ARTICLE

Sequential analysis and design of fixed-precision sampling ofLake Kariba fishes using Taylor’s power lawMeng Xu, Jeppe Kolding, and Joel E. Cohen

Abstract: Taylor’s power law (TPL), which states that the variance of abundance is a power function of mean abundance, hasbeen used to design sampling of agricultural pests and fish species. We show that TPL holds for means and variances ofabundance of accumulated fish samples in the fished and unfished areas separately of Lake Kariba (between Zambia andZimbabwe), measuring abundance indices by number and weight separately. We use TPL parameters estimated from sequen-tially accumulated samples to update a stopping line of fixed precision 0.1 after each new sample from a sampling day. In theseLake Kariba data, depending on the sampling area and abundance measure, our updated stopping-line method requires 21% to41% of the number of sampling days and 19% to 40% of the number of samples that are planned a priori and performed undersystematic sampling. Our novel method yields mean abundance estimates similar to those from systematic sampling andprovides a conservative approach to reaching a fixed sampling precision level with reduced sampling labor and time. Usingmixed-effect modeling for cumulative means and variances with either number or weight from both fished and unfished areas,we find that fishing increases the slope of TPL. This study provides the conceptual framework and an empirical case study forimplementing a sequential sampling method for fish assemblages of an inland lake. The possible limitations and applications ofour method for sampling in other environments are discussed.

Résumé : La loi de puissance de Taylor (LPT), qui énonce que la variance de l’abondance est une fonction puissance del’abondance moyenne, a été utilisée pour concevoir l’échantillonnage de ravageurs agricoles et d’espèces de poissons. Nousdémontrons que la LPT est valide pour les moyennes et variances de l’abondance d’échantillons de poissons accumulés dansdes zones exploitées et non exploitées prises séparément du lac Kariba (entre la Zambie et le Zimbabwe), mesurant séparémentdes indices d’abondance selon le nombre et la masse. Nous utilisons des paramètres de la LPT estimés à partir d’échantillonsaccumulés séquentiellement pour mettre à jour une ligne d’arrêt d’une précision fixe de 0,1 après chaque nouvel échantillond’une journée d’échantillonnage. Dans ces données pour le lac Kariba, selon la zone échantillonnée et la mesure d’abondance,notre méthode de ligne d’arrêt requiert de 21 % à 41 % du nombre de jours d’échantillonnage et de 19 % à 40 % du nombred’échantillons, qui sont planifiés a priori et réalisés par échantillonnage systématique. Cette nouvelle méthode donne desestimations de l’abondance moyenne semblables à celles obtenues par échantillonnage systématique et fournit une approcheprudente pour obtenir un degré de précision fixé de l’échantillonnage qui réduit l’effort et le temps d’échantillonnage. Enutilisant la modélisation des effets mixtes pour les moyennes et variances cumulatives, selon le nombre ou la masse, des zonesexploitées et non exploitées, nous constatons que la pêche accroît la pente de la LPT. L'étude fournit un cadre conceptuel et uneétude de cas empirique pour l’application d’une méthode d’échantillonnage séquentiel pour la caractérisation des assemblagesde poissons dans un lac intérieur. Les limites et applications possibles de cette méthode pour l’échantillonnage dans d’autresmilieux sont abordées. [Traduit par la Rédaction]

IntroductionSampling design in fisheries plays a pivotal role not only in

understanding the temporal and spatial distributions of fish pop-ulations, but also in estimating the total or relative populationsize of the fish stocks that guides conservation and sustainabilitypolicies. Traditional sampling approaches (e.g., simple randomsampling, cluster sampling, etc.) have been successfully imple-mented in fisheries (Cadima et al. 2005).

Sequential sampling, or adaptive sequential sampling, has beenused in various applications, such as in neural networks (Easonand Cremaschi 2014), medical studies (Hof et al. 2014), intelligencesystems (Chen and Chen 2011), ecology, and environmental sci-ences (Resh and Price 1984; France 1992; Salehi and Smith 2005;

Brown et al. 2008). The basic idea of a sequential sampling methodis that the number of samples is not fixed, but is determined by asampling criterion based on the collected samples. We apply andanalyze a particular sequential sampling method where samplingcontinues until the desired precision has been reached. Comparedwith traditional methods (e.g., simple random sampling or clustersampling), sequential sampling can reduce the number of sam-ples required.

Here, using experimental (not based on fishery) gillnet (see def-inition in Materials and methods) fish samples in the fished andunfished areas of Lake Kariba, we designed and tested a sequentialsampling method with an updated stopping line of fixed preci-sion. We compared the abundance estimate of fish assemblages

Received 7 March 2018. Accepted 9 August 2018.

M. Xu. Department of Mathematics, Pace University, 41 Park Row, New York, NY 10038, USA.J. Kolding. Department of Biology, University of Bergen, N-5020 Bergen, Norway.J.E. Cohen.* Laboratory of Populations, The Rockefeller University, 1230 York Avenue, New York, NY 10065, USA; Earth Institute and Department ofStatistics, Columbia University, New York, NY 10027, USA; Department of Statistics, University of Chicago, Chicago, IL 60637, USA.Corresponding author: Joel E. Cohen (email: cohen@rockefeller.edu).*Present address: Laboratory of Populations, The Rockefeller University and Columbia University, 1230 York Avenue, New York, NY 10065, USA.Copyright remains with the author(s) or their institution(s). Permission for reuse (free in most cases) can be obtained from RightsLink.

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Can. J. Fish. Aquat. Sci. 00: 1–14 (0000) dx.doi.org/10.1139/cjfas-2018-0091 Published at www.nrcresearchpress.com/cjfas on 15 August 2018.

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(in number or weight) and the empirical sampling precision be-tween our sequential method and a systematic temporal samplingmethod designed a priori. Throughout the paper, we defined theempirical precision level (Ct) after t days of sampling as the ratio ofthe estimated standard error of the mean to the mean

(1) Ct ��Vt/Nt

Mt, t � 1, 2, …, T

Here Mt and Vt were, respectively, the mean and variance of abun-dance (aggregate number or aggregate weight of all individuals ina sample) across accumulated samples in the first t sampling daysof a year. In both areas, a sample was defined as a single use of agillnet panel of a unique mesh size on a sampling date. Nt was thenumber of accumulated samples in the first t days. The total num-ber of sampling days in a year (T) was determined by the system-atic sampling designed a priori.

We used the variance function method to update the stoppingline of fixed precision in sequential sampling (Kuno 1969). A par-ticular variance function, called Taylor’s power law (TPL), de-scribes the variance of abundance as a power function of the meanabundance. The power-function form of TPL (variance = a(mean)b,a > 0) and its log-linear form (log(variance) = log(a) + b log(mean))have been confirmed for thousands of biological taxa. All loga-rithms are to base 10 in this paper. The exponent or slope b iscommonly between one and two (Taylor 1961; Taylor et al. 1978).TPL has been used successfully to describe the temporal and spa-tial fluctuations of abundance of fish populations or assemblages(Elliott 1986; Baroudy and Elliott 1993; Mellin et al. 2010). Green(1970) tested TPL with mollusc samples and substituted the esti-mated variance function in eq. 1 to construct a sampling stoppingline with fixed precision for independent samples. Mouillot et al.(1999) and Xu et al. (2017) tested TPL for marine and freshwater fishsamples, respectively, and used Green’s method to estimate theminimal number of samples required to achieve the desired pre-cision for the same samples. The stopping line of fixed precision Cbased on TPL (see eq. 7 in Green 1970) was

(2) log At �log�C2

a�

b � 2�

b � 1b � 2

log Nt

where At was the cumulative abundance of Nt samples in the firstt sampling days of a year. Equation 2 was derived from the preci-sion level formula (eq. 1), the mean–variance relationship of TPL,and Mt = At/Nt (Green 1970).

