Athree-dimensional representation of the beach described in the text, showing ..... ex- periment. This effort is seldom made. Most workers rely on a "feel" for a ...
8. Sampling Strategies David Thistle and John W. Fleeger Meiofaunal ecology has only recently made the transition from a qualitative to a quantitative science. As a result, many workers in the field have little training in statistics and are at a disadvantage when faced with designing sampling programs (Coull and Palmer, 1984). Below we introduce what we feel to be fundamental issues in sampling design. We also describe some statistical procedures in order to point out problems that arise in analysis. We do not attempt to duplicate the coverage of a formal text. Rather, we concentrate on principles critical to the collection and interpretation of sample data. In our presentation, we have attempted to minimize use of the vocabulary of statistics and to confine ourselves to relatively simple situations. In some of our comments, we have deliberately chosen to be conservative. Our presentation is not intended to substitute for the many textbooks on this subject, and the reader is encouraged to consul t such resources for a more complete understanding.
tion. (We use "population" in the statistical sense, to mean a group of objects or events (Tate and Clelland, 1957), not in the biological sense.) Only if the samples meet this criterion do they allow logical inferences to be made about the POPUlation from which they are drawn. Therefore, the population must be carefully specified before the sampling design is planned. In some circumstances, the population will be easy to define. For example, all the nematodes between mean low water and mean high water (i.e., in the eulittoral zone) on a beach is a well-defined population in the statistical sense (Figure 8.1). Properly taken samples would allow inferences about the nematodes of the eulittoral region of that beach. However, they would not allow statistical inferences about the nematodes of all beaches, of a similar adjacent beach, or even of the non-eulittoral portion of the beach being studied. Stated more abstractly, specification of the population defines the universe that must be sampled and the universe within which inferences are strictly valid. Although extrapolation of results to other, similar beaches may be ecologically useful, it is still extrapolation. In practice, specifying a population is not always simple. In some circumstances, the question being asked or logistic considerations may make it
Concepts at the Start Statistical analyses are based on the assumption that the samples have been collected in such a way that they represent, without bias, a defined popula-
Figure 8.1.--A three-dimensional representation of the beach described in the text, showing the rectangular sampling area in stipple. 126
CHAPTER 8: SAMPLING STRATEGIES
impractical to study a naturally bounded population. For example, a beach may extend along shore for kilometers without a natural break. It is likely to be impractical to study the entire beach. A region of any convenient shape can be selected for study and arbitrarily bounded. In these circumstances, although the population is bounded arbitrarily, it still exists. It is the population subject to sampling and to which inferences strictly apply.
•
•
• •
••
•
•
•
•
Figure 8.2.-A diagram of the sampling area as seen from above. The locations of randomly placed samples are shown. The placement was achieved by imposition of a Cartesian coordinate system on the beach with the along-shore axis (x axis) running from 0 to 100 and the axis perpendicular to shore (y axis) running from 0 to 50. Pairs of numbers were drawn from a random-number table and used as coordinates in the placement of samples. The procedure is as follows (see also Elliott, 1977). From the circumstances, one can decide the number of digits that are needed to make up a coordinate. The random-number table is entered at the beginning or where one previously left off (Snedecor and Cochran, 1967), and successive entries (using as many digits as necessary) are examined. If a value falls out of the range of values that could serve as coordinates (e.g., is larger than 50 for the y-axis), then the entry is not used. Successive entries are examined until the required number of coordinates is obtained.
The investigator takes samples to obtain unbiased information about a population. The way to accomplish this goal is to take samples randomly from the population (Tate and Clelland, 1957; Elliott, 1977). In the simplest case, the samples are taken such that every unit of the population has an equal opportunity of being selected (Snedecor and Cochran, 1967). To assure this condition, a mechanical randomization procedure, such as the use of a random-number table (see Elliott, 1977), must be involved in the selection of locations to be sampled. For example, if we wished to estimate the average (mean) density of nematodes living on the eulittoral beach described above, we would have to sample at randomly selected locations. This sample placement could be achieved by imposition of a coordinate system on the eulittoral region (the population of interest) (Figure 8.2); x and y coordinates could be selected by means of a random-number table to specify the location of each sample (see Elliott, 1977; Thistle, 1980; Ravenel and Thistle, 1981).
iII.
