Optimal allocation of effort in studies using the size-frequency method of estimating secondary production

Secondary production estimates based on the size-frequency method can be improved through the use of optimal sample allocation without increasing the total sampling effort. We advocate a two-step procedure. First, sampling dates should be chosen to minimize bias. Then, sample sizes on each date should be allocated in such a way as to minimize variance. Calculating the optimal sample allocation requires some prior variance information, but even rather imperfect information will usually result in improved estimates. An example is presented using published data on the mayfly Ephemerella dorothea. An estimate of secondary production is of little value without some idea of the potential bias and variability of the estimate. Estimation of aquatic macroinvertebrate secondary production has received substantial attention in recent years, and several aspects of bias and variability in estimation have been examined. Resh (1979) and Waters ( 1979) identified biological and mechanical factors which contribute to bias and variability. Krueger and Martin (1980) and Newman and Martin (1983) concentrated on quantifying the sampling variability of production estimates. Efforts to reduce the error have focused mainly on mechanical or biological problems, such as loss of small animals due to coarse screens, diurnal behavior patterns altering the animals’ likelihood of being caught, etc. (Resh 1979; Waters 1979). However, even if these problems were eliminated, secondary production estimates would still often have large sampling variances because they are based on small random samples from the parent population. We describe here a method of allocating effort which can substantially reduce the variance of size-frequency production estimates without increasing the overall study effort. The size-frequency approach is the method most commonly used to estimate macroinvertebrate production. A concise history and overview of the procedure is given by Krueger and Martin (1980). Krueger and Martin presented equations for estimating the variance of size-frequency production estimates; we have recast these equations so that the optimal allocation problem can be solved in a manner analogous to that used for stratified random sampling (e.g. Cochran 1977). We argue that sampling date selection is important in controlling bias and should be done as the first step in designing a study. Then the number of samples processed from each date should be chosen by optimal sample allocation so as to reduce variance. Our method requires some prior information about variances on each of the sampling dates, but will still be useful even if the variance information is quite rough (Cochran 1977, p. 115-l 17; Kish 1965, p. 94-95). We envision two general applications of the technique. Data from previous studies can be used to determine sample sizes to be taken in the field for a future study, or, in the absence of prior information, excess samples can be taken in the field, and a small preliminary sample counted to determine further effort allocation in the lab. We give an example that illustrates the first situation. The second approach is feasible because large numbers of samples can often be collected in the field with little effort; most of the effort in aquatic macroinvertebrate studies is in sorting, identifying, measuring, and enumerating samples in the lab (Resh and Price 1984). The technique is very flexible, and a sound design could incorporate both approaches. M. Butler stirred our interest in sampling problems in secondary production estimation. R. Newman, P. Wingate and R. Lake provided advice. The comments of anonymous reviewers improved the clarity of the paper. In the following considerations, each sampling date is indexed by i (i = 1, 2, . . . , n). The (random) samples within each sam-