- and post-exploitation populations and the catchi The estimate was made through the application of simultaneous equations and a maximum likelihood estimator formulation to changes in population composition resulting from selective removal. Biased data was shown to greatly affect the estimates of popula-tion size. A wide range of population estimates was derived from the various dichotomous characters u t i l i z e d * Graphical analysis of these estimates provided an overall estimate of the population, ^ . v The physical restrictions of the research area made this estimate of considerable value. Relative to most other population estimation techniques, the dichotomous method has wider potential use because of i t s less restrictive basic assumptions. This f l e x i b i l i t y can be accompanied, however, by a reduction in the accuracy of resultant estimates. i i i T A B L E O F C O N T E N T S Page INTRODUCTION 1 The Dichotomous Method ." ...... 1 Rivers Inlet Sockeye Salmon 2 METHODS 3 The Dichotomous Method 3 Rivers Inlet Survey 8 RESULTS 13 Rivers Inlet F i e l d Study ...; '.. •• 13 DISCUSSION 15 The Dichotomous Method 15 Rivers Inlet Sockeye Salmon Field Evaluation 25 Comparison o f Dichotomous and Mark -Recapture Techniques kk SUMMARY 1*7 LITERATURE CITED %9 APPENDIX 1. Derivation of N Q estimation equation 50 APPENDIX 2. Pre-exploitation population data ...... .. 51 APPENDIX 3. Commercial catch data .... 52 APPENDIX k. Post-exploitation population data 53 APPENDIX 5. Estimated N Q and i t s variance calculated from age groupings 5*+ APPENDIX 6. Estimated N Q and i t s variance calculated from 3 cm. length groupings 55 APPENDIX 7. Estimated N Q and i t s variance calculated from 8 cm. length groupings 57 APPENDIX 8. Estimated N Q and i t s variance calculated from 13 cm. length groupings *. 58 APPENDIX 9. Different estimates of N Q resulting from a number of treatments of the data 59 APPENDIX 10. Overall results 60 i v L I S T O F T A B L E S Page TABLE I. Demonstration of the effect of sampling bias using hypothetical data 19 V L I S T O F F I G U R E S Page FIGURE 1. Map showing location of various landmarks mentioned in 'the text ... 10 FIGURE 2. Changes i n the estimated pre-exploitation population size as a result of changes in the size of catch ....... 17 FIGURE 3. Isopleths of rate of net-induced mortality (K) plotted against natural^mortality (E) and resultant estimated N Q 22 FIGURE U. Gillnet selectivity as demonstrated in Rivers Inlet commercial fishery for sockeye salmon (0. nerka) 26 FIGURE 5. Graphical analysis for July 2-29. Estimates of N derived from use of a l l numbers, no extremes, omitting negatives, and absolute values estimates 32 FIGURE 6. Graphical analysis for July 2-8 and July 9-15. Estimates of W_ derived from use of a l l numbers, no extremes, omitting negatives, and absolute values estimates 33 FIGURE 7. Graphical analysis for July 16-22 and July 23-29. Estimates of N derived from use of a l l . numbers, no extremes, omitting negatives, and absolute values estimates ..- 3^  FIGURE 8. Distribution of estimated N Q and catch through time 37 FIGURE 9. Mean number of days to recovery site of f i s h tagged at the mouth of Goose Bay (x) ...... 39 FIGURE 10. Modal number of days to recovery site of f i s h tagged at the mouth of Goose Bay (x) ................... kO A C K N O W L E D G M E N T S The author would like to express his gratitude to Dr. N.J..Wilimovsky, who supervised the writing o f the thesis, for his invaluable criticism and advice. Thanks are due to Drs. J.F. Bendell, J.T. McFadden and J.E. Phillips who reviewed the manuscript and made many helpful suggestions. I wish to thank the Canada Department of Fisheries for providing the necessary financial assistance, f a c i l i t i e s and equipment, without which the problem could not have been undertaken. Thanks are also due to the commercial fishing companies and fishermen in the study area, in partic-ular British Columbia Packers Ltd. and The Canadian Fishing Co. Ltd., for their assistance and cooperation. I am indebted to a l l those persons who gave encouragement and assistance throughout the study. 1-INTRODUCTION The Dichotomous Method The determination of stock size is a general problem in a population study. A specialized case of indirect count method of population estimation is the "dichotomous" (Chapman, 195*0 or "change of composition" method. "If a population Is classifiable in two or more ways, and harvesting from i t is selective with respect to this classification, then i t is possible to make a population estimate from knowledge of the original composition, the final composition, and the composition of the harvested catch" (Ricker, 1958). Factors such as age, colour, sex, size, species, etc., may be used for classification of the population. The ease of classification by sex and, in many instances, age (i.e., juveniles or adults), and the marked selectivity of k i l l (as a result of legal restriction or animal behaviour) has permitted relatively wide use of this technique for population estimates of game birds and mammals. The procedure has been used mainly for pheasant (Allen, 19^2) and deer (Kelker, 19^0, 19^2; Rasmussen and Daman, I9U3J Riordan, l^kB; Petrides, 19^9j Dasmann, 1952) populations because their environments and modes of l i f e generally meet most of the requirements of this form of study. Although this system of population estimation is known among fishery biologists, there Is l i t t l e documented use of i t . The method has been used by Vaughn (1955) for estimating populations of chinook salmon on the Taku Rlverj i t has been tried also for racial studies of the Bristol Bay sockeye salmon stocks. As both of these studies had only limited success, i t was proposed to use the dichotomous principles to make a population estimate of the sockeye salmon (Oncorhynchus  nerka) stocks passing through Rivers Inlet. 2 Rivers Inlet Sockeye Salmon Work was carried out in 1965 in the Rivers Inlet area (see Figure l ) . This area was ehosen because the inlet i s relatively long (26 miles) and narrow (maximum of 2 miles), and as a l l sockeye in the inlet enter the Wannock River at the head of the inlet, they must migrate the f u l l length of the i n l e t . The commercial fishery during the sockeye season, by regulation, i s limited to gi l l n e t and t r o l l gear, the g i l l n e t fishery being the only one operable during the season. The location of the commercial fishery, predominantly within the inlet, permits sampling at or near the mouth of the inlet of what would be basically an unexploited population. The location of the fishing boundary 3.5 miles from the head of the inlet provided an area of pooling and mixing of schools from which the "after-fishery" population could be sampled. The short (3 miles) Wannock River flows from Owikeno Lake to the head of Rivers Inlet; most of the major spawning areas (except the Wannock River which i s a late stock) are located considerable distances from the outlet of the lake (closest i s 6 miles) and i t was planned therefore to sample near the outlet of the lake for a second sample of the post-exploitation population. METHODS The Dichotomous Method There are at present two basic procedures for making population estimates by change of composition method: simultaneous equations (Model I) and maximum likelihood estimators (Model I I ) . Both of these forms of estimation treat the populations as binomial populations. That i s , for any one individual there are two possible categories, only one of which i t can occupy. For example, the f i s h can be male or female, i t usually cannot be both. Although characters such as age and length give rise to a multinomial population, the method of treatment of data such that a single category i s compared to a l l other categories changes this to an essentially binomial situation. The estimation of population size through the use of simultaneous equations is achieved by equating the actual proportion of individuals in the escapement having a chosen character to the expected proportion as calculated from the catch and original population. With knowledge of the absolute size of one of these populations, generally catch, an estimate of the size of the other two populations can be made. The Model I method of population estimation can be demonstrated through the use of the following parameters of a hypothetical population: SEX CLASSES SAMPLE SEX SAMPLES % males' % females SIZE ORIGINAL POPULATION 50$ 50$ 2500 CATCH k0% 60f« 5000 ESCAPEMENT 6V?o k0% 2500 SIZE OF CATCH (500,000) 200,000 300,000 Assuming unbiased sampling, l e t X be the number of f i s h in the unexploited population. As removal (commercial catch) is 5 0 0 , 0 0 0 , then escapement is X - 500,000. Fifty per cent of the unexploited population is composed of males; the number of males in that population is .5X. The incidence of males in the escapEment is 60 per cent so the number escaping is .6(X - 500,000). As the number of males in the catch is 200,000, the number of males in the escapement must also equal ,5X - 200,000. Setting the values of male escapement equal to each other and solving for X provides an estimate of the size of the original population. .6X - 300,000 = .5X - 200,000 .IX . 