Chapter 8

Abundance, Biomass and Production

DANIEL B. HAYES¹, JAMES R. BENCE², THOMAS J. KWAK³, AND BRADLEY E. THOMPSON⁴

¹Department of Fisheries and Wildlife

Michigan State University

East Lansing, MI 48824-1222

Phone: 517-432-3781

FAX: 517-432-1699

E-mail: hayesdan@msu.edu

²Department of Fisheries and Wildlife

Michigan State University

East Lansing, MI 48824-1222

Phone: 517-432-3812

FAX: 517-432-1699

E-mail: jbence@perm.fw.msu.edu

³U.S. Geological Survey

North Carolina Cooperative Fish and Wildlife Research Unit

Box 7617, North Carolina State University

Raleigh, NC 27695

Phone: 919-513-2696

FAX: 919-515-4454

E-mail: tkwak@ncsu.edu

⁴Department of Fisheries and Wildlife

Michigan State University

East Lansing, MI 48824-1222

Phone: 517-353-6697

FAX: 517-432-1699

E-mail: thomp149@msu.edu

8.1 INTRODUCTION

Fisheries scientists face a challenge in that virtually all methods of fish capture or observation are selective. Further, most fish capture methods can only be applied to a fraction of the entire area of interest. Thus, measures such as catch per unit effort (CPUE) or catch per area (CPA) can only be regarded, at best, as being proportional to the true population abundance (see Chapter 7). The methods presented in this chapter are designed to address these problems and provide estimates of absolute abundance or "true" fish density. In general, these methods require additional sampling beyond that required to estimate relative abundance. As such, careful consideration should be given to whether relative measures of abundance are adequate, or if the need for estimates of absolute abundance justify the additional cost.

In many cases, relative abundance is sufficient to answer important research or management questions. One example is when the principal goal is to determine if abundance has changed over time. As long as vulnerability to the gear remains constant over time, trends in CPUE can accurately indicate changes in abundance (see Chapter 7). In such cases, the extra effort required to determine absolute abundance is better spent in sampling more sites. In general, estimates of absolute abundance are needed when catchability is likely to vary across time or between sampling sites, confounding comparisons of CPUE across space or time. Absolute abundance estimates are also important when harvest quotas are being computed, or when one needs to relate the population abundance to other populations.

Whether relative or absolute measures of abundance are desired, it is critical to carefully define the population of interest. In many cases, some part of the population is excluded from consideration because of limitations of the sampling gear. For example, population estimates of yellow perch in mid-summer using gill nets would likely not include age-0 fish because they would not be vulnerable to the gear. Similarly, care must be taken in defining the spatial extent of the target population. Sometimes one is interested only in the population in a particular stream reach, whereas in other situations the desired scale is an entire watershed, which would likely need to be sub-sampled.

Another consideration common to both relative and absolute measures of abundance is the precision and accuracy required for the task at hand. Accuracy, bias, and precision are defined in Chapter 1. Applying these concepts to population estimates, it is important to recognize that failures to meet assumptions often reduces both accuracy and precision. Therefore, we emphasize methods for checking assumptions in addition to the methods commonly used to provide point estimates and measures of variability.

The principal goal of this chapter is to provide guidance on how to apply some of the most common and effective methods used to estimate population size, biomass, and production. Although we can not provide comprehensive coverage of methods used in all situations, we have tried to provide a basis for building models applicable to more complex or specialized situations.

8.2 DIRECT OBSERVATION METHODS

In some situations, direct observation of all fish in a given area (sampling site) is possible, providing a complete census of the area searched. This approach has been applied in small streams (Hankin and Reeves 1988) or in other situations where fish are tightly constrained. Likewise, counts of fish in hydroacoustic surveys are often assumed to represent all individuals within the hydroacoustic beam path. In situations where counts are assumed to be accurate and complete, the total population is estimated as the product of the mean density in the sites sampled times the total area. The precision of total population estimates depends principally on the variability between sampling sites (Hankin and Reeves 1988) and the sampling design used (e.g., stratified random sampling). Methods of computing the variance for several sampling designs are presented in Chapter 3, and can be applied directly to data collected through complete censuses at selected sites. One specialized design not included in Chapter 3 is hydroacoustic surveys where counts are collected along the path of the boat (i.e., along a transect). If data are collected along a single transect, specialized statistical methods are necessary to calculate the variance of the population estimate because of the autocorrelation between counts at adjacent points. (Foote and Stefansson 1993, Vondracek and Degan 1995). If two or more randomly-placed transects are followed, however, each transect can be treated as a sampling site, and the methods described in Chapter 3 can be applied.

In many situations, visual observation misses some proportion of the population, even in situations where fish are constrained. Because of this, estimates of density for individual sites are imprecise, and contribute to the overall imprecision of total density estimates. In order to estimate the proportion observed within a sampling site, additional information needs to be gathered. The most commonly used method is to measure the distance that each animal observed lies off of the transect (i.e., the right-angle distance from each animal seen to the transect) or from the center of a fixed point of observation. Depending on the observation technique, this distance can be determined directly, or the distance and angle of departure from the transect can be determined, and the right-angle distance calculated by simple geometry. Generally, the proportion of fish present that are detected (i.e., sightability) declines further from the point of observation or from the transect surveyed (Figure 8.1). Assuming that fish are randomly distributed with respect to the transect, and sightability asymptotes at 100 percent near the center of the transect, the proportion observed can be estimated as a function of distance from the transect.

Critical assumptions for applying the direct observation approach include (1) fish are randomly and independently distributed, and movement of the observer does not attract or repel fish prior to observation, (2) distances are measured accurately, (3) fish are not counted more than once, (4) fish are detected at their original position with respect to the transect, and (5) sighting of each fish is independent of other fish, meaning that the likelihood of seeing an individual fish does not depend on the number of other fish in the vicinity (Seber 1982, Buckland et al 1993; Thompson et al. 1998). Carefully implemented field techniques can help insure that assumptions 1 to 4 are met. The assumption of independent sightings, however, depends on the behavior of fish and their schooling behavior and patchiness. When fish are sighted in groups, but the proportion of fish sighted is constant with fish density, the precision of population estimates is generally reduced, but the population estimate is not necessarily biased (Buckland et al. 1993). In such cases where the unit of observation is a school or other aggregation of animals, we refer the reader to Buckland et al. (1993) for methods for appropriately analyzing these data. When sightability varies as a function of density or school size, estimates of fish density are likely to be biased, and the applicability of this approach should be reconsidered.

For a single line transect survey, the general formula for density is (Buckland et al. 1993, pg 39):

When counts are conducted from a single fixed point (point plot survey), the area surrounding the point is observed, resulting in a circular search area. In this situation, the general formula for density is (Buckland et al. 1993, pg 41):

In applying these formulae, a critical component is estimating , the effective width of the transect or search radius from a point. Essentially is estimated to correspond to an equivalent transect where all fish out to are detected, and all fish beyond are not. In order to accurately estimate this quantity, it is necessary to select a function describing the pattern of sightability with distance. Many functions can be used to describe the sightability function; we illustrate the application of two of these functions and provide formulae for estimating total population abundance from density and the total area of the study site in Box 8.1. Buckland et al. (1993) provide a thorough discussion of various functions and methods for selecting among them. The variance for the density estimate (and population size) for a single transect within a site can be approximated based on the binomial distribution describing observed and unobserved fish (Box 8.1), and assuming that fish are randomly and independently distributed. When multiple transects or points are observed, the variance between transects should be determined based on the overall sampling design, following methods outlined in Chapter 3.

Specialized software packages are available to estimate population size based on distance sampling. DISTANCE (Thomas et al. 2001) is a comprehensive package, and is available on the web at the following web address: http://www.ruwpa.st-and.ac.uk/distance/.

8.3 POPULATION ESTIMATION - MARK-RECAPTURE METHODS

8.3.1 Closed Populations

The underlying concepts and assumptions of mark-recapture methods of population estimation have a long history in the fishery literature. Because of the extensive reviews available on this subject (Otis et al. 1978, Seber 1982, Burnham et al. 1987), we will emphasize common applications and methods that provide a base for specialized or particularly complex situations.

