North American Journal of Fisheries Management 3:66 77, 003 Coyright by the American Fisheries Society 003 Tag Reorting Rate Estimation: 3. Use of Planted Tags in One Comonent of a Multile-Comonent Fishery WILLIAM S. HEARN* Commonwealth Scientific and Industrial Research Organisation, Marine Research, Post Office Box 0, North Beach, Western Australia 600, Australia JOHN M. HOENIG Virginia Institute of Marine Science, College of William and Mary, Post Office Box 346, Gloucester Point, Virginia 306, USA KENNETH H. POLLOCK Deartment of Statistics, North Carolina State University, Raleigh, North Carolina 7695-803, USA DANIEL A. HEPWORTH Virginia Institute of Marine Science, College of William and Mary, Post Office Box 346, Gloucester Point, Virginia 306, USA Abstract. Tag return models are used to estimate survival and tag recovery rates. With additional information on tag reorting rates, one can searate the survival rate into its fishing and natural mortality rate comonents. One method of estimating the tag reorting rate is to secretly lant tags in fishers catches. However, if the fishery has more than one comonent, it may not be ossible to lant tags in all comonents. Nevertheless, it is ossible to estimate the reorting rates of all comonents in a multile-comonent fishery and the fishing and natural mortality rates, if at least one comonent has a known reorting rate and the catches are known for each comonent. We simulate a variety of tag return exeriments in which tags are lanted in one comonent of a multicomonent fishery. The simulations show that this method is most effective (i.e., rovides good recision of arameter estimates) when a sufficient number of tagged fish are lanted into a fishery comonent with a high reorting rate and with a high roortion of the total catch. It is also advantageous to encourage the reorting of tags in the fishery comonents without lanted tags. We rovide a method for testing various model assumtions when it is ossible to lant tags in more than one comonent. The estimation of total mortality rates from multiyear tagging data has a well-develoed theory. As described by Brownie et al. (985), such estimation does not require knowledge of tag reorting rates (i.e., the robability that a fisher who catches a tagged fish will reort the tag to the aroriate authorities), nor does it require that tag reorting rates be estimated from the data. However, when information about the tag reorting rate is available in a multiyear study, the mortality from natural causes and from one or more fisheries can be estimated searately (Pollock et al. 99; Brooks et al. 998; Hearn et al. 998; * Corresonding author: bill.hearn@marine.csiro.au Present address: Commonwealth Scientific and Industrial Research Organisation, Marine Research, Private Bag 5, Wembley, Western Australia 693, Australia. Received June 8, 00; acceted Aril 6, 00 Hoenig et al. 998a, 998b). Also, in a single year, the exloitation rate (the fraction of the stock resent at the start of the year that is harvested during the year) can be estimated from tagging data if the tag reorting rate is known. Two aroaches can be used to obtain information about tag reorting rates. The first is to make use of the information about the tag reorting rate that is imlicit in the tagging data (Youngs 974; Siddeek 989). This internal information comes from the contrast between exeriments and is generally quite weak (Hoenig et al. 998a), but it can be enhanced through secial design of the tagging study (see Hearn et al. 998; Frusher and Hoenig 00a, 00b). The second aroach is to conduct auxiliary studies to obtain information about reorting rates. These may involve conducting a creel- or ort-samling rogram (Pollock et al. 99), releasing secial, high-reward tags 66
TAG REPORTING RATE ESTIMATION 67 for which the reorting rate can be assumed to be 00% (Henny and Burnham 976; Nichols et al. 99; Pollock et al. 00, 00b), lacing observers on a ortion of the boats (Hearn et al. 999; Pollock et al. 00a), and lanting a known number of tagged fish in the catches and noting the fraction of tags returned by the fishers. We focus on the use of lanted tags to estimate the tag reorting rate. Although there are a number of examles of the use of this aroach (shrim: Costello and Allen 968; Cambell et al. 99; menhaden: Ruert et al. 984; sort fish: Green et al. 983; tunas: Hamton 997), this method has not been critically reviewed from the ersective of the required assumtions and study design. Secifically, it is frequently the case that only a ortion of the boats can be studied using lanted tags. For examle, an investigator may have access to boats from one country s fleet but not those from another country, or the rocessing rocedure for one fleet may be more conducive to the use of lanted tags than that of another fleet. We develo a method to estimate the reorting rate for all fishery comonents rovided that () the catches are known (or can be estimated) for each comonent and () the reorting rate of at least one comonent can be estimated by means of lanted tags. The roblem of multicomonent fisheries, in which some comonents offer no oortunity for samling, is very common. Hearn et al. (999) and Pollock et al. (00b) showed that the reorting, fishing mortality, and natural mortality rates can be estimated in a multile-comonent fishery in which catch is known by comonent rovided the reorting rate is 00% in one comonent. They treated boats with observers as the comonent with all tags reorted (which is analogous to the second requirement in revious aragrah). Thus, the use of lanted tags in this aer shows the generality of this aroach (i.e., it furnishes a case in which the reorting rate of at least one comonent can be estimated). We begin by describing a lanted-tag study in a general context and the structure of the tagging data that arises from this design. We then discuss the required assumtions and resent