PATCHY D I S T R I B U T I O N S : O P T I M I S I N G S A M P L E SIZE J.E. HEWITI', G.B. MCBRIDE, R.D. PRIDMORE and S.E THRUSH Water Quality Centre, Ecosystems Division, National Institute of Water and Atmospheric Research, P.O. Box 11115, Hamilton, New Zealand
(Received: January 1992; Revised: July 1992)
Abstract. A method for estimating sample size which does not require an a priori definition of desired precision, or the assumption that the population is normally distributed with constant variance, has recently been proposed. This paper discusses this method and presents five modifications which make the method easier to use and reduce the probability of estimating a larger sample size than is actually required, The method is extended and used to estimate the mean abundance of patchily distributed benthic organisms. The technique can be used to guide the design of any environmental sampling programme, be it physical, chemical or biological, where comparisons between times and/or locations are required. Trade-offs between numbers of replicates and numbers of levels/sites are discussed.
1.
Introduction
A crucial part of the design of ecological monitoring programmes and field experiments is selecting the sample size (i.e., number of replicates) which will optimize both the performance of statistical analyses and the amount of information obtained, when balanced against the cost of the study. This is generally done by comparing attainable precision (predicted from the results of a preliminary survey) for a range of sample sizes with the cost of collecting the samples. If the sampled populations follow a known distribution (e.g., normal or negative binomial) such comparisons are relatively straightforward (Elliott, 1983; Green, 1979). Determining the optimum sample size for benthic studies has long perplexed marine ecologists. Marine benthic studies often have a small (e.g., 3-10 samples) sample size, due to the difficulties involved in sample collection and the time required for sample analyses. As it is generally considered inappropriate to conduct normality tests with sample sizes of less than 10, and the power of goodness of fit tests on sample sizes under 30 is low, marine benthic studies often have difficulty determining, with any confidence, what distribution the data population follows. Benthic organisms commonly exhibit significant spatial aggregations (Jumars and Eckman, 1983; McArdle and Blackwell, 1989; Sun and Fleeger, 1991; Thrush, 1991), which typically result in data distributions being distinctly non-normal. It is therefore frequently recommended that a suitable transformation be applied to make the data conform with a normal distribution (Green, 1979; Millard and Lettenmaier, 1986). However, as previously mentioned, unless the sample size of the study is large, determining the distribution of the data (and hence which function should be used to transform the data) can not be done with confidence. Indeed, in some cases (e.g., sparsely distributed species whose sample populations contain large numbers of zeros) there may be no suitable transformation. While these species Environmental Monitoring and Assessment 27: 95-105, 1993. @ 1993 KIuwer Academic Publishers. Printed in the Netherlands.
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would usually be of little interest, in some studies (e.g., impact assessment) the change in density of these species could be of focal importance if they were known to be sensitive to a likely form of stress (i.e., indicator species). Finding a suitable transformation is not always easy even for more abundant species. For example, in a study of macrobenthic spatial patterns carried out on an intertidal sandttat, we collected 60 core (13-cm diam) samples from a 300 m 2 area. 24 of the 40 taxa collected were of low and extremely patchy abundance. Of the 16 species demonstrating reasonable abundances (mean of > 2 individuals per core), only two were considered normal by the EDF Kolmogorov-Smirnov test (SAS, 1985). When log, arcsin, and square root transformations were applied to the remaining species data, only five of the transformed populations were considered not significantly different from normal. For the above reasons, many recent marine benthic studies have used nonparametric statistical methods (Commito, 1982; Elmgren et al., 1986; Hines et al., 1990; Olafsson and Moore, 1990; Van Blaricom, 1982). Unfortunately, while most of the parametric methods have nonparametric equivalents, formulae for defining the sample size required by many nonparametric statistical tests (e.g., KruskalWallis) are not available. The best approach appears to be randomisation testing (e.g., Edgington, 1987; Noreen, 1989; Sokal and Rohlf, 1981). This technique makes no assumptions about the distribution of the data, but does take account of patchiness and skewness of populations. It was first proposed in 1935 (Conover, 1980 p. 328) but calculation time precluded its general utility. However, the general availability and speed of today's computers has resulted in an increasing use of randomisation techniques in ecological methodology (e.g., Krebs, 1989; Legendre et al., 1990). Recently, Bros and Cowell (1987) have proposed a randomisation technique for optimising sample size. In this paper, we present several modifications to the Bros and Cowell method which make it easier to use and reduce the risk of oversampling. We also present an application of the method to monitoring patchily distributed benthic organisms where precise estimates of means were required for temporal analyses.
2.
The Bros-Cowell Technique
In essence, data from a single intensive pilot survey are used to contrast decreasing estimates of the standard error with increasing sampling cost as sample size is increased. From a single intensive pilot survey at one site, a number of sample replicates (N) are collected. Combinations are taken from these data to represent subsamples of size n = 2, ..., N / 2 . If the pilot survey sample size N is small enough, all possible combinations of data for each subsample size may be analysed. For pilot surveys with larger sample sizes, a limited selection of all the possible combinations for each subsample size is made by repeated random draws, because of limitations
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on computational storage. Bros and Cowell (1987) recommended that the number of random draws made be, at least, equal to 10% of the pilot survey sample size N. For each random draw of subsample size n the standard error ( S E ) is calculated, being s/x/~, where s is the standard deviation of the draw. Curves of the maximum, mean and minimum standard error versus subsample size are produced. From these curves the minimum acceptable sample size (mass) is selected. The maximum possible sample size (mpss) is then calculated from the study budget as a function of money, time and materials. If mpss < mass the study should not be attempted as is and perhaps a less complex or intensive approach be adopted (e.g., reducing the variety of habitats sampled). If mpss > mass, then for each of the intervening sample sizes the maximum standard error is used to calculate the minimum detectable differences (as given by Zar (1981, 1984) for a parametric C-test). A trade-off is then performed between sampling cost and minimum detectable difference.