According to eq. 2, the slope of the stopping line is 0 if and onlyif b = 1 (as would arise from a Poisson distribution of abundancewith a parameter that varied from sample to sample), is +∞ if andonly if b = 2 (as would arise from an exponential distribution ofabundance with a parameter that varied from sample to sample),is negative if and only if 1 < b < 2, and is positive if b < 1 or b > 2.Assuming a > 0, the intercept of this stopping line is infinite if andonly if b = 2; otherwise, the intercept is finite. In principle, depend-ing on the values of a and b, we may expect to see stopping lineswith positive or negative slopes and positive or negative inter-cepts. As we shall see, all these possibilities arise from data. Inpractice, the sampling stops if and when the empirical cumulativeabundance curve (log(At) plotted against log(Nt)) intersects withthe stopping line (eq. 2), at which point the fixed-precision level C(0.1 in this paper) is reached. (Appendix A gives necessary andsufficient conditions for this intersection to exist.) Otherwise,the sampling continues until all samples designed a priori areexhausted.

The current work differs from the previous literature in that,first, we tested TPL and estimated its parameters for mean and

variance of abundance (in number or weight) from accumulatedsamples up to a certain date, which were consistent with themean and variance used in constructing Green’s stopping line(eq. 2) and the sampling precision formula (eq. 1). This is a subtlebut important modification because previous works used meanand variance for samples from a single date or area and did notconsider the between-date or between-area variations, which canlead to a biased estimate of empirical precision when using thestopping line. Second, in our method, as a new sampling date wasincluded, new TPL parameters (a and b) using samples accumu-lated before and on that date (not all samples from the systematicsampling) were estimated and used to update the stopping line offixed precision sequentially. Sampling workers can then get se-quential updates on whether and when the desired precision C isreached. Third, although the influence of variability in TPL param-eters between independent samples has been recognized andtested for insect and computer-simulated data (Trumble et al.1987, 1989), no previous literature, to our knowledge, addressedhow to incorporate the variability of TPL parameters in develop-ing efficient sampling methods for fisheries. Our work investi-gates this knowledge gap to understand better an application ofTPL.

The specific goals of this paper are as follows: (1) testing TPL forfish abundance of accumulated samples in the fished and un-fished areas of Lake Kariba separately; (2) using TPL parameterestimates to update a fixed-precision stopping line that gives con-tinuous feedback about empirical precision; (3) comparing themean abundance estimate and empirical precision between ournew method and the systematic temporal sampling designed apriori; and (4) examining the effects of fishing and year on theparameters of TPL. Goals 2–4 are achieved using the same fishgillnet sample of Lake Kariba as in goal 1. Our new sequentialsampling method is designed specifically to exploit the data col-lection and structure of the studied area. The usefulness of thismethod in other sampling scenarios is elaborated in the Discus-sion.

Materials and methods

Sampling area, method, and dataFish samples were collected from the Zimbabwean side (1958,

1960–1980, and 1982–2001) and the Zambian side (1980–1999) ofLake Kariba, the world’s largest manmade reservoir by volume(Kolding et al. 2003). The Zambian side has experienced heavyartisanal fishing since the impoundment of the Lake and wasdesignated as a fished area (Musando 1996). In contrast, the sam-pling location on the Zimbabwean side has experienced light orno fishing activities and was designated as an area closed to fish-ing (or unfished area) (Kolding et al. 2003).

Fishes were sampled by experimental multifilament gill nets innine stations of the fished area (Zambian side; see figure 1 ofKolding et al. 2003) on 3 consecutive days of each month in a year(1980–1999). Panels of 13 unique mesh sizes (from 25 to 178 mmwith �12.75 mm increment) were bottom-set at different stationsaround 1800 on the first day and hauled around 0600 the nextmorning. In the unfished area (Zimbabwean side), fishes weresampled almost weekly every year from 1958 to 2001 (except 1959,1966, and 1981) using multifilament gill nets at the Lakeside sta-tion (see figure 1 of Kolding et al. 2003). A single net consisting of12 panels of unique mesh sizes (from 38 to 178 mm with �12.73 mm in-crement) was set at 1500 on the first day and retrieved for fish sam-pling at 0800 the next morning. Detailed sampling information atfished and unfished areas are available in Musando (1996) andKolding and Songore (2003). In both fished and unfished areas,individual fishes caught by the gill nets were nearly all identifiedby species and were measured by fork length (mm), weight (g),sex, and gonadal maturity stage (Kolding et al. 2003). We call theaforementioned sampling procedure the “systematic temporal

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sampling method” because samples were collected on a regulartime schedule.

The total numbers of individual fish sampled over the years were123 533 and 224 556 in the fished and unfished area, respectively. Forindividuals with no weight measurements, their weights were esti-mated a posteriori using species-specific length–weight allometricequations (weight = �(length)�). 4899 individuals (4899/123 533 ≈ 4%)in the fished area and 2261 individuals (2261/224 556 ≈ 1%) in theunfished area had no weight records (either measured or esti-mated) due to the lack of length measurement or species identityand were excluded from the analysis when weight was the abun-dance measure. Fish abundance within a sample was measured intwo ways: (i) aggregate number of individual fishes per sample(regardless of species) and (ii) aggregate weight of individuals persample (regardless of species).

Taylor’s power law (TPL) analysisWe tested the cumulated mean and the cumulated variance of

fish abundance in a year for agreement with TPL. To be consistentwith the stopping line of fixed precision (eq. 2), we calculated themean and the variance of sample abundance (in number orweight) across accumulated samples over sampling days in eachyear. Specifically, in the fished and unfished area separately, oneach sampling date of a year, a pair of mean and variance ofabundance (in number or weight) per sample was calculatedacross all samples collected on and before the date in that year.The number of samples used to calculate each mean–variance pairwas at least 12 on average, since panels of at least 12 distinct meshsizes were used on each sampling date in each area. As moresampling dates are accumulated, the number of samples in-creased and exceeded 15, the minimum number of samples rec-ommended by Taylor et al. (1988). These means and variancesquantified, respectively, the central tendency and variation offish abundance among different sampling days, locations, andfishing gear (mesh sizes of nets). In the unfished area, the firstsampling day in 1967 and 1970 contained one sample only andwas merged into the corresponding second sampling day to gen-erate a nonzero sample variance estimate and a finite logarithmicvariance.

In each area, we fitted the log(variance) as a linear function ofthe log(mean) (log-linear form of TPL) across all sampling daysfrom each year using autoregressive-moving-average (ARMA) re-gression models to account for the temporal autocorrelationbetween sampling days within that year. We required that thenumber of mean–variance pairs was at least five in each fitting,following the recommendation by Taylor et al. (1988). The year of1968 (in the unfished area) was eliminated because it containedfour mean–variance pairs only. The number of autoregressiveterms and moving-average terms in an ARMA model ranged fromzero to three. The most complex (with three autoregressive termsand three moving-average terms) ARMA model that we consideredcan be written as

(3)log(Vt) � a � b log(Mt) � ut,

with ut � �i�1

3�iut�i��j�1

3�jt�j � t

Here ut was the temporally autocorrelated error with ARMA struc-ture, where ut�i (with coefficients �i) (i = 1, 2, 3) and t−j (withcoefficients �i) (j = 1, 2, 3) were, respectively, the autoregressiveand moving-average terms, and t was the normal error with zeromean and constant standard deviation. Model parameters wereestimated using the maximum likelihood method (Pinheiro andBates 2000). We also included a constant-intercept model withzero slope as a benchmark to compare with other ARMA models.In each year (20 years in the fished and 39 years in the unfishedarea, respectively), the total number of models examined was

17 (i.e., 4 (number of autoregressive terms) × 4 (number of moving-average terms) + 1 (constant-intercept model)). The model with theminimum Akaike information criterion corrected for small sam-ple size (number of mean–variance pairs) (AICc) was chosen as thebest model (Brockwell and Davis 2016).

