by
H.F. Henderson
Fisheries Department
FAO, Rome
It is quite clear that unless examples of the fish species or stocks under study can be caught, counted, or otherwise observed, little study can take place. However, catching fish should not be confused with sampling fish, and it is the latter rather than the former which is the real concern of the readers for which this book is intended. The essential difference between catching and sampling arises out of variability - the variability of the efficiency of the methods used to catch and observe the fish and the variability of the fish themselves both in structure and in behaviour. The difference between catching and sampling also arises from the wish of the observer to extrapolate what he finds in the catch to the corresponding characteristics of the “population” of fish which his “sample” is intended to represent. Sampling, and the statistical analysis which goes with it, are concerned primarily with this extrapolation.
It is assumed that the investigator knows and has clearly defined his objectives when formulating a sampling programme. An essential feature of scientific experiment and statistical inference is the formulation of specific hypothesis prior to collection of data. While it is unlikely that the data would be any different if the hypotheses are formulated a posteriori, but experiments which are subjected to statistical test only when the result is not obvious, or to test only “interesting” experiments would require quite different probability assumptions than those on which statistical arguments are usually based. This is not to suggest that formal experiments are the only ones worth doing. Almost all formal experiments are preceded by what might be called exploratory studies or experiments. The objectives of such studies should include the selection and careful formulation of formal hypotheses for more vigorous testing. They should also include refinement and standardization of the sampling method, and, as will be noted later, the estimation of the variability likely to be encountered during more formal testing.
This chapter is mainly concerned with guidelines or hints relating to the design of a sampling programme suitable for the objectives at hand. There is no attempt to describe the details of statistical procedures or the calculations to be made. For these, the reader should consult standard elementary texts (see for example the basic ideas of sampling which are treated in Chapter 2). But because the intent in this chapter is to assist the fishery investigator in the proper selection of statistical procedures, a discussion of some of the more troubling problems of elementary statistical and sampling terminology are also included here.
One source of confusion in the statistical analysis of fish catches is to be found in subtle differences between the biological concept of a population and the statistical concept. The biologist is most comfortable with an implicit definition of a population based on a description of the attributes of members, for example, all fish belonging to a particular species occurring in a given lake. The statistician, on the other hand, prefers an explicit definition based on a list of members. In the former case, the decision as to whether or not an individual (fish) taken from a sample is a member of the defined population can be taken a posteriori, that is by examining the fish after it is caught. The statistician would be happier, however, if the fish (that particular fish) had been labelled previously as a member of the population. While the latter a priori approach to population identification can be taken in some practical work, as in some kinds of marking experiments or in fish culture, it is meaningless in others, as in estimating the number of fish in a lake!
Normally there is little trouble over these differences in view point. The reader should beware, however, the temptation to let the samples that are taken define the population. In carrying out a monitoring programme in a lake with gill-nets, for example, it usually is not known how well the fish occupying various difference depths or regions of the lake are represented in the catches. The biologist, being practical, is apt to accept that the “population”, which inferences are to be made is whatever population is effectively sampled by his set of gill-nets, set in his way and at his stations. This approach may be a useful way to studying relative changes in the lake over time. The biologist should recognize, however, that “population” has little statistical meaning in such a context. The population so defined is logically identical with the samples!
A related problem sometimes arises in the use of acoustic gear to estimate the biomass of fish populations (see Chapter 9). As the acoustic method is relatively non-selective, responding to very small as well as to large fish, the population estimated may include very young as well as adult fish. If the real population estimated is not recognized, inferences made about the harvestable biomass, may be in considerable error.
Other problems arise in understanding variation. Variation is a fact of life. It is a fact of life in the sense that it cannot be stopped in either fish or in observers of fish. It is also a fact of life in the sense that living things, as opposed to non-living, have evolved to cope and to profit by it and, indeed, probably require it in a theoretical sense. Whereas the physical scientist tends to equate variation in his observations with the process of measurement, and thus tends to talk about error instead of variation, the biological scientist tends to assign variation in observations to the material or process studied, sometimes unaware or at least unconcerned that the variations may partly be errors of abservation. It is perhaps unfortunate that statistics (but not necessarily statisticians) confuses this distinction. Variation of the observed, and error of the observer, are lumped together in the measure called Variance1. In practice, the two levels of variances are difficult to separate, but they must both be recognized in interpreting results of sampling.