3.
Modification of the Technique
We propose five modifications to the Bros and Cowell (1987) technique:
1. Use the decrease in standard error with increasing subsample, n, as an index of increased information. By using the minimum detectable difference for assessing trade-offs with cost/effort, Bros and Cowell (1987) relate precision directly to a parametric test result. However, if it is visualised that parametric tests will not be appropriate for use on the data collected by the study, we suggest that the minimum detectable difference not be used in this way. We note that the standard error, predicted by the randomisation technique, does not related directly to any test result. However, recognising that the standard error serves as a measure of precision, we suggest that a trade-off between standard error and cost/effort is an appropriate approach, regardless of the form of test eventually used. 2. Use the 95th percentile of the distribution of the randomised standard errors for each subsample size, n, when making trade-offs between increased information and cost. Bros and Cowell (1987) accounted for the skewness in the statistical population by taking the maximum standard error of the randomised samples for each subsample size. In most cases this will lead to an overestimation of the attainable standard error for each sample size. Instead, we propose taking the 95th percentile of the randomised standard errors for each sample size as a compromise between unnecessary sampling effort and the risk of under estimation of the standard error. 3. Use the curve of the pooled standard error versus sample size for trade-off calculations for comparison of means. Where the study is designed to compare geographically distinct sampling areas (= sites), there is no reason to assume that variances at every site will be similar, and a preliminary survey of size N needs, therefore, to be carried out at every site. The standard error used to estimate
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precision must, then, becomes a pooled standard error as random sampling from one population only could give quite erroneous results. The pooled standard error is obtained from
s E ; = x/(4 +
(1)
where s 2 and s 2 are the variances of the two samples. The 95th percentile of the pooled standard error for each pair of sites can then be calculated from randomised draws carried out on the results of the surveys from each pair simultaneously. Note that the distribution of SEp for the random draws could be more narrow than that for the S E alone: having obtained one draw with an extreme S E it is most unlikely that the draw from the other sample will also give a value that extreme. For the case of an experiment involving different treatments at one location or where the means to be compared are from two different times at the same place, the SEp term could be calculated from paired randomised draws from a preliminary survey size N. While this pooled standard error term may often be little different from the single standard error term used by Bros and Cowell (1987), differences can occur for draws of small subsample size, n, and for very clumped or skewed populations. 4. Make a maximum of l O00 unique random subsampIe draws. For a reasonable sized pilot survey (say, sample size > 30), using all the possible draws can lead quickly to an excessive amount of computer time and storage being needed, as Bros and Cowell (1987) noted. They suggested that a minimum number of random draws equal to 10% of N be performed. For a sample size of 30 this would result in a minimum of 3 random draws being performed for each subsample size; which would not be enough to minimize any effects of skewness in the sample population. Edgington (1987) recommends the use of 1000 as an appropriate number of draws in randomisation tests. This number of random draws does not require an excessive amount of computer time and, in our case resulted in a larger 95th percentile, compared with using 100 random draws. We therefore recommend that when the total number of possible combinations for a particular subsample size is in excess of 1000, that 1000 unique random draws be made for that subsample size, and that, otherwise, all possible unique combinations be used. 5. Specify an explicit criterion for determining the minimum acceptable sample size. When the relationship of the 95th percentile of the standard errors with n gives a gently sloping curve it is difficult to select mass (as defined by Bros and Cowell, 1987) in a consistent fashion. To overcome this problem we recommend defining mass quantitatively. For our work we have defined mass to be the point at which the slope of the curve was less than 20% of the initial slope; at this point the very obvious gains in information for increasing sampling effort have occurred and it is necessary to start balancing cost/effort against precision.
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4. Application The application of this method is illustrated in the design of a monitoring program on the intertidal sandflats of Manukau Harbour. Manukau Harbour is a large (340 km 2) shallow inlet, adjacent to Auckland, on the west coast of the North Island of New Zealand. Intertidal sandflats comprise about 40% of the area of the harbour and are commonly used for recreation and food gathering. The monitoring programme was established to create a time series: which would provide managers with some stock-taking of the resource under stewardship ; and against which more process orientated/causal studies could be conducted. For these reasons we wanted to estimate adequately the mean abundance of benthic taxa occurring at the sites such that spatial variability would not confound the temporal sequence, and to compare the means of each site over time. Although this programme was not designed to compare means between the two sites, we will use the data collected in the preliminary surveys to show how the method could be applied if this comparison was required. A pilot survey of each of the sites was undertaken to assess the abundance of some polychaete, bivalve and crustacea taxa. Thirty six samples were taken from each site with a 13 cm diameter corer. The aim of the survey, apart from providing data for the estimation of the regular monitoring sample size, was to show if either of the sites exhibited large scale spatial patterns in the density of animals (e.g., gradients, homogenious patches, etc.). Spatial analysis of the survey showed that most of the common taxa were aggregated (Thrush et al., 1989). A study budget for sample sizes n = 3, 4, 6, 8, 10, 12, 16, 18, 20 was derived for the case where each site had the same sample size n. Once n became greater than 16, an extra day became necessary for the collection of samples from both sites, resulting in a large increase in costs. We therefore set mpss at 16 for each site.
