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Effects of measurement error on catch–effort estimation W.R. Gould, L.A. Stefanski, and K.H. Pollock
Abstract: We have investigated the consequences of using imprecise catch and effort estimates in closed-population catch–effort analyses using traditional regression techniques and maximum likelihood to estimate the catchability coefficient and population size parameters. Our simulation study involved adding known amounts of measurement error to error-free catch and effort data to determine the effects of using such estimates of catch and effort rather than the true, and in many cases unknown, quantities. Our results suggest that naive estimation using estimates of catch and effort as true values may bias estimates of population size and the catchability coefficient. In most cases, the effects of measurement error in catch and effort were to inflate the parameter estimates, the magnitude of inflation being dependent on the size of the measurement error variance. Maximum likelihood estimation proved to be the estimation procedure most robust to the errors in measurement, but still displayed the need for correction of the measurement-error-induced bias. A recently developed simulation–extrapolation method of inference (J.R. Cook and L.A. Stefanski. 1994. J. Am. Stat. Assoc. 89: 1314–1328) is suggested as a possible means for making bias adjustments. Résumé : Nous avons exploré les conséquences de l’utilisation d’estimations imprécises des prises en fonction de l’effort pour les analyses prises–effort portant sur une population fermée à l’aide des méthodes traditionnelles de régression et du maximum de vraisemblance pour estimer les deux paramètres suivants : coefficient du potentiel de capture et taille de la population. Notre étude de simulation comportait l’introduction d’une erreur de mesure de grandeur connue à des données prises–effort sans erreur pour déterminer les effets de l’utilisation de telles valeurs estimées des prises en fonction de l’effort au lieu de valeurs réelles et, dans bien des cas, inconnues. Nos résultats indiquent que l’estimation naïve réalisée en considérant les valeurs estimées des prises en fonction de l’effort comme étant des valeurs réelles peut fausser les estimations de la taille de la population et du coefficient du potentiel de capture. Dans la plupart des cas, l’erreur de mesure avait pour effet de gonfler les estimations des paramètres, l’importance de la surévaluation étant liée à l’importance de la variance de l’erreur de mesure. L’estimation du maximum de vraisemblance s’est révélée la méthode d’estimation la plus robuste pour ce qui est des erreurs de mesure, mais elle exige tout de même une correction du biais causé par l’erreur de mesure. Une méthode d’inférence par simulation–extrapolation récemment mise au point (J.R. Cook et L.A. Stefanski. 1994. J. Am. Stat. Assoc. 89 : 1314–1328) est proposée comme méthode possible pour corriger le biais. [Traduit par la Rédaction]
Introduction When both catch and effort are known for heavily depleted populations, estimates of population size and catchability are possible with a variety of estimation techniques known as catch–effort methods (Ricker 1975; Seber 1982; Hilborn and Walters 1992). Catch–effort methods, generalizations of the removal method (Moran 1951; Zippin 1956, 1958), rely on the principle that the size of a sample obtained from a population is proportional to the amount of effort expended in retrieving the sample, and the population size. Catch–effort estimation has generally seen application in large-scale commercial fisheries, where a significant proportion of the population is removed for economic purposes. However, catch–effort estimation may also be useful in recreational fisheries with Received June 23, 1996. Accepted October 3, 1996. J12739 W.R. Gould,1 L.A. Stefanski, and K.H. Pollock. Statistics Department, Box 8203, North Carolina State University, Raleigh, NC 27695, U.S.A. 1
Author to whom all correspondence should be sent at the following address: University Statistics Center, Department 3CQ, New Mexico State University, Box 30001, Las Cruces, NM 88003-8001, U.S.A. e-mail:
[email protected]
Can. J. Fish. Aquat. Sci. 54: 898–906 (1997)