The most important concern of this manual is with the estimation of the number or weight of fish available to a fishery, e.g. with population size. When sampling the population to derive such an estimate, variation in the numbers of fish observed in each sample can arise from:
Variation in the size of the population from one sampling time to the next
Variation in the behaviour of the fish from one sampling time to the next
Variation in the efficiency of the catching gear from one time to the next
Variation in the distribution of the fish from one sampling place to the next
Variation in the distribution of the fish from one sampling time to the next
Variation in the behaviour of the fish from one sampling place to the next
Variation in the efficiency of the sampling gear from one sampling place to the next
Errors in counting the fish in the sample
Errors in recording the numbers counted
Errors in reading the number recorded
etc.
It is worth noting, by reflection upon the elements of the above list, that the most probable sources of large variability in sample counts, and hence the greatest uncertainty in the population estimate eventually obtained, are usually to be found in the “variations” rather than the “errors”. Thus it would appear that the sampling method is rather less important in determining the uncertainty of population estimates than the variations that occur in the population itself. The sampling method is not unimportant, however, especially if there is thought to be some order or organization to the spatial distribution of the population of fish to be estimated. Schooling or other sources of patchiness (see Section 12.5.3) can produce different sorts of results according to the plan of sampling and the size and kind of gear used.
A number of kinds of problems should be considered when selecting a programme. One set of problems, of course, concerns the mechanical or logistic efficiency of the sampling, in other words the design and cost considerations of obtaining a sample of suitable size. These have been discussed in other chapters. Also important, however, are the questions concerning how many samples to take, when they should be taken, and where, in order to achieve the objectives of the programme.
Biologists, as well as other users of statistical tools, often pay little attention to the selection of appropriate confidence levels of tests of hypotheses, assuming that a 95 percent level is “good” and a 99 percent even “better”. The practical worker should remember that the penalty for failing to declare a difference, should it be real, may sometimes be as great as the penalty for declaring a difference that is false. If a simple change in gear shows the possibility (say with a 60 percent probability) of a 10 percent increase in catch in an inexpensive trial (experiment), it may not be worth a large increase in cost of experimentation to demonstrate that there is in fact a 95 percent probability of such an increase. This can be more readily demonstrated in actual use. Whether or not to take a particular management decision is often not as critical a choice as the one taken in regard to acceptance rejection of a finding as a base for further scientific work. While it is no doubt desirable to design studies to be as scientifically trustworthy as possible, cost limitations do have to be considered. The proof of a conclusion arrived at somewhat tentatively, may better be obtained in the course of application. Some types of fishing gear experiments are apt to be of this sort.
Similarly to estimate the biomass of fish in an unfished lake in order to assess its potential for the development of a fishery, an order-of-magnitude (factor of ten) estimate may be sufficient, in the first instance, until the fishery actually develops. In most cases, very crude sampling would suffice, provided the observer had some experience in judging productivity of that particular type of water body. When an active fishery is present, the fishery itself may be used to refine the estimate. But if one is required to estimate the change in population resulting from heating a lake, or the addition of some pollutant, it may be important to reliably detect small changes even at the start, especially if cumulative long-term effects are important.
In order to plan sampling programmes which will achieve a desired level of precision, it is necessary to obtain an estimate of probable variance beforehand. As a rule-of-thumb, the variance of the numbers and weight of fish caught by most kinds of fishing gear is the same as, or rather larger than, the mean catch. In fact, this tendancy for the variance to be proportional to the mean value seems characteristics of population sampling and has been reported for the concentration of contaminants in water (Eberhart, 1975). But this characteristic invalidates the assumption inherent in a number of statistical tests that the mean and variance are independent. For this reason, such data should be transformed. The logarithmic transformation (for an example, see Chapter 3) is routinely used for this purpose. The observed numbers or weights are replaced by their logarithms before calculating means and variances. If there are zero observations, the number 1 should be added to all observations2 before taking the logarithm.