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.I__ I.-__ ---
127
Survey Sampling A variety of circumstances can arise in which an investigator would wish simply to estimate a parameter of a population. In our example, the density of nematodes might be critical in estimating the biological productivity of the beach. No test of a hypothesis is involved; merely a description is required, but a population must still be identified and randomly sampled . Simple Random Sampling.--The scheme of sample placement described above for estimating nematode abundance on a beach by random selection of sample locations is an example of simple random sampling . It has the virtue of being easy to use, but it works best when the range of values of the parameter of interest (in our example nematode density) is small. In the example, if the number of nematodes per unit area were not too variable, then, even if the number of samples were small, a simple random sampling scheme would give an estimate of the mean density nearly equal to the true value for the population. However, if nematode density were markedly variable, then simple random sampling might do a poor job. Usually meiofaunal workers analyze small numbers of samples (5-20). If these are placed in a simple random manner, they may not sample the highs and lows of the distribution of nematode densities on the beach in their true proportions. The use of simple random sampling is legitimate in that every location on the beach has an equal probability of being sampled, but, in the face of much variability (as is usually the case with variables of interest to meiofauna ecologists (Fenchel, 1978», it will estimate a mean value close to the true value only if many samples are taken. Stratified Random Sampling. - - An alternative sampling scheme that works better when density (or another variable of interest) is markedly variable is stratified random sampling. In this scheme (Figure 8.3), the population is divided into regions (strata) before sampling. Because the scheme imposes a regularity on the sample coverage, the irregularities in the distribution of the parameter being measured (i.e., nematode density) are more likely to be sampled in their true proportions by a small number of samples. To present this approach, we chose an example in which the strata are of equal area. They need not be, as discussed below. In our beach example, preliminary data may suggest that nematode concentrations are much greater in the offshore portion than in the onshore region. Suitable strata would be bands of equal onshore-offshore dimension that paralleled the shore; each would receive the same
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INTRODUCTION TO THE STUDY OF MEIOFAUNA
number of samples. When the strata are of equal area and samples are apportioned in this manner, the calculation of the mean and variance are unaffected by the stratification.
.. •• •
•
-------------------------------------------------------------------
••
•
•
Figure 8.3.-A diagram of the sampling area as in Figure 8.2. Strata are indicated by horizontal lines. Sample placement was random with the constraint that the same number (two in this case) of samples be placed in each stratum. The procedure follows that described in Figure 8.2. Additionally. the pairs of coordinates must be examined for cases in which a stratum would receive more than its share of samples. Such coordinates are discarded. as are coordinates that would place a sample exactly on the boundary between two strata.
Designs with strata of equal area constitute a special case of the general scheme of proportional stratified-random sampling, in which samples are assigned to each stratum in proportion to its area. When the strata are unequal, the mean can be calculated as the average of the resulting values, but the variance (and other statistics) require new formulae, which are given by Elliott (1977) and more completely by Snedecor and Cochran (1967). Finally, in survey sampling and elsewhere, there is a tendency to fix attention on the mean because it is the value that is used to represent the population and is used in statistical tests for differences. For both ecological and statistical reasons, however, the variability around the mean is also important. If the variability is small, then most samples would yield values similar to the mean, so the mean characterizes the population well. If the variability is large, few of the samples will yield values similar to the mean, so the mean does not characterize the population in the same way; in this case, it is much more a statistical abstraction. Populations with large variability are likely to be heterogeneous in biologically important ways. Statistically, high variability decreases the investigator's ability to discern differences among populations. For both reasons, it is necessary that estimates of variability accompany reports of mean values. Elliott (1977) describes the calculation and use of several measures of variability, including the variance, the standard deviation, the standard error, and confidence intervals.