1. Calculation of sample size for estimating mean abundances. Each taxon at each site was treated separately. Following modifications 2 and 4, the 95th percentile of the standard errors from 1000 unique randomized draws for each of n = 3, 4, 6, 8, 10, 12, 16 were determined. Curves of the 95th percentile of the standard errors versus draw size n were plotted and the mass determined for each taxon at each site (Figure 1). Although the mass varied with taxa and site it was always less than or equal to 10 and therefore always less then the mpss of 16. None of our plots of standard error versus drawsize gave a curve that flattened out totally before the mpss of 16 was reached. We therefore had to examine the trade-offs between information (as determined by the standard error) and cost for each taxon at each site. This was done, taking into account not only the aims/cost of the proposed study, but also the likely natural variation in the density of the organisms through time. For example, a population of the large, long-lived polychaete Travisia olens could be expected to be inherently more consistent than that of the opportunistic, short-lived polychaete Heteromastusfiliformis and a sample
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5.
Discussion
A potential problem with any method that uses a pilot study to determine sample size is that it assumes that the population variance remains constant. That is, that the true variance of the population will not change with, for example, recruitment, or, in the case of experimental design, the experiment undertaken. This may or may not be a reasonable assumption: as yet, few studies have investigated factors affecting changes in the variability of populations (Botton, 1984; Fairweather, 1988; Schneider, 1978). Another assumption implicit in these methods is that the sample size is equivalent to the number of independent samples required. Therefore, any factors which may compromise sample independence (e.g., temporal and/or spatial autocorrelation) or affect the variance of the populations need to be accounted for in the design of the study. It is sometimes suggested that, rather than conducting a pilot survey to obtain the necessary data for estimating the preferred sample size, data from previous work within the study area or a similar habitat could be used. However, any differences between previous and proposed studies should be considered carefully. For example, different types of and/or sizes of samplers can affect estimates of abundance. Timing and scale of the previous study could also affect the appropriateness of the preferred sample size estimated. When considering the merits of conducting a pilot survey, they should not be seen only as a means of providing information for determining sample size. If designed correctly, the data collected may be used to provide a variety of ecological information useful in the study design and interpretation. The pilot survey discussed in the application section also gave us information on population density gradients, spatial patterns exhibited by organisms (Thrush et al., 1989), species associations and community structure (Pridmore et al., 1990). Once it has been decided to conduct a pilot survey it is important to match the sizes of the pilot survey and study area. For example, if the proposal is to establish a number of experimental plots within a site (i.e., area of apparent homogeneity), then pilot sampling should be directed to assessing both the density and the variation
PATCHY DISTRIBUTIONS: OPTIMISING SAMPLE SIZE
] 03
in density of populations both at the site size scale and the plot size scale. Then decisions can be made as to whether sampling can best be optimised: by collecting one sample from a number of replicate plots; or, if within-plot variation is high, by collecting many samples from a few replicate treatment plots. The decision to collect all the samples, for one experimental level, from one plot only (i.e., not to use replicate plots) should only be done after careful consideration of the hypothesis to be tested and the inferences which can be made. Hurlbert (1984) lists a number of problems with this approach; particularly that it can only demonstrate a difference between locations as opposed to demonstrating an effect, and that the actual experimental Type 1 error risk oz will be greater than that selected a priori: the latter appears to be a problem only if the sample size is increased during a study. Stewart-Oaten et al. (1986) and Millard and Lettenmaier (1986) dispute some of his criticisms, while Winer (1971) suggests that the advantages accrued by using replicate plots is dependent on the magnitude of replicate variation relative to treatment and experimental effects. Once the design of the study (i.e., sample size and replicate plots) has been decided upon, trade-offs between the number of replicate and number of levels/sites can be made. For example, for a study with a total m a s s of 21, the preferred number of independent samples for each level/site of the study may be seven, thus allowing only three experimental levels/sites. However, the increase in ecological information gained by having four levels/sites may outweigh the resulting decrease in power when the sample size drops to five. As Bros and Cowell (1987) point out, the ability to make this trade-off between power of statistical analyses and the possible increase in ecological information from an increased number of experimental levels/sites is one of the most useful aspects of this type of method for estimating sample size. Krebs (1989 p. 197) comments on the wastage in ecological research through poor study designs. This may occur either when insufficient information is gathered to enable patterns to be confidently identified or, more rarely, when too much sampling provides more accurate estimates than are needed. Where raw or transformed data conform to known distributions, exact calculations of detectable differences can be used to determine samples sizes. However, as we illustrate, marine macrobenthic populations, even when intensively sampled, frequently do not conform to normality (even after data transformations by common methods). Plots of standard errors, calculated from randomised draws performed on data collected by a pilot survey, versus sample size, provide a useful graphical way of helping to make decisions balancing information gained and cost.
Acknowledgements Field work for the examples in this paper was funded by the Regional Water Board, Auckland Regional Authority and we are grateful for their permission to publish this interpretation of their data. Thanks also go to D.S. Roper who helped with the
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field c o l l e c t i o n a n d identification o f species and to W i l l i a m Bros for his constructive criticism on an earlier draft.