In the simplest case, fish are randomly collected from a closed population and easily recognizable, permanent marks are applied to captured individuals. These individuals are then released and allowed to mix completely with the remainder of the population. A second sampling is undertaken, and the ratio of marked to unmarked fish can be used to estimate the total population. Assumptions in basic mark-recapture studies include (1) the population is geographically closed, with no immigration or emigration, (2) the population is demographically closed, with no birth or deaths, (3) no marks are lost or missed (4) marking does not change fish behavior or vulnerability to capture, (5) marked fish mix at random with unmarked fish, and (6) all animals have an equal probability of capture that does not change over time (Otis et al. 1978, Seber 1982). A number of formulae have been developed for this basic situation. In practical terms, all give similar results when reasonable numbers of marked fish are recaptured (e.g., at least 2-3, but preferably greater than 10; Chapman, 1951; Robson and Regier 1964). Because of its widespread use and theoretical basis, we recommend the use of the Chapman estimator (Seber 1982, page 60):

The variance of this estimator can be approximated as (Seber 1982, page 60):

Numerous approaches are available to develop confidence intervals for . Unfortunately, a diversity of methods occur because of different distributional assumptions and different approximations for small and modest sample sizes. As such, a single method has not yet been identified as being generally best. For large sample sizes (e.g., m₂ >50) the normal approximation is generally adequate (Seber 1982, page 63 ), and 95% confidence intervals can be calculated as:

When there are fewer than 50 recaptures, Chapman (1948, reproduced in Seber 1982, page 574) provides a table (Appendix Table 1) from which confidence intervals can be calculated based on the number of recaptured fish.

When multiple marking and recapture samples are collected over a short period (so that the population is closed with no immigration, emigration, recruitment, or mortality) population size can be estimated with the Schnabel method (Schnabel 1938, Seber 1982 page 139):

The variance of this estimator can be approximated as (Seber 1982 page 139):

The confidence interval for with the Schnabel method can be computed following the same recommendations for the Chapman method in a single mark-recapture experiment.

In many situations, a simple design using a single marking period and single recapture period or a Schnabel-type design is sufficient to estimate population abundance. The effectiveness of such designs, however, rests on adequately meeting the assumptions. Unfortunately, it is generally not possible to test these assumptions using the data collected during a single recapture period or when fish are simply marked as being previously caught. To test the assumptions underlying mark-recapture methods of population estimation, it is generally necessary to sample over multiple periods, and to have marks that allow for the capture history of individual fish to be determined (e.g., by using individually numbered tags).

For closed populations sampled with a design as described above, Otis et al. (1978) present a hierarchical suite of models intended to cover a range of situations where particular assumptions hold (Figure 8.2). The simplest, yet most restrictive model is where all assumptions listed earlier apply (M_o). In the next tier of models, three basic mechanisms causing unequal capture probabilities are addressed. In model M_t, the probability of capture is allowed to vary among different sample periods (time). Variability in capture over time may occur due to factors such as weather, or due to changes in the amount or type of fishing gear deployed. In model M_b, the probability of capture is allowed to vary due to behavioral response to prior capture (i.e., fish become more prone or less prone to capture after being caught, handled, and marked). In surveys of small mammals, for example, investigators find that marked animals may become "trap happy" or "trap shy," thus biasing population estimates if not considered. The final model, M_h, allows for heterogeneity in the capture probability of individual fish. This heterogenity may occur for a variety of reasons, including inherent features of each fish such as its size, or less obvious factors such as variation in the size of home ranges, resulting in different vulnerabilities to passive gears such as trap nets. Methods have been developed to estimate population size for each of these models, and are illustrated below. Because of the complexity of the required analyses, we strongly recommend the use of specialized software when applying these models. Program MARK (White and Burnham 1999) is a very flexible software package designed to analyze data from mark-recapture studies, and is available at the following web address:

http://www.cnr.colostate.edu/~gwhite/mark/mark.htm.

In the next tier of models, variations in probability of capture occur through combinations of two of the above factors (Figure 8.2). Thus, model M_bt represents the case where capture probability varies over time, as well as with the prior capture history of an animal (behavior). Estimation methods are also available for each of these models, however we refer the reader to software, such as MARK, specially designed to handle such situations. The final model in this hierarchy (M_bth) is where all three factors cause variation in capture probability. At present, no method has been developed to estimate population size and account for these three sources of variation simultaneously.

A central concept to estimating population size using these models is the capture history of an animal. Since the population is assumed to be closed, the number of animals in the population (N) remains constant over all sampling periods. As such, during each of the sampling periods (numbered 1 to t), an animal can either be caught or not. For convenience, the capture history of all animals observed can be recorded in a matrix (Table 8.1) where a 1 is used to indicate a capture, and 0 to indicate no capture during a particular sampling period.

The second concept central to estimating population size using these models is the likelihood function. Although this is the foundation for many methods of population estimation (it in fact is the basis for the Chapman and Schnabel estimators), likelihood functions may be unfamiliar to many readers. We provide a brief synopsis of this topic in the context of population estimators in Box 8.2. Readers should consult texts in mathematical statistics (e.g., Bickel and Docksum 1977; Rice 1995) for a more thorough treatment. In some cases, likelihood methods result in a formula for directly estimating population abundance. In most situations, however, there is no direct formula relating the data to the population estimate. Instead, the likelihood function is repeatedly evaluated at trial values of (or related parameters that determine ), until a value of is found that produces the maximum value of the likelihood function. This is chosen as the "best" or most likely population estimate. With modern computing power, a "brute force" solution can be found simply by starting with a trial value of set equal to the total number of distinct individuals caught (i.e., the minimum possible population), and then evaluating the likelihood function for each integer value of up to an arbitrary maximum. Generally, the logarithm of the likelihood function is evaluated because it is often easier to compute, and because it provides a useful basis for comparing between models and for estimating confidence intervals.

The log-likelihood function for model M_o (Otis et al. 1978 page 105) is:

Once has been calculated, the probability of capture (which is assumed constant over time and across all animals) is estimated as:

An asymptotic estimate of the variance for is (Otis et al. 1978 page 105):

Confidence intervals for can be obtained in a number of ways. The first method is to use the estimated variance of using equation 8.10, and to then assume a distribution (e.g., normal when more than 30 animals are recaptured) for and calculate upper and lower bounds using equation 8.5. An alternate method, discussed in Box 8.2, is to use the likelihood function itself to determine confidence intervals. Trial values of that produce likelihood values that differ from the maximum likelihood by more than 3.841, which is the critical value for a ² distribution with 1 degree of freedom, define the bounds of the confidence interval.

In model M_t, the capture probability for individual animals varies over time. As such, this model has two distinct sets of parameters: N, which is the population abundance, and p₁, p₂, ..p_t which are the time-specific capture probabilities, resulting in a total of t+1 parameters. The log-likelihood function for model M_t is (Otis et al. 1978, page 106):

The asymptotic variance of under model M_t is (Otis et al. 1978, page 106):

As with model M_o, confidence intervals for under model M_t can be estimated using the variance of and an assumption of normality, or through a likelihood-based approach as outlined in Box 8.2.

Model M_b (variability in capture probability due to changes in behavior after capture) has three parameters: , p, the probability of capturing an unmarked animal, and c, the probability of capturing an animal which was previously captured, marked and released. The parameter c can be estimated independently of N and p by (Otis et al 1978, page 107):

The likelihood function for model M_b is (Otis et al. 1978, page 108):

Once has been estimated, the probability of capture can be estimated via (Otis et al. 1978, page 108):

The asymptotic variance of is (Otis et al. 1978, page 108):

Estimation of the parameters for model M_h is more problematic than for model M_o, M_t, or M_b. The reason for this is that each fish (including unobserved fish) has its own individual catchability. A number of approaches have been taken to solve this problem, generally by making an assumption regarding the statistical distribution of catchabilities. For details of computation for this model, we refer the reader to Otis et al.(1978), and to program MARK (White and Burnham 1999).

An example applying models M_o, M_t, and M_b is given in Box 8.3. Beyond being able to estimate the parameters for each of these models, an important question is how to choose among them. The most common way of doing this is to compare the likelihood value for each model, and select the model with the highest likelihood. Because likelihood generally increases as more parameters are added, the likelihood from models with more parameters is typically "penalized" for the additional flexibility offered. The most widely-used adjustment to the likelihood function is Akaike's Information Criterion (AIC; Akaike 1973), which is calculated as:

After computing the AIC, one then selects the model that has the lowest AIC value (Box 8.3).