Updated stopping-line method of sequential samplingAs mentioned in the Introduction, we designed a new stopping-

line method using updated TPL parameters based on cumulativemeans and cumulative variances. The updated stopping lines pro-vide sequential feedback on whether the desired sampling preci-sion has been reached, so that fishery scientists can get timelyinformation on when to stop sampling, potentially saving sam-pling labor and time.

To use this stopping-line approach, one first builds an empiricalcumulative abundance curve sequentially. Namely, we calculatedand plotted the cumulative sample abundance (At) against thenumber of samples (Nt) on double-logarithmic scale. As more sam-ples were added, the cumulative abundance increased or stayedconstant (when the sample contained zero fish).

To build the updated stopping line with fixed precision (C = 0.1),we fitted cumulative means and variances across sampling daysaccumulated sequentially within each year. Specifically, in eachyear, we tested the 16 ARMA models and the constant-interceptmodel using five mean–variance pairs from the first 5 samplingdays in that year (following Taylor et al. 1988). On each of the first5 sampling days, one mean–variance pair was calculated across allsamples accumulated on and before that date. After the bestmodel with the minimal AICc was selected, parameter estimatesof the selected model (a and b of TPL) were substituted into thestopping line with fixed precision C = 0.1 (eq. 2). If this stoppingline intersected the empirical cumulative abundance curve (basedon samples from the first 5 sampling days), then the desiredprecision of 0.1 was reached and sampling would have stoppedaccording to our sequential procedure. Otherwise, an extra mean–variance pair from the sixth sampling date (based on the cumu-lated samples from the first six sampling dates combined) wasadded to the existing five mean–variance pairs. ARMA and constant-intercept models were again fitted to the six pairs, and the modelwith the minimal AICc was selected. A new stopping line based onparameter estimates of the newly selected model was drawn to-gether with the updated cumulative abundance curve (based onsamples from the first 6 sampling days) to check for intersection.This process was repeated sequentially at each succeeding sam-pling date until the updated stopping line intersected the updatedempirical cumulative abundance curve or the number of sam-pling days (determined by the systematic temporal sampling) wasexhausted in that year.

To evaluate the abundance estimate and efficiency of ourmethod, we compared the mean abundance estimate and empir-ical sampling precision obtained using our updated stopping-linemethod and using the systematic sampling performed a priori.Because only sample data were available, our comparison did notevaluate the sampling accuracy of the mean abundance estimateof either method.

Effects of fishing and year on TPL parametersTo evaluate the impact of fishing and year on the parameters of

TPL, using number and weight separately, we combined the cu-mulative means and variances from both fished and unfishedareas in all sampling years and fitted a linear mixed-effects modelwith log(variance) as the dependent variable and log(mean) as theindependent variable (Zurr et al. 2009). We chose area (fished orunfished, with fished area being the reference level) as the fixedeffect and year as the random effect. We considered the effects ofarea and year on both the intercept and the slope of the log(mean)–log(variance) relationship (TPL). To account for the temporalcorrelation among sampling days, we fitted multiple linear

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mixed-effects models with different ARMA correlation structures(up to three terms in either autoregressive or moving-averagecomponent). The general structure of our mixed-effects modelswas

(4) log(Vt) � a � b log(Mt) � �1A � �2[A:log(Mt)] � 1Y� 2[Y:log(Mt)] � ut

where A and Y stood for area and year, respectively. A:log(Mt)denoted the interaction term between area and log(Mt), andY:log(Mt) denoted the interaction term between year and log(Mt).�1 and �2 were, respectively, the coefficients of fixed intercept andfixed slope effects by area. 1 and 2 were, respectively, the coef-ficients of random intercept and random slope effects by year,

assuming �1

2� � N��0

0 �, ��int2 0

0 �slope2 ��. The variances �int

2 and

�slope2 of the random intercept and random slope effects were esti-

mated in model fitting. ut was the autocorrelated error defined ineq. 3.

Model fitting and selection were carried out in the followingsteps for each abundance measure (number and weight). First, wefitted the lumped mean–variance values using mixed-effects mod-els with the full fixed effects (area effect on the intercept and areaeffect on the slope of TPL) and different random effects (no randomeffect, pure random intercept effect, pure random slope effect, ran-dom intercept and random slope effects) and correlation structures(16 ARMA variants) using restricted maximum likelihood. The to-

tal number of mixed-effects models fitted was 64 (4 random effectforms × 16 correlation structures) for each measure. Models thatfailed to converge were eliminated from analysis. Second, fromthe remaining of 64 mixed-effects models we selected the mod-el(s) with an AIC weight of at least 5%. Third, for each of theselected model(s) in step two, we refitted the full fixed-effectsmodel and fitted additionally three corresponding mixed-effectsmodels with reduced fixed effects (intercept effect only, slopeeffect only, no effect) using maximum likelihood (Pinheiro andBates 2000). Lastly, we compared each set of four mixed-effectsmodel with various fixed-effect forms using AIC and AIC weights.We interpreted the effects of area (fished and unfished) and yearon the parameters of TPL based on the mixed-effects models withthe optimal random-effect form and optimal correlation structurein the last step. In this analysis we did not correct for small samplebias in AIC because of sufficient data points (2737 for number and2735 for weight).

ARMA models and mixed-effects models were fitted using the“gls” and “lme” function, respectively, in R 3.4.2 (R Core Team2017).

Results

Temporal trend of fish sample abundance and TPLMean abundance per sample calculated from accumulated sam-

ples over sampling days within each year (also known as meancatch per unit effort or mean CPUE, where the unit of effort isdefined as one sample) showed different trends among sampling

Fig. 1. Log(cumulative variance of number) against log(cumulative mean of number) across sampling days within each year in the fished area.Each dot represents a log(mean)–log(variance) pair calculated over accumulated samples on and before a sampling day of that year.

1995 1996 1997 1998 1999

1990 1991 1992 1993 1994

1985 1986 1987 1988 1989

1980 1981 1982 1983 1984

1.212 1.214 1.216 1.218 1.220 1.21051.21101.21151.21201.2125 1.209 1.210 1.211 1.2078 1.2080 1.2082 1.2066 1.2070 1.2074 1.2078

1.20 1.21 1.210 1.212 1.214 1.216 1.210 1.215 1.220 1.225 1.220 1.222 1.224 1.226 1.222 1.224 1.226

0.98 1.00 1.02 1.03 1.04 1.05 1.06 1.07 1.07 1.08 1.09 1.10 1.11 1.12 1.130 1.135 1.140 1.145 1.15 1.16 1.17 1.18 1.19

0.7 0.8 0.9 1.0 1.03 1.04 1.05 1.02 1.03 1.04 1.05 0.98 1.00 1.02 0.95 0.96

1.925

1.950

1.975

3.0

3.2

3.4

3.405

3.410

3.415

3.420

3.425

3.430

3.331

3.332

3.333

3.334

3.335

2.00

2.05

2.10

2.70

2.75

2.80

2.85

3.39

3.40

3.41

3.42

3.43

3.337

3.339

3.341

2.150

2.175

2.200

2.3

2.4

2.5

2.6

2.7

3.400

3.405

3.410

3.415

3.345

3.350

3.355

3.360

2.22

2.24

2.26

2.28

2.28

2.30

2.32

2.34

3.400

3.405

3.410

3.415

3.365

3.370

3.375

3.380

1.0

1.5

2.0

2.0

2.1

2.2

3.39

3.40

3.41

3.42

3.38

3.39

3.40

log(daily cumulative mean number)

log(

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years, across measures and areas. In the fished area, using eithernumber (refer to online supplementary material, Fig. S11) orweight (Fig. S31), mean abundance increased during some years(e.g., 1990), decreased during some years (e.g., 1995), or changedirregularly (e.g., 1984). In the unfished area, using number(Fig. S21) or weight (Fig. S41), mean abundance increased monoton-ically during some years (e.g., 1968, 1974), decreased monotoni-cally during some years (e.g., 1963, 1964), or exhibited nonuniformchanges (e.g., 1991, 1995). Although the trend between numberand weight measures in the same area looked similar in the sameyear for most years, in some years the trend differed drastically(e.g., 1987 in the fished area and 1971 in the unfished area). In thefished area, the variance of abundance showed all three patternsfor both measures and showed a consistently decreasing patternin each year from 1995 to 1999, regardless of measure (Figs. S51,S71). In the unfished area, the variance of number showed all threetrends and the variance of weight mostly decreased (Figs. S61, S81).