some simulation results that rovide information on exerimental design strategies. We show how to deal with the likely nonrandomness in the tag lanting rocedure as well as how to check some model assumtions when it is ossible to lant tags in more than one fishery comonent. We conclude with advice on the design of lanted-tag studies. Structure of a Tagging Study with Some Planted Tags As in a Brownie model, fish are tagged with normal tags (i.e., not lanted tags) at the start of each year i (i,...,i) and recovered during year j (j i,..., J). Suose that there are K comonents to a fishery (for examle, different fleets in a commercial fishery or a fishery with recreational and commercial comonents) and, of all the tagged fish catured in year j, the roortion catured by comonent k is jk. The robability that a tagged fish will be reorted in year j from fishery comonent k, given that it has been caught by comonent k, is jk. The exected number of tags reorted from comonent k involving fish that were tagged in year i and recovered in year j is with j Niu j jk jk S h, (j i) E(R ) ijk h i Nu, (i j) i j jk jk K jk k In the first set of equations, N i is the number of fish that were tagged and released at the start of year i, S h is the annual survival rate in year h, u j is the annual exloitation rate (i.e., the fraction of the survivors that is caught) in year j, and j h i S h is the fraction of the N i tagged fish that survive u to the beginning of year j. The cell robabilities of returned recatures for a two-comonent fishery are shown in Table for the case where the and arameters are held constant over years. The table is arameterized in terms of exloitation and survival rates (u and S), but in ractice we usually want to link these arameters. This can be accomlished by rearameterizing the model in terms of additive instantaneous rates of fishing and natural mortality, F and M. Then, survival rates can be exressed as S ex( F M). The exloitation rate is also a function of the fishing and natural mortality rates, but its nature deends on the timing of the forces of mortality. If all fishing occurs as a ulse at the start of the year (i.e., during a short eriod of time, which is a Ricker [975] tye fishery), then the exloitation rate is u ex( F).
68 HEARN ET AL. TABLE. Cell robabilities for a multiyear tagging study in which N i tagged fish are released in year i (i,, 3) and recatures are obtained from two fishery comonents over a 4-year eriod. In fishery comonent, N i tags are lanted in year i; S the annual survival rate; u the exloitation rate; the tag reorting rate; and the fraction of the tagged fish caught in a year that were caught by fishery comonent. Note that S i ex( F i M), where F i is the instantaneous rate of fishing mortality in year i and M is the instantaneous rate of natural mortality, which is secified to be constant over all years. For tye (ulse) fisheries (Ricker 975), the annual exloitation rate in year i is given by u i ex( F i ); for tye (continuous) fisheries, it is given by u i (F i )/(F i M)( S i ). Year of tagging Number tagged or lanted Fishery comonent Probability of tag recovery in year 3 4 3 3 3 4 N N N 3 N N N 3 N 4 u u ( ) S u S u ( ) u u ( ) S S u 3 S S u 3 ( ) S u 3 S u 3 ( ) u 3 u 3 ( ) S S S 3 u 4 S S S 3 u 4 ( ) S S 3 u 4 S S 3 u 4 ( ) S 3 u 4 S 3 u 4 ( ) If the ratio of fishing and natural mortality is constant over the year, which is a Ricker (975) tye fishery, then F u [ ex( F M)]. F M A formulation allowing for an arbitrary attern of fishing over the course of the year is given by Hoenig et al. (998a). This can be generalized to accommodate cometing fisheries with different timing throughout the year (Brooks et al. 998). We cannot observe the roortion of the tagged fish caught by fishery comonent k ( jk ), but under the condition of comlete mixing (see Assumtions below), we can assume that it is similar to the fraction of the total catch catured by comonent k, which can be estimated from known catch and effort statistics or survey data. In our analyses and simulations we treat jk as a known constant, but we advise how to incororate catch uncertainty into our method. We roose that in one or more comonents of the fishery, tagged fish are lanted in catches to determine the fraction of the fish reorted in the comonent(s). In some cases, the lanting of tags can be viewed as indeendent Bernoulli rocesses. For examle, menhaden have been tagged with coded wire tags. In the fish rocessing lants, the fish are cooked and ground u and the resulting mush asses by owerful magnets that recover the wire tags. Thus, fish can be lanted with known tag numbers in the rocessing stream to measure the robability of tag recovery. In general, however, tagged fish cannot be randomly lanted in the catches of all vessels over all years. Rather, reorting studies involve a two-stage or multistage rocess in which certain vessel tris are selected each year and then tagged fish are randomly lanted in the catch within these tris. If we ignore this samling structure and treat the lanted tags as reresenting random trials from a Bernoulli rocess, we will tend to overestimate the recision of our estimate of the reorting rate. We will consider this further in the Results and Discussion sections. For the urose of describing the lanted-tag study design, however, we treat the number of the returned lanted tags as a binomial random variable with two arameters: N, the number of lanted fish, and, the tag reorting rate. The cell robabilities are shown in Table for the case in which tags are lanted in only fishery comonent. Each batch or cohort of normally tagged fish (corresonding to a row in Table ) is viewed as a random samle from a multinomial distribution. As in a Brownie model, the likelihood is thus roortional to the roduct over all cells of the cell robability raised to the ower corresonding to the observed number of recatures in the cell. (Note that there is an imlicit column of cells for fish never recatured; this must be included in the likelihood.) Estimation can be accomlished as follows: For the fishery comonent with lanted tags, the value of can be estimated from the lanted tags; the value of is assumed known from the landings of all of the comonents. Therefore, the likelihood for the comonent with lanted tags is equivalent to the likelihood for an instantaneous-rates for-