8.3.2 Open Population Mark-Recapture Methods

Open populations are characterized by having immigration, emigration, mortality or recruitment occur during the study period. As in closed populations, general models developed to estimate abundance in open populations also make use of the encounter history matrix as the basis for maximum likelihood estimators, and assume that each fish is uniquely marked. Conceptually, the encounter history matrix is important because it defines which animals are observed at particular times. From this, we can also infer which time periods the animal is known to be alive even if it is not observed. For example, a fish marked at the beginning of an experiment, but not observed again until the end is known to have survived through all intervening sampling events. Open population models are also similar to the Otis et al. (1978) hierarchy of models in that numerous factors and assumptions can be represented in a suite of models applicable to open populations. Compared to closed population models, additional parameters describing losses and additions to the population are necessary for open population models. The additional parameters necessary to describe open populations often lead to a decline in the precision of population estimates. Further, allowing for an open population requires stronger adherence to some of the model assumptions. In particular, heterogenity in capture probability becomes increasingly important, and can lead to bias in population estimates.

In this section, we cover a basic model illustrating the computation underlying the broader range of possible models for open populations. Lebreton et al. (1992) and Seber (1982) provide in-depth coverage of this family of models, commonly referred to as Cormack-Jolly-Seber (CJS) survival models. The software program MARK, referred to earlier, can accommodate CJS models. The program RELEASE was also recommended by Lebreton et al. (1992) and can be found at: ftp://ftp.cnr.colostate.edu/pub/release/.

A commonly-used open population model represents the case where the abundance of animals changes over time due to births and deaths, and survival varies over time, but capture probability is constant over time and across all individuals in the population. As such, this model is analogous to model M_o, with the addition of time-varying population abundance and survival. In the CJS models, four basic sets of parameters are estimated: population abundance (N_i), capture probability (p_i), apparent survival (_i), and additions (births and immigrant) to the population (B_i). The term apparent survival is used instead of survival because in most cases it is impossible to distinguish any losses due to emigration from mortality. If the population is geographically closed, _i is an estimator for actual survival rate. Each of the above parameters are indexed by time, but care must be taken in understanding that _i indicates the survival rate from time i to i+1. Further, not all quantities are estimable; for example abundance at the beginning of the study (N₁) generally can not be determined. The application of this model is illustrated in Box 8.4. For simple models, closed-form equations exist to estimate population size and other necessary parameters. In more complex situations, an iterative approach is necessary to solve the likelihood equations.

8.4 POPULATION ESTIMATION-REMOVAL METHODS

8.4.1 Closed Populations

Like mark-recapture methods, removal methods rely on sequentially sampling the target population. During each sampling period, the number of fish captured are recorded, and captured fish are temporarily (e.g., during monitoring surveys) or permanently (e.g., in recreational or commercial fisheries) removed from the population. Through the reduction in the population, catch in subsequent sampling periods is reduced. The rate that catch declines gives a measure of the proportion of the original population that has been removed.

As with mark-recapture methods, removal methods generally rely on the population being closed, and for individuals in the population to have equal vulnerability to the sampling gear. Typically, equal amounts of effort are expended during each sampling period, and it is assumed that the capture probability is equal across all sampling periods. Historically, regressions relating CPUE to cumulative catch (Leslie method, Leslie and Davis 1939) or cumulative effort (De Lury method, De Lury 1947) were used to estimate population size in removal experiments. These methods are still commonly used, and often result in reasonable population estimates. Currently, there is a shift away from the regression-based methods to likelihood-based methods. The principal advantage of likelihood methods over regression methods is that they provide a means for testing some of the assumptions of the removal method, and creating models that can accommodate a relaxed set of assumptions. For example, the assumption of equal catchability over all sampling periods can be relaxed if a function can be used to describe how catchability changes over time.

In the simplest situation, all of the above assumptions hold, and sampling is done during two periods with equal effort. Catch (i.e., the number removed) is recorded for each sampling period. Zippin (1956, 1958, Seber 1982 page 318) showed that the maximum likelihood estimator in this situation is:

Because of the dependence on only two data points, the precision of population estimates in this situation is often poor. Moreover, variability in catches can result in higher CPUE in the second sampling period, resulting in no estimate of abundance. Heimbuch et al. (1997) present a method for adjusting population estimates in two-pass sampling when many sites are visited. In their method, the catch data are added across sites to allow an estimate of the population for all sites together. By adding data from many sites, variations in the catch at individual sites tend to cancel, resulting in a better estimate of average catchability. Heimbuch et al. (1997) also present extensions to this method where variability in individual catchability can be accounted for, analogous to model M_h of Otis et al. (1978).

When fish are removed during three sampling periods, Junge and Libosvárský (1965; cited in Seber 1982, page 315)) developed the following maximum likelihood estimators for catchability and population size:

Variance estimates and confidence intervals for are covered in the general case (below) of four or more removal passes.

When more than three removal passes are conducted, there is no closed-form equation available for directly estimating population size from the data using current maximum likelihood methods. As in the more complex mark-recapture situations, the relative likelihood of parameter values is calculated and numerical search methods are used to determine which combination of parameter values is most likely given the observed data. When catchability (q) is assumed to be constant over time, the results of this analysis are easy to portray graphically as a profile likelihood (see Box 8.2). In the maximum likelihood approach, catchability is generally estimated directly, and is calculated from the cumulative catch and estimated cumulative proportion of the population that this represents, based on the estimated catchability. The conditional likelihood function (dropping those parts of the function that are constants not affecting estimation) for estimating is (Gould and Pollock 1997):

Once has been estimated, is estimated by:

For large N, the asymptotic variance of and are (Seber 1982, page 312):

Confidence intervals can be obtained by assuming that and are normally distributed, or from the profile likelihood of (e.g., Box 8.2). Once and have been estimated, the goodness of fit of the estimates can be assessed by comparing the expected catches with the observed catches. Expected catch for each removal pass is predicted by:

Goodness of fit can then be assessed by a ² test by comparing the observed catches with the expected catches:

One of the more common violations of the assumptions in the removal method is that individual fish often differ in their catchability, analogous to model M_h for mark-recapture studies. Two approaches can be used to estimate population size in this situation. The first approach rests on the observation that removal studies can be viewed as a special case of mark-recapture model M_b, where the "response" to capture is removal from the vulnerable population (this is equivalent to setting c in model M_b equal to 1.0). Heterogenity in individual catchability can then be accounted for by fitting model M_bh in the Otis et al.(1978) hierarchy. The calculations for this model are complex, but program MARK includes this option.

The second approach for handling variations in catchability is to fit a time-varying function to . Because fish with higher catchability tend to be captured and removed earlier in the sampling process, the average catchability of the remaining population tends to decline as the population is depleted. Thus, additional parameters describing how declines with each sampling pass can be estimated. We refer the reader to Schnute (1983) for more detailed description of this approach.

Several software packages are available to estimate abundance from removal experiments. MARK (White and Burnham 1999) handles removal data well, and has the option of fitting alternate models as described above. MicroFish (Van Deventer and Platts 1989) is a software package available through the American Fisheries Society that is designed for removal studies. Its particular strength is that removal experiments from multiple sites and multiple species can be analyzed from a single data file.

8.4.2 Open Populations - Removal Methods

The application of removal methods to open populations is much more difficult than for closed populations because mortality and recruitment need to be estimated in addition to population size. Furthermore, removal methods are generally applied only to open populations when there is a fishery harvesting a substantial portion of the population. As such, the timing and magnitude of removals are often out of the fisheries scientist's control. Further, there is the potential problem of under (or over) reporting of catch, resulting in biased estimates of population size. This is not to dissuade readers from pursuing removal methods for open populations - this is often the only feasible approach given the data available. Rather, we emphasize that the particular details of the data and the fishery will determine which model is most appropriate. In this chapter, we present a relatively simple formulation requiring minimal data to illustrate the essence of these methods.