Plotting the log variance of abundance as a function of the logmean abundance revealed generally (but far from always) a posi-tive linear relationship for each combination of measure and area(Figs. 1, 2, S91, S101). In some years, the log(mean)–log(variance)relationship was highly irregular (e.g., 1977 for number in theunfished area, 1992 for number in the fished area, 1988 for weightin the unfished area, and 1982 for weight in the fished area).Nevertheless, the TPL model (with or without temporal correla-

tion) fitted the data better than the constant-intercept model forevery year in both areas and for both measures, except 1972 and1975 in the unfished area with weight. In the fished area, usingnumber, the uncorrelated TPL model (without temporal correlation)and the correlated TPL model (with temporal correlation) fittedthe mean–variance data well (smallest model AICc) in 1 of the 20and 19 of the 20 sampling years, respectively. Using weight, theuncorrelated TPL model (without temporal correlation) and thecorrelated TPL model (with temporal correlation) fitted the mean–variance data well (smallest model AICc) in 3 of the 20 and 17 of the20 sampling years, respectively. In the unfished area, using num-ber, the uncorrelated TPL model (without temporal correlation)and the correlated TPL model (with temporal correlation) fittedthe mean–variance data well (smallest model AICc) in 19 of the 39and 20 of the 39 sampling years, respectively. Using weight, theuncorrelated TPL model (without temporal correlation) and thecorrelated TPL model (with temporal correlation) fitted the mean–variance data well (smallest model AICc) in 15 of the 39 and 22 ofthe 39 sampling years, respectively. Among the best models,across areas and measures, the intercept estimates of TPL rangedfrom −7.1557 to 12.1848. The slope estimates of TPL ranged from−1.402 to 5.613. No clear trend of either parameter was observed.Model statistics are summarized in Figs. S111–S121 and Tables S11–S41. Captions of supplementary tables, including explanations of

1Supplementary data are available with the article through the journal Web site at http://nrcresearchpress.com/doi/suppl/10.1139/cjfas-2018-0091.

Fig. 2. Log(cumulative variance of number) against log(cumulative mean of number) across sampling days within each year in the unfishedarea. Definition of each dot was given in the legend of Fig. 1.

1997 1998 1999 2000 2001

1990 1991 1992 1993 1994 1995 1996

1983 1984 1985 1986 1987 1988 1989

1975 1976 1977 1978 1979 1980 1982

1968 1969 1970 1971 1972 1973 1974

1960 1961 1962 1963 1964 1965 1967

1.082 1.086 1.087 1.090 1.091 1.094 1.091 1.093 1.089 1.092

1.088 1.091 1.087 1.090 1.084 1.087 1.082 1.083 1.084 1.084 1.088 1.092 1.088 1.094 1.082 1.088

1.064 1.069 1.061 1.063 1.060 1.062 1.062 1.068 1.074 1.075 1.076 1.077 1.074 1.077 1.077 1.088

1.030 1.045 1.05 1.07 1.069 1.077 1.078 1.084 1.070 1.075 1.080 1.072 1.076 1.068 1.072

0.877 0.885 0.89 0.92 0.924 0.928 0.930 0.939 0.945 0.960 0.964 0.976 0.98 1.02

0.8 1.0 1.3 1.161.201.24 1.05 1.10 0.93 0.96 0.99 0.88 0.91 0.86 0.87 0.860 0.8751.96

1.98

2.00

2.02

2.402.412.422.432.44

2.4652.4702.475

2.4252.4302.4352.440

2.4512.4542.4572.4602.463

1.9651.9701.975

2.3902.3952.4002.4052.410

2.47252.47502.47752.4800

2.4162.4182.4202.4222.424

2.46502.46752.4700

1.98

2.00

2.02

2.04

2.36

2.37

2.38

2.39

2.442.452.462.47

2.42502.42752.43002.43252.4350

2.4502.4552.4602.465

2.4532.4542.4552.4562.4572.458

2.0252.0502.0752.1002.125

2.3402.3452.3502.355

2.440

2.445

2.450

2.430

2.435

2.4402.4422.4442.4462.448

2.4592.4602.4612.4622.463

2.15

2.20

2.25

2.3302.3322.3342.3362.338

2.4352.4402.4452.450

2.420

2.425

2.430

2.45002.45252.45502.4575

2.4632.4642.4652.4662.4672.4682.469

2.32.42.5

2.102.152.202.252.30

2.4452.4502.4552.460

2.4332.4362.4392.442

2.4582.4592.4602.4612.462

2.4622.4632.4642.465

1.21.62.02.42.8

2.022.032.042.052.06

2.44752.45002.45252.45502.4575

2.4452.4502.4552.460

2.4402.4452.4502.4552.4602.465

2.455

2.460

log(daily cumulative mean number)

log(

daily

cum

ulat

ive

varia

nce

of n

umbe

r)

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columns and entries, are in the online Supplementary Informa-tion text file.

Sequential sampling using updated stopping-line methodFollowing the updated stopping-line approach described in the

Materials and methods, we showed that sampling reached thedesired precision (C = 0.1) prior to the completion of systematicsampling in that year for 14 and 12 years using number (Fig. 3) andweight (Fig. S131), respectively, in the fished area, and for 31 and34 years using number (Fig. 4) and weight (Fig. S141), respectively,in the unfished area. Appendix A gives simple necessary and suf-ficient conditions for the existence of a unique number of samplesat which the desired precision is attained, using a simple, empir-ically defensible model of the relationship between the cumula-

tive number of samples and the cumulative abundance (howeverabundance is measured).

In the fished area, using number, our updated stopping-linemethod showed that the mean number of sampling days andmesh samples required to reach 0.1 precision were, respectively,21.57 days and 287.79 samples per year. Compared with the sys-tematic temporal sampling method with a mean of 57.5 samplingdays and 793.79 samples used per year, the updated stopping-linemethod saved about 62% (i.e., (57.5 − 21.57)/57.5) of sampling daysand 64% (i.e., (793.79 − 287.79)/793.79) of samples per year (Table 1).

The mean abundance estimates were very similar between thetwo methods (Fig. 5a), with a mean of 13.73 individuals per samplefor the updated stopping-line method and 14.42 individuals per

Fig. 3. Log(cumulative number of individuals) as a function of log(number of samples) (solid dots) for each of 14 sampling years in the fishedarea. Stopping lines were sequentially updated, and the first stopping line that intersected with the empirical cumulative abundance plot isshown (solid line). Desired sampling precision of 0.1 was reached at the intersection point.

