Comparison of sampling effort allocation strategies in a stratified random survey with multiple objectives

Guosheng Zhng,Jing wng,Ying Xue,b,Chongling Zhng,b,Binduo Xu,b,*,Yun Cheng,Yiping Ren,b

aCollege of Fisheries,Ocean University of China,Qingdao,266003,China

bLaboratory for Marine Fisheries Science and Food Production Processes,Pilot National Laboratory for Marine Science and Technology(Qingdao),Qingdao,266237,China

cOffshore Ecological Development Co.,Ltd.(Dalian),Dalian,116023,China

ABSTRACT

Keywords:

Fishery-independent survey

Stratified random sampling

Computer simulation

Sampling efforts allocation

1.Introduction

The latest and representative survey data,which contribute to the development of fisheries management and supporting stock assessment(Gallucci,Saila,Gustafson,&Rothschild,1996;Smith et al.,2011),is commonly provided by well-designed fishery-independent surveys(Blanchard,Jennings,&Maxwell,2008).The high quality of survey data from fishery-independent surveys is on the base of high cost of money and time(Scheirer,Chen,&Wilson,2004).In order to get the most accurate and precise data,the optimal survey design needs to be determined before conducting survey(Zhou et al.,2014).The optimal survey designs always achieve maximum accuracy and precision of survey data with a certain amount of budget or reduce costs with desirable accuracy and precision in fishery-independent surveys(Liu,Chen,&Cheng,2009).In some cases,however,the number of survey stations is limited because of some constraint conditions such as survey budget.

Several sampling methods are commonly used in scientific surveys,including simple random sampling(SRS),systematic sampling(SYS),cluster sampling(CS),stratified random sampling(StRS)and adaptive sampling(AS).StRS design is widely used in fishery-independent surveys in estimating abundance indices of target species,which divides the survey area into different strata and SRS is used within each stratum(Gavaris&Smith,1987;Stein&Ettema,2003;Cochran,1977).

There are mainly two ways to improve precision and accuracy of estimates of abundance indices in StRS design,including well-designed stratification scheme(Khan,Najmussehar,&Ahsan,2005;Miller,Skalski,&Ianelli,2007;Reddy,Khan,&Khan,2018)and rational sampling effort allocation schemes(Khan,Ahmad,&Prasad,2012;Khan&Ahsan,2003;Reddy,Khan,&Khan,2018;Swain,2016).The stratification was often determined based on some critical environmental variables,such as depth and substrate type(Ault,Diaz,Smith,Luo,&Serafy,1999;Gavaris&Smith,1987;Zhang,Brzezinski,Chang,Stepanek,&Chen,2011).A well-designed stratification scheme can make observations more homogenous within strata than between strata and achieve relatively precise and accurate estimates with reduced sample size comparing with other sampling methods(Miller et al.,2007;Reddy,Khan,&Khan,2018;Wang et al.,2018;Xu,Zhang,Xue,Ren,&Chen,2015).Xu et al.,(2015)and Wang et al.(2018)compared performances of different stratification schemes in a stratified random survey with multiple species.Rational sampling efforts allocation scheme is also vital to the improvement of precision and accuracy of estimates in stratified random surveys.A reasonable allocation of sampling efforts among strata will aid to get representative and accurate survey data for target species.The sample size among different strata should be allocated in accordance with the principle of optimal allocation strategy(Xu et al.,2015;Skinner,Holmes,&Holt,1994;Tillé&Wilhelm,2017).