Consider a population that is closed to immigration and emigration, but is open to natural mortality (M), fishery harvest (C), and recruitment (R). One representation of the dynamics of the population is (Collie and Sissenwine 1983):

In this model, E_t represents random variations in mortality that are not included in either catch or natural mortality (which is assumed to be constant). E_t reflects what is often called a "process error," meaning variation in the underlying dynamical processes that is unaccounted. Including this in the population dynamic equation 8.33 is important because process error actually influences system dynamics, and these process errors can accumulate over time. This model implicitly assumes that recruitment and fishery removals occur at the beginning of the year. Natural mortality operates at a constant rate for the remainder of the year, and a proportion (e^-M) survive to the beginning of the next year. Alternative formulations can be derived for populations where the fishery and recruitment occur throughout the year (see Ricker 1977).

For the model described above, information on harvest alone is insufficient to estimate population abundance. Additional information, in the form of relative abundance indices (e.g., CPUE) for the adult stock (n_t) and recruits (r_t) are also required. Age-structured measures of CPUE and population-dynamic equations can also be used, leading to methods such as virtual population analysis or statistical catch at age. We refer the reader to Ricker (1977), Edwards and Megrey (1989), and Hilborn and Walters (1992) for a detailed discussion of these extensions.

If we assume that the expected CPUE for adults and recruits (and ) is directly proportional to the "true" population size (N_t and R_t) and that all members of the population are equally vulnerable to the survey gear, we have:

A critical concept underlying equations 8.34 and 8.35 is that CPUE, which is based on samples from the entire population, is generally estimated with considerable variance. The variance associated with these estimates is often termed measurement error, and in the context of population modeling, implies that CPUE should not be treated as exact, but rather needs to be treated as being imprecise. Using equations 8.34 and 8.35 leads to the following dynamic equation describing the trajectory of the expected value for adult CPUE:

Here, _t = qE_t, and is the process error as it influences adult CPUE. The estimation procedure attempts to minimize these process errors as well as the measurement errors (see Box 8.6).

Generally, M is assumed to be known and constant over time. Under the additional assumption that the measurement errors are negligible (i.e., all are close to 1.0), equation 8.36 can be rewritten in a form where standard linear regression can be used to estimate q (and thereby N_t and R_t). However, as Collie and Sissenwine (1983) point out, n_t and r_t are generally both measured with substantial imprecision. Because of this, we recommend the methods of Collie and Sissenwine (1983; illustrated in Box 8.6) over a regression approach because the assumption of negligible measurement error is rarely credible.

We are not aware of any software program that handles the broad range of situations that are likely to occur when using removal methods in open populations. As such, practitioners must often develop the models and associated estimation routines in a general programming environment (e.g., C++, Visual Basic, SAS), a spreadsheet environment (e.g., Excel), or in a specialized programming environment designed for statistical parameter estimation (e.g., AdModel Builder (Otter Research)). Schnute et al. (1998) discuss some of the trade-offs faced in choosing software for such modeling.

8.5 BIOMASS AND YIELD ESTIMATION: SURPLUS PRODUCTION METHODS

In situations where a geographically closed population is subjected to a significant fishery (e.g., where the population has been substantially reduced by fishing; Hilborn and Walters 1992), it is sometimes possible to estimate biomass from the pattern of harvest and fishing effort over time. Conceptually, surplus production models (also known as biomass dynamic models, Hilborn and Walters 1992) are based on the idea that the biomass in a given year (B_t) depends on the biomass in the previous year (B_t-1) plus recruitment and growth, minus harvest and natural mortality. It is often convenient to group recruitment and growth into a single term representing processes that contribute to biomass. If this production is in excess of natural mortality, the surplus production will increase the biomass from one year to the next. Alternately, the surplus production may be harvested and still maintain the population biomass. Generally, surplus production is related to the standing biomass; at low biomass levels, surplus production is low due to limited recruitment. At high biomass levels, surplus production is also generally low due to density-dependent growth and/or recruitment. Surplus production typically peaks at intermediate levels of biomass.

Because of the relatively simple representation of population dynamics, surplus production methods do not require age-specific data. As such, these methods are often used in the analysis of difficult-to-age marine fish stocks. Moreover, simulation studies have suggested that management advice based on surplus production methods may be as robust as population estimates based on delay-difference type of age-structured analyses using only yield and effort data (Ludwig and Walters 1985). Although surplus production models have not been widely used in freshwater fishery analysis, they are likely to be applicable and beneficial in some situations where data are limited.

Hilborn and Walters (1992) provide a thorough review of the principal approaches for fitting surplus production models to data in order to estimate biomass, recruitment, and density dependence. We follow their advice that the best (but not the only) way to approach the problem is to develop a model of the biomass dynamics, use that model to predict fishery harvests over time, and then fine-tune the parameters of the model so that the predicted harvest (or CPUE) best fits the observed harvest time series. Although there are many variations of surplus production models, a common model is (Hilborn and Walters 1992, page 306):

Although it appears that equation 8.37 can be fit without information on effort, it is generally necessary to include this information to allow estimation of commercial catchability (q). The biomass at the start of the time series is also often estimated as a parameter in the model, allowing the iterative solution of equations 8.37- 8.39 in order to best fit the observed and predicted time series of CPUE. An example of the application of this approach is provided in Box 8.7. A useful software package for surplus production modeling is ASPIC, available at the following web address (as of July 16, 2001): http://shrimp.ccfhrb.noaa.gov/~mprager/ASPIC.html.

8.6 BIOMASS ESTIMATION

Most of the methods presented in this chapter produce estimates of numerical abundance. In some situations, however, biomass (i.e., weight of the population) may be a better measure of the "size" of a population. Generally, biomass is estimated indirectly by multiplying the numerical abundance by the mean weight, or by applying methods such as surplus production models that directly estimate biomass. In this section, we will cover indirect methods for estimating biomass.

In the simplest situation, biomass is estimated as:

In this equation, can be estimated using any of the methods presented earlier, and mean weight is estimated from a random sample representative of the size or age groups contained in (Anderson and Neumann 1996).

Assuming that the variance of is estimated through methods described earlier, and the variance of is also estimated, the variance of is approximated as:

This approximation (Kendall and Stuart 1964) is based on the assumption that and are estimated independently; an assumption that is reasonable in most cases.

Although equations 8.40 and 8.41 provide relatively simple means of obtaining point estimates of biomass and the associated variance, developing confidence intervals for is much more difficult because the distribution of must be known or assumed. We are not aware of any general guidance in the literature suggesting a suitable distribution for . Since is computed as the product of two random variables, the lognormal distribution is a reasonable choice (Aitchinson and Brown 1976). Assuming a lognormal distribution, approximate 95% confidence interval bounds for are:

In many situations, abundance and mean weight are estimated separately for different age or size classes. In such situations, biomass can be estimated as:

In this case, the variance of is:

and the 95% confidence interval can be computed following equation 8.42.

8.7 PRODUCTION ESTIMATION

8.7.1 Concepts and Definitions

Fish abundance parameters, such as density or biomass, are static measures of a population's status. That is, information on the state of the population is provided only for a single point in time. Conversely, dynamic population measures describe parameters as rate functions over time, and may be more descriptive and meaningful for applications in fisheries science. Examples of dynamic population parameters are rates of recruitment, growth, and mortality (Section II, Chapters 4-6). Production is the integration of static and dynamic population measures over time, wherein biomass, recruitment, growth, and mortality are synthesized into a single dynamic measure. As such, production is an indicator of ecological success, and is especially responsive to environmental change (Mann and Penczak 1986). Thus, production rate of a fish population can be a useful measure and comparative tool, with many valuable applications for fisheries research and management.

Production is defined as the rate of tissue elaboration over time, regardless of whether it survives to the end of a given period (Waters 1977). It is expressed in units of quantity/space/time, usually kg/ha/year for fish populations. Production rate represents the flow of energy through trophic levels, and may also be expressed in units of calories/ha/year.

The methods and terminology for estimating fish production have evolved to a generally accepted convention, and fish production estimates are routinely found in the literature, especially for fishes of small streams and salmonid species. However, many fisheries scientists rarely consider using this assessment tool, even though they may regularly gather the data required to estimate production. Presumably, this occurs because the computations can be complex and cumbersome, and are more so if precision of production estimates and related parameters is estimated. The development of computer software and availability of other technical resources to minimize computation effort and reduce calculation error associated with the process of estimating production, may increase the utility of this tool in fisheries science (Railsback et al. 1989; Kwak 1992).