100,000 X = 1,000,000 The original population size is equal to 1,000,000 fish. The estimate is based on the proportion of males in the population. Additional estimates of the size of that same population could be based on the proportion of females, different age classes, and length groupings. An indication of the reliability of the resultant estimates can be derived from the size of the variances of the individual estimates. A crude estimate of the variances of estimates derived from the simultaneous equations method is obtained by summing the variances of the three binomial populations involved. "The variance of a binomial population is equal to the product of the relative frequencies of successes and failures" (Li, 1964). For example: u = relative frequency of successes in the population n = sample size u variance original population catch .5 .k .6 .25 .2k ,2k escapement variance = u(l •— u) 5 estimate variance = .25 + .24 + ,2k = + 73% (including catch) = .25 + .24 = + k9% (excluding catch) As the absolute size of the removal i s known, i t s variance is not considered to contribute significantly to the variance of the overall estimate of variance, therefore i t is not added into the estimate variance. Rather than adding the variances i t may be considered desirable to estimate a weighted average of the variances, i.e., (n x - l ) ^ + (n 2 - l ) u 2 + (n~ - l ) u weighted average variance = — :—r : r y *—r * (n x - 1) + (ng - 1) + (n 3 - l ) Using the above data the weighted average variance is 47.7 per cent; excluding catch data i t i s 55 per cent. The simultaneous equations method is relatively simple, but with this simplicity comes reduced r e l i a b i l i t y i n the results. Failure to account for sample size and poor variance estimates are major weaknesses of this method. The second method of population estimation based on-the change of composition method u t i l i z e s equations derived as moment estimates from the product of the binomial representations of the pre- and post-exploitation population characteristics (Chapman, 1954). (See Appendix 1 for algebraic derivation. The following notation is used in these estimates: = population size at time t ^ ( i = 0, l ) made up of two classes X and Y which are defined by the characters upon which selection acts, NQ = the size of the original (unexploited) population N 2 = the size of the escapement population X.,Y. = size of classes X and Y at times t. i ' i I X 0,Y 0 = size of classes X and Y in the original population X.,Y. - size of classes X and Y in the escapement 6 = V * i = V N o p l • V w i R X • x o - x i - removal (catch) from class X R y • Y o " y i - removal (catch) from class Y R = R + R = to t a l removal (catch) x y n i ' = size of random samples taken at times t ^ no ' = size of random samples of original population n l -• size of random samples of escapement x I ' y i -= the numbers in classes X and Y respectively in sample i Vyo = = the numbers in classes X and Y respectively in the original population x l ' y l = ^ e n u m ^ e r s ^n classes X and Y respectively in the escapement The maximum likelihood estimators, then, are: X. = u x * — = 500,000 0 n i x 0 - n o X l n0(n,R - x.R) NQ = _ _ - = 1^ 000,000 nixo " noxi Y Q = N Q - X Q = 500,000 X. = X - R = 200,000 1 0 x ' Yx = Y Q - Ry = 300,000 i f the following parameters are used: CLASSES TOTAL MALES (Y) FEMALES (X) SAMPLE SEX SAMPLE % y± % x± SIZE ORIGINAL POPULATION 50 1250 (y 0) 5*0 1250 (x Q) 2500 (n Q) ESCAPEMENT (N - R) 60 1500 (y^ kO 1000 (a^) 2500 (x^) CATCH (R) kO 2000 (my) 60 3000 (mx) 5000 (m) SIZE OF CATCH (R) 200,000 (R ) 300,000 (R ) y x An asymptotic variance-covariance matrix computation provides an equation with which to estimate the variance of each individual estimate of the original population size. This estimate permits weighting of the.mean of a group of population estimates, which improves the accuracy of the mean since the effects of strongly divergent estimates (large variance) are reduced. S2(x0) . -2° 1 "1 <p0 - V , 2 , „ > n 0 " l r(B„) = For the preceding numerical example, the estimate of variance of X^ is 1,600,000,000 and N Q is 12,400,000,000. A necessary extension of this variance estimate is to account for variance caused hy use of sub-samples of the commercial catch rather than total samples. This is accounted for by Chapman (1955) by some additions to the basic variance equation. X0 Y0 , X1 Y1 , "*x R y / , 3 P 0 ( l - V , 3 F l ( l - V q 2 r w , n0 n l r 2 ( P 0 - P l ) 2 \ n0 n l S ( M ; ( p o - p i > The variance estimate for the preceding numerical example is S 2(N Q) = 12,401,200,000. Chapman (1955) presents an equation for the estimation of NQ which includes two correction factors for unreported losses (or gains): the fi r s t (K) is for loss proportional to catch (net-induced mortality); the second (E) is for loss which is not proportional to removal (natural mortality). 8 If a large sample size is assumed (n^*-> N^), then the sample st a t i s t i c s can be replaced by the population parameters (x^ becomes X^, y^ becomes Y^ , etc.) and the estimate of N Q converges in probability to (NAR - X.R) - ER 0 0 ( N ^ - XQRKI + K) - EX Q where unreported removals are: for X, KR^  for N, KR + E and actual removals are: for X, R + KR + E . x x for N, R + KR + E Using previous data, assume K = .01 and E = 1000. The estimate of N Q from these previous data is 99^ *000• There are a number of assumptions upon which the change of composition population estimate is based: (1) A closed population in which i t is assumed that negligible emigration, immigration, recruitment, natural or any other unaccounted mortality is occurring at the time of removal (exploitation) is necessary. (2) The population must be composed of at least two distinguishable classes of individuals which are unequally vulnerable as a result of the method of removal (gillnet selection for medium sized fish) or of local laws regarding removal (hunting regulations which limit k i l l to males). (3) Population samples taken before and after the removal proeess are random with respect to the specified classes of individuals. Rivers Inlet Survey In sampling at the mouth of the inlet for the pre-exploitation population and at the head of the inlet for the post-exploitation population, a 36-foot chartered commercial drum seiner was used. A seiner was chosen because i t causes minimal physical damage to f i s h (Hartt, 19^ 3)> and because 9 of i t s efficiency and relative non-selectivity. The seine was 250 fathoms in total length hy 16 fathoms in t o t a l depth, i t s body was made of 3-inch tarred cotton web and there were kO fathoms of 1^-inch herring web on the bunt. (The web size and the bunt mesh were specially chosen for juvenile salmon studies, therefore a l l adults should have been caught.) A seine ski f f was used to maneuver the outer end of the seine. Early i n the fishing season, test fishing was conducted at various locations at the mouth of the i n l e t . The only site found to be suitable was near the mouth of Goose Bay (see Figure l ) . Sites in Darby Channel were strongly influenced by tides, sites near Sharbau Island, Major Brown Rock, Dimsby Point and Addenbroke Point were open to the sea wind, which made fishing in these areas possible at only certain times. The distances involved in travelling between these sites, the speed of the seiner (8 m.p.h.) and the relative size of the catch led f i n a l l y to the selection of an area around Cow Island (mouth of Goose Bay) as the major fishing s i t e . A l l f i s h caught in the mouth of the inlet were measured (nose to fork length, accurate to .5 cm.) and sampled for sex (estimated from external characteristics); a scale sample was taken (to be used in age determinations). Also, the f i s h were tagged with a "spaghetti"-type tag, this type having been chosen because of i t s non-selectivity to giII nets (Davis, 1959). The tags used in this experiment were made of l/l6-inch tubing (resinite tubing, Borden Co. Chemical Division) with a legend printed on the tubing; no transparent covering tube was used. Tags were applied with a stainless steel rod with one end thin enough to f i t inside of the tag end the other end sharpened in a three-sided point (so that i t would cut i t s way through instead of ripping). The tag was slipped over the thin end of this rod, sewed through the fi s h , and held in; place by 10 FIGURE 1. Map showing location, of various landmarks mentioned in the text. an over-hand knot. The f i s h were tagged in order to provide an indication of timing, and a stock of f i s h upon which to try a "change of composition" estimate. Fishing at the head of the inlet for a post-exploitation (escapement) sample was concentrated at the extreme head of the inlet in the hope that sampled f i s h would not d r i f t back across the commercial boundary. Spaghetti-tagged f i s h caught in the sample were to be used in making a population estimate. Length measurements (nose-fork), scale samples and estimates of sex were taken; f i s h were tagged with large (22 mm.) Petersen disc tags, the purpose of which was to provide an indication of the d r i f t of f i s h back across the commercial boundary, and a means of estimating population size (by recovery in the lake). Sampling near the outlet of Owikeno Lake was carried out to provide a second sample of the exploited population, as well as to recover the Petersen disc tags and spaghetti tags applied ea r l i e r . Sampling was originally to be by beach seine, but high water conditions did not permit this, so the beach seine was modified to a small purse seine. Nose-fork and orbital-hypural lengths were measured, scales taken and sexes estimated. The commercial catch was sampled as i t was being delivered to f i s h packers. Samples were taken mainly at Wadhams and Sampson IV (near Dawson's Landing), but also at Good Hope (see Figure l ) , and on collecting boats. Both nose-fork and orbital-hypural lengths were measured. Since nose-fork lengths were taken in the tagging operations, because of the ease of measuring live f i s h , and since previously-collected data are predominantly in orbital-hypural lengths, because they were taken i n spawning ground surveys, both measurements were taken in this experiment in order to provide data from which a conversion factor could be calculated. Scale samples were taken to provide an estimate of the age composition. Estimates of sex were recorded f i r s t (based on external characteristics) and then, i f the captain of the packer would permit i t , f i s h were dissected and the actual sex was recorded. Comparison between estimated and actual sex provided a means of quantifying the accuracy of the sex estimates. The commercial fishery in 1965 operated from 6 :30 p.m. Sunday u n t i l 6 :30 p.m. Thursday and from June 2k through July 2 1 . On July 2 0 , a 3-day fishing week was declared (Thursday-July 22 was omitted). A t o t a l closure of 18 days was declared the following week; the fishery reopened August 8 . The major tag returns were of the spaghetti tags<* A reward of 50 cents for each tag was offered to stimulate return of tags. A canvas of a l l the major f i s h camps on two days of each closed period was conducted to recover tags found by the fishermen and to distribute tag return envelopes. A l l the collecting boats operating in the inlet were provided with tag return instructions and given a supply of tag return envelopes to distribute to fishermen. When the fleet moved to the Fraser River, a canvas of boats fishing that river was conducted in a further effort to recover tags. RESULTS Rivers Inlet Field Study As the change in composition method of population estimation requires samples of the pre-exploitation population, f i s h removed by exploitation, and the post-exploitation population, analysis by this method was limited to a central l+-week period of the migration by the lack of one or more of these samples (see Appendices 2 to k). The f i r s t week of fishing lacks a post-exploitation sample, which is to be expected as f i s h have not yet reached the head of the i n l e t . The second week has the necessary samples but the size of the post-exploitation sample is questionably small (l8). The third, fourth and f i f t h weeks have complete sets of data with sufficient sample sizes. At the end of