TAG REPORTING RATE ESTIMATION 69 mulation of the Brownie models (as described by Hoenig et al. 998a); the only difference is the inclusion of the and factors, which are known and estimable, resectively. Hence, the values of the fishing and natural mortality rates can be estimated from the data from the fishery comonent with the lanted tags. For the other fishery comonents, is known from landings data and estimates of F, M, and u are available from the data for the comonent with lanted tags. Therefore, all that remains is to relate the observed number of recatures to the exected number by aroriate choice of the reorting rate, k, for fishery comonent k. Let the comonent of the fishery with the lanted tags be denoted as comonent. Assuming that each lanted fish reresents an indeendent Bernoulli trial, the maximum likelihood estimate of the tag reorting rate for this fishery comonent in year j is R j N j ˆ, () j where N j is the number of tags lanted in the catch from comonent in year j and R j is the number of tags reorted from the lantings. Given this estimate of the reorting rate, the reorting rate from any other comonent in year j can be estimated by the equation (see Aendix ) j j Rijk Rijk j i j j i j j jk j jk Rij Rij i i R ˆ jk ˆ j, () N where R ijk is the number of tags reorted by comonent k during year j from fish that were tagged in year i. (Essentially, the middle factor in the right-hand side of the above equation is the ratio of all recatures from the two comonents in year j.) Assumtions All the assumtions required for use of an instantaneous-rates formulation of the Brownie models (Hoenig et al. 998a) are required for the multile-comonent model with lanted tags. These assumtions have been reviewed by Pollock et al. (00): () The tagged samle is reresentative of the oulation being studied. This imlies that fish are thoroughly mixed, so that all fishery comonents have the same catch rate of tagged fish er unit of catch (tags/catch). () There is no tag loss from fish. (3) Survival rates are not affected by tagging. (Short-term tag loss and tag-induced mortality can be evaluated by means of holding exeriments, e.g., Latour et al. 00.) With resect to assumtions and 3, it is noted that there are sometimes differences in the roficiency of taggers (Hearn et al. 99) that are ideally mitigated by strict tagging rotocols. (4) The fishery comonent and time of recature of each tagged fish is reorted correctly, that is, the recature is tallied in the correct cell of the recovery matrix. (Sometimes tags can be returned several years after the fish are recatured.) (5) The fate of each tagged fish is indeendent of the fates of other tagged fish. (6) All tagged fish within a release cohort have the same annual survival and recovery rates. (7) The instantaneous fishing and natural mortality risks are additive. Five additional assumtions are needed for the lanted-tag method with multicomonent fisheries: (8) The catch data for each comonent of the fishery are accurate. (In articular, there is no underreorting of the catch in some comonents.) (9) The tags are lanted surretitiously and fishers do not see any tags being lanted (otherwise the behavior of fishers might be altered). This almost certainly recludes the use of lanted tags in recreational fisheries because there would robably be no oortunity to lant the tags without being seen. This assumtion does not aly if the tags are detected automatically by machine. (0) The lanted tags are identical to the normal tags and are laced in such a way as not to look unusual (otherwise the behavior of fishers might be altered). For examle, using lanted tags with sequential numbers would be roblematic. () Planted tags are rare in any individual fisher s catch (the occurrence of an unusually high number of tags might change the behavior of fishers), and they cover all years. () The tags are lanted early enough after fish are caught that no art of the rocess for finding and reorting normal tags is omitted. Matlock (98) describes scientists secretly
70 HEARN ET AL. lanting a tag into a fish in creels. Most fishers were contacted later and few knew or susected that the tags were lanted. However, this validation rocess would make fishers aware of scientists real urose, so it would not be feasible for multiyear tagging rojects. Test of Assumtions If tagged fish are lanted in all comonents of a fishery and their catches are given, the assumtions of the tagging model can be tested. The intuitive argument is as follows: The tags lanted in each fishery comonent can be used to estimate that comonent s tag reorting rate. For each comonent, the tag recovery rate er fish landed can then be converted into the rates of tagged fish catured er fish landed (tags/catch) by dividing by its estimated reorting rates. For examle, if tagged fish are reorted er,000 fish landed and the estimated tag reorting rate is 0.5, then we estimate that 4 (i.e., /0.5) tagged fish were caught er,000 landed. The estimate of tags/catch for comonent k in year j is therefore j R ijk R jk i jk jk jk jk ˆ ˆT jk, (3) ˆ C N where ˆ jk is the tags returned/catch (i.e., not adjusted for the reorting rate) and C jk is the catch. Assuming that the normally tagged fish are fully mixed in the catch, the catches are correctly tabulated, and the tag lanting is imlemented correctly in all comonents, in a articular year j the estimates of Tˆ jk should be the same for all K comonents. It is imortant to note that equation (3), which is used for the estimation of Tˆ jk, requires only data that is entirely collected within comonent k (i.e., indeendently of other comonents). This allows us to examine the Tˆ jk residuals, as shown in Aendix. If some of the residuals are large, this casts doubt on the assumtions. Simulation Studies We used the SURVIV rogram (White 983) to simulate observations from fisheries with secific characteristics. In this aroach, the user secifies a formula for the exected values for each cell of the recovery matrix and the number of lanted tags recovered; the rogram then generates,000 samles from multinomial distributions with the secified arameters. We simulated tag recoveries from a study with 3 years of tagging data and 4 years of recature data, assuming a tye fishery. The fishery consisted of two comonents. Parameters held constant over all scenarios were as follows: F F F 3 F 4 0.3/year; M 0./year; N (the number of fish with lanted tags each year) 50; and N (the number of normally tagged fish each year),000. The tag reorting rates,, and the fraction of the total catch taken by the first comonent,, were held constant over time but varied among scenarios as follows: Exeriment. The values of and were held constant at 0.4 and 0.3, resectively; the value of was varied from 0. to.0. Exeriment. The values of and were held constant at 0.8 and 0.4, resectively; the value of was varied from 0. to.0. Exeriment 3. The value of was held constant at 0.4, and the exected number of tags recovered from the first fishery comonent was held constant ( 0.4) as the value of was varied among scenarios from 0.4 to.0. Exeriment 4. The values of and were held constant at 0.8 and 0.3, resectively; the value of was varied from 0 to.0. For each of the,000 samles in every scenario, we estimated all four fishing mortality rates, the natural mortality rate, and both tag reorting rates. We knew a riori that the maximum likelihood arameter estimates were asymtotically unbiased, and in all our simulations excet one the means of the estimated arameter values were close to the actual arameter values. The one case with bias was due to estimating a arameter close to the boundary of the arameter sace (i.e., estimating when it was close to.0); we discuss this in Results. Therefore, we focus on the coefficients of variation (CVs) of the arameter estimates, which are defined as 00 SD/mean. Results For exeriment, the CVs of all the arameter estimates are lotted against (Figure ). Similarly, the CVs from exeriments, 3, and 4 are lotted against,, and in Figures, 3, and 4, resectively. The CVs of all arameter estimates decline with increases in the value of (Figure ), (Figure ), and (Figure 3), and (Figure 4), with one excetion: the CV of the estimates of in exeriment first decreases and then increases as increases (Figure ). In the simulation results from exeriment 4, a
TAG REPORTING RATE ESTIMATION 7 FIGURE. Effects on the coefficients of variation of the arameter estimates that result from varying the tag reorting rate of fishery comonent ( ), the comonent with lanted tags. Other variables are defined as follows: M the natural mortality rate er year of fish in the fishery; F F 4 the fishing mortality rates of fish recatured in years 4; and the tag reorting rate of fishery comonent. bias occurred in the estimate of when the actual value of was close to.0. For examle, at an actual value of, the mean of the estimates from the simulation was 0.967, which reresents a negative bias. This came about because the estimate of (from equation ) was less than.0 in about one-half of the simulated data sets while it was equal to.0 in the others (a reorting rate greater than.0 was not ossible). Thus, the mean estimate was areciably less than.0. This also resulted in a reduction of the coefficients of variation of the estimates of when was close to.0 (Figure 4). In normal fisheries we do not exect the reorting rates to be close to.0, so this does not aear to be a major roblem. Note that in exeriments and 3 the estimates were not biased when they were set equal to.0 (Figures, 3). This is because is estimated directly from the lanted tags (equation ), so that if.0 the estimate of will always be.0. In summary, the difference between the bias characteristics of the two reorting rate estimates is FIGURE. Effects on the coefficients of variation of the arameter estimates that result from varying the fraction of the total catch that is taken by fishery comonent ( ). due to the fundamental roerties of their estimating equations and the data analyzed. For the case with 0.8, 0.4, and 0.3, which was common to all of the exeriments, we ran the simulated exeriment with 5 lanted tags er year (i.e., one-half the lanted tags). The CV of increased by 38%, which would be exected because is directly estimated from the lanted tags (equation ) and the number of observations was halved. However, the CVs of F to F 4, M, and increased by less than 9% comared with those derived from the exeriment with 50 lanted tags er year. Discussion Design Considerations Based on the Simulations The recision of the mortality and tag reorting rate estimates increased markedly as the reorting rate of the fishery comonent with lanted tags ( ) aroached.0 (Figure ). Similarly, recision imroved for all arameters excet as the fraction of the total catch taken by comonent aroached.0 (Figure ). An intuitive exlanation for these results is that the normal tags returned from the second comonent rovide infor-