Many methods have been developed to allocate sampling efforts among strata in stratified random surveys(Neyman,1934;Dalenius,1953;Yates,1953;Kokan&Khan,1967;Chatterjee,1967;Gre'n,1964;Wywia?,1988;Holmberg,2002).For surveys targeting single species,(Neyman,1934)allocated the sampling efforts among strata by comprehensively considering the stratum variance and stratum size in a stratified random survey.Yates(1953)allocated sampling efforts among different strata based on the variances of target indicators in different strata.However,allocation scheme for single species may be suitable for one species and not proper for others especially when the distribution characteristics of many species are of conflicting nature in the same waters.The optimal allocation schemes of sampling efforts in stratified random surveys might vary among different fish species,and it was necessary to find a suitable compromise scheme in multispecies surveys.The allocation scheme was determined by minimizing the survey cost in some studies.For example,Dalenius(1953)determined allocation scheme by linear equation of survey cost facing multivariables.Kokan&Khan,(1967)used a non-linear equation of cost to deal with the multivariate problem of sampling efforts allocation.New allocation schemes was examined by modifying or optimizing the traditional methods used for single species survey.Chatterjee et al.(1967)used Yates allocation method in multivariate surveys.Gre'n(1964)used Neyman method in multivariate surveys by simply averaging all optimal univariate Neyman allocation schemes in each stratum.In addition,mathematical method was also applied to deal with the sampling efforts allocation question.Wywia?(1988)solved the allocation program by matrix calculation,and determined sample sizes among strata by minimizing the spectral radius of sample variance-covariance matrix.Sample sizes in the surveys mentioned above were usually large enough.In some situations,however,the limited survey funds only could afford fishery-independent surveys with low sampling efforts.Facing this situation,traditional method might need adjustment to allocate sampling efforts among strata for multispecies surveys.

In this study,different potential allocation schemes of sampling efforts including the current scheme were evaluated in stratified random multispecies surveys with relatively small sample size.The objectives of this study are:(1)to develop a framework to compare the performances of allocation schemes of relatively small sample size among strata in a stratified random survey targeting multiple species with various spatial distribution patterns;(2)to find the optimal allocation scheme for multiple species by comparing the performances of different allocation methods;(3)to evaluate the consistency of performances of different allocation schemes over different sampling seasons in stratified random surveys.

2.Materials and methods

2.1.The stratified random survey in the Haizhou Bay

Haizhou Bay,located in the western part of central Yellow Sea,is an important spawning,nursery,and feeding ground for many commercially important fishes.It was ever an important fishing ground(Wan&Jiang,1998).Some important fish stocks have severely declined in abundance or even have been depleted in the bay because of highfishing pressure and environmental degradation.It is urgent to get high quality data in fishery-independent surveys for evaluating the status of those fish stocks and conducting stock assessment and fisheries management.

The stratification scheme for the survey was established in the Haizhou Bay,and the bay was divided to five strata based on the oceanographic,geological,regional,and biological characteristics that played a key role in influencing the spatial and temporal distribution offish species(Wang et al.,2018;Xu,Zhang,Xue,Ren,&Chen,2015).This survey area was divided into 10 min×10 min grids(sampling units)with 76 grid cells in total.In the surveys 3,5,3,9,4 sampling units in stratum A,B,C,D and E were selected,respectively(Fig.1).The bottom trawl surveys with 24 sampling stations were conducted in spring(May)and autumn(October)in 2011 in the bay(Wang et al.,2018;Xu et al.,2015).

The bottom trawl survey was conducted in the daytime using a 220 kW otter trawl vessel,and the towing speed was at 2-3 knots and the trawling lasted about 1 h at each sampling station.The opening width of the trawling net was 25 m and the cod-end mesh size was 17 mm.The catch was identified to species,and the species composition and abundance in number and weight of each species were recorded.

2.2.Computer simulation based on ordinary kriging interpolation

Ordinary kriging interpolation(OKI)is one of the most commonly used geostatistical interpolation method to simulate the status of unsurvey area based on the survey data.The geostatistical interpolation method is based on the spatial autocorrelation analysis,which can get interpolated data of unsurvey sampling units according to the spatial variability pattern(Van Beers&Kleijnen,2003).Each grid cell(sampling unit)was divided into twenty-five 2 min×2 min sub-units,and there were 1900 sub-units in total in the bay.By using OKI,abundance indices of target fish species in each sub-unit in spring and autumn were predicted by the original survey data using the combination of weights,which could be estimated according to semi-variance(Oliver&Webster,1990).The simulated results of ordinary kriging interpolation were regarded as ‘true’values of target species distribution,which were used to evaluate the precision and accuracy of the subsequent allocation schemes.