the f i f t h week, the commercial fishery closed for 18 days, thereby removing the selective process. Sampling in Owikeno Lake to provide a secondary post-exploitation sample provided no data usable in population estimation because the samples did not span the necessary time period and were taken with a different type of gear, therefore under a different selective regime. The weekly periods have been divided so that the pre-exploitation sample is taken at the f i r s t of the week and the commercial catch and post-exploitation samples are taken later in the week. Based on the modal rate of migration, the three samples should a l l be from the same stock of f i s h . The data collected permitted population estimation based on age and length characteristics. Using age as estimated from scale analysis, N Q (original population size) was estimated using three age groups (see Appendix 5)« Analysis of length data for an estimate of was carried out with three different length groupings: 3 cm., 8 cm. and 13 cm. (see Appendices 6, 7 and 8). The estimates resulting from the age and length analyses were grouped in a number of different manners. Each estimate was weighted with the reciprocal of i t s variance. A graphical analysis in which individual estimates of N Q were plotted against their variances permitted a visual estimate of the central or most l i k e l y estimates (see Appendix 9) • Corrections for periods not sampled, catch outside the sampling site and unaccounted mortality are applied in calculation of f i n a l estimates (see Appendix 10). DISCUSSION The Dichotomous Method In a l l population estimation techniques the d i f f i c u l t y of meeting the basic assumptions limits the accuracy of estimates. The grosser the assumptions, the more d i f f i c u l t they are to meet. The assumptions of the dichotomous population estimation method are relatively easy to meet. If the time between the "before" and "after" samples is minimized, the effects of emigration, immigration, recruitment, and natural mortality should also be minimal. In many cases populations tend to overlap, such that animals removed from these areas of overlap cannot be c l a s s i f i e d as belonging to one population or the other; this situation may be considered a form of immigration. Alternatively, a portion of the population's range may be inaccessible to the form of exploitation; i f there i s l i t t l e mixing within the population, the inaccessible portion may be treated as a separate population. The assumption of selectivity of removal appears to be quite secure as most forms of exploitation tend to show some selectivity, usually by size. "A selective process in fishing may be defined as any process which gives rise to differences in the probability of capture among members of the exploitable f i s h " (Parrish, 1963)* "From general inspection of the size frequency distribution i n catches taken by gillnets of different mesh sizes, i t has long been believed that the typical selection curve is similar to the normal distribution. The fraction of the number of f i s h of a certain length which encounter the net and are retained by i t is thus highest at a certain length, and decreases symmetrically to zero both above and below that length" (Holt, 1963). Peterson (195*0 demonstrated g i l l n e t selectivity by size; also g i l l n e t selection by sex i n Pacific salmon as'ra result of secondary sexual characteristics (i.e., male pink salmon, Oncorhynchus gorbuscha, selected for more highly than females). Sport fishing and hunting tend to be selective by regulations applied to them, usually a minimum size, sex, or both. Behaviour of animals may change at different ages or states of sexual maturity, resulting in responses which make the animal more or less susceptible to the form of removal. This i s especially true in sport fishing and hunting. Qualitative characteristics, such as age or sex, have the advantage over quantitative characteristics, such as weight, in that boundaries of the classes of characteristics are naturally defined. Any changes in the distribution of individuals of an age class relative to length results in no change in the population estimate provided the proportion of the age class to the t o t a l population does not change. Generally, qualitative characteristics are readily grouped, therefore they lead to a limited amount of sampling bias. Quantitative characteristics form an in f i n i t e array of sub-groups. The necessity of measuring these sub-groups permits i n f i l t r a t i o n of considerable amounts of bias. To use the data i n a change of composition type of population estimation, the data must be grouped into a usable number of groups. This necessitates using either a number of groups of random size or a series of groups of equal size defined by either the numbers in the group or a fixed span of the measured character (i.e., 3 cm. groupings). Both of these forms of grouping offer much potential bias. There is a linear relationship between the sitse of catch (R) and estimate of the original population size. The rate of change of the estimated N Q as a result of changes in R is equal to the slope of the line in Figure 2. If the center of selection remains constant the more marked : the selection is (smaller m /m), the smaller the catch w i l l be, and the smaller the slope w i l l be. As a result, the smaller the slope gets, the 17 (OOO'OOI *) ON 0.3±VWI±S3 FIGURE 2. Changes in the estimated pre-exploitatlon popu-lation size as a result of changes1' in the size of catch. Composition data remaining constant. Based on data from previous example calculation on page 6. Catch and estimated N n in numbers of fish. 18 less effect catch, regardless of i t s size, has on the estimate of N^, therefore the more accurate the estimation of N Q. If the fate of selectivity i s within the control of a fishery manager wanting to use a dichotomous population estimation method, i t would he very advantageous to maximize the selection. In any case, the relationship between the original population size and the catch (R) makes i t v i t a l that accurate estimates of the size of catch be used in computations i f accuracy of resultant estimations of N Q is to be achieved. Changes in the aforementioned "slope factor" due to sampling bias can cause a gross bias in the estimate of N^. I f length i s the selected character, a constant measuring bias can cause marked effects on the estimate of N Q. In Table I, the result of reducing f i s h lengths by one centimeter i s shown in the column headed "Commercial catch, biased". This type of error could result because of untrained personnel or faulty equipment and could occur in a l l the samples, resulting i n a t o t a l average bias of 1 cm. Using 6 cm. length groupings of the three stocks, estimates of the original population size were calculated using actual and biased lengths. The estimates were a l l made using a Model II type of computation. The marked differences between the biased and actual estimates of the original population emphasize the importance of the assumption of unbiased samples of the selected character. When the proportion of class X in the pre-exploitation population exceeds the proportion of Glass X in the post-exploitation population, an increased value for the catch of class X results in an increased estimate of N Q (PQ-> P-^  J ^> mx = ^ >NQ). This situation i s probably the most common occurrence in my data. It occurs three times in Table I. 19 TABLE I. Demonstration of the effect of sampling bias using hypothetical data. PRE-EXPLOITATION COMMERCIAL CATCH Actual Biased POST-EXPLOITATION N, LENGTH (x0)(Po) (n^Hmx/m) (m^On^/m) ( X l ) ( P l ) 1 • 35 1 1 36 1 21 .01+3 1 .002 1 .002 20 .080 37 5 5 38 6 5 39 8 9, 40 3' 1+ 1+1 1+ 1 2 1+2 1 12 .025 2 .001+ 3 •006 8 .032; 1*3 2 1+1+ 1 2 1+5 \ 1 1+6 I s 1+7 3 • 1+8 3 M .093 8 120 .229 196 , -375 1 17 .068 k9 9 17 3 50 17 35 1+. 51 16. 56. 9, 52 1+1 77 19 53 50 61 3*+ 51+ 53 298 .616 50 261+ .505 205 .392 1+6 180 .723 55 53 26 h3 56 72 23 2l+ 57 29. 27, Hi 58 21" 18 6" 59 19 19 3 60 ll+ 81 .167 21+ 111 .212 101 .193 3 19 .076 61 ll+ 12 2 62 1+ 20 5 63 9. 18. 61+ 5" 8" 3 65 1+ 5 2 66 11 27 .056 1+ 25 .01+8 17 .oi+o 5 .020 67 1+ 5 68 l 3 69 2. * l+81+( 523 (m) 21+9(3 )^ Actual Biased 181,712 181,712 328,002 305,9^ 6 559,762 1,061+,3H I7l+,661+ 264,998 128,088 110,090 66,303 29,539 20 When the proportion of class X in the pre-exploitation population is less than the proportion of class X in the post-exploitation population, increased values for the catch of class X results in a decreased estimate of N Q (P 0< P 1 : > mx =<N Q). These conditions occur relatively frequently and can he expected to occur in length data analysis, in an area of displace-ment of the escapement curve as a result of selectivity (i.e., i f escapement is displaced two units to the right of the pre-exploitation population curve, the use of the data in the area of displacement may result in the above-mentioned condition (PQ<£P^).).' If the proportion of class X in the post-exploitation population exceeds the proportion of class X in the catch but not in the pre-exploitation population, negative values for the estimate of N Q result. This will occur when using data from an area of a frequency distribution with moderate numbers of individuals but with very low rate of removal. This situation does not occur commonly. If the proportion of class X in the post-exploitation population exceeds the proportion of class X in the pre-exploitation population but not in the catch, negative values of the estimate of\N Q result. These conditions are found when using data from an area of a frequency distribution which is not heavily selected for, in a population with a low rate of exploitation. When the proportion of class X in the pre-exploitation population equals this same proportion in the catch and also in the post-exploitation population, no selection has occurred; the 'dichotomous methods of population estimation do not function and the value of equals zero. This situation could occur almost anywhere in a frequency distribution, but Is most likely to occur in the area of selective displacement of the escapement curve. This situation could also occur i f the range of sampling for the frequency 0.'-••• . •'. ••; . • ' • '-'v -;):•'.