References Bros, W.E. and Cowell, B.C.: 1987, 'A Technique for Optimizing Sample Size (replication)', J. Exp. Mar. Biol. Ecol. 114, 63-71. Botton, M.L.: 1984, 'Spatial Distribution of Three Species of Bivalves on an Intertidal Flat: The Interaction of Life-History Strategy with Predation and Disturbance', Veliger 26, 282-287. Commito, J.A.: 1982, 'The Importance of Predation by Infaunal Polychaetes in Controlling the Structure of a Soft-Bottom Community in Maine, USA', Mar. Biol. 68, 77-81. Conover, W.J.: 1980, Practical Nonparametric Statistics, Wiley, New York, 2nd edition, 493 pp. Edgington, E.S.: 1987, Randomization tests, Marcel Dekker Inc., New York, 2nd edition, 341 pp. Elliott, J.M.: 1983, 'Some Methods for the Statistical Analysis of Samples of Benthic Invertebrates', Freshwat. Biol. Ass. Sci. Publ. 25, 1-167. Elmgren, R., Ankar, S., Marteleur, B. and Ejdung, G.: 1986, 'Adult Interference with Postlarvae in Soft Sediments: The Pontoporia-Macoma Example', Ecology 67, 827-836. Fairweather, P.G.: 1988, 'Predation Can Increase Variability in the Abundance of Prey on Seashores', Oikos 53, 87-92. Green, R.H.: 1979, Sampling Design and Statistical Methods for Environmental Biologists, Wiley, Toronto, 257 pp. Hines, A.H., Haddon, A.M. and Wiechert, L.A.: 1990, 'Guild Structure and Foraging Impact of Blue Crabs and Epibenthic Fish in a Subestuary of Chesapeake Bay', Mar. Ecol. Prog. Ser. 67, 105-126. Hurlbert, S.H.: 1984, 'Pseudoreplication and the Design of Ecological Field Experiments', Ecol. Monogr. 54, 187-211. Jones, M.L.: 1978, 'Three New Species of Magelona (Annelida, Ploychaeta) and a Rediscription of Magelona pitelkai Hartman', Proc. Biol. Soc. Washington 91, 336-363. Jumars, P.A. and Eckman, J.E.: 1983, 'Spatial Structure within Deep-Sea Benthic Communities', in: Grove, G.T. (Ed.), The Sea. Deep-Sea Biology, Vol. 8, Wiley, New York, pp. 399-451. Krebs, C.J.: 1989, Ecological Methodology, Harper & Row, New York, USA, 654 pp. Legendre, P., Oden, N.L., Sokal, R.R., Vaudor, A. and Kim, J.: 1990, 'Approximate Analysis of Variance of Spatially Autocorrelated Regional Data', J. Class. 7, 53-75. McArdle, B.H. and Blackwell, R.G.: 1989, 'Measurement of Density Variability in the Bivalve Chione stutchburyi Using Spatial Autocorrelation', Mar. Ecol. Prog. Ser. 52, 245-252. Millard, S.P. and Lettenmaler, D.P.: 1986, 'Optimal Design of Biological Sampling Programs Using the Analysis of Variance', Estuarine Coastal ShelfSci. 22, 637-656. Noreen, E.W.: 1989, Computer Intensive Methods for Testing Hypotheses, Wiley, New York, 229 pp. Olafsson, E. and Morre, C.G.: 1990, 'Control of Meiobenthic Abundance of Macroepifauna in a Subtidal Muddy Habitat', Mar. Ecol. Prog. Ser. 65, 241-249. Pridmore, R.D., Thrush, S.E, Hewitt, J.E. and Roper, D.S.: 1990, 'Macrobenthic Community Composition of Six Intertidal Sandflats in Manukau Harbour, New Zealand', N.Z.J. Mar. Freshwat. Res. 24, 81-96. SAS: 1985, SAS Users Guide: Statistics, Version 5 Edition, SAS Institute Inc., Cary, North Carolina, 956 pp. Schneider, D.: 1978, 'Equalisation of Prey Numbers of Migratory Shorebirds', Nature 271, 353-354. Sokal, R.R. and Rohlf, F.J.: 1981, Biometry, Freeman & Co., New York, 859 pp. Stewart-Oaten, A., Murdoch, W.W. and Parker, K.R.: 1986, 'Environmental Impact Assessment: "Pseudoreplication" in time?', Ecology 67, 929-940. Sun, B. and Fleeger, J.W.: 1991, 'Spatial and Temporal Patterns of Dispersion in Meiobenthic Copepods', Mar. Ecol. Prog. Ser. 71, 1-11. Thrush, S.E: 1991, 'Spatial Patterns in Soft-Bottom Communities', Trends Ecol. Evol. 6, 75-79. Thrush, S.E, Hewitt, J.E. and Pridmore, R.D.: 1989, 'Patterns in the Spatial Arrangement of Polychaetes and Bivalves in Intertidal Sandflats', Mar. Biol. 102, 529-535.
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Van Blaricom, G.R.: 1982, 'Experimental Analysis of Structural Regulation in a Marine Sand Community Exposed to Ocean Swell', Ecol. Monogr. 52, 283-305. Winer, B.J.: 1971, Statistical Principles in Experimental Design, McGraw-Hill Book Co., New York, 907 pp. Zar, J.H.: 1981, 'Power of Statistical Testing: Hypotheses About Means', Am. Lab. 13, 102-107. Zar, J.H.: 1984, BiostatisticalAnalyses, Prentice-Hall, New Jersey, 2nd ed., 718 pp.