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1980

0

1

2

3

0.0 0.5 1.0 1.5

1983

0

1

2

3

0.0 0.5 1.0 1.5 2.0

1984

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1985

2

3

4

0 1 2

1986

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1987

0

1

2

3

0.0 0.5 1.0 1.5 2.0 2.5

1988

1

2

3

4

0 1 2

1990

1

2

3

0.0 0.5 1.0 1.5 2.0 2.5

1991

1

2

3

4

0 1 2

1993

0

1

2

3

0.0 0.5 1.0 1.5 2.0

1994

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1995

2

3

0.0 0.5 1.0 1.5 2.0 2.5

1996

0

1

2

3

0.0 0.5 1.0 1.5 2.0

1997

log(

cum

ulat

ive

num

ber o

f ind

ivid

uals

)

log(number of samples)

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6 Can. J. Fish. Aquat. Sci. Vol. 00, 0000

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Fig. 4. Cumulative number of individuals as a function of the number of samples (on log–log scales, solid dots) for each of 31 sampling yearsin the unfished area. The stopping line (solid line) was defined in the legend of Fig. 3.

1

2

3

0.0 0.5 1.0 1.5 2.0

1961

1

2

3

0.0 0.5 1.0 1.5 2.0

1962

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1963

1.2

1.4

1.6

1.8

2.0

2.2

0.0 0.2 0.4 0.6

1967

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

1972

0

1

2

3

0.0 0.5 1.0 1.5 2.0

1973

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0

1974

2.0

2.5

3.0

0.0 1.0 2.0 0

1975

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1976

1

2

3

0.0 0.5 1.0 1.5 2.0

1977

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1978

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0 2.5

1979

1

2

3

0.0 0.5 1.0 1.5 2.0

1980

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1982

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1983

1.6

2.0

2.4

2.8

3.2

0.0 0.5 1.0 1.5 2.0

1984

1.0

1.5

2.0

2.5

3.0

0.0 1.0 2.0

1985

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

1986

1

2

3

0.0 0.5 1.0 1.5 2.0

1987

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1988

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

1989

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

1991

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1992

1.0

1.5

2.0

2.5

0.0 1.0 2.0

1993

1

2

3

4

0 1 2

1994

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1996

1

2

3

0.0 0.5 1.0 1.5 2.0

1997

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

1998

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

1999

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0

2000

1.0

1.5

2.0

2.5

3.0

0.0 0.5 1.0 1.5 2.0

2001

log(

cum

ulat

ive

num

ber o

f ind

ivid

uals

)

log(number of samples)

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sample for the systematic sampling method. The empirical preci-sion level (error-to-mean ratio) based on required samples fromthe updated stopping-line method was 0.0828 on average per year,compared with 0.0644 from systematic sampling (Fig. 6a).

Similar results occurred when weight was the abundance mea-sure. The mean number of sampling days and samples required bythe updated stopping-line method were, respectively, 22.58 days and182.92 samples per year, saving about 59% (i.e., (54.75 − 22.58)/54.75)and 60% (i.e., (459.75 − 182.92)/459.75) of the 54.75 days and 459.75samples used per year by the systematic sampling (Table S51). Themean abundance estimates were 2418.39 and 2567.34 grams persample using updated stopping-line and systematic samplingmethods, respectively (Fig. 5b). The empirical precision levels(error-to-mean ratios) were 0.0818 by the updated stopping-linemethod and 0.0592 by the systematic sampling method (Fig. 6b).

In the unfished area, using number, the mean number of sam-pling days and samples required by the updated stopping-linemethod were, respectively, 14.16 days and 153.84 samples per year,saving about 70% (i.e., (47.71 − 14.47)/47.71) and 70% (i.e., (520.52 −153.84)/520.52) of the 47.71 days and 520.52 samples used per year bythe systematic sampling on average (Table 2). The mean abundanceestimates were 15.06 and 13.35 individuals per sample using theupdated stopping-line and the systematic sampling methods, re-spectively (Fig. 5c). The empirical precision levels (error-to-meanratios) were 0.0963 by the updated stopping-line method and0.0568 by the systematic sampling method (Fig. 6c).

Using weight, the mean number of sampling days and samplesrequired by the updated stopping-line method were, respectively,9.76 days and 89.06 samples per year, saving about 79% (i.e.,(46.24 − 9.76)/46.24) and 81% (i.e., (458.53 − 89.06)/458.53) of the46.24 days and 458.53 samples used per year by the systematicsampling (Table S61). The mean abundance estimates were 6563.41and 5785.94 grams per sample using the updated stopping-lineand the systematic sampling methods, respectively (Fig. 5d). Theempirical precision levels (error-to-mean ratios) were 0.0874 bythe updated stopping-line method and 0.0443 by the systematicsampling method (Fig. 6d). TPL parameter estimates for the se-quentially updated stopping lines are given in Tables S71–S101.

Mixed-effects models of TPLFollowing the model selection described in Materials and meth-

ods, under number and weight separately, the mixed-effects modelwith random intercept and random slope, full fixed-effect form, andorder-one autoregressive correlation was selected since it had 100%AIC weight compared with other models with different random-effects forms and correlation structures (Tables S111 and S121). Com-pared with the other three models with reduced fixed-effectsforms (identical random-effects form and correlation structure),the model with full fixed-effects form had an AIC weight of 100%and 83.04% with number and weight measure, respectively. Underweight, reduced fixed-effects models with slope, intercept, and noeffect of area only (fished versus unfished) carried 10.43%, 6.44%,and 0.08% AIC weights, respectively (Table 3).

Under number, the selected mixed-effects model with full fixed-effects form showed a negative effect of fishing on the intercept ofTPL and a positive effect of fishing on the slope of TPL (Table 3).Random intercept and random slope effects of year accounted for,respectively, 58.45% (i.e., 1.16742/(1.16742 + 0.88772 + 0.42522)) and33.80% (i.e., 0.88772/(1.16742 + 0.88772 + 0.42522)) of the variation inlog(variance), after the full fixed effects were used to explain thevariation in the response variable (Table 3).

Under weight, three selected mixed-effects models with at leasta 5% AIC weight (of various fixed-effects forms) showed similarstandard deviation estimates of the random effects (from 3.8175 to3.8705 for the random intercept effect and from 1.0546 to 1.0679for the random slope effect) and similar residual standard errors(from 0.3848 to 0.3911; Table 3). For the full fixed-effects model,besides the fixed effects, random intercept and random slopeT

able

1.Sa

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ng

met

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o.of

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No.

ofsa

mp

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(req

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Mea

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tim

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requ

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sam

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s

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sam

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s

Prec

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mp

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No.

ofda

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ll)

No.

ofsa

mp

les

(all

)

Mea

nes

tim

ate

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ng

all

sam

ple

s

Var

ian

cees

tim

ate

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all

sam

ple

s

Prec

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nu

sin

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lsa

mp

les

1980

2520

010

.175

192.

316

0.09

627

217

10.3

5519

4.59

10.

092

1983

980

11.5

2512

4.32

90.

108

4951

87.

938

56.0

700.

041

1984

2523

97.

678

32.6

230.

048

6574

59.

208

72.1

170.

034

1985

813

910

.187

123.

443

0.09

372

1680

12.3

6228

7.88

10.

033

1986

2456

414

.844

380.

199

0.05

593

1534

14.3

0526

2.02

40.

029

1987

1215

613

.038

159.

082

0.07

711

614

5918

.702

1423

.234

0.05

319

8822

269

15.2

6020

0.58

10.

057

6666

620

.059

2673

.920

0.10

019

9056

707

25.1

6545

79.0

870.

101

5670

725

.165

4579

.087

0.10

119

9121

300

11.9

2725

8.95

10.

078

3345

012

.229

560.

279

0.09

119

9353

532

12.7

8259

1.42

90.

082

5958

517

.056

4789

.927

0.16

819

9411

143

15.9

1613

73.2

040.