8.7.2 Production Methods

Five methods to estimate production rate of aquatic animal populations have been developed, refined, and accepted by ecologists (Waters 1977; Bagenal 1978; Chapman 1978). Some of these methods were originally intended for estimating aquatic macroinvertebrate production, but were readily adapted for use with fish populations. The five methods include two iterative summation methods, (1) the removal summation and (2) increment summation methods; (3) the instantaneous growth rate method and a graphical representation, (4) the Allen curve; and (5) the size-frequency method. Three of the methods (increment summation, instantaneous growth rate, and size-frequency) have been refined for application to fish populations, and variance estimators for all parameters associated with those methods have been derived. All methods except the size-frequency method are cohort-based, meaning that information on the age structure of the fish population is required.

Sampling requirements to estimate fish production are a series of absolute density and biomass estimates (Sections 8.3, 8.4, 8.6) for a population within a one-year period, with the first and last sampling dates occurring approximately one year apart to estimate annual production. Cohort-based methods, require stratification and separate estimates by cohort; thus, data must be collected on population age structure. In general, production is estimated by individual cohorts for a single time interval, then those partial estimates are summed for all cohorts to yield a production estimate for the entire population during the specific interval. The production estimate for a one-year period (annual production) is the sum of the production estimates for intervals within the annual period. When the size-frequency method is used, individual losses from one size group to the next are summed from mean values over the annual period to yield a production estimate.

8.7.2.1 Summation methods

Summation methods to estimate production stem from the concept that tissue lost from, or accumulated by, a population over a series of time increments is equivalent to an estimate of production. The removal summation method involves estimating the number of individuals lost, by mortality or other removal, from a cohort over a time interval and coupling those data with biomass information, which results in an estimate of production for that cohort during that interval. The summation of the production estimates for that cohort are summed for intervals over the annual period to estimate annual production. Removal summation is not typically applied to fish populations, and algorithms to estimate associated variance are not readily available. As such, we recommend the use of the more commonly used increment summation method in preference to the removal summation method.

Similar to the removal summation method, but quantifying accumulation, rather than loss, the increment summation method sums the growth increments of a cohort over time. The growth increment is quantified as the increase in mean individual weight over a time interval for each cohort, and this increment is multiplied by the density of the cohort to obtain a production estimate for the cohort during that interval. Production for each cohort is summed for the population, and production for each interval is summed for an annual estimate.

Explicit formulae for increment summation production estimation and associated variance estimators for fish populations were developed by Newman and Martin (1983) as

, (8.45)

where P = production for a given cohort within a specified interval,

= arithmetic mean cohort density from time t to t+1, and

= change in mean weight of individuals in the cohort from time t to t+1 (i.e., - ).

Sampling variance of the production estimate [V(P)] can be estimated as

, (8.46)

where V()and are the variances of and , respectively.

Algorithms to estimate variance of and are found in Newman and Martin (1983) and require estimates of variance for each density and mean weight estimate (Sections 8.3, 8.4). Pop/Pro statistical software (Kwak 1992; http://www4.ncsu.edu/~tkwak) includes a module to estimate fish production by the increment summation method according to cohort and time interval, including related parameters and associated variances.

8.7.2.2 Instantaneous growth rate and Allen curve methods

The instantaneous growth rate method was initially developed to estimate production of fish populations (Ricker 1946; Allen 1949). By this method, production is simply the product of the instantaneous growth rate and mean biomass as

P = G, (8.47)

where P = production for a given cohort within a specified interval,

G = instantaneous growth rate for the cohort from time t to t+1
(i.e., log_e

- log_e), and

= arithmetic mean cohort biomass from time t to t+1.

From Newman and Martin (1983), the variance of the production estimate [V(P)] may be estimated as

V(P) = V()G² + ²V(G), (8.48)

where V() and V(G) are variances of the mean biomass and instantaneous growth rate, respectively. The variance of mean biomass is estimated as

V() = [V(B_t) + V(B_{t + 1})]/4, (8.49)

where V(B_t) and V(B_{t + 1}) are the variances of biomass at times t and t+1, respectively.

The variance of the instantaneous growth rate may be estimated as

V(G) = V(log_e

) + V(log_e), (8.50)

where V(log_e

) and V(log_e) are variances of the natural logarithms of mean weights of individuals of the cohort at times t and t+1, respectively. By using a Taylor series expansion (delta method; Seber 1982; Cone and Krueger 1988), V(log_e) can be approximated as

V(log_e

) = V()/. (8.51)

Mean annual density or biomass estimates (and their variances) computed from multiple intervals of different duration must be weighted according to the number of days in each interval (i.e., equation 8.48 must be modified) and should be computed following formulae in Newman and Martin (1983). Fish production can be estimated by the instantaneous growth rate method using Pop/Pro statistical software (Kwak 1992; http://www4.ncsu.edu/~tkwak) according to cohort and time interval, including related parameters and associated variances. Additional algorithms are available in the software documentation. An example calculation of fish annual production estimated by the instantaneous growth rate method is presented in Box 8.8.

Allen (1951) extended the instantaneous growth rate method to a graphical form to estimate production in what has become known as the Allen curve method. The Allen curve is a growth-survivorship curve for a given cohort, wherein the number of surviving individuals is plotted against the mean weight of those individuals (Figure 8.6). Following this configuration, the biomass of the cohort can be estimated at any point in time on the curve as the corresponding product of number of individuals (or density) and their mean weight. Likewise, the area under the curve may be calculated in corresponding units as production of the cohort during the specific interval plotted. The Allen curve is rarely presented in recent literature, most likely because explicit variance estimators have not been derived for it, and the quantitative form of this concept, the instantaneous growth rate method, is more precise and relatively easy to apply using software applications. Nonetheless, examination of Allen curves can be an instructive means to visualize and elucidate production dynamics of a fish population.

8.7.2.3 Size-frequency method

The size-frequency method to estimate production was first developed by Hynes (1961) as an alternative means to estimate production when cohort identification was not possible; it was formerly also referred to as the Hynes method. Conceptually, the method is similar to the removal summation method, except that the size-frequency method sums tissue losses between successive size groups, rather than over time intervals. Originally, the method was used to roughly approximate production for multi-species assemblages of aquatic animals, but subsequent criteria and assumptions, established to improve the method, restrict its practical application to single species or closely related species with similar life histories (Waters 1977; Krueger and Martin 1980).

Production is estimated by the size-frequency method for fishes as (Garman and Waters 1983):

, (8.52)

where P = production for a given population or multi-species group within a specified interval,

= mean density (arithmetic mean of estimates) for a specific length group,

= mean weight (arithmetic mean of estimates) of individuals in a specific length group,

k = index for length groups,

c = number of length groups, and

CPI = the cohort production interval (average maximum age of fish in the population or
multi-species group in years).

Variance of the production estimate [V(P)] is computed as:

(8.53)

where V() and V() are variances of mean density and mean weight, respectively, for length groups. If the production period spans more than a single time interval (more than two samples), then mean density and mean weight estimates (and associated variances) must be weighted by interval length (days) according to algorithms provided by Garman and Waters (1983). An example calculation of fish annual production estimated by the size-frequency method is presented in Box 8.9.

8.7.3 Production to Mean Biomass (P/) Ratio

The annual production to annual mean biomass (P/) ratio is of special interest as an ecological index (also referred to as the turnover ratio) and as a simple conversion factor to approximate production (Waters 1977). Based on the premise that P/ is relatively constant for an organism or related organisms, it has been suggested that production may be approximated from a biomass estimate using the P/ ratio as a multiplier. The ratio of P/ for most fish species ranges from 0.2 to 4.0. However, P/ can be quite variable within and among species (Waters 1977, 1999; Mann and Penczak 1986; Elliott 1994) and may vary with the number of cohorts (or life span) of a population (Waters 1992; Kwak and Waters 1997). Thus, this method should be applied carefully and considered an imprecise approximation of production. The exact P/ ratio to employ for such estimates should be species-specific and may be refined further if the number of cohorts in a population is known (Waters 1992; Kwak and Waters 1997). Estimates of mean annual biomass should be weighted by interval duration if more than a single interval is included. Newman and Martin (1983) present formulae for estimating mean annual biomass and its variance, as well as a variance estimator for the P/ ratio.