2.3.The allocation strategies of sampling efforts among strata

As conducted in 2011,in total 24 units out of 76 grids were selected in the stratified random surveys during spring and autumn in this simulation study.The grid cell(unit)was randomly selected from each stratum,and then one sub-unit from 25 sub-units was randomly chosen as a sampling station.We conducted one trawl randomly in each sampling unit,and sub-unit here represented the sampling variation and randomness in the unit.

Fig.1.Sampling grid cells(sampling units)in a stratified random survey and bathymetric contours in the Haizhou Bay.Solid circle and triangle indicate the sampling station in May and October,respectively.

In most cases of stratified random sampling,at least two sampling units are drawn in each stratum to guarantee the statistical analysis like sample mean and variance(Cochran,1977).Firstly,10 sampling units were distributed equally among five strata to ensure at least 2 sampling units in each stratum.Secondly,the remaining 14 sampling units were allocated among strata using the following allocation schemes,which considered different factors like the stratum size,variance within stratum and survey cost.In order to distinguish them from traditional methods,we called them adjusted methods in this study.

In this study,the adjusted Neyman allocation method was scheme A.Scheme A allocated the remaining sampling units among strata using Neyman allocation method,in which the sample size was allocated to stratum based on the variance within stratum and stratum size.In this study,we supposed that the survey cost of each trawl in all stratum was equal.The sample size in stratumhwas proportional to the number of sampling units within stratumh(Nh)and standard deviation(Sh)of abundance index within stratumh.Neyman allocation method was defined by the formula:

wherenhwas the sample size in stratumh,nwas the total sample size in all strata.Nhwas the number of units within stratumh.Shwas the standard deviation of abundance index of target fish species in stratumh.

Neyman allocation is usually applied to allocate sample size among strata in fisheries-independent surveys for single target species,and the Neyman allocation schemes for different target species may differ greatly.The standard deviation of abundance index within stratum was calculated based on the simulated data and was used to scheme A to get Neyman scheme for the target species.The target species in the surveys included small yellow croaker(Larimichthys polyactis,LP),fat greenling(Hexagrammos otakii,HO),whitespotted conger(Conger myriaster,CM)and pinkgray goby(Amblychaeturichthys hexanema,AH)in this study.There were four Neyman schemescorresponding to LP,CM,AH and HO,respectively.

The adjusted average-Neyman allocation method was scheme B.Scheme B considered schemecomprehensively(Greń,1964),which was called average method in this study.Thenhof scheme B was the average ofnhfor schemeTheof scheme B was calculated by:

where,nhwas the sample size in stratumhallocated in scheme B;were the number of sampling stations in stratumhallocated by schemerespectively.

The adjusted Yate allocation method was scheme C.Scheme C was a simplification of Yate allocation method.The traditional Yate allocation method is an optimal design of sampling efforts allocation with fixed total sample size to obtain presupposed accuracy of estimation.In this study,the optimal allocation method was to achieve maximum accuracy and precision with a certain amount of budget,which was different from the goal of regular Yate allocation.The sample size in stratumhof Scheme C allocation method was calculated by:

The adjusted proportional allocation method was scheme D.SchemeD used proportional allocation method for the remaining sampling efforts.Sample size in stratumhnhis proportional to the stratum sizeNh.This method ignores the composition of target species biomass and the distribution patterns of all target species.The advantage of proportional allocation scheme is easy to be conducted.However,this scheme is sometimes far from the optimal scheme and is difficult to meet the requirements in stratified random surveys targeting multiple species(Groves et al.,2011).The sample size in stratumhin Scheme D was calculated by:

Table 1Sample size in each stratum determined by different allocation schemes in spring and autumn in the Haizhou Bay in this simulation study.

where,Nhwas the number of units in stratumh;Niwas the sum of number of units in all stratum;nwas the total sample size in all strata.

Scheme E was the current sampling effort allocation method used in the fishery-independent trawl survey design in the Haizhou Bay,with 3,5,3,9,4 sampling stations in stratum A,B,C,D and E,respectively(Xu,Zhang,Xue,Ren,&Chen,2015).