Sampling in Support of Hypothesis
Testing
In testing a hypothesis, the investigator wishes to determine whether some treatment brings about a change in the population. Statistical techniques that allow such questions to be answered in a rigorous fashion require erection of a null hypothesis appropriate to the situation. The null hypothesis takes the form "the treatment has no effect." If such a null hypothesis is true, the values measured in samples drawn at random from the portion of the population that was exposed to the treatment and those drawn from the portion that was not exposed will be the same except for natural (that is, chance) variation. Because samples only estimate the true population values, the two sets of samples will differ even if the treatment has no effect. Statistical procedures allow one to examine the size of the observed difference and to state how probable it is that such a difference would arise if the null hypothesis were true. Generally, if a difference as large as or larger than the observed difference would arise by chance only 1 time in 20 or fewer (that is, at the 5% significance level), it is conventional to reject the null hypothesis of no difference and to accept the alternative hypothesis that the treatment has an effect. (See Green, 1979, for more discussion of the formation of appropriate null and alternative hypotheses.) Tests of hypotheses occur under two fundamentally different circumstances. In one case, the treatment is a condition that has been identified by an investigator; in the other, the treatment is applied by an experimenter. Tests for Differences among Conditions.--From theory, observations, or preliminary data, an ecologist may suspect that two (or more) regions differ on a variable of interest. We introduce techniques appropriate to testing such a supposition using an example (see also Cochran, 1983). Assume that preliminary samples suggest that nematode density varies with tide level on the eulittoral portion of a beach. To test this working hypothesis, the ecologist will compare densities in four along-shore bands (Figure 8.4). If the ecologist is confident, based on preliminary data, that the variation in nematode density within bands is small compared to the differences among the bands, then simple random sampling is appropriate. Equal numbers of samples are assigned to each band. Within each band, the samples are located at random, so that every location within a band has an equal probability of being sampled (Figure 8.4). It is not appropriate to take the samples from a single location or a small number of adjacent locations within each band if inferences are to be made about the differences
CHAPTER
between the bands over the entire eulittoral portion of the beach.
•
.
--------•
~----------------
• ------------------------•• ~------------------~----• • Figure 8.4.-A diagram of the beach as in Figure 8.2 but with the four regions that are the treatments in the example shown as horizontal bands. Samples are placed at random, equal numbers in each band.
The data from this simple random sampling scheme are appropriate for analysis by one-way analysis of variance. Analysis of variance is a statistical procedure that allows one to test for differences among treatments, for example, the differences in nematode density among the bands described above. Either a parametric or a nonparametric test can be used. (When only two treatments are to be compared, Student's T test can be used.) Parametric statistical procedures entail a number of assumptions (see Sokal and Rohlf, 1969), and these merit checking; statistics texts (e.g., Sokal and Rohlf, 1969, 1981) describe procedures. If the data do not satisfy the assumptions, the analysis may still be able to produce reliable tests (see Green, 1979; Underwood, 1981), or it may be possible to transform the data to meet them (see Sokal and Rohlf, 1969; Green, 1979; Underwood, 1981). These data could also be analyzed by means of nonparametric statistics. Nonparametric procedures differ from parametric in that they require no particular assumptions about the form of the population distributions (Tate and Clelland, 1957), making this approach particularly suitable for the type of data usually available to meiofaunal ecologists (i.e., small numbers of samples from populations with unknown distributions). The nonparametric equivalent of the one-way analysis of variance is the Kruskal-Wallis test (Sokal and Rohlf, 1969). The nonparametric equivalent of Student's T test is the rank test (Tate and Clelland, 1957), which is also called the Wilcoxon T test and the Mann-Whitney U test (Sokal and Rohlf, 1969). Meiofaunal ecologists usually wish to know which conditions (or treatments) differ from which others (in the example, which along-shore bands differ from which others in nematode density). The results from the analysis of variance (except in the two-
.,;:~"
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treatment case) will not provide this answer directly. If the meiofaunal ecologist has planned to test for a specific difference among treatments before the experiment is carried out, the procedures for a priori testing are appropriate (see Sokal and Rohlf, 1969, for methods). However, it is often the case that neither theory nor preliminary data are available to suggest specific comparisons of interest. In these circumstances, an a posteriori multiple-comparison procedure should be used to decide which treatments differ. For parametric analyses, a variety of procedures exists, many of which are given by Sokal and Rohlf (1969). For nonparametric analyses, the multiple comparisons procedure based on the Kruskal-Wallis test is available (Hollander and Wolfe, 1973). If both along-shore and onshore-offshore differences were of interest, a different sampling scheme would be needed, one in which there were two treatments. In the beach example, to assess along-shore differences, the investigator should divide the beach into equal bands perpendicular to the shore. The result is a grid of sampling units (Figure 8.5). Samples are located at random within each cell. This design is appropriate for two-way analysis of variance, a statistical procedure that allows the effects of two treatments to be tested simultaneously. Although a two-way analysis of variance can be done without replication, a good design will have two or more samples per cell in order to permit the calculation of the interaction between the two treatments. A discussion of interaction and other complexities that arise in the two-way analysis of variance is beyond the scope of this chapter. Pitfalls exist, however, and a thorough familiarity with the procedure should be achieved before this design is employed (see, for example, Sokal and Rohlf, 1969; Underwood, 1981).