195

5773

413

.743

562.

320

0.06

419

9514

182

11.1

3263

.750

0.05

354

701

11.0

6062

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0.02

719

9614

336

15.5

5420

8.15

20.

051

2455

615

.086

157.

319

0.03

519

978

182

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2221

6.77

30.

064

3456

114

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136.

993

0.03

4

No

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explained, respectively, 92.05% (i.e., 3.87052/(3.87052 + 1.06792 +0.39112)) and 7% (i.e., 1.06792/(3.87052 + 1.06792 + 0.39112)) of thevariation in log(variance) (Table 3). The model with the full fixedeffects again showed a negative effect of fishing on the interceptof TPL and a positive effect of fishing on the slope of TPL. Themodel with slope effect of area only showed a positive effect offishing on the slope of TPL. The model with intercept effect of areaonly showed a positive effect of area on the intercept of TPL(Table 3).

DiscussionThe current work develops a sequential fixed-precision stopping-

line method with continuous feedback and provides an exampleof its implementation using fish samples from an African lake.Compared with systematic sampling with fixed dates and set-tings, the sequential stopping-line method applied to long-termgillnet fish data saved about 60%–80% of sampling effort (see Re-

sults) and yielded comparable mean abundance estimates (exceptin 1967 in the unfished area using weight; see Fig. 5d), at an em-pirical precision level close to but slightly lower (better!) than thedesired value (except in 1967 in the unfished area using number orweight; see Figs. 6c, 6d), for the fished and unfished area sepa-rately. The extreme values obtained from 1967 in the unfishedarea were probably due to the small sample size within each earlysampling day of that year (each of the first 5 sampling days con-tained one sample only), leading to large errors in the cumulativemeans and variances and TPL parameters. The empirical precisionlevel obtained was lower than the desired precision level (C = 0.1)because, in some years, the updated stopping line intersectedwith the empirical cumulative abundance curve before the re-quired samples were collected (see Fig. 3 year 1993, Fig. S131 year1984, and Fig. S141 year 1982 for examples). Since the empiricalprecision (eq. 1) tends to decrease as more samples are collected,our updated stopping-line method provided a conservative ap-

Fig. 5. Comparison of the estimated mean abundance � using number (a, c) and weight (b, d) in the fished (a, b) and unfished (c, d) areasbetween updated stopping-line method and systematic sampling method. Solid lines are 1:1 reference lines.

10 15 20 25

1015

2025

Number

��us

ing

requ

ired

sam

ples

1000 2000 3000 4000

1500

2000

2500

3000

3500

4000

Weight

5 10 15 20 25

1015

2025

��using all samples

��us

ing

requ

ired

sam

ples

5000 10000 15000 20000 25000

010

000

2000

030

000

4000

0

��using all samples

(a) (b)

(c) (d)

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proach to obtaining desired sampling precision with reduced la-bor and time, therefore offering fishery scientists an alternativesampling method with sequential feedback and improved effi-ciency.

Despite relationships of cumulative means and cumulative vari-ances of abundance that were highly irregular in some years(Figs. 1, 2, S91, S101), TPL (with uncorrelated and correlated errorstructures) described the data robustly well, regardless of area andmeasure. For means and variances calculated from sequentiallyaccumulated samples, TPL (with or without temporal correlation)was superior to the constant-intercept model in 98.50% (920/934)of sampling days in the fished area with number (Table S71), in99.33% (1785/1797) of sampling days in the unfished area withnumber (Table S81), in 98.72% (922/934) of sampling days in thefished area with weight (Table S91), and in 94.31% (1692/1794) ofsampling days in the unfished area with weight (Table S101).

A common problem with sequential analysis is that varyingsample size can cause bias. In other words, it is possible thatpopulation parameters were over- or underestimated in the pro-cess so that the sampling stops at a precision level different fromthe desired value. In this work, we evaluated the bias in two ways.First, we compared the mean abundance estimated from our se-quential method and from the systematic sampling (with fixednumber of samples). Except for the unfished area using number,where the mean estimate was systematically overestimated (relativeerror = (mean estimate of sequential sampling − mean estimate ofsystematic sampling)/mean estimate of systematic sampling =(15.06 − 13.35)/13.35 ≈ 12.81%; see Table 2 and Fig. 5c), our methodgenerated an estimate of the mean comparable to that of thesystematic sampling. The relative errors were (13.83 − 14.42)/14.42 ≈ −4.09% in the fished area using number, (2418.39 −2567.34)/2567.34 ≈ −5.80% in the fished area using weight, and

Fig. 6. Comparison of the empirical precision estimate C using number (a, c) and weight (b, d) in the fished (a, b) and unfished (c, d) areasbetween updated stopping-line method and systematic sampling method. Solid lines are 1:1 reference lines. Dashed lines showed desiredprecision level C = 0.1.

0.00 0.05 0.10 0.15

0.00

0.05

0.10

0.15

Number

C�u

sing

requ

ired

sam

ples

0.00 0.05 0.10 0.150.

000.

050.

100.

15

Weight

0.00 0.05 0.10 0.15 0.20

0.00

0.05

0.10

0.15

0.20

C�using all samples

C�u

sing

requ

ired

sam

ples

0.00 0.05 0.10 0.15 0.20 0.25

0.00

0.05

0.10

0.15

0.20

0.25

C�using all samples

(a) (b)

(c) (d)

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Table 2. Sampling statistics of the updated stopping-line method and systematic sampling method in the unfished area using number as abundance measure.

Updated stopping-line method Systematic sampling method

YearNo. of days(required)

No. ofsamples(required)

Mean estimateusing requiredsamples

Variance estimateusing requiredsamples

Precision usingrequiredsamples

No. ofdays (all)

No. ofsamples (all)

Mean estimateusing all samples

Variance estimateusing allsamples

Precisionusing allsamples

1961 21 103 14.583 153.775 0.084 83 469 12.318 126.367 0.0421962 20 125 12.912 204.258 0.099 90 723 7.602 74.110 0.0421963 24 234 6.859 86.757 0.089 85 828 5.072 52.669 0.0501967 5 5 29.000 140.000 0.183 20 41 26.707 604.762 0.1441972 18 211 14.782 394.638 0.093 19 225 15.476 445.938 0.0911973 16 183 14.749 394.552 0.100 27 312 12.571 277.320 0.0751974 8 95 22.663 442.779 0.095 26 310 24.284 439.686 0.0491975 7 84 24.738 390.147 0.087 26 312 18.946 310.624 0.0531976 9 108 17.944 389.305 0.106 52 624 15.566 213.193 0.0381977 8 96 17.625 346.995 0.108 51 611 13.642 314.230 0.0531978 9 108 15.352 240.193 0.097 51 612 13.678 249.135 0.0471979 28 336 6.378 114.815 0.092 48 576 9.797 502.173 0.0951980 20 234 11.380 358.752 0.109 50 594 10.572 324.448 0.0701982 10 120 10.967 135.041 0.097 46 524 10.334 132.724 0.0491983 10 109 12.321 143.535 0.093 45 511 9.501 93.827 0.0451984 10 120 11.875 150.463 0.094 48 556 11.183 165.599 0.0491985 9 91 11.209 116.545 0.101 41 457 11.228 168.137 0.0541986 13 151 20.914 580.293 0.094 50 591 16.926 351.733 0.0461987 15 180 16.850 241.849 0.069 44 524 12.313 142.231 0.0421988 9 107 12.430 142.776 0.093 39 467 13.525 245.220 0.0541989 22 250 17.400 328.570 0.066 47 538 17.952 453.923 0.0511991 19 227 10.771 180.222 0.083 50 598 10.105 209.957 0.0591992 9 108 12.037 186.933 0.109 49 587 10.296 154.461 0.0501993 7 83 8.072 70.458 0.114 48 575 12.184 262.081 0.0551994 35 407 19.688 696.998 0.066 54 634 16.787 537.242 0.0551996 10 102 9.324 95.409 0.104 51 561 7.264 53.766 0.0431997 20 219 19.662 759.087 0.095 45 495 16.448 480.839 0.0601998 9 106 20.189 369.221 0.092 48 543 15.818 333.024 0.0501999 13 155 19.768 508.543 0.092 50 597 14.027 270.013 0.0482000 16 192 12.708 270.668 0.093 51 612 11.002 200.476 0.0522001 10 120 11.600 136.393 0.092 45 529 10.665 173.140 0.054