high fishing pressure. Catch is a general term describing the number of fish harvested plus fish released (Malvestuto 1983). Effort is the amount of time spent fishing usually measured in trot-lines per day or boat-hours per day for commercial fisheries and angler-hours or angler-days in creel surveys. Since it is impractical to census most fisheries, sampling studies of one form or another are sometimes performed by fisheries personnel, such as capture–recapture or large-scale removal studies, to collect demographic information. These studies take significant amounts of time, financial, and personnel resources. Creel or angler surveys are routinely performed as a part of recreational fisheries management to provide managers with information on estimated catch, fishing pressure (effort), and other characteristics of a fishery (Best and Boles 1956; Van Den Avyle 1986; Guthrie et al. 1991), but such data collection efforts have not been used to formally estimate population size via catch–effort analysis (Pollock et al. 1994). If catch and effort data can be obtained from individuals who routinely use the fishery for recreational or commercial purposes, which can lead to the estimation of population size, etc., then fisheries personnel could divert their time and money toward more productive endeavors. Some commercial fishery sampling programs attempt to completely enumerate the catch and effort of those using the fishery upon returning to the port, while others may only sample © 1997 NRC Canada
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the population of fishers. These types of surveys, along with roving creel surveys and access point surveys in recreational fisheries, are referred to as on-site surveys (Pollock et al. 1994). Telephone or mail surveys rely on a different sampling frame and are termed off-site surveys. The specific sampling schemes for these angler surveys (Neuhold and Lu 1957; Malvestuto 1983; Robson and Jones 1989; Guthrie et al. 1991) are not the primary focus of this paper, but the fact that samples are taken rather than a complete enumeration of catch and effort is important in discussing catch per unit effort (CPUE) as an index of stock size (Ricker 1975; Prouzet and Dumas 1988). The effect of sampling the catch and effort from a fishery generates some level of uncertainty in the estimates of total catch and effort, which is to say that there is measurement error in these estimates. Richards and Schnute (1992) presented two models for estimating CPUE from catch and effort data but did not address the implications of using estimated CPUE in catch–effort analyses. Through simulation, Gould and Pollock (1997) evaluated the effectiveness of traditional catch–effort regression methods (Leslie and Davis 1939; DeLury 1947) and maximum likelihood (ML) estimation (Seber 1982) when catch and effort are known. Here, we examine the estimation of population size and the catchability coefficient based on catch and effort data contaminated with error. Walters and Ludwig (1981) assessed the effects of errors in numbers of spawners and recruits on stock–recruitment relationships. Collie and Sissenwine (1983) accounted for measurement error in relative abundance data in estimating fish abundance for an open population using filtering theory (Kalman 1960). However, no one to our knowledge has evaluated the effects of errors in catch and effort on catch–effort estimation of closed populations. By adding measurement error into otherwise error-free catch and effort data, we were able to evaluate the effects of errors in measurement on catch–effort estimates of the catchability coefficient, k, and population size, N. Initially, the errors in catch and effort were assumed to be independent, but they were later correlated in both a moderate and strong fashion to determine the effects of correlations in errors on the estimates.
Measurement error models Linear regression example To better understand the structure of measurement error models, it is best to first describe a simpler model with and without measurement error, namely, a simple linear regression model. Simple linear regression without measurement error is of the form Yi = α + βui + εi ,
εi ~
(0,σ2)
,
where Yi (the response variable) is a random variable and ui (the explanatory variable) is assumed to be fixed and known. The population parameters α and β, the intercept and slope, respectively, are assumed fixed and unknown, and the errors εi are assumed to be independent with mean zero and variance σ2. Often, the errors are assumed to be normally distributed to aid in the construction of confidence intervals for the population parameters. The errors-in-variable model, or measurement error model, generalizes the linear regression model above and considers the observed explanatory variable ni to be some
combination of the true (unobserved) ui, referred to as a latent variable, and some error term δi such that ni = ui +δi. Models with fixed ui are referred to as functional models, whereas models with random ui are termed structural models (Kendall and Stuart 1973). The simplest case assumes that the errors in measurement are independently and identically distributed normally with mean zero and variance σ2δ and are independent of the εi. Now, the regression model is Yi = α + βui + εi ;
ni = ui + δi , δi ~ N(0,σ2δ) , εi ~ N(0,σ2) , cov(δi,εi) = 0 .