• :v. c t i i A a n d m t e r n . c t . ; .on distribution was such that i t encompassed a l l selection and interaction within the sample and n u l l i f i e d the selection. A l l of the above-mentioned situations can, and do, occur in data, especially with small samples. Many of the "undesirable" situations can be avoided or eliminated by the choice of a sample range in the frequency distribution which provides the "desirable" situations. The arbitrariness of this choice of sample range could lead to a conscious or unconscious biasing of the N Q estimate. It seems probable that the best way to eliminate this bias i s to use a fixed sample range (e.g., each range is defined as containing the number of s e r i a l length groupings which contains as near to 100 individuals as possible), or sample range (e.g., every range covers 6 cm.). If a large number .of independent estimates are desired, a running mean of a sample range (1st range = 50-55; 2nd = 51-56; etc.) may be used. In general, catch could be expected to be under-evaluated because of unaccounted mortality in the form of unknown natural mortality (not proportional to catch) and net-induced mortality (proportional to catch). The effects of these two forms of unaccounted mortality on the estimated size of the original population (N Q) are shown i n Figure 3» With K (net-induced mortality) acting by i t s e l f (E = 0), the lower rates of K cause more change per percentage of mortality in the estimated than do the higher rates. The rate of decrease in estimated N Q decreases with an increase in the rate of net-induced mortality (Figure 3). When E (natural mortality~a number and not a rate) acts alone, the rate of decrease in estimated NQ decreases with increased natural mortality (Figure 3). At low values of K, the E to N Q relationship remains basically the same as when K equals zero. At a K value of just below .2, changes in E cause no changes in the estimated N N, that i s , the isopleth 400 450 500 550 ESTIMATED N 0 600 650 Isopleths of rate of net-induced mortality (K) plotted against natural mortality (E) and resultant estimated K . Data based on B = 605,000, = 301,000, K = 695,000, X n = 35^ ,000, variable K and E. of K at this rate of net-induced mortality is a straight l i n e . With K larger than this value, increased E tends to increase estimated N Q towards i t s actual value. At low values of E, the effects of K are rapidly diminished, hut at an ever decreasing rate. When E i s very large, the effect of K is very small; the converse i s also truei If the fishery manager has the a b i l i t y to control the amount of net-induced mortality, the effects of natural mortality could be eliminated by fixing K at the value which produced the straight line v e r t i c a l isopleth and then making an adjustment for K. The biasing effects resulting from unaccounted net-induced mortality (K) can be minimized by selection of the location at which the pre-exploltation population sample is taken. Net-induced mortality occurring between the pre- and post-exploitation sampling sites tends to bias the estimated N Q such that an inflated N Q estimate results. Net-induced mortality as a result of another fishery operating before f i s h reach the pre-exploitation sampling site tends to bias the estimate of N Q such that a deflated value results. The rate of net-induced mortality can be assumed to be proportional to catch and constant over the entire fishing area. If the catch distribution over the fishing area i s known, then sampling so that one-half of the catch occurred inside and the other outside of the pre-exploitatibn sampling site would result i n an inter-action between the net-induced mortality i n these two halves which should result in mutual cancellation, thereby eliminating the bias due to unaccounted net-induced mortality. The presence of net-induced mortality (K) and natural mortality (E) must be accepted as fact. The problem of quantifying either K or E is extremely d i f f i c u l t and in most cases must be satisfied by hardly more than a poorly informed guess. The value of K should be easier to estimate than E, because i t is proportional to catch and proportional to the number of f i s h escaping the nets and l i v i n g . To arrive at a value of K i t is necessary, as with E, to estimate i t , but with K there are some guide l i n e s . With knowledge of the incidence of gi l l n e t marked f i s h in the escapement, the rate of migration of the f i s h , the distribution of effort over the migration route and the calculated rates of exploitation, i t should be possible to back-calculate an approximate rate at which f i s h escape gillnets and l i v e . Significant estimates of f i s h escaping gillnets have ranged from 0 to 30 per cent of the t o t a l number of f i s h entering the net. This value w i l l vary considerably with weather conditions, f i s h size, type of twine of net, etc. It seems reasonable that the value of K should remain proportional to the t o t a l number of f i s h f a l l i n g out of the net provided the set is of reasonable length. Dasmann (1952) points out that "Small errors in determining the sex ratio may be magnified many times when this ratio i s used to compute tot a l numbers. Such magnification is most marked in l i g h t l y hunted herds where the sex ratio approaches unity". As the relative catchability of X to Y approaches 1, (E /X )/(R /Y ) = 1, the sample size must approach WQ to remain representative. The more marked the selectivity,, the more accurate the population estimate; also, the smaller the necessary sample size required. The desirability of minimal variance of such that i t provides the most reliable population estimate leads one to try to adjust factors, such as sample size, to achieve this end. Chapman (1955) demonstrates that through elementary calculus the equation for the ratio of n^ to n Q is n^/n^ = (X^Y^/XQYQ)2 when NQ i s to be found with variance minimized and n.. + n n fixed (i.e., fixed t o t a l sample size). From this equation i t can be seen that the less selectivity exhibited, the larger the n^ sample required to minimize the variance. The optimum sample sizes for the demonstration data (page 6) are n Q = 3355.7 and n 1 = 1 6 4 4 . 3 . Rivers Inlet Sockeye Salmon Field Evaluation The assumption of selection of sockeye salmon by the gi l l n e t fishery in Rivers Inlet i s met (see Figure 4 ) . The main portion of the catch i s from the l e f t of the peak frequency of the pre-exploitation stocks. Removing f i s h from the left-hand side of the peak tends to move the peak to the right; as a result, the post-exploitation peak is displaced to the right of that of the original population. The distance that this peak is displaced is a function of the rate of exploitation and the location of maximum selection; even though the catch in Figure 4 i s only slightly displaced to the l e f t of the pre-exploitation population, a high rate of exploitation can move the escapement curve relatively far to the right. Selectivity centered near the peak of the original population curve with a high exploitation rate w i l l cause the escapement curve to move farthest and also may result in the formation of another mode on the escapement curve i f the range of selectivity i s narrow enough. As the f i s h smaller than 45 cm. are relatively unexploited, their actual numbers do not change but their contribution to the tot a l population increases. This can be seen in the difference between the pre- and post-exploitation curves in the smaller than 45 cm. range. If the 5z= 45 cm. = 10 (10$) and the > 4 5 cm. = 90 (90$) and these undergo 50 per cent exploitation, the results are: ^ 4 5 cm. = 10 ( 2 0 $ ) ; > 45 cm. = 40 ( 8 0 $ ) . The smaller than 45 cm. range contained only 10 per cent of the population before exploitation, but as a result of selection contained 20 per cent after exploitation. FISH LENGTH (CMS) FIGURE U. Gillnet selectivity as demonstrated in Rivers Inlet commercial fishery for sockeye salmon (0, nerka). 27 The relatively heavy removal of the greater than 58 cm. fi s h resulted in a major reduction in their percentage contribution to the, post-exploitation population. The sampling schedule used in the f i e l d study was poor because i t was not representative of post-exploitation or pre-exploitation populations. Early in the fishing period, the pre-exploitation samples were taken during fishery closures in an area in which f i s h had already undergone considerable selection in the outer fishery. As a result, during this period the amount of selection applied to these stocks previous to sampling was over-estimated such that is estimated at a lower than i t s true value. Sampling only during the fishing period each week at the post-exploitation sampling site should tend to under-evaluate the selection of these stocks. The pooling effect at the head of the inlet should cause a damping effect on selection which I would expect to cause an underestimation of NQ, especially on weekly estimates. As a result of the sampling schedule and escapement pooling, the sum of the weekly estimates of N Q should be lower than the real value. The catch statistics are based on a whole s t a t i s t i c a l area; there-fore, they had to be adjusted to be used in these computations. The adjustment used was effort, in the form of boat distribution (i.e., proportion of boats inside and outside sampling s i t e ) . A comparison of expected catch calculated in this manner with estimated catch distributions made independently by a fisheries o f f i c e r showed good correlation. It was proposed to use sex as a selected character but evaluation of the v a l i d i t y of the sex estimations i n the immature sockeye showed that the data were unreliable. The sources of error in sex estimation arise from the immature condition of the f i s h sampled and the necessity of returning f i s h to the water unharmed (i.e., actual sex could not be ascertained by 28 dissection). In samples in which actual and estimated sex could he taken, the accuracy of sex estimation was found to he 76.51 per cent. This degree of inaccuracy would cause large biases in the estimated N^; therefore, sex was not used as a selected character. Using ages as estimated from scale analyses, N Q was estimated using each age group. The data for 3-year-old f i s h have the advantage of being representative at low sample sizes because of the very low removal, but they have a major disadvantage in that estimates of N Q are extremely sensitive to bias of catch as a result of the large slope factor that occurs in this age group. The 4-year-old f i s h comprise the major part of the stock (60-80$) and i t might be expected that this age group would be the most representative because of large samples, but this abundance may contribute to the damping effect between areas of high and low selection within the sample. This internal interaction as a result of sample sizes i s probably what affeeted the estimate of N Q based on overall 4-year-old data (see Appendix 5). The relatively large slope factor resulting from this age group comprising a large part of the catch makes the estimate of N Q sensitive to bias of catch (R). Five-year-old data may have the same problems as the other two age groups. A