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(6563.41 − 5785.94)/5785.94 ≈ 13.44% in the unfished area usingweight. For the last case, excluding 1967 (the year with smallnumber of samples), the relative error was (5395.00 − 5135.89)/5135.89 ≈ 5.05% (Tables 1, S51, and S61; Figs. 5a, 5b, 5d). Thesecomparisons indicated that our sequential method gave reason-able estimates of mean abundance, but at a much lower samplingcost. Second, since we performed stopping-line analysis in eachyear for multiple years, the sampling bias in precision was re-duced with overestimation in some years balancing underestima-tion in other years (Schou and Marschner 2013).

Throughout this work, we used a desired sampling precisionof C = 0.1, meaning that the standard error of the mean abundanceestimate was 10% of the corresponding mean estimate. Weadopted the 0.1 precision level following the suggestion bySouthwood (1966). In sampling practice, the desired precisionlevel depends on the actual sampling budget and goal. Our up-dated stopping-line method can be tested similarly at other preci-sion levels (e.g., 0.25, 0.05).

To implement our sequential method, one sets up multiplesamples at a single location or time, then repeats sampling (eithersystematic sampling or simple random sampling) at different (atleast five recommended) locations or times. Samples accumulatedover time or space can be used to calculate a mean–variance pairwhen a new location or time is added. An advantage of ourmethod is that it does not require a large number of samples at agiven location or time, because it utilizes accumulated (currentand previous) samples that automatically enlarge the number ofsamples as more samples are added. On the other hand, specialstatistical treatment is needed to describe the relationship be-tween means and variances of abundance because they are basedon samples that are overlapping and correlated in time or space(e.g., the ARMA model we used to analyze TPL).

Our mixed-effects models showed that the parameters of TPLvaried from year to year and that fishing increased the slope ofTPL, regardless of the abundance measure used. One possible bi-ological cause of this difference is that annual lake level fluctua-tions affect recruitment variation (Henderson 1985; Allen et al.2003; Kolding and van Zwieten 2012). Karenge and Kolding (1995)documented a high correlation between lake level change andchange in CPUE in Lake Kariba. Using multiple regression models,Kolding et al. (2003) showed that lake level fluctuations had agreater effect on the CPUE in the fished area than in the unfishedarea. They argued that the species abundance became more vari-

able and sensitive to environmental forcing under increased fish-ing pressure. Statistical models can test whether the lake level hasa significant effect on TPL’s slope. Another possible biologicalcause of this difference is that the fish populations in the fishedareas may have had lower synchrony (average correlation overtime in the sizes of all pairs of populations at different locationswithin each area), because fishing activity exerts external noisethat can reduce strong correlation between time series. Cohenand Saitoh (2016) showed that increased synchrony lowered theslope of TPL in a dynamic metapopulation model. This effect holdsmore generally; although changed synchrony can invalidate TPL,greater synchrony typically lowers the slope of TPL (Reuman et al.2017). We leave tests of these two and other possible mechanismsfor future research.

This work is not the first to evaluate the effect of fishing on TPLand its parameters. Cohen et al. (2012) and Fujiwara and Cohen(2015) developed theoretical models to study how fishing mayaffect the variance function of population abundance. Cohen et al.(2012) showed that TPL’s slope was uniform across a range offishing intensity under balanced harvesting (Garcia et al. 2012;Law et al. 2012). Fujiwara and Cohen (2015) found that differentmodes of density dependence and different fishing strategiescould generate diverse variance functions of fish population den-sity, including but not limited to TPL. Kuo et al. (2016) built mul-tiple linear regression models to analyze the effect of fishing onTPL’s slope b using long-term data. Two major differences distin-guish the work of Kuo et al. (2016) from the current work. First,Kuo et al. (2016) considered the fishing effect by studying ex-ploited and unexploited species separately and estimated TPLslopes for each species. In the current work, TPL was tested andcompared for fish assemblage abundance (regardless of species) indesignated fished and unfished areas. Second, Kuo et al. (2016)developed multiple regression models with TPL’s b as the re-sponse variable and life history trait and fishing (including theirinteraction) as explanatory variables. By contrast, our mixed-effects models incorporated random effects of year and temporalautocorrelation, therefore accounting for the temporal variationof TPL parameters. Despite these differences, our results and thoseof Kuo et al. (2016) both showed that TPL’s slope increased whenpopulations were fished. However, while Kuo et al. (2016) sug-gested that fishing may increase the spatial aggregation potentialof a species, likely through degrading their size structure or agestructure, the Kariba fishery is more balanced across sizes and

Table 3. Mixed-effects models (eq. 4) with various fixed effects, optimal random effects structure, and order-one autoregressive correlation (seeResults).

Fixed effects Random effects

Measure Structure Intercept log10(mean) Area_unfishedlog10(mean):area_unfished Intercept

log10

(mean) Residual AICAICweight

Number Slope + intercept −1.1819 (0.2169) 2.9358 (0.1650) 1.1085 (0.1161) −0.7088 (0.1034) 1.1674 0.8877 0.4252 −7453.04 1Slope −0.4082 (0.2077) 2.2577 (0.1620) NA 0.2703 (0.0146) 1.1970 0.9681 0.4506 −7367.17 <0.0001Intercept −0.6301 (0.2045) 2.4430 (0.1561) 0.3214 (0.0163) NA 1.1762 0.9333 0.4477 −7409.40 <0.0001None −0.4752 (0.2124) 2.5033 (0.1703) NA NA 1.2425 1.0217 0.3959 −7050.34 <0.0001

Weight Slope + intercept −1.0312 (0.6755) 2.2435 (0.1864) 0.8970 (0.3693) −0.2683 (0.1026) 3.8705 1.0679 0.3911 −7823.90 0.8304Slope −0.4659 (0.6263) 2.0838 (0.1725) NA −0.0195 (0.0056) 3.8206 1.0555 0.3848 −7819.76 0.1043Intercept −0.4188 (0.6252) 2.0699 (0.1721) −0.0673 (0.0201) NA 3.8175 1.0546 0.3848 −7818.79 0.0644None −0.3526 (0.6159) 2.0368 (0.1688) NA NA 3.7572 1.0351 0.3922 −7810.04 0.0008

Note: Abundance was measured by number and weight separately. The fixed effect of area (unfished versus fished, with fished area being the reference level) onthe intercept or slope of TPL is shown under the “Fixed effects” column, with parameter estimate followed by standard error (in parentheses). Standard deviationestimates of the random slope and random intercept effects and the model residual standard error are shown under the “Random effects” column. These valuesmeasure the amount of variation explained by the respective random effect and left unexplained, after the corresponding fixed effects have been used to model thevariation in the response variable (log(variance)). Model AIC and AIC weights are given in the last two columns. “NA” means not applicable. For example, usingnumber, the model with “slope + intercept” fixed-effects structure shows the intercept estimate (Intercept) and slope estimate (log10(mean)) of TPL, with the areaeffects on the TPL intercept and slope indicated as “Area_unfished” and “log10(mean):area_unfished” respectively. Values under the “Random effects” column showthe maximum likelihood estimate of the standard deviation associated with the (intercept or slope, denoted by “Intercept” and “log10(mean)”, respectively) randomeffect of year. For example, in the first model using number (with “slope + intercept” fixed-effect structure), the values 1.1674 and 0.8877 quantify, respectively, thevariation (as standard deviation) in the intercept and slope of TPL among years. Residual values give the standard deviation estimate of the response that is leftunexplained (unexplained variation) after the fixed and random effects are included in the model.