(Received: January 1992; Revised: July 1992)
Abstract. A method for estimating sample size which does not require an a priori definition of desired precision, or the assumption that the population is normally distributed with constant variance, has recently been proposed. This paper discusses this method and presents five modifications which make the method easier to use and reduce the probability of estimating a larger sample size than is actually required, The method is extended and used to estimate the mean abundance of patchily distributed benthic organisms. The technique can be used to guide the design of any environmental sampling programme, be it physical, chemical or biological, where comparisons between times and/or locations are required. Trade-offs between numbers of replicates and numbers of levels/sites are discussed.
1.
Introduction
A crucial part of the design of ecological monitoring programmes and field experiments is selecting the sample size (i.e., number of replicates) which will optimize both the performance of statistical analyses and the amount of information obtained, when balanced against the cost of the study. This is generally done by comparing attainable precision (predicted from the results of a preliminary survey) for a range of sample sizes with the cost of collecting the samples. If the sampled populations follow a known distribution (e.g., normal or negative binomial) such comparisons are relatively straightforward (Elliott, 1983; Green, 1979). Determining the optimum sample size for benthic studies has long perplexed marine ecologists. Marine benthic studies often have a small (e.g., 3-10 samples) sample size, due to the difficulties involved in sample collection and the time required for sample analyses. As it is generally considered inappropriate to conduct normality tests with sample sizes of less than 10, and the power of goodness of fit tests on sample sizes under 30 is low, marine benthic studies often have difficulty determining, with any confidence, what distribution the data population follows. Benthic organisms commonly exhibit significant spatial aggregations (Jumars and Eckman, 1983; McArdle and Blackwell, 1989; Sun and Fleeger, 1991; Thrush, 1991), which typically result in data distributions being distinctly non-normal. It is therefore frequently recommended that a suitable transformation be applied to make the data conform with a normal distribution (Green, 1979; Millard and Lettenmaier, 1986). However, as previously mentioned, unless the sample size of the study is large, determining the distribution of the data (and hence which function should be used to transform the data) can not be done with confidence. Indeed, in some cases (e.g., sparsely distributed species whose sample populations contain large numbers of zeros) there may be no suitable transformation. While these species Environmental Monitoring and Assessment 27: 95-105, 1993. @ 1993 KIuwer Academic Publishers. Printed in the Netherlands.
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would usually be of little interest, in some studies (e.g., impact assessment) the change in density of these species could be of focal importance if they were known to be sensitive to a likely form of stress (i.e., indicator species). Finding a suitable transformation is not always easy even for more abundant species. For example, in a study of macrobenthic spatial patterns carried out on an intertidal sandttat, we collected 60 core (13-cm diam) samples from a 300 m 2 area. 24 of the 40 taxa collected were of low and extremely patchy abundance. Of the 16 species demonstrating reasonable abundances (mean of > 2 individuals per core), only two were considered normal by the EDF Kolmogorov-Smirnov test (SAS, 1985). When log, arcsin, and square root transformations were applied to the remaining species data, only five of the transformed populations were considered not significantly different from normal. For the above reasons, many recent marine benthic studies have used nonparametric statistical methods (Commito, 1982; Elmgren et al., 1986; Hines et al., 1990; Olafsson and Moore, 1990; Van Blaricom, 1982). Unfortunately, while most of the parametric methods have nonparametric equivalents, formulae for defining the sample size required by many nonparametric statistical tests (e.g., KruskalWallis) are not available. The best approach appears to be randomisation testing (e.g., Edgington, 1987; Noreen, 1989; Sokal and Rohlf, 1981). This technique makes no assumptions about the distribution of the data, but does take account of patchiness and skewness of populations. It was first proposed in 1935 (Conover, 1980 p. 328) but calculation time precluded its general utility. However, the general availability and speed of today's computers has resulted in an increasing use of randomisation techniques in ecological methodology (e.g., Krebs, 1989; Legendre et al., 1990). Recently, Bros and Cowell (1987) have proposed a randomisation technique for optimising sample size. In this paper, we present several modifications to the Bros and Cowell method which make it easier to use and reduce the risk of oversampling. We also present an application of the method to monitoring patchily distributed benthic organisms where precise estimates of means were required for temporal analyses.
2.
The Bros-Cowell Technique
In essence, data from a single intensive pilot survey are used to contrast decreasing estimates of the standard error with increasing sampling cost as sample size is increased. From a single intensive pilot survey at one site, a number of sample replicates (N) are collected. Combinations are taken from these data to represent subsamples of size n = 2, ..., N / 2 . If the pilot survey sample size N is small enough, all possible combinations of data for each subsample size may be analysed. For pilot surveys with larger sample sizes, a limited selection of all the possible combinations for each subsample size is made by repeated random draws, because of limitations
PATCHY DISTRIBUTIONS: OPTIMISING SAMPLE SIZE
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on computational storage. Bros and Cowell (1987) recommended that the number of random draws made be, at least, equal to 10% of the pilot survey sample size N. For each random draw of subsample size n the standard error ( S E ) is calculated, being s/x/~, where s is the standard deviation of the draw. Curves of the maximum, mean and minimum standard error versus subsample size are produced. From these curves the minimum acceptable sample size (mass) is selected. The maximum possible sample size (mpss) is then calculated from the study budget as a function of money, time and materials. If mpss < mass the study should not be attempted as is and perhaps a less complex or intensive approach be adopted (e.g., reducing the variety of habitats sampled). If mpss > mass, then for each of the intervening sample sizes the maximum standard error is used to calculate the minimum detectable differences (as given by Zar (1981, 1984) for a parametric C-test). A trade-off is then performed between sampling cost and minimum detectable difference.