The sample sizes among strata determined by different allocation schemes in spring and autumn in stratified random surveys in the bay were shown in the Table 1.

2.4.Simulation procedure

The ‘true’spatial distribution patterns for each target species were generated by the ordinary kriging interpolation and ‘true’values of mean abundance index for each species were calculated for comparisons among different allocation schemes of sampling efforts.For each allocation scheme,the interpolated abundance index data of the target species at a defined sample size of 24 sampling stations were resampled for 1000 times and,for each simulation run,the means of the abundance index were estimated using the resampled data.Suitable performance indices were used to compare the estimated abundance index and the ‘true’value,and the steps above were repeated 100 times to get the distribution of performance indices.The different allocation schemes were compared by performance indices.The relative estimation error(REE)and relative bias(RB)were chosen to measure the performances of different sampling effort allocation schemes in this simulation study.

REE was used to quantify the accuracy and precision of estimated indices,which was calculated as below:

where,Ytruewas the ‘true’value of mean abundance index calculated from the simulated data by ordinary kriging interpolation,Yiestimatedwas the mean abundance index value estimated from the resampled data in theith runs for different allocation schemes,andRwas the total number of runs of simulation for each scenario.

RB was used to evaluate the accuracy of estimates of abundance index of target species from different allocation schemes,which was calculated as:

The means of REE or RB for 100 times simulations were used to evaluate precision and accuracy of the allocation schemes for each target species.The sum of mean REE or RB for all species was applied to evaluate the precision and accuracy of each allocation scheme.

where,meanREEiis the mean of REE for one scheme with 1000 times resampling for speciesi.totalREEcomprehensively evaluated the precision of one scheme for all species.

where,meanRBiis the mean of RB for one scheme with 1000 times resampling for speciesi.totalRBevaluated the accuracy of one scheme for all species.

The algorithm of this study was described in the following flowchart(Fig.2).

3.Results

3.1.Spatial distribution patterns

The simulated spatial distribution patterns of relative abundance for four fish species in spring and autumn using ordinary kriging interpolation method were shown in Fig.3.Abundance index of LP increased with the water depth increasing in spring,and it mainly distributed in the southeastern waters in the bay.AH mainly distributed in the coastal shallow waters in both spring and autumn,and it relatively widely distributed in the bay in autumn.In spring,the abundance of HO was low,distributed especially in the shallow coastal waters.In autumn,the abundance of fat greenling mainly concentrated in the 20-30 m waters.CM was low in abundance and randomly distributed in the bay in spring.In autumn,it mainly distributed in the southern waters from 20 m to 30 m in depth.

3.2.Comparison of REE

REE values of estimates of abundance indices for the same fish species varied under different allocation schemes,and REE values for the same allocation scheme also showed large variation among different species in spring and autumn(Fig.4).The REE values of estimates of abundance indices for LP in spring and for CM,AH,HO in autumn were relatively low and similar under different allocation schemes.The REE values for LP in autumn and for CM,AH,HO in spring were relatively high and different under five allocation schemes.

Scheme A had small REE value of estimates of abundance index for one species,but it was sometimes not suitable for other species.For example,allocation schemeAHOperformed best in estimating abundance index of HO,but the REE values were relatively high when it applied to other fish species.Scheme A was the worst allocation scheme in multispecies survey.

Among the schemes B,C,D and E,scheme D showed relatively high REE values,and scheme C had relatively small REE values in most cases.The performance of scheme B was the most stable with relatively low REEs.The scheme E performed fairly well,which was better than scheme D and worse than scheme B in terms of REE values.

Fig.2.Flowchart of comparisons of sampling efforts allocation schemes in a fishery-independent survey targeting multiple species.

3.3.Comparison of RB

The relative bias(RB)of estimates of abundance indices under different allocation schemes varied with different species.The RB values of estimates of abundance indices for each species showed slight change among different allocation schemes.In many cases,the RB distributions of all allocation schemes were around zero,ranging between-3%and 3%in spring and autumn,showing that the estimates of abundance indices of these species were unbiased.However,RB of the estimates of abundance index for HO in spring and LP in autumn,were positively biased varying from 0 to 6%(Fig.5).