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does not exhibit a significantly degraded size structure, but doesshow a significantly decreased standing biomass (Kolding et al.2016). Thus, the increased slope of TPL is more likely a result ofrecruitment variation and intensified fishing on small-sized fish.

Our method has several limitations and needs modificationsbefore it can be applied to broader areas. First, the temporal auto-correlation in our TPL models was relatively simple and did notincorporate seasonal effects. Data used in our case study werecollected at a regular time interval (weekly or monthly), whichmay not be the case under other sampling scenarios. Second, ourmethod was designed for data from a single inland water bodywith homogeneous habitat and climatic condition. In marine eco-systems, sampling quality and data structure may hinge greatlyon various environmental factors besides year and fishing thatwere considered in our model (e.g., water depth, season, current,etc.). Extending our method to data from heterogeneous condi-tions requires incorporating various spatial or environmentalfactors into the model. This will further increase the modelcomplexity and may potentially impair its practicality. Third,our method was tested using samples collected by gill nets, oneof the most commonly used sampling methods in freshwatersystems. In other aquatic environments, different samplingmethods may pose difficulty in calculating the statistical terms(mean and variance of abundance require samples from multi-ple locations or times) necessary for the estimation of TPL pa-rameters and the stopping line. Lastly, data used in this casestudy were collected systematically a priori, presenting a pos-sibly unique opportunity to compare the efficiency and preci-sion of our new method with those of the traditional samplingmethod. Practitioners who plan to use our method as the sam-pling protocol may find it challenging to determine the accu-racy of our method unless they have background data tocompare with for a quality check. Overall, to apply our methodsuccessfully and to give confidence to fisheries scientists inusing our method, we believe that further empirical testing ofTPL and the method itself in different sampling environmentsis required. This goal requires a collective and long-term effortin future research.

AcknowledgementsThe data used in this study were collected by the Lake Kariba

Fisheries Research Institute (Zimbabwe) and by Lake Kariba Re-search Unit and the Sinazongwe Fisheries Training Center (Zambia).Much of the empirical data compilation and background analyseswere performed during the Zambia–Zimbabwe SADC Fisheriesproject, sponsored by Norad (Norway) and Danida (Denmark) anda 4-year research project on freshwater fisheries development inthe SADC area (“Management, co-management or no manage-ment? Major dilemmas in southern African freshwater fisheries”)funded by the Norwegian Research Council. We are indebted to allthese institutions for help and assistance. JEC was partially sup-ported by National Science Foundation grant DMS-1225529. JECthanks Priscilla K. Rogerson and Roseanne Benjamin for assis-tance. JK and JEC thank Michael J. Plank and Richard Law forhospitality in New Zealand and the support of the Marsden Fundin 2012, which led to this collaboration. The authors declare thatthey have no conflict of interest in this work.

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Appendix A. Necessary and sufficient conditions forthe updated stopping-line method to yield a uniquenumber of samples

The purpose of this appendix is to identify simple necessary andsufficient conditions for the updated stopping-line method toyield a unique stopping point in the positive quadrant of the planewith horizontal axis x = log Nt and vertical axis y = log At, where Nt

is the number of samples up to an including time t in a given year,and At is the cumulative abundance of fishes in all samples up toand including time t in a given year.

If we compare the graphical tests of TPL in Figs. 1, 2, S91,and S101

with the respective graphical displays in Figs. 3, 4, S131, and S141 ofthe yield–effort curve, we see that straight lines are at least asgood approximations to the data points (x, y) (solid dots) in thelatter figures as the straight lines posited by TPL are to the datapoints (log Mt, log Vt) in the former. In every panel in Figs. 3, 4,S131, and S141, y increases roughly linearly with x, so it is reason-able to assume that the cumulative abundance At can be reason-ably approximated as an increasing power-law function At = hNt

k,h ≥ 1, k > 0, of the number Nt of samples.

If population abundance were stationary in time and indepen-dent from one sample to another, one would expect proportion-ality between the cumulative abundance and the cumulativenumber of samples, or k = 1. A careful visual examination ofFigs. 3, 4, S131, and S141 shows that k < 1 sometimes, k > 1 some-times, and generally k is close to 1. Taking logarithms of both sidesof the power-law model of the yield–effort curve and recalling thenotation x = log Nt, y = log At yields

(A1) y � log h � kx

By our assumption (based on the data in Figs. 3, 4, S131, and S141)that h ≥ 1, we have log h ≥ 0. If we define

(A2) HÍlog�C2

a�

b � 2, KÍ b � 1

b � 2

then H may be positive or negative (depending on the values of a,b, or C) and eq. 2 in the main text becomes

(A3) y � H � Kx

Now we have two linear equations (eqs. A1 and A3) in two un-knowns x and y. We assumed already that eq. A1 has nonnegativeintercept log h and positive slope k, so that for all x ≥ 0, we havey ≥ 0 according to eq. A1. Then eqs. A1 and A3 will have exactly onepoint of intersection in the positive quadrant (x, y) ≥ (0, 0) if andonly if the line with a lower intercept has a higher slope. In otherwords, under the assumptions already made, a necessary and suf-ficient condition for the existence of a unique number of samplesNt that attain the desired precision C in the positive quadrant isthat

(A4) (log h H and k � K) or (log h � H and k K)

Empirical illustrations of both cases in eq. A4 appear in Fig. 3. Forexample, in 1986 and 1987, log h < H and k > K, while in 1988 and1990, log h > H and k < K.

When eq. A4 and the other assumptions are satisfied, even ifthere are too few samples collected in a year to reach an intersec-tion in the positive quadrant with the stopping line (eq. A3), ourmethod identifies the number of samples that would be requiredto attain the desired precision C.

It is possible in principle that k = K, in which case the two linesare parallel and that, in addition, log h = H, in which case the twolines coincide. In the space of statistical estimates of the slopes kand K, the event that k = K has measure zero and is of little prac-tical interest. Nevertheless, if the lines are parallel but have dif-ferent intercepts, no number of samples will give the desiredprecision.

If 1 < b < 2, then, from the definition in eq. A2, we have K < 0.Empirical illustrations of this case occur in Fig. 3 in 1980, 1983,1984, and 1993, for example. In this case, since k > 0 by assumptionand with empirical support, we have k > K. Therefore, in this case,according to the first alternative considered in eq. A4, a necessaryand sufficient condition for the existence of a unique number ofsamples that attain the desired precision is that log h < H (i.e., thatthe yield–effort curve has a lower intercept than log–log form ofTPL). The examples in Fig. 3 in 1980, 1983, 1984, and 1993 illustratethis conclusion. However, empirically, it is not always true that1 < b < 2 or that K < 0.

This analysis is conditional on modeling the cumulative abun-dance as an increasing power-law function of the cumulativenumber of samples and on modeling the cumulative sample vari-ance as a power-law function of the cumulative sample mean.Here, these choices are motivated and justified by our data. Othermodels can be analyzed similarly to determine the conditionsunder which our method will or will not yield a unique estimateof the number of samples required to attain a desired precision.

Pagination not final (cite DOI) / Pagination provisoire (citer le DOI)

14 Can. J. Fish. Aquat. Sci. Vol. 00, 0000

Published by NRC Research Press

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