7 HEARN ET AL. FIGURE 3. Effects on the coefficients of variation of the arameter estimates that result from varying and while the roduct is held constant. A constant roduct imlies that the number of tags recovered from comonent is constant. FIGURE 4. Effects on the coefficients of variation of the arameter estimates that result from varying. Note that the coefficient of variation of at 0is0/0 (i.e., undefined). mation on the survival rate (or equivalently, the sum F M) but extremely little information on tag reorting rate (and thus extremely little information on how to aortion the total mortality rate to its comonents). By contrast, the tag returns from the first comonent rovide information on the sum F M as well as information on how to aortion the total mortality to its comonents. Therefore, the larger the fraction of the normal tags returned by comonent (because either or or both are high), the better the recision. It was reviously noted that the CV of the estimates of in exeriment first decreases and then increases as increases (Figure ). This can be established analytically from equation (). Intuitively, it is because estimation of deends on there being tag returns for both fishery comonents; the number of normal tags returned by comonent becomes small when is close to zero, and the number of normal tags returned by comonent also becomes small when is close to.0. Our third exeriment held the roduct constant. This imlies that the exected number of tags recovered from comonent was held constant. Here, recision varied considerably as and varied, even though the roduct was constant (Figure 3). This shows that the size of is more imortant than that of in determining the recision of the estimates. Thus, it is generally better to lant tags in the comonents with high reorting rates. This might aear to go against common sense, as one might intuitively exect it to be more effective to secretly check on the bad guys (the comonent with the low reorting rate) than the good guys (the comonent with the high reorting rate). By way of illustration, consider a comonent with 00% reorting, though this is unknown to scientists. Planted tags in that comonent will readily and efficiently reveal that fact to a high recision. At the other extreme, consider a comonent with a 0% reorting rate. That fact will be known without lanted tags, but it brings no information to the estimation of the mortality arameters. In exeriment 4, we varied the value of while holding everything else constant. The higher the tag reorting rate from the fishery comonent
TAG REPORTING RATE ESTIMATION 73 without the lanted tags, the better was the recision of the estimates (Figure 4). Even though it is ossible to estimate F, M, and u without considering the normal tagging data from the comonents with no lanted tags (see CVs of arameters in Figure 4 when the reorting rate by comonent is 0%), the additional data from these comonents will imrove the recision of the estimates. This is because the data from the comonents without lanted tags rovide information about the survival rate, S, or equivalently, about the sum F M. The simulation results lead us to the conclusion that the method described in this article will work best when () the tag reorting rate ( ) in the fishery comonent with the lanted tags is close to.0; () the comonent with the lanted tags comrises a large fraction of the total fishery ( is close to.0); and (3) the tag reorting rate in the comonent without the lanted tags is close to.0 (though this is a secondary consideration). Mixing Assumtion All fishery comonents must have the same catch rate of tagged fish (tags/catch). For examle, the exected catch rate of tagged fish for all fisheries might be tagged fish er 0,000 fish caught. This imlies that the tagged fish are randomly distributed over the oulation, so that a decision by the fleet catains of one comonent to fish in a articular area has no influence on the catch rate of tagged fish er landed fish. (Obviously, the catch of fish er unit effort will be affected.) This assumtion can be met if fish are tagged throughout the area inhabited by the stock in roortion to their local abundance. Local abundance can be judged in terms of local catch er unit effort. For examle, if secimens are obtained with a trawl, it would be aroriate to tag 0% of the catch from each tow but not to tag 0 fish from each tow. Another way to hel assure that the assumtion is met is to tag fish well before the start of the fishing season so that tagged fish have a chance to mix randomly throughout the oulation (though even then such factors as schooling behavior might imede thorough mixing). Still another way is to assume that mixing occurs after a delay, in which case the data analysis method would be adjusted as described in Hoenig et al. (998b). Model Checking Using Residuals If tagged fish are lanted in two or more comonents of a fishery, the tagging model can be checked for violations of the assumtions. In Aendix, we derive a method to check for the equality of tags/catch in the various comonents that have lanted tags. A large discreancy between the estimates could be due to several factors. The first is the failure of tagged fish to mix throughout the oulation. This would cause bias in any model that does not allow for nonmixing, such as a Brownie model. The second is incorrect tabulation of catches of the various fishery comonents. For examle it may be due to a oor data collection and rocessing rocedure, deliberate decetion by fishers, illegal fishing, or ghost fishing by lost nets. This would not cause bias in Brownie models because such models do not use catch data. However, in the instantaneous-rates models of this aer or Hearn et al. (999), an error that affects one fishing comonent more than another will lead to bias in the estimate of (and hence in the estimates of the mortality rates). Catch errors would also be of concern in the assessment and management of the stock (e.g., allocation of quota). A third factor that would lead to biased estimates is oor imlementation of the tag lanting rocedure in some of the fishery comonents. For examle, fishers in one comonent might return all tags whenever observers or lanters are resent but return few tags at other times. In an extreme case, all lanted tags might be returned when only 50% of the normal tags