large slope factor can make this group very susceptible to bias i n catch. Sample size can cause problems; i f i t i s small i t may be poorly representative, i f i t i s large i t may hide selection within i t s range. The choice of the "best" estimate of N Q based on age is d i f f i c u l t . The aberrancy of the 4-year-old estimate eliminates i t as a potential best estimate unless a sum of weekly estimates i s used instead of the overall estimate. The higher variance of the 3-year-old estimate might lead to rejection of i t in favour of the 5-year-old estimate. The 3-year-old overall estimate (89^ ,363) and the summed weekly 5-year-old estimate (722,385) are probably the best estimates of N Q based on age (Appendix 5). A l l of the other estimates are too small i n relation to the catch (1+76,890) and the distribution and intensity of effor t . Analysis of length data for an estimate of NQ was carried out with three different length groupings; 3 cm., 8 cm. and 13 cm. The 3 cm. length grouping analysis provided thirteen estimates (12 independent, 1 dependent) of K Q.for the four weekly periods and the overall time period (see Appendix 6). A l l of the estimates based on the 32-3*+ cm. length grouping are aberrant. The July 2-8 estimate is zero because of lack of data i n the post-exploitation sample in the length range at this time period. The other weekly estimates and the overall estimate are a l l negative. The sample size in this size range is so small that random chance capture of one of these individuals can greatly bias these estimates. The low variance of the July 16-22. estimate is a result of no f i s h of that length in the commercial sample in that time period (see variance equation). The N Q » G estimates of the July 2-8 sample are a result of lack of data in that length grouping. The NQ estimates for July 23-29 appear to be quite divergent. This condition i s probably a result of the relatively small size of the pre-exploitation sample. Sampling of f i s h less than 52 cm. was sparse as well as sporadicj correspondingly, estimates based on small f i s h are the most aberrant. The negative values of WQ for the 68-70 cm. group for the July 2-29 (overall) and July 16-22 periods are probably a result of the small sample sizes in this length grouping. The negative value in the 56-58 cm. length grouping on July 2-29 is a result of using data from the area of escapement curve displacement. In this area, small changes in the length groupings used can make major differences in N N estimates. The weighted means of N Q for the five time periods are also presented in Appendix 6. The means axe weighted by the use of the reciprocal of their variance. The July 2-29 estimates range from -2,289,751 to 6,267,544. The best estimates appear to be those in the 38-40 cm. group; the sum of the weekly estimates (716,622) is good, as i s the overall (July 2-29) estimate (8l2,54l). The extreme estimates can be rejected on the grounds of being smaller than the commercial catch or being so large as to have received an extremely low rate of exploitation. Five estimates of N Q for five time periods were provided by the 8 cm. length grouping analysis (see Appendix 7). The individual estimates for each time period appear to be better grouped in the 8 cm. grouping analysis than in the 3 cm. grouping. This could be a reflection of the increased sample size and the probable decrease in the effect of random variation. The range of estimates of for the July 2-8 period could be p a r t i a l l y attributed to the small sample sizes in this period. The 40-47, cm. length grouping is made up of individuals from the area between the main body of the 3-year-olds and 4-year-olds. As a result, the sample i s small and subject to composition changes as a result of changes in i t s size. Any deviation from the mean of the time period by estimates in this length grouping may be attributed to this p o s s i b i l i t y . Similarly, the 56*»63 cm. length grouping encompasses the area between the peak numbers of 4- and 5-year-olds and is therefore subject to this same possible source of bias. The most l i k e l y estimates of original population size are the 32-39 cm. overall estimate and the sum of the weekly 48-55 cm. estimates. A l l other estimates are highly 1 divergent and can be disregarded on the basis of size of catch and intensity of effort . Three estimates of N Q (original population size) were provided by the 13 cm. length grouping analysis (see Appendix 8). The boundaries of these three groups are relatively similar to those of the age grouping analysis and one would expect comparable estimates. This, however, is not the case. A l l the estimates in this analysis were widely divergent from the estimates of the other three analyses. It seems probable that one of the main causes of this divergence is within-sample interaction between areas of high and low selection covering or creating selection. The estimates of N Q are generally larger than would be expected. The most reasonable estimates are the summed weekly 32-kk cm. (962,915) and weekly weighted mean (983,173) estimates. The choice of a "one best, estimate" for each of the five time periods i s very d i f f i c u l t . The weighted means of the age, 3 can., and 8 cm. analyses showed some agreement for the July 2-8, 9-15 and 23-29 time periods. There is no indication of agreement i n the July 16-22 time period. The weighted means of the 13 cm,, analysis.are a l l markedly divergent from the weighted means of the other analyses. The variances of the results of the 13 cm. analysis are very much lower than those of the other analyses; as a result, an overall weighted mean may be pulled strongly toward the 13 cm. estimates (July 16-22 and 23-29, overall weighted mean; see Appendix 9)« A weighted mean of the age, 3 cm. and 8 cm. estimates brings the July 16-22 and 23-29 estimates back into line, but a l l the estimates are small in relation to adjusted catch (Appendix 9). This same weighted mean, with negative values omitted, gave larger estimates, but these were s t i l l small in relation to catch. A graphical analysis of the estimated was attempted by plotting the individual estimates and their variances on a graph with variance and N axes (see Figures 5, 6 and 7). The purpose of this analysis is to get 32 IO'V ui < EE < > IO10 ° IO8 < 10" 2 10 10 200 400 • Positive ° Negative + All numbers = 810,000 ; '05x10 O No extremes - 765,000 ; -9 x IO10 S Omitting neg. s 915,000 •, -6 x IO10 A Absolute values = 920,000 ; I x IO11 600 800 N„ 1000 1200 In thousands 1400 1600 isoo 2000 July 2 - 2 9 — • Exceeds and negative — o Exceeds and positive FIGURE 5. Graphical analysis f o r July 2-29 ( o v e r a l l ) . Estimates of N Q derived from use of a l l numbers, no extremes, omitting negatives, and absolute values estimates. 33 12 O |0 <t E < o I0 e tu l -< 2 10" W 4 w 10 10 10 UJ g i o ' 2 < > IOV 10 5 UJ 10' -10 S A • 10 50 • Positive ° Negative + All numbers = 31,000 •, 2 x 10 • No extremes = 28,500 •, 3 x IO9 S Omitting neg. = 33,000 ; 6 x 10* A Absolute values = 36,500 ; 7 x IO9 20 30 40 50 60 70 80 90 N 0 in thousands July 2 - 8 + All numbers = 210,000 D No extremes = 275,000 S Omitting neg. = 250,000 A Absolute values = 257,000 _L_ 100 150 200 250 300 in thousands 350 400 450 July 9-15 — • Exceeds and negative — o Exceeds •• and positive 100 10 9,x IO8 4 x 10 9 I x 10 9 500 FIGURE 6. Graphical analysis for July 2-8 and July 9-15. Estimates of N derived from use of a l l numbers, no extremes, omitting negatives, and absolute value s est imates. 3h 10" UJ 2 IO12 or > 10'° u. O 10 LU 5 10s s h- • CO UJ , 0 4 14 10 ce < X I0 K I0 C £ IO6 I -UJ | 0 « 10' + All numbers 75,000t 4x10 a No extremes = 174,000; 6 x 10 S Omitting neg. = 480,000; 10" A Absolute values = 366,000 4 x10 100 200 300 400 500 600 700 "8tX N 0 In thousands July 16-22 900 s A°S. + All numbers = 4,000 a No extremes = I 7,000 S Omitting neg. = 33,000 A Absolute values = 30,000 20 40 60 80 100 120 140 160 180 N 0 in thousands • Positive ° Negative July 2 3 - 2 9 — • Exceeds and negative — o Exceeds and positive FIGURE 7. Graphical analysis for July 16-22 and July 23-29. Estimates of derived from use of a l l numbers, no extremes, omitting negatives, and absolute values estimates. a visual estimate of a central, or most l i k e l y , estimate of the original population size. The use of this analysis i s j u s t i f i e d in that other methods of combining data to a single estimate only take into account the variance inherent i n the mathematics of the estimation. Allowance i s made for other sources of variance in the graphical analysis, especially i n the "no extremes" estimate. Using an overall two-dimensional mode, the " a l l data" estimates of Appendix 9 resulted. Omitting a l l negatives (Appendix 9) produced not much better estimates. On a similar plot the center of a clumped area was used as an estimate (see "no extremes" in Appendix 9()« These estimates appear to be reasonable, relative to catch, but probably are s t i l l small (except July 2-29 estimates). Of the eleven July 2-29 estimates, that based on age appears to be the best (601,400). There are two groups of estimates in this time period, the 400,000-500,000 and the 800,000-900,000 groups. The larger groups might be the best estimate i f unaccounted mortality i s relatively high; i f i t is not, the middle numbers are most l i k e l y . An estimation of the numbers of f i s h entering before and after the period in which-complete samples were taken must be based on same method other than a dichotomous technique. An indication of relative abundance may be calculated from the catch per unit effort of the commercial fishery or the tagging boat. The commercial fishery should be the better estimator early in the season; after the closure of the commercial fishery, the tagging boat i s the only indicator. The catch per unit effort for the period before July 10 should be a function of the to t a l numbers of f i s h available to the fishery. The effort during this period i s relatively low; therefore, i t should be reasonable to assume l i t t l e or no gear interaction. Correlation between catch per unit effort and estimated NQ is poor; there axe too few points to establish any relationship. If i t is assumed that fish entering the inlet axe normally distributed through time, a symmetrical curve can be drawn over estimated weekly NQ'S. Graphical analysis can be used on this curve to approximate the number of fish entering the inlet in the time periods before and after the dichotomous estimates were possible. The relatively small proportion of the ran contri-buted by these " t a i l " portions of the curve should make the assumptions of their shape and size relatively insignificant. A graphical plot of average daily catch