There is evidence that the standard deviation in the linear dimension of vertebrates, for a group of individuals of identical age, is from 5–10 percent of the mean for the group (Simpson, Roe and Lewontin, 1960; Ricker, 1958 & 1959). Such information is useful, for example, in deciding whether a particular mode in a frequency distribution of lengths of fish should be accepted as a group of a single age.
Sometimes, the probability model used as basis for calculating confidence limits, provides an estimate of the expected variance. Marking experiments for example, are usually based on binomial or Poisson probabilities. In binomial distributions, the variance is a definite fraction of the mean 3, while in the Poisson the variance is equal to the mean.
3 The fraction is equal to the probability of failure of a desired event to occur in a single trial.
Previous experiments and observations can also provide estimates of expected variance. Lacking other evidence, exploratory experiments (Section 12.1) may be undertaken to decide how much effort will be needed to achieve a given level of precision. If the variance of n observations is v, then the expected variance of N observations from the same population is n.v/N, that is, the variance is inversely proportional to the number of observations.
Calculation of confidence limits or other probability statements about the results of sampling may help very little or even be misleading if the biological facts do not match the assumptions upon which the probability statements are based. Balon (1974), for example, used cove sampling with rotenone to estimate the standing stock and production of fish in Lake Kariba, Zambia. For practical reasons the method did not sample effectively in deep water. Evidence from echosounding indicated few fish in the deeper water. He therefore took the portion of the lake shallower than 20 metres as the target area which confined the population, considering the estimated biomass for this shallow portion to be a fair estimate for the lake as a whole. On the other hand, he noted again from echosounding, that the fish were more concentrated in coves and bays, so that his samples from coves may have significantly over-estimated the population for all areas less than 20 m deep. This illustrates how a proper consideration of the target population (Section 12.2) and of stratification (Section 12.5.2) is essential to the overall sense of the results.
The theory of sampling is based on the assumption that replicate samples provide independent estimates of the desired population characteristic. This may not be true for many reasons. It is generally accepted that the members of a school of fish tend to be more alike in size than would be the case among an arbitrary group of individuals in the population. If so, then a sample of 1,000 fish from, say, a purse seine haul which has captured members of two schools should not be expected to be equivalent to a random sample of 1,000 fish. A sample of a schooling fish will probably not give the same precision of estimate of, say, the mean size of individuals in the population, as the same sized sample of a catch of a non-schooling species. This problem is thought to be a main source of difficulty in obtaining stable estimates of the size distribution of the clupeid Stolothrissa tanganyikae from Lake Tanganyika. When a population is naturally divided into relatively more homogeneous groups, such as schools, or localized sub-populations, it is a more efficient use of each measurement (of length, say) to take a few individuals from each of the groups than a large number from a few of them. See the discussion of stratified sampling (Sections 2.3.2, 12.5.2).
The simple “statistical” populations which are modelled in fundamental probability distributions such as the normal, the Poisson. the binomial, etc. assume “homogeneity”. A population can be considered homogeneous if the same probability statement is as appropriate for any subgroup of the population that might be selected as for the population as a whole. It is this assumption that makes it possible to regard a sample as representative of the targeted population. Consider the measurement of the length-weight relationship for a population of some species in some lakes. Do samples of females exhibit the identical relationship and the same variance as those of males? If so, the population would seem to be homogeneous, with respect to the length-weight relationship and sex can be disregarded in its application. If not, the population would seem not to be homogeneous and to be really two populations. If, in practice, the sexes are easy to distinguish, it will be easiest to treat these two populations as separate and distinct, both in the sampling and in the calculation. But if the sexes are not easily distinguished other strategems may be needed.