Naive estimation would simply treat the observed data as the true, unobserved data. It can be shown in large samples that the expectation of the naive slope estimate, given the observed explanatory variables, is ^ )= E(β Y|n
σ2u σ2u + σ2δ
β,
where σ2u/(σ2u + σ2δ) is sometimes referred to as the reliability ratio (Fuller 1987) or heritability in the genetics literature. The effects of measurement error are to underestimate the absolute value of the true slope, sometimes referred to as the attenuation of the slope (Fuller 1987; Rawlings 1988). Among other effects, the coefficient of determination (R2), a measure of the fit of a model, is also attenuated in the presence of measurement error. Assuming that the reliability ratio is known, an unbiased estimate of the true slope can be constructed; however, this assumption is not likely to be met in practice. The use of measurement error models in modelling fisheries dynamics has been proposed before (Ludwig and Walters 1981; Schnute et al. 1990; Schnute 1994), but none has suggested their use in catch–effort estimation of closed populations. Catch–effort regression methods The catch–effort regression methods proposed by Leslie and Davis (1939) and DeLury (1947) are simple linear regression models, regressing some function of CPUE against prior cumulative catch (xi) or prior cumulative effort (Fi). Viewing the true sample sizes removed (nti) from the population as conditional binomials with probability of capture pi, E (nti | xi) = (N − xj)pi where i−1
xi = ∑ ntj . j=1
The regression techniques make the assumption that sampling is a Poisson process with respect to the true effort, fti, so that pi = 1 – e−kft . An approximation for the capture probability is used in the formulation of the linear models, so that pi ≈ kfti for small kfti. The relationship between CPUE and prior cumulative catch is i
nti E = kN − kxi . f ti © 1997 NRC Canada
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DeLury’s method has a slightly different structure, relating the natural logarithm of CPUE to prior cumulative effort: nti E ln = ln(kN) − kFi f ti where i−1
Fi = ∑ ftj . j=1
Catch and effort are commonly recorded in creel surveys and commercial fisheries, but in many cases being based on samples and not censuses, they are both estimated with error. Thus, the observed catch noi = nti + εn and the observed effort foi = fti + εf , where nti and fti are the true unobserved values of catch and effort, respectively, lend themselves to catch–effort analysis, but are random variables with measurement error. As such, the structural relation with Leslie’s model takes the form
in catch–effort estimation via ML, an alternative estimation procedure that has received attention in recent years (Seber 1982; Dupont 1983; Gould and Pollock 1997). Defining pi as the capture probability of an individual in the population at the ith sample with fti units of effort, sample sizes are distributed binomially as N g(nt1) = n pn1t1 (1 − p1)N − nt1 t1 N − nt1 n g(nt2 | nn1) = p t2 (1 − p2)N − nt1 − nt2 n 2 t2 such that the joint distribution of the sample nti values is s
N − x i n g {n~ti} = ∏ p t (1 − pi)N − x + 1 nti i i=1
i
i
i
nti E = α + βxj + εi ; f ti
i
where i−1
xi = ∑ntj .
noi = nti + εni ,
j=1
foi = fti + εf ,
Rearranged, the joint distribution becomes the multinomial distribution
i
εn ~ (0,σ2n) , i
N!
εf ~ (0,σ2f ) ,
∏ nti!(N − xs + 1)!
εi ~ (0,σ2) where α = kN, β = –k, and the εi values are the random errors that result even with knowledge of the true data. The structural relation with DeLury’s model relates the natural logarithm of CPUE to prior cumulative effort, nti E ln = α + βFi + εi ; f ti
noi = nti + εn , i
foi = fti + εf , i
εn ~ i
p1nt1(q1p2)nt2 . . .