with a smoothed curve interconnecting the points results in an asymmetrical curve (Figure 8). By superimposing on the catch curve a-symmetrical curve with parameters the same as the left-hand portion of the catch curve, Figure 8 results. In over half of the groups of estimates in Appendix 9, the above-described graph of N Q closely parallels the symmetrical catch curve. Close parallels between these two curves give weight to the reliability of the WQ distri-bution. The divergence of the catch curve from the symmetrical could be a function of secondary catch and of change of effort. If the symmetrical catch curve is taken to represent the actual migration into the fishing area from the ocean and is representative of incoming stocks then the numbers of fish entering the inlet could also be estimated. As the fishery in Rivers Inlet is solely a gillnet fishery, rough water conditions could be expected to result in a relatively high loss of fish from the nets. If fish are dead in the net and f a l l out they are lost. Of the live fish escaping the nets, some die shortly after; the number dying could be affected by the depth to which the fish sink before recovering. The mortality of fish lost from the nets should be directly, proportional to the number of gillnet marked fish in the escapement. It should also be direct,ly proportional to the catch, i f environmental 37 X li. I i . o m 5 Z 90,000 80,000 70,000 60 ,000 50,000 40,000 30 ,000 20 ,000 10 ,000-Adjusted catch Estimated N 0 Catch curves X I ' I I L "V J _ l _ l L. 26 28 30 2 JUNE 8 10 12 14 16 18 20 22 24 26 28 30 JULY DATE i FIGURE 8. Distribution of estimated WQ and catch through time. Estimated N i s based on the no extremes data from Appendix 9 . Dotted line represents an hypothetical catch curve. The symmetrical curve i s hypothesized as approximating N ; divergence of this curve from the actual catch curve i s a result of secondary catch. conditions (mainly weather) are constant. However, nothing more than arbitrary estimation is possible in putting a value to K (net-induced mortality); therefore, K must be disregarded in computations but remembered in the final interpretation in.that i t will bias estimates upwards i f not taken into account. Unaccounted natural mortality may cause a bias to the estimate of WQ. In the last three years, an increase in the value of hair seal hides has resulted in a decrease in the numbers of this natural predator. Seals were observed only in Owikeno Lake and the upper Wannock. River. Observa-r tions of sea lions were limited to outside the inlet; numbers observed were small. Killer whales were observed irregularly in the area; a pod of 25 was the largest observed. The main body of fish appear to migrate through the inlet rapidly (3 to k days) (see Figures 9 and 10) and then mill at the head of the inlet. This rapid migration coupled with the apparent high rates of exploitation and low incidence of natural predators, leads me to believe that natural mortality is relatively insignificant in this ease. A l l my calculations of estimated H Q disregard the effects of natural mortality. Rivers Inlet sockeye typically mili about in the head of the inlet for variable lengths; of time before migrating upstream. Often, there appears to be a build-up of fish in the head of the inlet, after .which almost a l l the fish migrate upstream within a 12-day period. In the period of build-up, fish tend to move in and out with the tides. When the build-up has reached certain proportions, tidal movements take the fish back and forth across the commercial boundary, thereby making them susceptible to the fishery a second time. The fishery tends to show an increasing con-centration along the boundary as the season progresses, such that up to 39 FIGURE 9 . Mean number of days to recovery site of fish tagged at the mouth of Goose Bay (x). Interconnecting lines represent possible migration pattern of sockeye salmon through the inlet. 1+0 FIGURE 10. Modal number of days to recovery site of f i s h tagged at the mouth of Goose Bay (x). Figure is representative of the rate of migration of the bulk of the f i s h . one-half of the fleet i s fishing as close to the boundary as possible i n the last week of fishing. This high intensity of boundary fishing i s usually coordinated to some extent with the maximum build-up of sockeye in the head of the inlet; as a result, secondary catch of these f i s h can be very high. Catch data are assumed to be limited to primary catch; secondary catch tends to inflate catch data and thereby inflate the estimated original population size The secondary selection resulting from secondary catch w i l l tend to deflate estimated N Q. When secondary catch i s small i n relation to total catch, the sampling regime w i l l tend to cause an under-estimation of W Q . Conversely, when secondary catch i s large in relation to t o t a l catch, the sampling regime w i l l tend to cause over-estimation of N Q. The over-estimation i s reduced or reversed by the action of secondary selection which accompanies high rates of secondary catch. Seasonal increase i n secondary catch i s expected to cause a progressive biasing of the estimated K^. If, over the season, inf l a t i o n of WQ by secondary-catch and deflation of N Q by secondary selection do not cancel each other, there i s bias of N Q from these sources which cannot readily be quantified. Some indication of secondary catch was derived from return of tags applied to f i s h i n the extreme head of the i n l e t . From this source, an estimated minimum rate of secondary catch of 10.7 per cent was calculated. There i s a potential bias resulting from differences in the rates of migration throughout the season. The main bulk of the f i s h appear to travel at approximately the same rate of speed. The mode of the rate of migration from the tagging site to the commercial boundary was approximately three days. Using the mean, the rate of migration appeared to- accelerate throughout the season (early season, 5*1 days; mid-season, k.3 days; l a t season, 3.7 days). The effects of the different rates of speed would he that faster-moving f i s h would he available for exploitation for a shorter time than slower-moving f i s h . If a l l the f i s h at the end of the run are moving slowly, they w i l l show a very high rate of exploitation. Conversely, the f i s h from early in the run may be predominantly fast-moving f i s h , and therefore w i l l show l i t t l e exploitation. This factor, coupled with the progressive increase of effort throughout the season, could result in marked effects in the estimated i f f i s h size or the selected character is not randomly distributed over the t o t a l time period as i t appears to be in my data. The various estimates of in Appendix 10 should, where possible, be corrected for the aforementioned sources of error. Using the graphical technique described above, values for the l e f t - and right-hand t a i l s of the curves can be estimated. Calculations of escapement (N^) using data from the left-hand t a i l have the adjusted catch for the time period removed. A l l estimates of N Q are based on adjusted values of catch. This adjustment was necessary to account for f i s h caught outside the pre-exploitation sampling s i t e . There were 162,910 fi s h caught outside the sampling sit e ; this number should be added to the estimated to get a true pre-exploitation population size. Unaccounted mortality cannot be quantified with the data available. Assuming that no natural mortality occurs, variations i n the rate of net-induced mortality can be applied to the estimated of WQ to demonstrate the possible results. In Appendix 9, K (net-induced mortality) values of 0, 10 and 20 per cent are presented. It appears to be impossible to correct for any bias resulting from secondary catch or secondary selection. The use of the overall (July 2-29) data may eliminate this effect. The secondary catch selection may be one of the reasons that the July 2-29 estimate i s almost always larger than the sum of i t s component estimates. Bias due to different rates of migration throughout the season, i f i t exists, may be minimized by the use of overall data rather than weekly data. One problem with using overall data is that the size of the sample, although more representative, may permit interactions between parts of a sample to damp strong selection; I have assumed no natural mortality and a rate of net-induced mortality of approximately .10. With an allowance for net-induced mortality which occurred between the sampling site and spawning grounds, the escapement estimates are further lowered. I think that the "excluding extreme values" estimates (clumped) from Appendix 9 axe probably the single best estimates. When the assumed rates of unaccounted mortality are applied to these estimates, a probable spawning ground escapement of approximately 175,000 sockeye is achieved. This escapement is from an estimated original population of 900,000 sockeye. Applying the assumed rates of unaccounted mortality to the other estimates results i n estimations of spawning ground escapements of from -194,971 to 952,961 sockeye. If negative escapements are regarded as impossible, the remaining escapement estimates form a central clump ranging from about 100,000 to 350,000 sockeye. The estimated original population size from which these escapements are achieved ranges from 820,000 to 1,120,000. The approximate center of this clump of estimates is at about 250,000 sockeye, considerably higher than the "excluding extreme values" estimate. 