It is worth noting that homogeneity could exist in the length-weight relationship even though the means and variances of length alone or weights alone differed between the sexes. This is another way in which the biologists “population” and the statisticians “population” differ! The statistician is primarily concerned with sets of measurements or observations, these can easily be kept separate from other sets of observation. The biologist is primarily concerned with the individual animals themselves, any group of which are certainly not likely to be homogeneous in the ordinary, non-statistical sense.
The fact that some properties of a set of samples may show homogeneity while others do not is important to problems of selectivity in sampling. In the example quoted above, where the length-weight relationship is studied and there are differences between the sizes of males and females in the population, that selectivity sampling gear favours some sizes over others, will yield observations that are biased unless care is taken.
Some sampling methods used for studies of fish are more selective than others, but all methods are somewhat selective. Further, non-representative samples may result from other sources than the properties of the gear. The investigator may exert an unconcious selectivity in taking sub-samples. Or different behaviour among the individual fish sampled may make some individuals more available or vulnerable to the gear than others (see Hamley, 1975). It is perhaps useful to distinguish between selectivity based on attributes (species, behavioural types, etc.), and that based on more easily quantifiable dimensions (length, weight, etc.). While it is difficult to establish corrections for size selectivity of fish sampling gear (Hamley and Regier, 1973; Hamley, 1975), it is still more difficult to satisfactorily correct, say the ratio of the abundances of two different species for differing selectivity. Short of a complete enumeration of the population, the best that can usually be done is to base such estimates on a composite sample of several kinds of gears, on the assumption that each will favour rather different species. This will give a qualitatively truer picture of species composition than a single sample of the same size.
Given a size selective sampling method, estimates of most characteristics of a population will be biased unless: (a) the characteristic or variable is closely correlated with the length of the fish, or (b) the characteristic or variable does not depend at all on length, i.e., is completely uncorrelated with length. The latter is rare but may be approximately the case over a large part of the life cycle for some variables (e.g., number of fin rays or other meristic characters). But for most characteristics, size-selective sampling will give biased estimates of the population characteristic, that can only be corrected if the degree of selectivity can be quantified. The appropriate strategy is to stratify the sample catches onto sub-samples (see Section 12.5.2) according to size.
When there is lack of homogeneity in the population the appropriate procedure is to break down the population into sub-groups that are more homogeneous (and less variable) than the population. Ricker (1975) points out the “general rule that (biological) experiments or observations which seem simple and straightforward will prove to have important complications when analysed carefully - complications that stem from the complexity and variability of the living organism, and from the changes that take place in it, continuously, from birth to death.” He suggests that any body of data should first be divided into various categories (e.g., size, age, sex) for study, and then on the basis of time (e.g. days, seasons, etc.), in order to identify such complications.
It has been shown that the standard deviation (and variance) of overall population values is minimal, for a given total number of samples, when the samples are distributed among the strata in proportion to the standard deviations within the strata. Commonly, estimates of population abundance and similar quantities are stratified by geographic subdivision (statistical areas), but such divisions may not be as efficient as depth, or other ecologically effective criteria. In estimating such quantities as egg production by the adult stocks, stratification by size of individual is an obvious way of improving the precision of the overall estimate. Even where there is no a priori reason to suspect that stratification would improve the efficiency of sampling, breaking down the population into sub-sets as suggested above, is a useful precaution unless the system under study is very well known.
Again, it is clear from what has been said in previous sections that the overall process of sampling should be regarded as one of gradual improvement in both design and results. Only preliminary studies can provide evidence that criteria selected for stratification are satisfactory and then also provide estimates of the relative standard deviation (or variance) among the selected strata. The investigators' experience in similar problems or experience recorded in the literature can, however, be of great help in deciding what criteria to use and how to distribute sampling effort. For example, in estimating catch by fishermen in lakes, Bazigos (1974) has found stratification by geographical area and size of village or landing site, and a weighting of sampling effort by numbers of fishermen within each stratum to be satisfactory and efficient. For fish stock estimates, geographic stratification and depth of water are usually effective, while weighting the sampling effort by area is apt to give efficient results.