are returned. For this comonent, the estimated number of recatured tagged fish would be underestimated by 50%. Note that researchers may attemt to lant tags in several (or all) fishery comonents and later it may be found that assumtions 9, which ertain to lanted tags, cannot be (or have not been) comlied with for some comonents. In such a case the entire study is not ruined, you just need it to work for one comonent (if you know the catches by comonent). However, then one cannot check for assumtion violations as just described. Bernoulli Assumtion of Planted Tags The Bernoulli assumtion is not likely to be met. For examle, if two tags are lanted at the same lace and at about the same time, their robabilities of being found and returned are likely to be deendent. In the extreme case, in which the tags are either both returned or both not returned, it is as if only one tag were lanted (i.e., as if only one-half of the tags were lanted). In the Results section a case was considered in which half the number of tags was lanted, and it was found that aart from the CVs of the arameters increased
74 HEARN ET AL. by no more than 9%. It is clear from Figures 4 why this is so. We lanted a sufficiently high number of tags that the CV of is substantially less than those of the other arameters. Thus, the variance of has only a weak effect on the variances of the other arameters. This insures that an incorrect statistical model for will have minimal effect, or conversely, that a correct model will not result in arameter variances that are too high for meaningful oulation inferences. As the homogeneity of reorting rates across boats cannot normally be assured, few tags should be lanted in many catches rather than many tags in few catches. Also, as a logistical matter, fishers are less likely to detect the surretitious lanting of tags if only a very few tags are lanted at a time so that the number of tags encountered does not rise dramatically. To construct an adequate statistical model for, it is recommended that auxiliary information, such as the date, lace, vessel, and ersonnel, be collected when lanting and recovering tags. In reality, lanted-tag studies involve a multistage samling rocess. That is, within each year vessel tris are selected within weeks, fishing sets are selected within tris, and tags are lanted within sets. Because there can be enormous variability in fisher behavior among boats, the actual variability in tag reorting rate will be greater than that redicted from the binomial model assumed in this aer. The user of lanted tags should therefore lan to lant more tags than the number called for under the Bernoulli assumtion. Other Discussion Points In our simulations we lanted tags in each year, but we have assumed that is the same for all years. In a field study, if tags are lanted in all years, the standard likelihood ratio test will allow testing for differences in between years. Use of lanted tags is a owerful method but is difficult to imlement in sufficient numbers. The requirement for secrecy in lanting tags is the greatest obstacle and robably recludes its use in recreational fisheries. If the tags are automatically detected by a machine, then secrecy is not required. Another difficulty in commercial fisheries is that the catching, handling, and rocessing of fish is often a comlex multistaged rocess. Unless tagged fish are lanted in catches before the fish are first insected by eole, some comonent of the reorting rocess will be ignored. If normal tags that are found before tags are lanted are all returned, then the overall reorting rate will be underestimated. If some or all of those tags are not returned, it is ossible for the overall reorting rate to be overestimated. Many fisheries are age structured, and this should be taken into account as described in Hearn et al. (999). For our method, this imlies that the age of each lanted fish needs to be determined, say by measuring its length and using an age length key or taking scales to allow direct aging. A major roblem with the method we described lies in the assumtions that the catch information is accurate, which imlies no bias (which we have reviously discussed) and that the statistical uncertainty in estimating the catch (and hence ) is negligible. The latter is a wider roblem that iminges on stock assessments and the method of estimating the reorting rate from tags found by scientific observers (Hearn et al. 999). Collecting catch information often involves a multistage samling rocess, which needs to be taken into account. However, if variance information is available on catches it may be incororated into our method. Where.0, we note that the estimate of from equation () is identical to that for an observer rogram (Hearn et al. 999 [equations 5 and 6]). This means that the technique of exressing the likelihood as a roduct of likelihoods involving reorting rates, catches, and mortalities also alies to our model (Pollock et al. 00a). Pollock et al. (00a) discuss how to incororate catch variances into the rocedure for estimating reorting and mortality rates. Another connection to the observer aroach is that the agents lanting tags could ossibly serve as observers so that the number of tagged fish er,000 fish landed (i.e., tags/catch) could be estimated. However, comared with the lanted-tag information, this contribution to increased recision would robably be trivial. It might, however, detect serious bias, as might a modest high-reward rogram. Acknowledgments We thank P. Eveson, J. Nichols, C. Stanley, C. Wenner, and an unknown referee for their insightful and helful comments. This is Virginia Institute of Marine Science contribution 58. References Brooks, E. N., K. H. Pollock, J. M. Hoenig, and W. S. Hearn. 998. Estimation of fishing and natural mortality from tagging studies on fisheries with two user grous. Canadian Journal of Fisheries and Aquatic Sciences 55:00 00.