3.
Modification of the Technique
We propose five modifications to the Bros and Cowell (1987) technique:
1. Use the decrease in standard error with increasing subsample, n, as an index of increased information. By using the minimum detectable difference for assessing trade-offs with cost/effort, Bros and Cowell (1987) relate precision directly to a parametric test result. However, if it is visualised that parametric tests will not be appropriate for use on the data collected by the study, we suggest that the minimum detectable difference not be used in this way. We note that the standard error, predicted by the randomisation technique, does not related directly to any test result. However, recognising that the standard error serves as a measure of precision, we suggest that a trade-off between standard error and cost/effort is an appropriate approach, regardless of the form of test eventually used. 2. Use the 95th percentile of the distribution of the randomised standard errors for each subsample size, n, when making trade-offs between increased information and cost. Bros and Cowell (1987) accounted for the skewness in the statistical population by taking the maximum standard error of the randomised samples for each subsample size. In most cases this will lead to an overestimation of the attainable standard error for each sample size. Instead, we propose taking the 95th percentile of the randomised standard errors for each sample size as a compromise between unnecessary sampling effort and the risk of under estimation of the standard error. 3. Use the curve of the pooled standard error versus sample size for trade-off calculations for comparison of means. Where the study is designed to compare geographically distinct sampling areas (= sites), there is no reason to assume that variances at every site will be similar, and a preliminary survey of size N needs, therefore, to be carried out at every site. The standard error used to estimate
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precision must, then, becomes a pooled standard error as random sampling from one population only could give quite erroneous results. The pooled standard error is obtained from
s E ; = x/(4 +
(1)
where s 2 and s 2 are the variances of the two samples. The 95th percentile of the pooled standard error for each pair of sites can then be calculated from randomised draws carried out on the results of the surveys from each pair simultaneously. Note that the distribution of SEp for the random draws could be more narrow than that for the S E alone: having obtained one draw with an extreme S E it is most unlikely that the draw from the other sample will also give a value that extreme. For the case of an experiment involving different treatments at one location or where the means to be compared are from two different times at the same place, the SEp term could be calculated from paired randomised draws from a preliminary survey size N. While this pooled standard error term may often be little different from the single standard error term used by Bros and Cowell (1987), differences can occur for draws of small subsample size, n, and for very clumped or skewed populations. 4. Make a maximum of l O00 unique random subsampIe draws. For a reasonable sized pilot survey (say, sample size > 30), using all the possible draws can lead quickly to an excessive amount of computer time and storage being needed, as Bros and Cowell (1987) noted. They suggested that a minimum number of random draws equal to 10% of N be performed. For a sample size of 30 this would result in a minimum of 3 random draws being performed for each subsample size; which would not be enough to minimize any effects of skewness in the sample population. Edgington (1987) recommends the use of 1000 as an appropriate number of draws in randomisation tests. This number of random draws does not require an excessive amount of computer time and, in our case resulted in a larger 95th percentile, compared with using 100 random draws. We therefore recommend that when the total number of possible combinations for a particular subsample size is in excess of 1000, that 1000 unique random draws be made for that subsample size, and that, otherwise, all possible unique combinations be used. 5. Specify an explicit criterion for determining the minimum acceptable sample size. When the relationship of the 95th percentile of the standard errors with n gives a gently sloping curve it is difficult to select mass (as defined by Bros and Cowell, 1987) in a consistent fashion. To overcome this problem we recommend defining mass quantitatively. For our work we have defined mass to be the point at which the slope of the curve was less than 20% of the initial slope; at this point the very obvious gains in information for increasing sampling effort have occurred and it is necessary to start balancing cost/effort against precision.
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4. Application The application of this method is illustrated in the design of a monitoring program on the intertidal sandflats of Manukau Harbour. Manukau Harbour is a large (340 km 2) shallow inlet, adjacent to Auckland, on the west coast of the North Island of New Zealand. Intertidal sandflats comprise about 40% of the area of the harbour and are commonly used for recreation and food gathering. The monitoring programme was established to create a time series: which would provide managers with some stock-taking of the resource under stewardship ; and against which more process orientated/causal studies could be conducted. For these reasons we wanted to estimate adequately the mean abundance of benthic taxa occurring at the sites such that spatial variability would not confound the temporal sequence, and to compare the means of each site over time. Although this programme was not designed to compare means between the two sites, we will use the data collected in the preliminary surveys to show how the method could be applied if this comparison was required. A pilot survey of each of the sites was undertaken to assess the abundance of some polychaete, bivalve and crustacea taxa. Thirty six samples were taken from each site with a 13 cm diameter corer. The aim of the survey, apart from providing data for the estimation of the regular monitoring sample size, was to show if either of the sites exhibited large scale spatial patterns in the density of animals (e.g., gradients, homogenious patches, etc.). Spatial analysis of the survey showed that most of the common taxa were aggregated (Thrush et al., 1989). A study budget for sample sizes n = 3, 4, 6, 8, 10, 12, 16, 18, 20 was derived for the case where each site had the same sample size n. Once n became greater than 16, an extra day became necessary for the collection of samples from both sites, resulting in a large increase in costs. We therefore set mpss at 16 for each site.