s
i
(0,σ2n)
,
εf ~ (0,σ2f ) , i
εi ~ (0,σ2) where α = ln(kN) and β = –k. We have chosen a reasonable measurement error model for our investigation, one that makes specific assumptions about the error structure, e.g., no systematic bias exists in the measurements. Any conclusions to be drawn are dependent on this model and would likely change for cases requiring alternative measurement error models. Both of these models contain two sources of measurement error in the response variable (CPUE). In addition, the measurement error is cumulative in the explanatory variable. Clearly, the regression problem has become somewhat more complex and leads one to question whether or not it is necessary to account for the measurement error. We have attempted to answer that question through simulation by assessing the effects of various levels of measurement error on the estimates produced by the regression methods described earlier. Maximum likelihood estimation Our simulations also illustrate the effects of measurement error
i=1
(q1q2. . .qs − 1ps)nt (q1q2. . .qs)N − x s
s
+1
,
with pi = 1 – e−kft and qi = 1 – pi. A likelihood function of the parameters can be constructed from the observed set of catches and efforts and the ML estimates of N and k can be found numerically. However, when catch and effort are estimated quantities (noi and foi, respectively) rather than known values, then an assumption about the error variances must be made to allow for the identifiability of the population parameters. The more classical method of resolving the identifiability problem assumes knowledge of the ratio of variation owing to random error and measurement error. In most situations, knowledge of this type of information is unrealistic and in past applications has been misapplied with poor results. Ludwig and Walters (1981) and Collie and Sissenwine (1983) assumed a ratio of error variances to be 1, but they admitted a lack of knowledge of the actual variances associated with each error type. Ludwig and Walters (1981) concluded their report by stressing the importance of obtaining estimates of the measurement error variance but did not give any suggestions for doing so. Other possible sources of information exist that can lead to parameter estimation. Some applications may make use of external replicates, a second set of measurements on the subjects from an independent study. Instrumental variables, correlates to the explanatory variable, may also provide sufficient information leading to estimation in measurement error problems. Of course, knowledge of measurement error variance or at least a good estimate of it is the most obvious piece of information that allows for estimation and is one most suited to catch–effort analyses, since the specific sampling design used in creel surveys or commercial fisheries will provide an estimate of the measurement error variance. In all but the most i
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straightforward cases, explicit solutions for the parameters will not exist and must be solved for numerically using some maximization procedure.
Simulations We used simulations to examine the extent to which measurement error in both catch and effort affected estimates of the catchability coefficient k, and population size, N, for a closed population. Fisheries with initial population sizes of 10 000 and 100 000 fish, representing two sizes of recreational fisheries, were sampled on s = 5 or s = 10 occasions with the sequence of true efforts fi = 7.133, 8.133, 7.133, 6.133, 7.133 for s = 5 occasions (repeated for s = 10). In combination with the effort quantities, the catchability coefficient, k = 0.05, produced capture probabilities at a given sampling period ranging from 0.26 to 0.33. The total proportion removed from the population was 83% for s = 5 occasions and 97% for s = 10 occasions. True catches were generated randomly without measurement error as a series of conditional binomials. For every individual remaining in the population, a uniform random number between 0 and 1 was compared with the capture probability for that period to determine the capture or noncapture of the animal. Measurement errors were generated randomly by assuming a lognormal distribution for the observed catch noi, given the true catch nti, with the respective conditional expectation and variance nti and λσnt 2 :
produced an estimate of the catchability coefficient, its vari^ from ML, Leslie’s, and ance, and a population size estimate N DeLury’s methods. Five hundred replications (B = 500) were used in most cases to characterize the distribution of estimates, although some runs were made with 2000 replications to provide greater precision for the mean parameter estimates.
Simulation results We plotted the magnitude and direction of the relative bias for the average population size and catchability coefficient estimates from ML estimation, and the two regression approaches, as a function of the magnitude of the measurement error variance (λ = 0.0, 0.25,. . ., 1.0, 1.5, 2.0). The average estimate without measurement error is evident when λ = 0.0. Extreme values, those estimates that were larger than the true parameter value by a factor of 5 or were less than zero, were considered failures and excluded from the calculations of location (mean) and spread. The number of failures and the percentage of instances in which the total cumulative removal exceeded the population size estimate have been tabulated to evaluate the efficacy of catch–effort estimation when estimates of catch and effort are used. Note that failures and cases in which the cumulative removal exceed the population size estimate occur even in the absence of measurement error (Gould and Pollock 1997). The mean square error (MSE) is also presented as a measure of the overall performance of the point estimators.