8.7.4 Production Estimates in Practice

In general, if age data are available, the instantaneous growth rate method is the preferred approach to estimate fish production and associated parameters; otherwise, the size-frequency method may be used. Estimating production using the P/ ratio should be used only when data are lacking for application of more precise methods. Computer software is available for using the increment summation or instantaneous growth rate methods (Kwak 1992; http://www4.ncsu.edu/~tkwak), and a spread-sheet application can facilitate calculations by other methods. Estimates of variance (precision or sampling error) should be reported for all estimates of production and related population parameters as approximate 95% confidence intervals.

Many assumptions and criteria for applying these methods have been defined and should be considered with application (Waters 1977; Newman and Martin 1983). We suggest that subjective decisions encountered when estimating fish production should be resolved to be conservative, so that the direction of error will be clear, and estimates may be considered minimums. The biomass estimate of age-0 fish, newly recruited into the population, at first sampling should be considered a conservative estimate of production for that cohort during that interval. Negative estimates of production, resulting from negative growth (i.e., weight loss) should be interpreted as no production (zero) for that cohort and interval when using summation methods or the instantaneous growth rate method. However, negative losses (i.e., increase in numbers between size groups) should be included in the sum when applying the size-frequency method.

Generally, the greater the number of fish population estimates (density and biomass) that are integrated into an annual production estimate, the more accurate that estimate will be. The minimum number of two population estimates will yield a less accurate production estimate than one based on more estimates within the annual period. A reasonable, general approach to estimating populations over a one-year period for an annual production estimate is to conduct one estimate prior to spawning (e.g., spring for many temperate fishes), another near the end of the primary growing season (e.g., fall for temperate areas), and a third, one year after the first estimate (e.g., spring or fall).

Fish production estimates are valuable statistics for understanding population dynamics and elucidating ecological relationships and have great potential for improving fisheries management. Waters (1992) reviewed and proposed the application of annual production, annual P/ratio, and ecotrophic coefficient (annual angler harvest/annual production) to management of stream-dwelling trout fisheries. Incorporating production dynamics into fish assessment and monitoring may provide a broader perspective on the dynamics of harvested fishes. Thus, regulation and assessment of harvest as a proportion of fish tissue produced on an annual basis provides an alternative to the standard approach, based solely on fish density or biomass. 8.8 FUTURE DIRECTIONS

In many studies of fish populations, information is often available beyond that needed to apply the methods outlined in this chapter. In particular, information on the age-structure of the population is often collected. When the abundance of a population is estimated on an annual basis, knowledge of the prior age composition is helpful in constraining estimates. As a simple example, the abundance of a cohort can not be larger than the abundance in the prior year (in a closed population). The constraints imposed by age-structure relationships can help improve the precision and accuracy of population estimates. Application of auxillary information to population estimation opens up a rich, yet complex, diversity of models. Powerful statistical catch-at-age models (Hilborn and Walters 1992) are an example of a framework that incorporates the extensive information that is often available on intensively-studied fish populations. Because of the complexity of such models and their intensive data needs, such methods are generally applied to marine fish stocks and some stocks in large inland waters (e.g., Great Lakes) where the cost of data collection and analysis is commensurate with the value of the fishery.

Even within the scope of the methods presented in this chapter, there are potential gains to be made by combining data from different sources. In particular, the combination of removal methods with marking fish holds promise for improving population estimates. The methods illustrated here for analyzing mark-recapture data do not make direct use of measures of sampling effort. Removal methods, on the other hand, explicitly assume effort is constant or accommodate changes in effort by standardizing catch to CPE.

The estimation of the variance and confidence intervals for population estimates is an area where substantial improvements need to be made. Although the likelihood methods presented here have a long history of use, and provide a strong statistical basis for estimation, many of the formulae are strictly valid only for large sample sizes, or are approximations to the "full" formulae. In many applications, the target population itself may be small (e.g., the number of fish in a 100-m stretch of stream) or the number of marked or recaptured fish is small to moderate (i.e., less than 30). In situations like these, variance estimates and confidence intervals should be interpreted with caution.

Another trend we see emerging is the incorporation of a Bayesian approach to data analysis. In many situations, researchers and managers have knowledge from prior experience that is pertinent to the population being studied. Incorporating the experience and beliefs of "experts" can improve population estimates in many cases. The Bayesian approach, however, presents several practical concerns regarding how best to represent prior information.

8.9 CONCLUSIONS AND SUMMARY

In this chapter, we illustrate several approaches for estimating fish abundance, biomass and production. A foundational concept is that additional information beyond CPUE is generally needed to provide accurate population estimates. The incorporation of this information inevitably entails making assumptions about the sampling regime, and creating models of this idealized process. Therefore, it is important to test assumptions where possible, and apply models that best represent the data obtained. Some assumptions can be relaxed by applying alternate models, but some are essential to obtain any valid estimate. The key to meeting critical assumptions is often the judicious planning of the sampling program and the careful application of field methods. Given the wide array of sampling challenges facing fisheries scientists, this chapter should be viewed as an entry into some of the more common and basic methods. Every investigation poses its own set of challenges, but often these problems are not unique. By building on the base developed here, we hope to provide readers with the confidence to face the diversity of situations they are likely to encounter in their professional work.

Box 8.1 Estimating abundance and density using distance sampling

We define the following variables:

n = number animals observed

N = total population in reach

A = total area of reach (m²)

D = density of fish (number/m²)

L = length of transect (m)

y = right angle distance from transect for each animal (m)

w=effective strip width

V() = variance of estimated population

Using the assumption that sightability drops off exponentially with distance from the transect, and that fish are independently distributed in the reach, we have the following (Seber 1982):

If sightability drops off exponentially, the estimated population is 78 with an approximate confidence interval of 42 to 114. Note that the density (and hence total abundance) using a half-normal distribution is approximately half that obtained with an exponential, highlighting the need to test the assumed sightability function (see Buckland et al. 1993 for these methods).

Box 8.2 What is a Likelihood?

Applying this formula for each possible outcome, we can see that the outcome with the highest probability (i.e., the most likely outcome) is 24 fish captured. Equations of this type are known as probability functions for discrete distributions, or probability density functions for continuous distributions.

When estimating population size by maximum likelihood, we reverse the role of parameters and data. We know our data (or in this case datum, i.e., n=20) and ask what is the most likely population size that would have produced our observation. For simplicity in this example we assume q=0.4 and is known. Now, we can write:

Note that this is mathematically identical to the previous equation. However, we now refer to P(N = 60|n=20,q=0.4) as the likelihood. When using the likelihood, we generally take the view that the parameter we are estimating () can be varied to maximize this likelihood (this is called profiling the likelihood). The fundamental concept of statistical likelihood is that our observations (the data) occur through a stochastic or random process with a defined probability structure. Through this process, we are likely to observe data in proportion to their probabilities as described in the formulae above.

As shown above, the likelihood is maximized for N=50. Our estimate is less than the true value of 60 because in our one pass we happened to capture somewhat less than one might typically capture. Note that although N=60 does not maximize the likelihood, it has a likelihood that is reasonably high. Thus, we cannot rule out N=60 as it could have reasonably generated the observed data. On the other hand, the likelihood is very low for N=100. If we repeated the process of sampling, sometimes our population estimates would be above and other times they would be below the true value, but would very rarely be above 100. As such, the likelihood is a measure of how consistent the data are with different population sizes.

In this simple example we assumed that q was known. If we had not, we could not have computed a unique solution (e.g., our data could have resulted from a combination of smaller q and larger N). As indicated in the introduction to this chapter, we generally need more information than CPUE from a single sampling event to estimate true abundance. In our simple example, the additional information we need is the probability of capture with a single pass (i.e., catchability).