In dealing with selectivity in the sampling method (Section 12.5.1), stratifying the sample catches (sub-sampling) with respect to the selecting factor (size and sex are usually most important), is normally advisable. In this case, the size of the sub-samples (number of individuals examined) should, for best efficiency, be approximately proportional to the standard deviation of the group4. But considerable care must be taken to select the sub-sample representatively to avoid the bias resulting from the investigation selectivity. Because of this latter problem, it is often more practical to stratify the sample for purposes of calculation but not to sub-sample. Because of the variety of circumstances and purposes for which stratified sampling is appropriate, the reader should refer to the various practical statistical texts for details of the calculation of population values from various stratified sampling designs (e.g. Simpson, Roe and Lewontin, 1960; Raj, 1972).
One of the most troublesome aspects of sampling populations of fish and other organisms, especially in respect to their number or total weight is the tendency for individuals to arrange themselves in space in relation to each other or to features of the environment, rather than at random. Similarly the timing of biological events is often coordinated among individuals either by signals between them or through joint response to signals in their environment. Thus the normal assumption of independence among sample observations, is violated. The problem can be dealt with in various ways depending on the objectives of the study.
Usually the problem will be to minimize the effects of patchiness or the clustering of individuals on the variance of the desired estimates. One way to do so is to carefully randomize the sampling locations. While straightforward in principle, there are many practical limitations. Some sampling gears can only be operated at certain depths or certain kinds of habitats. The work and travel costs involved are apt to be higher than for regular sampling, though if the time of sampling does not need to be randomized, travel time may be reduced by planning an efficient route among randomly preselected locations. There are probably very few fishery studies in which a strictly randomized selection of sampling sites has been achieved. Recalling the discussion in Section 12.2, a clear definition of the domain of the population, the region within which sampling is permitted, is essential to randomization but is often not a trivial problem.
There are other reasons for adopting a more determinate or regular sampling scheme. A completely random sampling design will usually not be the most efficient. That is, other designs can usually be found which would reduce the overall variance for a given amount of sampling effort. As the reader should now expect, stratification is a probable solution. But before considering the problem further, factors of relative scale of patchiness and of gear effectiveness need to be examined.
Consider a population of schooling fish, each keeping at a somewhat definite distance from his neighbours. Let these schools be somewhat randomly distributed in a region of upwelling where the food organisms are abundant. Within a school, successive or independent catches with gear that samples a fraction of the school at a time (without disrupting the spacing) will tend to be rather uniform the number of fish caught. With gear that is large with respect to the schools but small compared to upwelling area, the catches will seem to vary in size randomly, while at a still larger scale, the patchy distribution of the food organisms (of upwelling regions) will be apparent. Thus, in general, the best sampling strategy to adopt will depend on the relationship between the scale of patches of the fish, the scale of the sampling gear, and the scale of the domain of the target population as a whole.
If the size of the sampling gear is small relative to the size of “patches”, decreasing the size of each sample and increasing the number of samples should reduce the overall variance. Owing to the usual proportionality of mean and variance in estimating the number of fish caught in a sample (Section 12.4.2), estimates of population size can as well be made with small samples as with large, provided there is not a preponderance of zero catches.
As long as the investigator wishes to make inferences about the population as a whole, and the pattern of arrangement of the population is assumed to be constant over time (even though individuals may move about), special probability models, such as the negative binomial distribution, may be found which fit the data well and can be used to calculate more reliable confidence intervals.
The investigator may, on the other hand, wish to describe the patchiness or other kind of organization of the fishes in the habitat. To do so quantitatively will require that the sampling be stratified by lake area or transect interval (or if organization in time is a problem, in regular time intervals). In a patchy distribution variance will be maximized when the strata are of the same order of size as the patches (Kershaw, 1964). Computing variances within and among strata, and among strata combined in pairs or in larger combinations, is one effective way of describing a patchy distribution. The reader who is interested in these problems should consult Kershaw (1964); Simpson, Roe and Lewontin (1960); and for methods of calculation, texts on analysis of variance.