1* Comparison of Dichotomous and Mark-Re capture Techniques Disregarding relative cost, "The tag sample procedure is under a l l circumstances more efficient than the change of composition estimation method" (Chapman, 1955). The mark-recapture estimation procedure yields more information for the same amount of effort hut i t operates on broader assumptions than the change of composition method. The following is a tabulation of the assumptions of the dichotomous and mark^recapttire methods of population estimation: DICHOTOMOUS METHOD MARK-RE CAPTURE METHOD closed population recruitment is insignificant during study period Marks do not affect natural * mortality selective removal of at least marks are not lost * one of the population classes removal data are correct a l l marks are recognized and * returned samples before and after marks randomly mix with non marks removal are random marks do not affect vulnerability * to gear Of the mark-re capture technique assumptions, four in particular (asterisked in the foregoing tabulation) are extremely difficult to warrant in many studies, especially those dealing with commercial fisheries. The condition that a l l marks are recognized and returned is often not met because commercial fishermen often fear that stricter regulations will result from a high return of tags. The reward offered for the return of tags often is so small as not to stimulate any effort toward return of tags. Both of these problems could be remedied by a public relations campaign coordinated with higher tag return rewards. In some species the assumptions of no tag loss or tag-induced mortality are justified for only extremely short time periods. Combined with this, the assumption of no gear selectivity relative to the marks adds a major potential bias. The signi-ficance of these assumptions makes the use of the mark-recapture technique more susceptable to bias than the dichotomous technique in a l l but ideal situations. If the assumptions of both methods can be met, the mark-recapture procedure provides data from which rate of recruitment, survival rate over a time period, rate of exploitation and population size can be calculated. Individual and group migrations and distributions can be followed through use of the mark-recapture procedure. The change of composition procedure can provide rate of recruitment (if i t approaches "knife-edge recruitment"), survival, rate of exploitation and population size, but estimates from this procedure have much larger variances than those from the mark-re capture procedure. The major advantage of the change in composition procedure is that the assumptions upon which i t is based are considerably more diminutive than those of the mark-re capture procedure. One advantage of the mark-recapture estimation method is that the estimates are not affected by secondary catch, secondary selection, time lag, etc. The relative cost of the two types of procedure may affect the choice of procedure. "The change of composition method must always be better i f tagging is pro-hibitively costly", but " i f tagging is no more costly than classification, from the large sample point of view, the tag sample procedure is under a l l circumstances more efficient than the change of composition estimation method" (Chapman, 1955) . Two different estimates of population size can be obtained for the purposes of comparison by conducting both markrrecapture and change of composition studies. It is possible to combine the two procedures and get what is probably the best single estimate. If the number of individuals tagged is of such size that at least a moderate number of tags are recovered, the tagged population can be treated as a "population" upon which to apply the change of composition method. This approach can be used provided the tags are not selected for by the method of removal. It has the advantage that the parameters of the tagged population are known and not estimated. If there is reasonable agreement between the mark-recapture and change of composition estimates, the results of both tests may be combined to form a single test. The many sources of error inherent in both the indirect count and the mark-recpature methods of population estimation may tend to make the use of these estimates for management purposes questionable. "However, management will always be faced with the necessity of making population estimates, a good guess must be rated higher than none at a l l . If estimates are guided they will be better than estimates made without such evidence" (Dasmann, 1952) . hi SUMMARY A primary estimate of pre-exploitation population size can be made with knowledge of the compositions of the original and f i n a l populations, and the harvested catch. The estimate can be made through the use of simultaneous equations or a maximum likelihood estimator formulation. Using a derivation of the formula for the inverse of the asymptotic variance-covariance matrix, an estimate of the asymptotic variance can be made. Allowance for additional variance resulting from the use of sub-samples of catch can be made by some additions to the basic variance equation. Estimated original population size is a function of the size of the catch. The rate of change of the estimated original population size relative to catch is determined by a slope factor which is composed of the proportions of the selected character in the pre- and post-exploitation populations and the catch samples; i t is therefore a representation of selective removal. Biased data can greatly affect the estimates of original population size. The amount and direction of bias are determined by the slope factor. Unaccounted mortality in the form of net-induced mortality and natural mortality cause a definite bias in the estimation of the original population size. If the rate of net-induced mortality can be held at a relatively low level (through fishery regulation), changes in the number of fis h lost to natural mortality do not result in any changes in the estimated popu-lation size. In this manner, effects of natural mortality can be eliminated. If the catch distribution over the fishing area of a fishery for migrant species is known, and the rate of net-induced mortality can be assumed to be constant and proportional to catch over the entire fishing area, then taking the pre-exploitation sample from the middle of the fleet (so that half the catch occurs on each side of the sampling site) causes an interaction between the net-induced mortality inside and outside the sampling site such that one cancels the other. In this manner, effects of net-induced mortality can be eliminated. In most cases, the two samples are of approximately equal size, but the accuracy of the estimate can be improved and its variance reduced by adjusting the sizes of the pre- and post-exploitation samples relative to the intensity of selection occurring in the fishery. Assuming no natural mortality, a net-induced mortality in Rivers Inlet of 10 per cent and a latent net-induced mortality which occurs in fresh water of 10 to 20 per cent, approximately 175,000 sockeye escaped to the spawning grounds from an original population of approximately 900,000. The assumptions upon which the change of composition population estimation procedure is based are considerably more diminutive than those of the mark-recapture procedure. The change of composition procedure is less efficient, relative to the final data available, than the mark-recapture procedure. The mark-recapture procedure is not affected by factors which affect the change of composition procedure, such as secondary catch, secondary selection, or time lag. 49 LITERA.TURE CITED Allen, D. L. 1942. A pheasant inventory method based upon k i l l records and sex ratios. Trans. N. A. Wildl. Conf. 7 : 329-333. Chapman, D. G. 1954. The estimation of biological populations. Ann. Math. Statist. 2 5 : 1-15. 1955. Population estimation based on change of composition caused by a selective removal. Biometrika 42: 279-290. Dasmann, R. P. 1952. Methods for estimating deer populations from k i l l data. Calif. Fish and Game.- 3 8 : 225-233. . Davis, W. S. 1959. Field test of Petersen, streamer, and spaghetti tags on striped bass, Roccus saxatilis. Calif. Fish and Game 8 8 : 319-329. Hartt, A. C. 1963. Problems in tagging salmon at sea. I. C. N. A. F. Spec. Pub. 4: 144-155. Holt, S. J. 1963. A method for determining gear selectivity and its applications. I. C. N. A. F. Spec. Pub. 5 : I O 6 - H 5 . Kelker, G. H. 1940. Estimating deer populations by a differential hunting loss in the sexes. Proc. Utah Acad. Sci. A. L. 17: 65-69. Kelker, G. H. 1942. Sex-ratio equations and formulas for determining hunting loss in the sexes. Proc. Utah Acad. Sci. A. L. 19 : I89-I98. L i , J. C. R. 1964. Statistical Inference. Vol. I. Edwards Bros. Inc. ' Mottley, C i McC. 1942. Modern methods of studying fish population. Trans. N. A. Wildl. Conf. 7 : 356-368. Parrish, B. B. 1963. Some remarks on selection processes in fishing operations. I. C. N. A. F. Spec. Pub. 5 : 166-170. Peterson, A. E. 1954. The selective action of gillnets on Fraser River sockeye salmon. I. P. S. F. C. Bull. 5 , 101p. Petrides, G. A. 1949. Viewpoints on the analysis of open season sex and age ratios. Trans. ,N. A. Wildl..Conf; 1 4 : . 3 9 1 - 4 1 0 . Rasmussen, D. I., and E. R. Daman. 1943. Census methods and their appli-cation in the management of mule deer. Trans. N. A. Wildl. Conf. 8 : 369-38O. Ricker, W. E. 1958. Handbook of computations for biological statistics of fish populations. Fish. Res. Bd-i Can. Bull. 119, 300p. Riordan, L. E. 1948. The sexing of deer and elk by airplane in Colorado. Trans. W. A. Wildl. Conf. 13 : 4 0 9 - 4 3 0 . Vaughn, A. E. 1955* Estimation of a biological population which is subject to biased mortality. PhD. Thesis,.Stanford University, 44p. 50 APPENDIX 1. DERIVATION OP N Q ESTIMATION EQUATION Vaughn (1955) demonstrated the derivation of an estimation equation for N Q using only simple algebra. I have transformed her symbols to Chapman's (1955) equivalents to provide continuity. xo/no = proportion of class X individuals in pre-exploitation population. ^(XQ/HQ) = expected number of individuals i n class X i n the pre-exploi-tation population, m /m = proportion of class X individuals in the commercial catch. R(m /m) = R = expected number of individuals in class X in the catch. X X N Q - R = N^ = number of f i s h in the post-exploitation population. N Q(x 0/n 0) - R(mx/m) = number of f i s h i n class X in the post-exploitation population. (N^x^n^) - R(mx/m))/(NQ - R) = proportion of class X in the escapement. n 1(x Q/n )(N ) - (m /m)(R) x- = solving for N. this becomes: 1 NQ - R 0 xiKo - x i R = YoVno " V V m ) R . X1 N0 " niwoVno = X 1 R * n T » ) R ^ ^ o " nixo^ = " n1(mx/m)R) no^xiR " ni(mx/m)R) nofciR - niR x^ 0 " noxi ni x o " V i " nixo This formulation i s basically the same as Chapman's in that the sign reversal is negated by sign cancellation to provide the same answer as Chapman's equation. Vaughn (1955) also demonstrates that this equation i s the maximum likelihood estimator of N Q, Deriving the equation in this manner results in / „ N _ - X 1 R ) 0 - n ; L x 0 - n0x± which i s identical to the aforementioned equations. 51 APPENDIX 2. PRE-EXPLOITATION POPULATION DATA The daily sample size (outer sample) and the number of sets necessary to catch this sample (effort) are pre-sented. Rate of tag return from this sample i s given as a percentage (percent return) and the calculated mean and modal rates of migration (mi./day) and time between tagging and capture (days out) are presented. DATE OUTER SAMPLE EFFORT (SETS) MEAN MODAL PER CENT RETURN mi./day days out mi./day days out 26/6 2 2 27 1 3 28 1 3 29 0 1 30 7 2 1/7 3 4 2 1 3 3 11 2 7.4 2.5 7 3 18.2 4 2 3 5 9 1 4.4 5.5 8 3 22.2 6 33 2 4.7 3.6 5 2 36.4 7 8 9 121 5 5.1 3.9 7 3 33.9 10 413 1 5.8 3.2 12 2 38.5 11 460 3 5.7 2.6 21 1 45.9 12 13 14 15 41 1 3.6 4.8 5 4 30.0 16 272 4 5.9 3.2 7 3 31.3 17 270 3 6.9 2.6 11 2 31.5 18 487 2 7.4 1.8 9 1 35.5 19 20 21 22 156 3 4.6 3.8 5 4 6.4 23 11 3 7.5 3.0 7 3 36.4 2k 97 4 9.1 2.1 21 1.5 18.6 25 1 1 26 27 28 29 30 54 3 31 0 3 1/8 19 3 2,472 65 APPENDIX 3 . COMMERCIAL CATCH DATA The daily commercial catch (total comm. catch), effort (comm. effort) and catch per unit effort (c/f) are presented. The sample size (comm. sample size) and the size of the catch adjusted for the number of f i s h caught outside the outer sample site (adjusted comm. catch) are also presented. COMM. TOTAL ADJUSTED COMM. DATE EFFORT COMM. CATCH c/f COMM. CATCH SAMPLE SIZE 28/6 188 874 4 . 