To be somewhat more realisitic, assume that q is unknown and we apply a depletion sampling experiment (see Section 8.4) to the stream, and catch 24 fish in the first sampling pass, 17 fish in the second sampling pass, and 8 fish in the third sampling pass. We will also make the usual assumptions that the population is closed, and that all fish have equal vulnerability, and that this is constant over time. The details of the likelihood function for the removal method are presented in Section 8.4, equation 8.24. Note that the log_e of the likelihood is often used to make the computations more tractable. Applying the formula to various levels of catchability from 0 to 1.0, we can profile the likelihood for these data as shown in the figure above.

From this graph, it is apparent that it is possible that catchability (q) is equal to 0.89, but it is not very likely relative to other possible values of q. Likewise, q could be 0.01, but that too is not very consistent with our observations. In this example, the value of q that is most consistent with our observations is 0.40. Thus, we term this the maximum likelihood estimate of q. Since we sampled the population 3 times, the cumulative proportion of the population removed is 1-(1-q)³ = 1-(0.6)³ = 0.784. Given that we caught a total of 49 fish, the most likely estimate of N is 49/0.784=62.

There are several ways to estimate variances and confidence intervals associated with maximum likelihood estimates. One way is to consider how the likelihood changes when parameters move small distances away from the maximum likelihood value. The first derivative of the logarithm of the likelihood measures how quickly the likelihood changes relative to a change in the parameter, and is zero at the maximum. Variance is estimated by taking the negative of the reciprocal of the second partial derivative of the logarithm of the likelihood with respect to each parameter (Seber 1982 page 4). The second partial derivative measures the curvature of the log-likelihood portrayed in the above figure. If the magnitude of the second derivative is large, this means that the likelihood falls off rapidly as we move the parameters away from the maximum likelihood estimate; the estimated variance would be relatively small because alternative values very far from the estimate are unlikely. Confidence intervals can be constructed from the variance estimated above, assuming a distribution (often normal) for the estimate. The profile likelihood can also be used to construct confidence intervals directly by determining values for the parameters that give a log-likelihood value that differs from the maximum value of the log-likelihood by 3.841. This method is based on the fact that, under the null hypothesis, this difference approximates a ² distribution with 1 degree of freedom, and that 3.841 is the 5% critical value for the ² distribution with 1 degree of freedom. As shown in the above figure, the maximum likelihood estimate of N is 62, with a 95% confidence interval of 51 to approximately 1650.

Both approaches for computing variances and confidence intervals produce approximations based on asymptotic (i.e., large sample) statistical properties, and require relatively large sample sizes to be accurate. The profile likelihood method often performs better because the shape of the likelihood profile is examined and no assumption of normality is made. The better performance of the profile likelihood method comes at the cost of greater computation, however.

Box 8.3 Estimation of Population Abundance for a closed population with Otis et al. 1978 Mark-Recapture Models

Starting with model M_o, we compute the log-likelihood for a trial values for by applying formula 8.8. Two examples for trial values are 30 and 23. Using these values, we obtain:

When model M_t is employed, the log-likelihood for the same trial values is:

The maximum of the log-likelihood is -10.854 at = 23.

When model M_b is employed, the log-likelihood for the same trial values is:

The maximum of the log-likelihood for model M_b is -11.247 at = 24. The AIC for each model is:

Based on the AIC, we would choose model M_o as the best model among those considered. The likelihood for this model is not substantially lower than for M_t and M_b, but it requires fewer parameters, resulting in a more parsimonious model.

Box 8.4 Estimation of abundance with a Cormack-Jolly-Seber (CJS) model for open populations.

From these data, we define the following components of the CJS model for an open population:

From these components, we can estimate the following quantities:

As this example illustrates, estimating abundance for open populations is much more difficult than for closed populations, with several important population parameters not being estimable.

Box 8.5 Estimation of abundance using the removal method

In order to estimate the abundance of brown trout in a 50-m section of stream below a culvert, a fishery manager conducts a three-pass removal experiment. Fish cannot move upstream because of the culvert, and the manager places a block net on the lower section of the study reach to insure that the population is geographically closed. All three sampling passes are conducted during the same day, using a backpack electrofishing unit. During sampling, 24 brown trout were caught in the first sampling pass, 17 in the second sampling pass, and 8 in the third sampling pass.

Although the population size can be estimated applying formula 8.23, we illustrate the application of the more general likelihood formula 8.24. Using a trial value for catchability (q) of 0.2 is:

A search across a range of from 0.01 to 0.99 in steps of 0.01 indicated the most likely value of was 0.40, with a log-likelihood value of -49.832. From this, is estimated as:

Confidence intervals are typically obtained from the profile likelihood of . From the search across values of ranging from 0.01 to 0.99 (in 0.01 increments), the log-likelihood values for q=0.65 and q=0.01 differed from the log-likelihood at =0.4 by 3.841 or more (which is the critical value for the ² distribution with 1 degree of freedom). The population sizes corresponding to these values of q are 51 and 1650, and represent approximate 95% confidence intervals for .

Box 8.6 Estimation of abundance using the removal method in an open population.

To proceed, we need initial values for the measurement errors (_t for the adult CPUE index, _t for the pre-recruit CPUE). A good initial guess would be to set all values to 1.0, but we show the table below with the final estimates. From these initial guesses, we then fill in the columns for the expected indices by dividing the observed CPUE values by these multiplicative errors. The final column in the table below is a forecast of the adult survey index at time t+1, from the expected index at time t, ignoring process error. Note that these forecast adult survey indices turn out to be the same as the input data to the number of digits displayed, and that no prediction is made for the first year of the time series (1985) because survey indices are not available for the year prior. To fill in this column, a value for q (catchability) is required. This is unknown (to be estimated) but an initial guess is needed to get started for this quantity also. One approach is to hazzard a guess regarding exploitation rate in recent years, say 25%, and thus approximate N, say, as four times C. Then your initial value for q would be N/n, perhaps based on an average of such values over years. We illustrate calculations, however, with the final estimate for this parameter also. Once a value of q is available, forecast values were obtained by application of equation 8.36, dropping the process error and denoting the forecast value as . For example, for 1986 assuming q=0.00011 and M=0.2 is:

Year	t	= expected adult survey index	t	= expected recruitment survey index	= forecast Adult survey index
1985	1.076	46.42	1.020	11.46
1986	1.032	39.68	1.007	8.04	38.88
1987	1.132	33.63	1.063	15.06	30.14
1988	0.964	31.68	0.979	18.76	33.18
1989	1.111	38.97	1.033	10.71	35.36
1990	0.969	33.32	0.984	17.28	34.68
1991	1.115	38.91	1.031	9.82	35.21
1992	0.810	25.14	0.914	12.85	31.46
1993	1.222	28.99	1.198	25.41	24.07
1994	0.709	26.29	0.902	11.20	37.46
1995	0.879	20.15	0.910	15.52	23.29
1996	1.089	25.59	1.050	13.89	23.76
1997	1.315	36.51	1.040	3.64	27.99
1998	1.017	29.07	1.013	21.39	28.78
1999	0.972	37.01	0.995	6.56	38.25
2000	1.183	37.90	1.050	9.19	32.21
2001	1.000	34.31	1.000	9.01	34.51

Assuming M is known to be 0.2, the above model has three sets of parameters: q which applies to both adults and pre-recruits, _t which encapsulates measurement errors for the adult index, and _t which encapsulates measurement errors in recruitment index. Parameter estimation is accomplished by minimizing the following sum (representing the sums of squared deviations): :

The first two components are directly calculated from parameter estimates that are also entries in the above table. The depend jointly on all the estimated parameters, and are calculated as . Thus, q, the _t's, and the _t's are adjusted through an iterative search procedure from the initial guesses so as to minimize SSE. Note that underlying this mimimization are assumptions that the process errors are normal, the logarithms of the measurement errors are normal (i.e., the original terms are lognormal) and the variances for each of these are equal. The terms in this sum could be weighted to represent different variances for each type of error (Collie and Sissenwine 1983)

Outputs of the model include estimates of the above parameters, as well as the annual population size and number of recruits entering the population. Using an excel spreadsheet to do the calculations, and the solver function to minimize the SSE by changing the parameters, we obtained the following estimates for population size and recruitment:

Collie and Sissenwine (1983) provide details on how to calculate the variance of the parameter estimates.