TAG REPORTING RATE ESTIMATION 75 Brownie, C., D. R. Anderson, K. P. Burnham, and D. S. Robson. 985. Statistical inference from band recovery data: a handbook. U.S. Fish and Wildlife Service, Resource Publication 56. Cambell, R. P., T. J. Cody, C. E. Bryan, G. C. Matlock, M. F. Osborn, and A. W. Green. 99. An estimate of the tag reorting rate of commercial shrimers in two Texas bays. U.S. National Marine Fisheries Service Fishery Bulletin 90:6 64. Costello, T. J., and D. M. Allen. 968. Mortality rates in oulations of ink shrim, Penaeus duorarum, on the Sanibel and Tortugas grounds, Florida. U.S. National Marine Fisheries Service Fishery Bulletin 66:49 50. Frusher, S. J., and J. M. Hoenig. 00a. Estimating natural and fishing mortality and tag reorting rate of rock lobster from a multi-year tagging model. Canadian Journal of Fisheries and Aquatic Sciences 58:490 50. Frusher, S. J., and J. M. Hoenig. 00b. Strategies for imroving the recision of fishing and natural mortality estimates from multi-year tagging models: a case study. Marine and Freshwater Research 5: 649 655. Green, A. W., G. C. Matlock, and J. E. Weaver. 983. A method for directly estimating the tag reorting rate of anglers. Transactions of the American Fisheries Society :4 45. Hamton, J. 997. Estimates of tag reorting and tagshedding rates in a large-scale tuna tagging exeriment in the western troical Pacific Ocean. U.S. National Marine Fisheries Service Fishery Bulletin 95:68 79. Hearn, W. S., G. M. Leigh, and R. J. H. Beverton. 99. An examination of a tag-shedding assumtion, with alication to southern bluefin tuna. ICES Journal of Marine Science 48:4 5. Hearn, W. S., T. Polacheck, K. H. Pollock, and W. Whitelaw. 999. Estimation of tag reorting rates in age-structured multicomonent fisheries where one comonent has observers. Canadian Journal of Fisheries and Aquatic Sciences 56:55 65. Hearn, W. S., K. H. Pollock, and E. N. Brook. 998. Pre- and ost-season tagging models: estimation of reorting rate and fishing and natural mortality rates. Canadian Journal of Fisheries and Aquatic Sciences 55:99 05. Henny, C. J., and K. P. Burnham. 976. A reward band study of mallards to estimate band reorting rates. Journal of Wildlife Management 40: 4. Hoenig, J. M., N. J. Barrowman, W. S. Hearn, and K. H. Pollock. 998a. Multiyear tagging studies incororating fishing effort data. Canadian Journal of Fisheries and Aquatic Sciences 55:466 476. Hoenig, J. M., N. J. Barrowman, K. H. Pollock, E. N. Brooks, W. S. Hearn, and T. Polacheck. 998b. Models for tagging data that allow for incomlete mixing of newly tagged animals. Canadian Journal of Fisheries and Aquatic Sciences 55:477 483. Latour, R. J., K. H. Pollock, C. A. Wenner, and J. M. Hoenig. 00. Estimates of fishing and natural mortality for subadult red drum in South Carolina waters. North American Journal of Fisheries Management :733 744. Matlock, G. C. 98. Nonreorting of recatured tagged fish by saltwater recreational boat anglers in Texas. Transactions of the American Fisheries Society 0: 90 9. Nichols, J. D., R. J. Blohm, R. E. Reynolds, R. E. Trost, J. E. Hines, and J. P. Bladen. 99. Band reorting rates for mallards with reward bands of different dollar values. Journal of Wildlife Management 55: 9 6. Pollock, K. H., W. S. Hearn, and T. Polacheck. 00a. A general model for tagging on multile comonent fisheries: an integration of age-deendent reorting rates and mortality estimation. Journal of Environmental and Ecological Statistics 9:57 69. Pollock, K. H., J. M. Hoenig, W. S. Hearn, and B. Calingaert. 00. Tag reorting rate estimation:. an evaluation of the high-reward tagging method. North American Journal of Fisheries Management :5 53. Pollock, K. H., J. M. Hoenig, W. S. Hearn, and B. Calingaert. 00b. Tag reorting rate estimation:. use of high-reward tagging and observers in multilecomonent fisheries. North American Journal of Fisheries Management :77 736. Pollock, K. H., J. M. Hoenig, and C. M. Jones. 99. Estimation of fishing and natural mortality when a tagging study is combined with a creel or ort samling. Pages 43 434 in D. Guthrie, J. M. Hoenig, M. Holliday, C. M. Jones, M. J. Mills, S. A. Moberly, K. H. Pollock, and D. R. Talhelm, editors. Creel and angler surveys in fisheries management. American Fisheries Society, Symosium, Bethesda, Maryland. Ricker, W. E. 975. Comutation and interretation of biological statistics of fish oulations. Fisheries Research Board of Canada Bulletin 9. Ruert, D., R. L. Reish, R. B. Deriso, and R. J. Carroll. 984. Otimization using stochastic aroximation and Monte Carlo simulation (with alication to harvesting of Atlantic menhaden). Biometrics 40: 535 545. Siddeek, M. S. M. 989. The estimation of natural mortality in Irish Sea laice, Pleuronectes latessa L., using tagging methods. Journal of Fish Biology 35(Sulement A):45 54. White, G. C. 983. Numerical estimation of survival rates from band-recovery and biotelemetry data. Journal of Wildlife Management 47:76 78. Youngs, W. D. 974. Estimation of the fraction of anglers returning tags. Transactions of the American Fisheries Society 03:66 68.