1. Calculation of sample size for estimating mean abundances. Each taxon at each site was treated separately. Following modifications 2 and 4, the 95th percentile of the standard errors from 1000 unique randomized draws for each of n = 3, 4, 6, 8, 10, 12, 16 were determined. Curves of the 95th percentile of the standard errors versus draw size n were plotted and the mass determined for each taxon at each site (Figure 1). Although the mass varied with taxa and site it was always less than or equal to 10 and therefore always less then the mpss of 16. None of our plots of standard error versus drawsize gave a curve that flattened out totally before the mpss of 16 was reached. We therefore had to examine the trade-offs between information (as determined by the standard error) and cost for each taxon at each site. This was done, taking into account not only the aims/cost of the proposed study, but also the likely natural variation in the density of the organisms through time. For example, a population of the large, long-lived polychaete Travisia olens could be expected to be inherently more consistent than that of the opportunistic, short-lived polychaete Heteromastusfiliformis and a sample
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101
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5.
Discussion
A potential problem with any method that uses a pilot study to determine sample size is that it assumes that the population variance remains constant. That is, that the true variance of the population will not change with, for example, recruitment, or, in the case of experimental design, the experiment undertaken. This may or may not be a reasonable assumption: as yet, few studies have investigated factors affecting changes in the variability of populations (Botton, 1984; Fairweather, 1988; Schneider, 1978). Another assumption implicit in these methods is that the sample size is equivalent to the number of independent samples required. Therefore, any factors which may compromise sample independence (e.g., temporal and/or spatial autocorrelation) or affect the variance of the populations need to be accounted for in the design of the study. It is sometimes suggested that, rather than conducting a pilot survey to obtain the necessary data for estimating the preferred sample size, data from previous work within the study area or a similar habitat could be used. However, any differences between previous and proposed studies should be considered carefully. For example, different types of and/or sizes of samplers can affect estimates of abundance. Timing and scale of the previous study could also affect the appropriateness of the preferred sample size estimated. When considering the merits of conducting a pilot survey, they should not be seen only as a means of providing information for determining sample size. If designed correctly, the data collected may be used to provide a variety of ecological information useful in the study design and interpretation. The pilot survey discussed in the application section also gave us information on population density gradients, spatial patterns exhibited by organisms (Thrush et al., 1989), species associations and community structure (Pridmore et al., 1990). Once it has been decided to conduct a pilot survey it is important to match the sizes of the pilot survey and study area. For example, if the proposal is to establish a number of experimental plots within a site (i.e., area of apparent homogeneity), then pilot sampling should be directed to assessing both the density and the variation
PATCHY DISTRIBUTIONS: OPTIMISING SAMPLE SIZE
] 03
in density of populations both at the site size scale and the plot size scale. Then decisions can be made as to whether sampling can best be optimised: by collecting one sample from a number of replicate plots; or, if within-plot variation is high, by collecting many samples from a few replicate treatment plots. The decision to collect all the samples, for one experimental level, from one plot only (i.e., not to use replicate plots) should only be done after careful consideration of the hypothesis to be tested and the inferences which can be made. Hurlbert (1984) lists a number of problems with this approach; particularly that it can only demonstrate a difference between locations as opposed to demonstrating an effect, and that the actual experimental Type 1 error risk oz will be greater than that selected a priori: the latter appears to be a problem only if the sample size is increased during a study. Stewart-Oaten et al. (1986) and Millard and Lettenmaier (1986) dispute some of his criticisms, while Winer (1971) suggests that the advantages accrued by using replicate plots is dependent on the magnitude of replicate variation relative to treatment and experimental effects. Once the design of the study (i.e., sample size and replicate plots) has been decided upon, trade-offs between the number of replicate and number of levels/sites can be made. For example, for a study with a total m a s s of 21, the preferred number of independent samples for each level/site of the study may be seven, thus allowing only three experimental levels/sites. However, the increase in ecological information gained by having four levels/sites may outweigh the resulting decrease in power when the sample size drops to five. As Bros and Cowell (1987) point out, the ability to make this trade-off between power of statistical analyses and the possible increase in ecological information from an increased number of experimental levels/sites is one of the most useful aspects of this type of method for estimating sample size. Krebs (1989 p. 197) comments on the wastage in ecological research through poor study designs. This may occur either when insufficient information is gathered to enable patterns to be confidently identified or, more rarely, when too much sampling provides more accurate estimates than are needed. Where raw or transformed data conform to known distributions, exact calculations of detectable differences can be used to determine samples sizes. However, as we illustrate, marine macrobenthic populations, even when intensively sampled, frequently do not conform to normality (even after data transformations by common methods). Plots of standard errors, calculated from randomised draws performed on data collected by a pilot survey, versus sample size, provide a useful graphical way of helping to make decisions balancing information gained and cost.
Acknowledgements Field work for the examples in this paper was funded by the Regional Water Board, Auckland Regional Authority and we are grateful for their permission to publish this interpretation of their data. Thanks also go to D.S. Roper who helped with the
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field c o l l e c t i o n a n d identification o f species and to W i l l i a m Bros for his constructive criticism on an earlier draft.