Box 8.7 Application of surplus production modeling

With B₀=800,000 kg, K=4,000,000, q=0.0001, and r= 0.17 as initial guesses for the parameters of equations 8.37 and 8.39, we can predict catch and CPUE as follows:

Recursive application of equations 8.37 and 8.39 result in the following time series of predicted values:

Note the discrepancy in the trend between observed CPUE and predicted CPUE, indicating our initial guesses for parameter values were not very good. We used the Solver function in Excel to perform a nonlinear search across the parameter values (i.e., B₀, K, q, and r were used as the "change cells" in Solver) to find the combination of parameters that minimize the sum of squared deviations between predicted and observed CPUE. Solver returned an estimate of B₀=732,506, K=1,160,771, q=0.0001484, r=0.4049 with a sum of squared deviations of 1616.17. Using these parameter values as the "best" estimates, the predicted biomass and CPE over time is:

Note that the trend in predicted CPUE matches the observed trend in CPUE quite closely after obtaining the "best" estimates for B_0,, K, q, and r.

Box 8.8 Production estimation by the instantaneous growth rate method.

Sampling date: 8 March 1974

___________________________________________________________________________
Age Density Mean weight Biomass

Class (N) V(N) (in grams) V() (B in grams) V(B)

___________________________________________________________________________

1 277.85 1,336.05 6.86 0.13 1,905.34 75,455.97

2 157.54 317.71 28.56 0.79 4,499.83 222,126.45

3 36.11 34.00 107.23 19.89 3,872.13 469,764.75

4 11.17 13.16 170.05 42.09 1,898.89 350,364.34

Total 482.67 1,700.92 12,176.19 1,117,711.50

___________________________________________________________________________

Sampling date: 29 July 1974

___________________________________________________________________________
Age Density Mean weight Biomass

Class (N) V(N) (weight) V() (B in grams) V(B)

___________________________________________________________________________

1 276.45 553.56 24.27 0.62 6,709.17 306,074.53

2 68.08 94.58 77.31 2.49 5,262.90 278,582.66

3 9.76 7.64 146.18 100.67 1,427.00 167,558.97

4 8.12 1.11 194.72 1.67 1,582.12 30,259.81

Total 362.41 656.89 14,981.19 782,475.97

___________________________________________________________________________

To estimate production for the age-1 cohort during this interval, we follow Equation 8.47 as

P = G

P = (log_e24.27 - log_e6.86)[(6,709.17 + 1,905.34)/2]

= 5,442 g.

To estimate the variance of P, we begin by estimating the variance of

from Equation 8.49 as

V(

) = [V(B_t) + V(B_{t + 1})]/4

V(

) = (75,455.97 + 306,074.53)/4

= 95,383 g.

Then we estimate the variance of G from Equation 8.50 as

V(G) = V(log_e

) + V(log_e), expanded by incorporating Equation 8.51 as

V(G) = V(

)/ + V()/

V(G) = 0.13/6.86² + 0.62/24.27²

= 0.003815.

Now we may employ Equation 8.48 as

V(P) = V()G² + ²V(G)

V(P) = (95,382.62)(log_e24.27 - log_e6.86)² + [(6,709.17 + 1,905.34)/2]²(0.003815)

= 223,058 g

The sampling area was 0.181 ha, so to convert our production estimate to a standard area unit (ha) we divide by the area, and to convert to a standard mass unit (kg), we divide by 1,000 as

P = 5,442.36 g/0.181 ha/1,000

= 30.1 kg/ha.

Whenever you multiply a statistic by a constant, you multiply the variance of that statistic by the constant squared. Thus, to convert the variance of our production estimate to standard units, we divide by the area (0.181 ha) squared and divide by 1,000 squared as

V(P) = 223,058/0.181²/1,000²

= 6.8 kg/ha.

Our production estimate ± approximate 95% confidence intervals [± 2V(P)^0.5] for this cohort during this interval is 30.1 ± 5.2 kg/ha. This procedure is then repeated for the other age classes to estimate production for the population (rounded to the nearest tenth) during this interval as 62.8 ± 7.4 kg/ha, as below. ____________________________________

Age Class Production (kg/ha ± 95% CI)

____________________________________

1 30.068 ± 5.219

2 26.856 ± 4.380

3 4.536 ± 2.718

4 1.303 ± 0.876

Total 62.763 ± 7.388

____________________________________

Note that confidence intervals are not additive, and variances should be summed to compute a confidence interval for a total. The entire procedure is then repeated for the other two intervals within the annual period to estimate annual production and its confidence interval.

Box 8.9 Production estimation by the size-frequency method.

___________________________________________________________________________ Length Mean density Mean weight Mean biomass

group ( per ha) V() (in grams) V() ( in g/ha)

___________________________________________________________________________

1 260.2 2,653.5 3.2 0.2 832.6

2 281.7 1,491.4 6.9 0.1 1,943.7

3 144.9 182.5 12.6 0.1 1,825.7

4 88.8 145.9 27.9 1.4 2,477.5

5 67.7 49.5 49.9 5.1 3,378.2

6 43.1 19.5 75.7 45.0 3,262.7

7 55.9 601.4 109.5 24.8 6,121.0

8 26.9 61.2 158.6 11.8 4,266.3

9 19.8 104.2 196.0 39.9 3,880.8

10 15.0 0.2 260.8 13.1 3,912.0

___________________________________________________________________________

To estimate annual production for the population, we follow Equation 8.52, using 3 years for the cohort production interval(CPI) as

P = 0.5(10)[3.2(260.2 - 281.7) + 6.9(260.2 - 144.9) + 7 other summation terms + 260.8(19.8 - 15.0)](1/3)

= 5[-68.80 + 795.57 + 17,589.87 + 1,251.84](0.333)

= 32,614.13 g/ha/year.

Variance of P is then estimated according to Equation 8.53 as

V(P) = [0.5(10)]²{(3.2 + 6.9)²(2,653.5) + 0.2(260.2 - 281.7)² + [(3.2 - 12.6)²(1,491.4) + 0.1(260.2 - 144.9)²

+ 7 other summation terms] + (196.0 - 260.8)²(0.2) + 13.1(19.8 - 15.0)²}(1/3)²

= 25{270,683.5 + 92.4 +[131,780.1 + 1,329.4 + 6,201,690.4] + 41,733.2 + 301.8}0.11

= 18,465,585.9 g/ha/year.

To convert our production estimate to a standard mass unit (kg), we divide by 1,000 as

P = 32,614.13/1,000

= 32.614 kg/ha/year,

and its variance is converted as

V(P) = 18,465,585.9/1,000²

= 18.466 kg/ha/year.

Thus, our annual production estimate ± approximate 95% confidence intervals [± 2V(P)^0.5] for this population and year is 32.614 ± 8.594 kg/ha/year.

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Table 8.1 Example of a capture matrix for 20 fish observed across 4 sampling periods in a mark-recapture study. Each cell of the matrix (X_ij) is referenced by fish_i in row i, and sample period_j in column j. An entry of 1 in the matrix indicates that a fish was caught, and a 0 indicates that the fish was not caught during that sampling period. Fish 1, for example was caught in all four sampling periods, whereas fish 4 was only caught in the first sample period.

Figure Legends

Figure 8.1 Example of animals sighted in a transect survey. The histogram depicts the relative frequency of observations within 0.1-m intervals. The shaded area depicts the effective width of the transect. Open circles indicate fish that are not sighted and closed circles indicate fish that are sighted. Figure modified from Thompson et al.(1998).

Figure 8.2 Hierarchical organization of models for capture-recapture methods of population estimation as described by Otis et al. (1978). Notation for each model is defined in the text. Figure modified from Otis et al. (1978).

Figure 8.3 Probability of capturing n fish from a population with 30 individuals, each with a 40% chance of capture.

Figure 8.4 Log-likelihood as function of q and N for data in box 8.2.

Figure 8.5 Likelihood as function of N for Otis et al. hierarchy. An approximate confidence interval for N under model Mo is where the x-axis crosses the log-likelihood curve.

Figure 8.6 A generalized Allen curve depicting growth-survivorship for a fish cohort (after Allen 1951; Waters 1977). Production for the time interval is estimated as the area under the curve (shaded), and biomass at any point in time (cross-hatched area) is the product of the number of individuals (or density) and their mean weight.