76 HEARN ET AL. Aendix : Derivation of Estimators for the Tag Reorting Rate in a Two-Comonent Fishery Consider a fish oulation just before fishing begins. The size of the oulation is P, and there are R tagged fish in the oulation from revious tagging events; both P and R are unknown arameters. Let u and u be the finite exloitation rates for the (assumed) two comonents of the fishery, and let and be the corresonding tag reorting rates. The data collected are as follows: C, C the catches in the two fishery comonents R, R the number of tagged fish reorted N R in the two comonents the number of tagged fish lanted in the first comonent the number of lanted fish reorted from the first comonent [ ] N R N R R ( ). The likelihood for all of the data is the roduct of the above three likelihoods. Conditional Likelihood If we condition the likelihood on the total catch, then the catch for each comonent is binomial with arameters C C (known) and u /(u u ). Thus, the likelihood for the two catches given that the total catch is binomial is C C u u [ ] u u u u C C C Similarly, we can condition on the total number of recatures. In that case the number of recatures from each comonent is binomial with arameters R R (known) and u /(u u ), so that the likelihood is R R u u [ ] u u u u R R R The likelihood for the lanted tag recoveries does not deend on the other arts of the likelihood and remains binomial: [ ] N R N R R ( ). Full Likelihood The moment and maximum likelihood estimators are based on the following equations: Under the assumtion that the actual catches are random variables governed by a multinomial distribution with arameters P, u, and u, the like- E(C û C C ) (C C ) (.) û û lihood of obtaining catches C and C is multinomial: û ˆ E(R R R ) (R R P (u ) C(u ) C( u u ) P C C. [ C, C ] ) (.) û ˆ û ˆ E(R ) N ˆ (.3) The number of reorted recatures of reviously Equation (.3) imlies that tagged fish is also multinomial, with arameters R R, u, and u. Thus, the likelihood is ˆ. N R (u ) R(u ) R( u u ) R R R. From (.) [ R, R ] û C The number of lanted tags that are reorted is binomial, with arameters N and, û C. Thus, the likelihood is and from (.) û ˆ û ˆ R. R Therefore, if ˆ is greater than.0, the likelihood should be maximized subject to the constraint that ˆ.0. û R CR ˆ ˆ ˆ. û R C R..
TAG REPORTING RATE ESTIMATION 77 Aendix : Model Checking Using Residuals of Fishery Comonent Tags/Catch Rates One aroach to the testing roblem is to standardize the estimated tags/catch for each fisheries comonent so that it has aroximately a standard normal residual and a standard normal table can be used to see if the residual is unusual. (For examle, if the model were valid, a residual larger than.96 in absolute value would occur aroximately time in 0; similarly, a residual larger than.57 would occur time in 00.) From equation (3) in the text, the estimate of the tags/catch common to all fisheries comonents is j R ijk R i jk ˆ jk C N jk jk ˆ jk ˆT. jk Note that here jk is estimated from equation () ˆ rather than equation (), with k relacing. This is because in this instance comonent k has lanted tags, whereas in develoing equation () it was assumed that comonent k (k ) had no lanted tags. Define the standardized residual to be K m Tˆ Tˆ K jk jm ˆQ jk, var(t ˆ ) jk where m refers to the mth fishery comonent and var( ˆ jk) var( ˆ jk) var(t ˆ ) (T ˆ ) jk jk, with ( ˆ ) ( ˆ ) var( ˆ jk ) ˆ jk ( ˆ jk )/C jk and var( ˆ jk ) ˆ jk ( ˆ jk )/N jk. jk This test can be readily adjusted for the case in which tags are lanted in two or more comonents but not all comonents. However, it cannot be used to test the assumtions ertaining to comonents with no lanted tags. jk