3.4.Comparison of totalREE and totalRB

4.Discussion

This simulation study evaluated the performances of different allocation strategies of sampling efforts among strata and identified the optimal allocation scheme to acquire abundance index estimates with high level of precision and accuracy for multiple species in stratified random surveys with relatively small sample size.The allocation schemes influenced the precision and accuracy of estimates of abundance indices of the target fish species.In general,different schemes ranked in the following order in terms of REE for four fish species:scheme B,C and D,E,and A.Regarding RB,the rank of different schemes from high to low accuracy was:scheme B,C,D,E and A.In many cases,the estimates of abundance indices under different allocation schemes were unbiased.However,RB distribution showed that abundance indices estimates for LP in autumn,and for AH and HO in spring were biased with overestimated abundance indices.

Aggregated spatial distribution patterns and low abundance were observed for LP in autumn,and for AH and HO in spring in the surveys(Fig.3).It might be the reason for biased estimation of abundance indices of LP in autumn,of AH and HO in spring.The stratified random sampling was not appropriate for target species with aggregation distribution characteristic(Cochran,1977;Groves et al.,2011).In these conditions,the assumption that the abundances within strata were more homogenous than between strata was violated.Adaptive sampling might be suitable for the species with aggregation distribution and would markedly increase the precision of estimates of abundance indices of fish species in this case(Francis,1984).

This study focused on the allocation strategy of sampling efforts among strata and improvement of precision of estimating fish abundance indices through sampling efforts allocation.Allocation scheme of sampling efforts was usually designed considering the stratum size and variance within stratum(Van Beers&Kleijnen,2003;Khan,Sehar,&Ahsan,2005;Olaniyi Mathew,2013).REE showed scheme A had excellent performances for single species surveys(Fig.4),which was consistent with previous studies(Ansari,Varshney,Najmussehar,&Ahsan,2011;Ghufran,Khowaja,&Ahsan,2011;Khowaja,Ghufran,&Ahsan,2012;Kozak,2006).However,the values ofshowed that scheme A was not suitable for multispecies surveys in spring and autumn(Table 2).Considering all the target species,scheme B was the most accurate scheme in estimating abundance indices.Each target species was equally important in scheme B,though different fish species had quite different biomass distribution.Scheme C allocated the remaining sampling efforts among strata considering stratum size,standard deviation within strata and the original biomass of target species.Scheme C improved the precision of abundance index estimates,but it might have effects on estimates of abundance indices for those fish species with low biomass in the multispecies survey.Scheme D mainly allocated the remaining sampling efforts according to stratum size,which ignored the variance of abundance index within stratum.It was not suitable for those fish species with highly heterogeneous distribution among strata,which mainly distributed in the shallow waters,deep waters or waters with specific substrate type in the bay.Scheme E adopted the same allocation scheme in spring and autumn,which reduced stability and precision of estimates of abundance indices of target species.

Fig.3.Spatial distribution of abundance index for target species(g/h)using ordinary kriging interpolation method based on the original survey data from 2013 in the Haizhou Bay.LP,small yellow croaker(Larimichthys polyactis);HO,fat greenling(Hexagrammos otakii);CM,whitespotted conger(Conger myriaster);AH,pinkgray goby(Amblychaeturichthys hexanema).(For interpretation of the references to colour in this figure legend,the reader is referred to the Web version of this article.)

Fig.4.The relative estimation error(REE)of estimates of abundance indices for the target fish species with different allocation schemes of sampling effort among strata in stratified random surveys in spring and autumn in the Haizhou Bay.LP,small yellow croaker(Larimichthys polyactis);HO,fat greenling(Hexagrammos otakii);CM,whitespotted conger(Conger myriaster);AH,pinkgray goby(Amblychaeturichthys hexanema).