i
noi | nti ~ lognormal(nti,λσn 2) . ti
In this way, the measurement error is unbiased with mean zero, but with a variance that is dependent on the variance of the true unknown catch. In a similar manner, given the true effort fti, the observed effort, foi, is foi | fti ~ lognormal (fti,λσft 2) , i
where λσnt and λσft are specified below. The measurement error variances λσnt 2 and λσft 2 were assumed to be directly related to the true catches nti and effort fti, respectively, in a Poisson-like manner, such that σnt 2 = nti and σft 2 = fti (i.e., variance proportional to the mean). The effect of the magnitude of the measurement error variance was demonstrated by systematically changing the value of the multiplying constant λ over a range of values (λ = 0.0, 0.25,. . ., 1.0, 1.5, 2.0). Initially, the errors in catch and effort were assumed to be uncorrelated (ρ = 0); however, some creel survey estimation procedures suggest possible correlations in the catch and effort estimates. For instance, catch may be estimated as the product of effort and success rate (equivalent to CPUE) in roving creel surveys or sports fisheries (Havey 1960; Rose and Hassler 1969), implying a positive correlation between the errors in catch and effort. We generated errors with two levels of positive correlation (ρ = 0.5, 0.9) and one negative correlation (ρ = –0.9) in addition to the uncorrelated case to determine the effects such correlation may have on the estimates. The program SURVIV (White 1983) performed the ML estimation using a conditional likelihood approach to estimation (Gould and Pollock 1997). We used SAS Institute Inc. (1988) to estimate the regression parameters for both DeLury’s and Leslie’s methods. Each repetition of the simulation 2
i
2
i
i
i
i
i
Population size Figure 1 displays the relative bias in the average population size estimate as a function of the measurement error variance for ML, Leslie’s method, and DeLury’s method for four levels of correlation between the errors in catch and effort (ρ = 0, 0.5, 0.9, –0.9). Measurement error inflated the population size estimates for all three estimation methods, in some cases by more than 20%. In general, ML outperformed the two regression approaches by producing less biased estimates on average with a smaller MSE. This observation is consistent with findings in the absence of measurement error (Gould and Pollock 1997). DeLury’s method, which is negatively biased for estimating population size without measurement error (Gould and Pollock 1997), became less biased with an increase in measurement error variance until a positive bias began. This observation does not purport to solicit the use of DeLury’s method when catch and effort estimates are contaminated with measurement error. Rather, it can be argued that DeLury’s method was the least robust to measurement error as evidenced by the steeper slope of the relative bias. Both Leslie’s and DeLury’s regression methods produced estimates with larger MSEs compared with ML (Fig. 2) and exhibited relative biases that were more sensitive to measurement error (Fig. 1). Positive or negative correlation between the errors in catch and effort did not alter the direction or magnitude of bias in the population size estimates (Fig. 1) or MSE (Fig. 2) from any of the estimation methods. This result is surprising considering the effect that correlation between errors can have in other measurement error problems. Table 1 displays the percentage of failures and cases in which the cumulative removal exceeded the population size estimate for ML and the regression methods. The number of failures increased as the measurement error variance increased © 1997 NRC Canada
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Fig. 1. Plot of the relative bias in population size estimates (true N = 10 000; B = 2000 replications) versus the magnitude of measurement error variance (indicated by λ) obtained with Leslie’s and DeLury’s methods and ML for four levels of correlation (ρ) between the errors in catch and effort.
Fig. 2. Plot of the MSE of population size estimates (true N = 10 000; B = 2000 replications) versus the magnitude of measurement error variance (indicated by λ) obtained with Leslie’s and DeLury’s methods and ML for four levels of correlation (ρ) between the errors in catch and effort.
for all three methods; however, correlations between errors did not noticeably affect the failure rate. Although the relatively low failure rates (