-1-
1
.1.
.1
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Figure 8.5.--A diagram of the beach as in Figure 8.4 but with the five along-shore bands that constitute the second condition in the example indicated as vertical bands. Equal numbers of samples are placed at random within each sampling unit.
There are many phenomena
that can only be
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INTRODUCTION
TO THE STUDY OF MEIOFAUNA
studied by means of testing for differences among conditions, particularly those involving the impact of pollution on the environment However, because the conditions are chosen rather than applied by the investigator, results can be influenced by variables not part of the study that could contribute to the detected difference (Hurlbert, 1984). For example, if fuel oil from a spill washed half-way up the beach, in a subsequent impact study, any differences detected between the oiled lower part and the nonoiled upper part might be caused by the pollution, but they could be caused by any other differences that existed between the two regions. In our example, the mean density of nematodes on the beach decreases onshore. This difference between regions will exist together with the difference caused by the oil. So, although the oil might cause a major decrease in the density of nematodes, the effect might not be detected if the upper portion of the beach was used as an unaffected, comparison site because the reduction on the lower part of the beach might only be to densities approximating the ordinarily low density on the upper portion of the beach. Green (1979) and Hurlbert (1984) discuss this problem. Stewart-Qaten et al. (1986) provide an approach for cases when it is possible to sample before and after the condition of interest (e.g., the oil spill) occurs. Tests for Differences among Experimentally Applied Treatments.--A variety of criteria exist that guide and regulate the design of manipulative experiments. We present those we feel to be most important to meiofaunal ecologists by using an example. Imagine the following hypothetical experiment. On the beach described above, an experimenter wishes to determine the impact on nematode recolonization of disturbances of different sizes. Assume the investigator knows from preliminary work that raking a plot will create a nematode-free patch. The experiment planned has four treatments: unraked plots of 0.25 m2 and 1.0 m2 and raked plots of 0.25 m2 and 1.0 m2• Five plots of each type will be laid out, making a total of 20 plots. The population about which inferences are to be made is to be the entire eulittoral portion of the beach, so the plots must be located at random in this region to make each location equally likely to be included in the experiment. The plots must also be independent of each other; that is, observations made on one plot must not be influenced by proximity to another plot. For example, plots must be sufficiently far apart that raking a plot does not increase or decrease the availability of nematodes to colonize nearby plots. Further, the plots should not affect one another during the experiment. For example, if two 1.0-m2 raked plots were close together, the nematode
population in the surrounding sediment available for colonization could be decreased below that normally available while the plots are being recolonized. As a result, these plots could recover at a lower rate than they would have if they had been isolated (i.e., independent>. (Underwood, 1981, gives a more general account of this issue.) An actual determination of the minimum spacing required for independence would require a preliminary experiment. This effort is seldom made. Most workers rely on a "feel" for a minimum spacing and decide during the planning that, in assigning plot locations at random, if the random numbers drawn would' place a plot less than a specified minimum distance from a plot already located, then a new set of random coordinates will be drawn. Although this intuitive procedure is better than no consideration at all of the independence of the plots, it does not guarantee independence. The requirement for independence suggests that plots be located as far as possible from one another. However, if environmental conditions that can affect the outcome of the experiment (e.g., number of nematodes available for recolonization) are distributed with large differences in the study site, then the random placement of plots may not efficiently detect differences among treatments because of the added variation contributed by this background heterogeneity. Specifically, if variations in the density of nematodes are large, then the rate of recolonization of the plots could be affected by this variability as well as by that caused by differences in plot size. To minimize the effect, the experimenter can lay the plots out in a randomized