Fig.5.The relative bias(RB)of estimates of abundance indices for target fish species with different allocation schemes of sampling effort among strata in stratified random surveys in spring and autumn in the Haizhou Bay.LP,small yellow croaker(Larimichthys polyactis);HO,fat greenling(Hexagrammos otakii);CM,whitespotted conger(Conger myriaster);AH,pinkgray goby(Amblychaeturichthys hexanema).

Table 2Thet otalREE and totalRB of estimates of abundance indices for all fish species for different allocation schemes of sampling efforts among strata in stratified random surveys in spring and autumn in the Haizhou Bay.

The simulated distribution patterns of target species were relatively reliable in this study.For instance,HO would prefer the rocky area in the shallow waters(Tong&Guo,2009).AH mainly distributed in the coastal estuary and adapted to the low salinity environment in spring(Zheng,Yu,&Zheng,2012).The simulated distribution patterns for HO in autumn and AH in spring were consistent with the previous studies(Fig.3).Fish species showed seasonal changes in the spatial distribution because of individual growth,migration and different life history stages(Carlsson,Kanneworff,Folmer,Kingsley,&Pennington,2000;Tong&Guo,2009;Wan&Jiang,1998;Xu&Chen,2009;Zheng et al.,2012).May is spawning period and October is the feeding period for LP.The feeding migration of LP changes the distribution pattern in study area(Xu&Chen,2009).

Stratified random sampling has been commonly used in fishery-independent surveys with different sampling objectives,such as estimation of abundance index,mean size and size structure,and age-or length-class proportions for commercially important fish species(Horppila&Peltonen,1992;Liu et al.,2009).Optimizations of sampling design in the multispecies surveys have been conducted to improve the precision and accuracy of estimates of abundance indices offish species by ascertaining the number of strata and redrawing the boundaries of all strata(Miller et al.,2007;Wang et al.,2018;Xu et al.,2015).Ault et al.(1999)improved survey design of estimating pink shrimp population abundance in Biscayne Bay by replacing simple random sampling with quarterly stratified random sampling.Miller et al.(2007)changed the total sample size and sample size in each stratum and assessed gains and losses for each sampling objective tofind optimal allocation scheme by comparing performances of all schemes.Khowaja et al.(2012)tended to find the optimal allocation scheme in multivariate stratified surveys by minimizing the sum of variances of the estimated parameters of target species to a quadratic cost constraint.Kozak(2006)compared five allocation methods of sampling efforts among strata for multispecies survey to find the best one in a simulation study.However,the allocation scheme of sampling efforts among strata for multispecies surveys was different from those studies because the sample size was relatively small,which required a modified two-step allocation method in this study.

The ‘true’distribution of abundance index of the target species was simulated using ordinary kriging interpolation based on the original survey data in 2013 in this study.However,the spatial distribution offish species might change annually because of the effects of environmental variability and fishing activities,which might further affect the variance in stratum and spatial distribution of fish species.The dominant fish species or main fisheries species may also change under the stresses of environmental variation and anthropogenic activities in the specific ecosystem.Therefore,if possible,continuous survey data in long-term survey program should be used to reflect the spatial distribution patterns of many potential target species and reduce the effect of possible inter-annual variability on allocation schemes of sampling efforts in the further study.

5.Conclusion

In this study we developed a framework to compare the performances of different allocation strategies of sampling efforts among strata in a multispecies fishery-independent trawl survey with relatively small sample size.For four target species with different spatial distribution patterns in spring and autumn,the adjusted average-Neyman allocation method(scheme B)was the most accurate scheme for the stratified random survey targeting multiple species in two seasons.Overall,scheme B,C,D and E had the similar precision in estimating abundance indices of the target species in spring and autumn.In general,the allocation of sampling efforts in stratified random survey design with relatively low sample size for estimating abundance indices of multiple species should comprehensively consider the variance of abundance in stratum and the seasonal variations.

Acknowledgements

We are grateful to all scientific staffand crew for their assistance in data collection during the surveys.This work was funded by the National Key R&D Program of China(2018YFD0900904),the National Natural Science Foundation of China(31772852)and the Fundamental Research Funds for the Central Universities(No.201562030,No.201612004).