References Bros, W.E. and Cowell, B.C.: 1987, 'A Technique for Optimizing Sample Size (replication)', J. Exp. Mar. Biol. Ecol. 114, 63-71. Botton, M.L.: 1984, 'Spatial Distribution of Three Species of Bivalves on an Intertidal Flat: The Interaction of Life-History Strategy with Predation and Disturbance', Veliger 26, 282-287. Commito, J.A.: 1982, 'The Importance of Predation by Infaunal Polychaetes in Controlling the Structure of a Soft-Bottom Community in Maine, USA', Mar. Biol. 68, 77-81. Conover, W.J.: 1980, Practical Nonparametric Statistics, Wiley, New York, 2nd edition, 493 pp. Edgington, E.S.: 1987, Randomization tests, Marcel Dekker Inc., New York, 2nd edition, 341 pp. Elliott, J.M.: 1983, 'Some Methods for the Statistical Analysis of Samples of Benthic Invertebrates', Freshwat. Biol. Ass. Sci. Publ. 25, 1-167. Elmgren, R., Ankar, S., Marteleur, B. and Ejdung, G.: 1986, 'Adult Interference with Postlarvae in Soft Sediments: The Pontoporia-Macoma Example', Ecology 67, 827-836. Fairweather, P.G.: 1988, 'Predation Can Increase Variability in the Abundance of Prey on Seashores', Oikos 53, 87-92. Green, R.H.: 1979, Sampling Design and Statistical Methods for Environmental Biologists, Wiley, Toronto, 257 pp. Hines, A.H., Haddon, A.M. and Wiechert, L.A.: 1990, 'Guild Structure and Foraging Impact of Blue Crabs and Epibenthic Fish in a Subestuary of Chesapeake Bay', Mar. Ecol. Prog. Ser. 67, 105-126. Hurlbert, S.H.: 1984, 'Pseudoreplication and the Design of Ecological Field Experiments', Ecol. Monogr. 54, 187-211. Jones, M.L.: 1978, 'Three New Species of Magelona (Annelida, Ploychaeta) and a Rediscription of Magelona pitelkai Hartman', Proc. Biol. Soc. Washington 91, 336-363. Jumars, P.A. and Eckman, J.E.: 1983, 'Spatial Structure within Deep-Sea Benthic Communities', in: Grove, G.T. (Ed.), The Sea. Deep-Sea Biology, Vol. 8, Wiley, New York, pp. 399-451. Krebs, C.J.: 1989, Ecological Methodology, Harper & Row, New York, USA, 654 pp. Legendre, P., Oden, N.L., Sokal, R.R., Vaudor, A. and Kim, J.: 1990, 'Approximate Analysis of Variance of Spatially Autocorrelated Regional Data', J. Class. 7, 53-75. McArdle, B.H. and Blackwell, R.G.: 1989, 'Measurement of Density Variability in the Bivalve Chione stutchburyi Using Spatial Autocorrelation', Mar. Ecol. Prog. Ser. 52, 245-252. Millard, S.P. and Lettenmaler, D.P.: 1986, 'Optimal Design of Biological Sampling Programs Using the Analysis of Variance', Estuarine Coastal ShelfSci. 22, 637-656. Noreen, E.W.: 1989, Computer Intensive Methods for Testing Hypotheses, Wiley, New York, 229 pp. Olafsson, E. and Morre, C.G.: 1990, 'Control of Meiobenthic Abundance of Macroepifauna in a Subtidal Muddy Habitat', Mar. Ecol. Prog. Ser. 65, 241-249. Pridmore, R.D., Thrush, S.E, Hewitt, J.E. and Roper, D.S.: 1990, 'Macrobenthic Community Composition of Six Intertidal Sandflats in Manukau Harbour, New Zealand', N.Z.J. Mar. Freshwat. Res. 24, 81-96. SAS: 1985, SAS Users Guide: Statistics, Version 5 Edition, SAS Institute Inc., Cary, North Carolina, 956 pp. Schneider, D.: 1978, 'Equalisation of Prey Numbers of Migratory Shorebirds', Nature 271, 353-354. Sokal, R.R. and Rohlf, F.J.: 1981, Biometry, Freeman & Co., New York, 859 pp. Stewart-Oaten, A., Murdoch, W.W. and Parker, K.R.: 1986, 'Environmental Impact Assessment: "Pseudoreplication" in time?', Ecology 67, 929-940. Sun, B. and Fleeger, J.W.: 1991, 'Spatial and Temporal Patterns of Dispersion in Meiobenthic Copepods', Mar. Ecol. Prog. Ser. 71, 1-11. Thrush, S.E: 1991, 'Spatial Patterns in Soft-Bottom Communities', Trends Ecol. Evol. 6, 75-79. Thrush, S.E, Hewitt, J.E. and Pridmore, R.D.: 1989, 'Patterns in the Spatial Arrangement of Polychaetes and Bivalves in Intertidal Sandflats', Mar. Biol. 102, 529-535.
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Van Blaricom, G.R.: 1982, 'Experimental Analysis of Structural Regulation in a Marine Sand Community Exposed to Ocean Swell', Ecol. Monogr. 52, 283-305. Winer, B.J.: 1971, Statistical Principles in Experimental Design, McGraw-Hill Book Co., New York, 907 pp. Zar, J.H.: 1981, 'Power of Statistical Testing: Hypotheses About Means', Am. Lab. 13, 102-107. Zar, J.H.: 1984, BiostatisticalAnalyses, Prentice-Hall, New Jersey, 2nd ed., 718 pp.