7 219 71 29 208 1,319 6 . 3 330 196 30 200 1 ,748 8 . 7 438 5 1/7 2 Q 348 2 968 6 . 9 624 296 J 5 333 14,757 41 .3 6 ,093 246 6 409 14,305 2 6 . 7 5,722 535 7 419 12 ,844 3 0 . 7 5,138 187 8 Q 814 28,888 3 4 . 3 10,958 495 y 10 i i 12 921 122,053 132.5 86,481 499 13 879 70,383 8 0 . 1 49,268 501 14 827 64,238 77 .7 44,967 513 15 1,503 72,077 4 7 . 9 50,099 499 16 17 18 19 1,119 85,770 5 0 . 0 77 ,648 525 20 1,114 42,402 3 8 . 1 38,162 500 21 1,586 44,277 2 8 . 0 39,927 500 22 23 2k 25 26 749 31,171 4 0 . 8 31,171 421 27 620 17 ,864 2 8 . 1 17 ,864 224 28 664 13,262 2 0 . 0 13,392 281 12,901 641,200 478,501 6,494 APPENDIX k. POST-EXPLOITATION POPULATION DATA The size of the daily escapement samples and lake samples are tabulated, as are the rates of tag return and the effort . ESCAPEMENT EFFORT PER CENT LAKE DATE SAMPLE SIZE (SETS) RETURN SAMPLE SIZE 7/7 3 2 8 15 2 9 10 11 12 13 93 3 Ik 308 1 15 16 17 18 19 20 399 1 21 2*4-9 1 22 23 2k 25 26 If 12 1 27 311 1 28 99 2 29 93 3 30 31 1/8 2 93 3 3 127 3 83 3 5 6 7 8 6.1+ 3.6 17 3.8 13 2.1+ 10 115 58 69 11 8.3 1 2.6 12l+ 85 *+7 79 97 69 23 6o 38 27 12 60 25 2,285 26 1,01+0 APPENDIX 5. ESTIMATED N Q AND ITS VARIANCE CALCULATED FROM AGE GROUPINGS Calculations are broken down into four weekly periods and one overall period. JULY 2-8 JULY 9-15 JULY 16-22 JULY 23-29 JULY 2-29 (overall) 3-YEAR-OLD N o 1+8,077 269,11+9 .68622131+xlO8 -1+15,672 -5^,H9 89^,363 Variance .38761+9660xl09 .111+89722X1012 . 3 2 9 0 9 9 3 x l 0 9 . 6 5 3 P 9 1 1 x l 0 1 0 ^-YEAR-OLD N o Variance 100,1+35 .23902088X10 1 1 301,591 .72551900xl0 9 -135,610 . 2 5 3 6 5 2 8 8 x l 0 9 18,159 .30296l7l+xl0 8 - 2 , 37*+, 550 .21306238X1011* 5-YEAR-OLD N o Variance 6,678 .36028930xl0 9 210,91+3 . 8 0 0 5 9 6 l 7 x l 0 9 21+2,517 . 9 8 2 8 5 3 1 5 x l 0 1 0 262,21+7 .I+223398OXIO11 587,595 .33503990xl0 1 0 WEIGHTED MEAiT 27,192 267,522 156,536 12,229 691,1+00 APPENDIX 6 . ESTIMATED N Q AND ITS VARIANCE CALCULATED FROM 3 CM. LENGTH GROUPINGS weighted means of the four weekly and the overall periods are weighted with the reciprocal of variance. JULY 2-8 JULY 9-15 -JULY 16-22 JULY 23-29 JULY 2-29 (overall) 32-34 CM. N 0 Variance 35-37 CM. N 0 Variance 38-40 CM. N 0 Variance 41-43 CM. N 0 Variance 44-46 CM. N o Variance 47-49 CM. N 0 Variance 50-52 CM. N 0 Variance 00000 . 00000000x10' 35 46,273 .78242633xl0 9 146,653 .63092627x10' 0000000 .22262354x10 30,996 .50088713x10' 54,427 .19935244x10' .12 11 8 10 242,927 .33385263x10' 13 14 -1 ,250,253 .54368676x10' 274,998 .30055725x10^ 265,411 , 1 3 8 l 8 8 8 x l 0 9 249,284 , l 4 6 l l 6 6 4 x l 0 9 156,624 .46501329x10' 307,790 .23396358xlO" 626,706 .25873307x10' 11 10 .11 -15,152 . 2 8 8 3 3 9 0 3 x l 0 9 1 ,294,885 . 8 8 4 2 1 5 9 x l 0 1 3 337,271 .84259212x10--266,235 .11582154x10-10 1 2 24,450 . l6233225xl0 ' 731,052 .20342368x10" 683,608 .12924835x10-.14 .12 12 - 2 5 , 8 8 9 .61855844x10 10 - 8 2 , 4 4 2 .47313858xlO-10 -32,713 . 1 7 8 l 3 2 4 9 x l 0 9 - 2 6 5 , 4 2 2 ,94708354x10 12 23,505 .10275891x10' - 6 8 9 , 2 4 4 .10 ,13 .296607x10' -23,697 Q .68380692X100 -169,667 .12306127X10' 11 836,430 .12005855X10 11 812,541 .74454587X10 10 786,890 .20652878xl0-11 2,663,664 .33550728xl0 1 5 1,421,400 ,51183145x10-.11 2,564,123 . 1 0 3 3 0 6 6 4 x l 0 x : 5 Continued VJ1 ON OJ I CVJ . ^ r« o 3 CVJ H o ir\ co •» ON CM ON •» H CVJ ON CO -4-H IfN t -•v ON CO CM •* CM I CM CO ON co c--o IA X * VO ON CO 0- CO ro IA H e» tfN CO a ON C- H NO NO CVJ O •» H NO CO CO O £ ti H C -•\ ITN ON CM r>. N§ CO SI o l o ON CVJ CO I CO 3 CO H NO NO 3 ON CVJ I CO OJ o H CM CM I CO NO a NO CM ITN IA H O NO CO CVJ NO O NO O CO El ro NO ON CO CO o CM ITN ON C -•\ rH 3 NO IfN CO •$ CVJ -3-CVJ 2 o ITN H 3 S •> NO IA CVJ C- l>-00 H CO ITN CO co 3 CVJ co ON H ON 0 3 0 H r-j CO .* X ON OJ «3CO •\ CO -=f- t -0 ON H OJ H ON NO 0 CVJ CVJ CO co CO ITN « CM • ° 2 » ti ITN ON CVJ 3 CO IfN - * CM O I IfN O H 5? NO O H O ti H H NO O •\ ON •V NO CO NO ITN CO 1 BN CO NO * ITN H I ON ITN ON NO «\ ON o-CO o ON NO CM CO CVJ O H O H TT 3 3 O CO q t - q <i °. -=r 0 NO rH 0 rH CO rH H rH t - H -=r X 0- X 0 X t>- X NO X rH CO 00 * CVJ •\ ON n H 3 ON t - CVJ CO CO IA ITN CO ON -=T ITN ON CVJ ON rH CO ITN rH NO CO CO -=t- H CO CVJ ON CM ^ cvi CO 1 0- CO -* ON H ON CO ITN ON CVJ H • ITN • H • CO • • CO I CVJ t l ON o I NO 00 NO -3" s CO CO CVJ CO 3 ti CO ITN NO CO CO CO NO ON O o ti CO ON ^ & O NO -=J" t -ITN 00 o o o o o o H NO ON 3N NO CO CO ON 1 CO •x CO 00 CVJ 2L o o o o o o i ITN 81 CVJ o NO CO cb~ •S H CVJ i r v a > LTN I CO ITN Cl) O . s o S 00 a > IfN 1 NO IfN 0 O 9 • r l O JS H a > NO I ON IfN a o 8 o j3 NO cvi NO d> o 8 ^ o S > I IA NO CO o a -rt o S O S !> C -I 00 NO Q 3 H a APPENDIX 7 . ESTIMATED N Q AND ITS VARIANCE CALCULATED FROM 8 CM. LENGTH GROUPINGS Weighted means of the four weekly and the overall periods are weighted with the reciprocal of variance. JULY 2-8 JULY 9-15 JULY 16-22 JULY 23-29 JULY 2-29 (overall) 32-39 CM. wo Variance 59,478 . 2 2 1 5 7 2 9 8 x l 0 1 0 283,901 . 2 0 5 8 4 0 2 6 x l 0 9 2 ,048,207 .2O78l046xl0lif -50,744 . 4 l 0 9 6 4 0 0 x l 0 9 1 ,046,991 .19110152X1011 k0-k7 CM. No 32,263 267,712 -887,688 .13394448X10111 - 2 7 , 6 3 1 553,597 Variance .27275895xl0 9 . 4 l 4 2 1 3 2 1 x l 0 9 . 7 2 0 0 1 1 2 3 x l 0 9 .12717505X10 1 1 48-55 CM. wo Variance -95 ,980 . 1 3 5 4 4 7 7 9 x l 0 1 2 344,208 . 1 3 3 1 3 0 5 0 x l 0 1 0 909,156 . 3 3 3 8 0 7 8 5 x l 0 1 2 13,397 Q .11951154x10 249,127 . 3 3 8 i 7 9 0 1 x l 0 1 3 56-63 CM. No Variance 14,249 •35979525x10 s 131,593 . 1 2 8 7 0 9 9 3 x l 0 1 0 -1 ,455,880 .15315108x10 H 3 , 5 6 l . 3 4 3 8 3 3 5 8 x l 0 1 0 26,435 . 7 7 2 1 0 1 0 1 x l 0 1 0 64-71 CM. No 19,855 117,508 1 ,979,018 29,472 3 ,180,897 Variance .67082272XIO9 .10665773xl0 1 2 . 3 2 1 7 4 3 3 8 x l 0 1 2 . 2 3 5 9 2 0 9 5 x l 0 9 . 6 4 7 3 5 0 4 8 x l o 1 2 WEIGHTED MEAN N Q 17,063 270,997 1,400,495 12,139 406,869 APPENDIX 8 . ESTIMATED N Q AND ITS VARIANCE CALCULATED FROM 13 CM. LENGTH GROUPINGS Weighted means of the four weekly and the overall periods are weighted with the reciprocal of variance. JULY 2-8 JULY 9-15 JULY 16-22 JULY 23-29 JULY 2-29 (overall) 32-1+4 CM. Ko 9H,759 48,044 659 1 2,453 1 ,243,704 Variance .37217814X1011 . 2 5 2 0 3 2 5 4 x l 0 1 0 . 4 2 l 2 4 7 8 4 x l 0 4 . 2 1 2 0 5 6 4 4 x l 0 6 . 8 2 4 9 5 9 8 0 x l 0 9 45-57 CM. N 0 1,060,618 43,165 2,166 2,701 1 ,991,404 Variance . 2 2 4 9 8 2 1 3 x l 0 1 2 . 2 5 9 l 4 4 5 9 x l 0 1 2 .60984239x10 .26028221X107 . 8 5 8 3 7 4 8 3 x l 0 1 0 58-70 CM. No - . 0 0 0 0 47,210 991 2,523 - . 0 0 0 0 u Variance -.OOOOOOOOxlO"38 .I4l01525xl010 4 .29501120x10 .1024O694X10 6 - . 0 0 0 0 0 0 0 x 1 0 ' 3 8 WEIGHTED MEAN N Q 932,889 47,494 284 2,506 1 ,309,263 VJI CO 59 APPENDIX 9 . DIFFERENT ESTIMATES OF N Q RESULTING FROM A NUMBER OF TREATMENTS OF THE DATA SOURCE JULY 2-8 JULY 9-15 JULY 16-22 JULY 23-29 JULY 2-29 (overall) ADJUSTED CATCH 27,9H 230,814 155,737 62,427 476,890 AGE WEIGHTED MEAN 27,192 267,522 156,536 12,229 691,400 3 CM. WEIGHTED MEAN 21,878 259,146 36,061 - 6 , 7 4 3 466,618 8 CM. WEIGHTED MEAN 17,063 270,997 1 ,400,495 12,139 406,869 13 CM. WEIGHTED MEAN 932,889 47,494 284 2,506 1 ,309,263 OVERALL WEIGHTED MEAN 20,295 311,471 285 2,544 947,556 OVERALL-13 CM. WEIGHTED MEAN 20,393 272,359 101,011 13 ,390 527,175 OVERALL-13 CM. NO NEGATIVES WEIGHTED MEAN 20,479 272,452 141,573 16,513 538,002 GRAPHICAL MODE ALL DATA 32,500 157,000 225,000 4 , 0 0 0 810,000 ABSOLUTE #'S GRAPHICAL MODE 36,500 250,000 240,000 30,000 920,000 NO EXTREMES GRAPHICAL MODE 28,500 255,000 174,000 17,000 765,000 OMITTING NEGATIVES GRAPHICAL MODE 31,000 238,000 245,000 2 3 , 0 0 0 915,000 APPENDIX 10. OVERALL RESULTS ESTIMATED N FROM APP. 9 INCLUDING TAILS OF CURVE ALLOWANCE FOR ADJUSTMENT OF R K = .10 K = .20 AGE wos 463,479 482,479 645,389 617,819 597,558 N i s -13,411 3,978 3,978 -39,883 -76,435 N0T 691,400 710,400 873,310 825,021 787,492 N1T 214,510 231,899 231,899 167,319 113,499 3 CM. Nos 310,342 318,342 481,252 468,601 460,777 N i s -166,548 -160,159 -160,159 -189,101 -213,216 N0T 466,618 474,618 637,528 610,671 591,007 N1T -10,272 -3,883 -3,883 -47,031 -82,986 8 CM. Nos 1,700,694 1,717,754 1,880,664 1,740,791 1,626,954 N i s 1,223,804 1,239,253 1,239,253 1,083,089 952,961 N0T 406,869 423,929 586,839 564,591 548,766 N1T -70,021 -54,572 -54,572 -93,111 -125,227 13 CM. Nos N i s K0T W1T 983,173 506,283 1,309,263 832,373 1,034,457 555,956 1,360,547 882,046 1,197,367 555,956 1,523,457 882,046 1,119,611 461,909 l , 4 l 6 , 0 6 l 758,359 1,057,540 383,547 1,329,292 655,299 Continued APPENDIX 1 0 . Continued. ESTIMATED N INCLUDING TAILS: ALLOWANCE FOR FROM APP. 9 OF CURVE ADJUSTMENT OF R K = .10 K = .20 OVERALL WEIGHTED MEAN Nos 33*+,595 3^0,59^ 503,501+ 1+88,805 1+79,322 N i s -1^2,291 -137,902 -137,902 -168,879 -194,971 "oT 9^7,556 953,556 1,116,1+66 1,01+6,071 990,122 "IT lr70,066 V 7 M 5 5 l+7l+,l+55 388,369 316,129 OVERALL-13 CM. Nos 1+07,153 1+23,653 586,563 56l+,3l+l 5j+8,537 WEIGHTED MEAN N i s -69,737 -5»+, 81+8 -5l+,8l+7 -93,361 -125,1+56 N 0 T 527,175 5^3,675 706,585 673,1+51 61+8,555 "IT 50,285 65,171+ 65,171+ 15,71+9 -25,1+38 OVERALL-13 CM. "os 1+51,017 1+69,267 598,899 605,807 586,51+8 OMITTING NEGATIVES " i s -25,873 -9,23!+ -9,23l+ -51,895 -87,1+1+5 "OT 538,002 556,252 719,162 68l+,89l 659,01+2 "IT 61,112 77,751 77,751 27,187 -1^,951 ALL DATA GRAPHICAL MODE "os 418,500 1+36,500 599,^10 576,021 559,292 " i s -58,390 -1+2,001 -1+2,001 -81,681 -n.1+,701 N 0 T 810,000 828,250 991,160 932,151 885,692 "IT 333,110 3^9,^99 31+9,1+99 27I+, 1+1+9 211,699 Continued APPENDIX 10. Continued ESTIMATED N Q FROM APP. 9 INCLUDING TAILS OF CURVE ALLOWANCE FOR ADJUSTMENT OF R K = .10 K = .20 ABSOLUTE VALUES N 0 S 556,500 584,750 747,660 710,791 682,792 OF ALL DATA N 1 S 79,610 106,249 106,249 53,089 8,799 N 0 T 920,000 948,250 1,111,160 l,o4l,246 985,692 W 1 T 443,110 469,749 469,749 383,544 311,699 EXCLUDING EXTREME VALUES Nos 474,500 497,500 660,410 631,481 609,992 N 1 S -2,390 18,999 18,999 -26,221 •64,001 N 0 T 765,000 788,000 950,910 895,565 852,162 288,110 309,499 309,499 237,863 178,169 OMITTING NEGATIVES Hos 537,000 561,000 723,910 689,201 662,992 N 1 S 60,110 82,499 82,499 31,499 -11,001 NOT 915,000 939,000 1,101,910 1,032,838 977,992 N 1 T 438,110 460,499 460,499 375A36 303,999 


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