A fundamental premise of this chapter is that all fishery investigators should have a clear understanding of the philosophy, assumptions and methods of statistical inference. It is, of course, not possible to achieve such an understanding by reading these few pages. Fortunately there are quite a number of good texts available for those who wish either to pursue a general study of the statistics of sampling or who need guidance on specific matters. Many of these are listed in the bibliography. But, while the basic methods are quite straightforward, sampling programmes that are really efficient statistically as well as economic are not easy to achieve, and there are also many pitfalls and problems in reliable analysis of the results, only a few of which have been discussed here. It is thus a matter of considerable importance that the assistance or advice of professional statisticians be solicited at the beginning of the planning for major sampling programmes. It should be clear from the discussion given, that any savings in sampling effort and many worthwhile gains in quality of the results, must be achieved, if they are to be achieved at all, before sampling begins.
It is also rather important that fishery workers recognize that the large variability in the size of the catches with all kinds of fishing gear is a rather fundamental characteristic of their problems with sampling, and, most importantly, that it cannot be cured by improving the sampling gears though some improvement may be achieved this way. It is a property of the fish populations themselves, and particularly of the way in which individuals distribute themselves in their environment. The attainment of substantially better precision can only be gained through more intensive sampling or improved designs of sampling programmes.
Bagenal, T.B., 1972 The variability of the catch from gill nets set for pike, Esox lucius. Freshwat.Biol., 2(1):77–82
Balon, E.K., 1972 Possible fish stock size assessment and available production survey as developed on Lake Kariba. Afr.J.Trop.Hydrobiol.Fish., 2(1):45–73
Bazigos, G.P., 1974 Applied fishery statistics. FAO Fish.Tech.Pap., (135):165 p. (Issued also in French and Spanish)
Bazigos, G.P., 1974a The design of fisheries statistical surveys - inland waters. FAO Fish. Tech.Pap., (133):122 p. (Issued also in French and Spanish)
Bazigos, G.P., 1976 The design of fisheries statistical surveys - inland waters. Populations in non-random order, sampling methods for echo surveys, double sampling. FAO Fish.Tech.Pap., (133)Suppl.1:46 p.
Eberhart, L.L., 1975 Some methodology for appraising contaminants in aquatic systems. J. Fish.Res.Board Can., 32(10):1852-9
Elliot, J.M., 1971 Some methods for the statistical analysis of samples of benthic invertebrates. Sci.Publ.Freshwat.Biol.Assoc.U.K., (25):144 p.
Hamley, J.M., 1975 Review of gill net selectivity. J.Fish.Res.Board Can., 32(11):1943-69
Hamley, J.M. and H.A. Regier, 1973 Direct estimates of gill net selectivity to walleye (Stizostechon vitreum vitreum). J.Fish.Res.Board Can., 30(6):817-30
Kershaw, K.A., 1964 Quantitation and dynamic ecology. London, Edward Arnold, 183 p.
Raj, D., 1972 The design of sample surveys. New York, McGraw-Hill, 390 p.
Ricker, W.E., 1969 Effects of size-selective mortality and sampling bias on estimates of growth, mortality, production and yield. J.Fish.Res.Board Can., 26(3):479–541
Ricker, W.E., 1975 Computation and interpretation of biological statistics of fish populations. Bull.Fish.Board Can., (191) :382 p.
Simpson, G.G., A. Roe and R.C. Lewontin, 1960 Quantitative zoology. New York, Harcourt, Brace and World, Inc., 440 p. Rev.ed.
Sokal, R.R. and F.J. Rohif, 1969 Biometry. San Francisco, Freeman, 776 p.
Stuart, A., 1962 Basic ideas of scientific sampling. London, Griffin, 99 p.
Tomlinson, P.K., 1971 Some sampling problems in fishery work. Biometrics, 27:631-41
Williamson, M., 1972 The analysis of biological populations. London, Edward Arnold, 180 p.
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