complete-block arrangement. The motivation for the randomized completeblock design is to reduce the impact of background variability. In this procedure, "blocks" of experimental material are chosen that are expected to be relatively homogeneous. In most field situations, the expectation is that places close together will be more similar to each other than locations that are far apart, so a block is typically an area that is large enough to accommodate one plot of each treatment type. Each block must be located at random on the eulittoral portion of the beach, so that the beach remains the population. Block orientation should be randomized. However, if a strong gradient is known to exist, then blocks can be oriented in a consistent manner relative to that gradient so that trends in the population will affect each block similarly (Figure 8.6). Within each block, the treatments are assigned to plots at random. In the example, the two sizes of unraked plots and two sizes of raked plots are the treatments. One plot of each treatment will be present in each block. If 5 replicates of each plot are to be set up, 5 blocks
CHAPTER
will be created and located at random on the beach. Blocks should be separated sufficiently to be independent Given the known onshore-offshore gradient on beaches, the blocks are probably best oriented parallel to shore with the 4 treatment plots side by side separated by a distance sufficient to assure their independence. Within each block, the relative positions of the treatment plots should be randomized. If at the end of the experiment a single sample is taken at random from each plot, the data can be analyzed as a randomized, complete-blocks design (Sokal and Rohlf, 1969) for the parametric case. The Friedman rank-sums procedure (Hollander and Wolfe, 1973) is the equivalent nonparametric procedure. If two or more samples are taken from each plot, the data can be analyzed by two-way analysis of variance (Sokal and Rohlf, 1969), an approach advocated by Underwood (1981).
8: SAMPLING
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131
artificially slowing the recolonization). If the plots had been interspersed, an effect on only one treatment could not have occurred. Hurlbert (1984) points out that simple random designs can result in an arrangement of treatments among plots that is largely segregated. Blocked designs circumvent this problem. Contrast the above design with a similar one in which four plots are placed at random locations, each receiving one of the treatments (chosen at random), and where five samples are taken at random locations from each plot (Figure 8.7), to yield a total of 20. The population has not changed, but if the nun hypothesis to be tested is the original one of no difference among the treatments, such a design is "pseudoreplicated" (Hurlbert, 1984) because differences among the locations cannot be separated from differences among treatments. Only additional applications of the treatments can allow the differences that necessarily exist among locations to be separated from differences caused by the treatments. More generally, it is the number of repetitions of the manipulation, not simply the number of samples, that is of importance in statistical tests.
• Figure 8.6.-A diagram of the beach as in Figure 8.2 showing a possible layout of an experiment designed as a randomized complete-block design. Squares indicate the locations of plots. Filled squares indicate the raked plots; unraked plots are open squares. Blocks are groups of four adjacent plots, one of each treatment. The placement of treatments within a block is randomized. Blocks are placed at random on the beach, but separated from each other by a minimum distance to assure independence.
The treatment plots in the randomized complete-block design are "interspersed" in Hurlbert's (1984) terms. He argues that interspersion in space or time is a requirement of good experimental design because it protects the experimenter from confusing treatment effects with natural variability in experimental material or changes that occur in experimental conditions during the experiment that only affect a subset of the experimental units. As an example of how such effects could arise, assume in the above experiment that the plots for each ireatment type were grouped in different areas of the beach. If a flock of shore birds arrived on the beach, fed in the area that happened to contain the small raked plots, and then were frightened away, they could have affected the nematode densities in the small raked plots and potentially changed the results of the experiment (by
ill,
_----------------.~-------------------------
o
•
o
Figure 8.7.--A diagram of the beach as in Figure 8.2 with symbols as in Figure 8.6. The plots were located at random on the beach. In this pseudoreplicated design, five samples are taken from each plot.
Conclusions based on manipulative experiments are less likely to be incorrect than those based on unmanipulated experiments because the random assignment of treatments to locations results in an equal (on the average) contribution of any extraneous variables to each treatment. However, Hurlbert (1984) describes situations where even well-designed manipulations would yield incorrect answers. Power Consider a simplified version of the beach example of Figure 8.3 in which, rather than five onshore-offshore bands, only two bands are considered. A simple random sampling scheme
132
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INTRODUCTION
TO THE STUDY OF MEIOFAUNA
followed by a one-way analysis of variance (or a t test) would allow the testing of the null hypothesis that the bands did not differ in nematode density. Test results indicating a significant difference at the usual (5%) significance level mean that the probability that as extreme or more extreme a result would arise by chance alone is 1 in 20. By convention, this level of confidence that chance is not causing the observed differences is considered adequate to reject the null hypothesis of "no difference between the bands." If the result were not significant, the null hypothesis would not be rejected. This result is much less useful because it is not legitimate to claim that in the absence of a statistically significant difference there is no difference between the bands (Cochran, 1983). The reason that accepting a null hypothesis is of limited utility is that the sensitivity of the statistical test affects the ability of the test to detect differences. The sensitivity (power) of the test is a function of the size of the difference to be detected, the significance level chosen, the number of samples taken, and the variability present. If the difference to be detected or the number of samples taken is small or the variability is large, then the power of the test may not be sufficient to detect a difference that really exists (the null hypothesis will not be rejected even though it is false). Questions of power should be considered when a sampling scheme is designed. In particular, good procedure would include an attempt to determine the ability of the design to detect the likely differences. It is best to obtain preliminary samples to get rough estimates of the difference between treatments and the variability of the population. With this information, one can use power tables or statistical procedures (see Elliott, 1977, or Green, 1979) to estimate the number of samples required to detect expected differences. It may be important to be able to assert that the treatments differ negligibly. As discussed above, the lack of a significant difference between the treatments does not provide statistical support for such an assertion. However, such support can be obtained by means of a procedure described by Rotenberry and Wiens (1985). Briefly, one can estimate the smallest difference between treatments that would have had a 95% chance of being detected given the design; any undetected difference must be smaller. If this smallest detectable difference is sufficiently small in biological terms, then the fact that no difference was detected implies that any difference that might exist is also small in biological terms. Therefore, one can assert, with statistical support, that there is no biologically important difference between the treatments. However, if the smallest detectable difference is large in biological
terms, then the fact that no difference was detected is uninformative because an undetected difference could be large, and no assertion can be made about the difference or lack of one between treatments. Multiple
Testing
Circumstances arise frequently in which a number of hypotheses are tested in the same data set. For example, if 20 species are counted from the samples and their abundances correlated with five environmental measures, 100 correlation coefficients can be calculated. Some of the correlations may be significant at the 5% level. In fact, from the definition of the significance level, it can be anticipated that, even if there were no relationship between any species and any environmental variable, on the average, 5 of the 100 coefficients would be significant. That is, even if the original data were thrown away and the data set reconstructed by drawing values from a random-numbers table, on the average 5% of the correlations would be significant. As a consequence, when a group of tests is performed, the significance level of the tests taken as a group differs from the nominal 5% level, and precautions must be taken in order to avoid increasing the probability of rejecting a true null hypothesis (see Underwood, 1981). In analysis of variance, the a posteriori multiple-comparison procedures correct for the multiple testing done, for example, when all possible treatments are compared. When procedures other than analysis of variance are used and multiple tests are made, the Bonferroni procedure (Brown and Hollander, 1977) should be followed. That is, the nominal significance level should be divided by the number of tests to yield a corrected (but conservative) significance level for the group of tests (see Thistle, 1980, and Taghon, 1982). For example, if one wanted a 5% significance level for a group of 10 tests, each individual test would be examined for significance at the 0.05/10 = 0.005 level.
Comments at the Finish Designing a proper sampling scheme is an exacting, time-consuming process. However, the time spent in planning is very cost effective if it saves the investigator from an after-the-fact salvage job, from having uninterpretable results, or from being mentioned in reviews of improper design or analysis (e.g., Hurlbert, 1984). Green (1979) gives recommendations. We would emphasize two. First, if at all possible, consult a statistician about the proposed design. Second, the information obtainable from preliminary samples can be crucial to arriving at a proper design. For example, one can establish
CHAPTER 8: SAMPLING STRATEGIES
that the sampling gear works as anticipated (see Green, 1979:31, for a sobering example). One can obtain estimates of means and variances with which to gauge the number of samples required to obtain a desired degree of precision (see Elliott, 1977, and Green, 1979). One can obtain information on the minimum spacing required for treatments to be independent The results of such preliminary investigations should permit adjustment of the design to increase the resolving power for the effort invested. Acknowledgments The manuscript has been improved by the comments of T. Chandler, P. Culley, F. Dobbs, P. Jumars, D. Meeter, K. Sherman, A. Thistle, and R. Varon.
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