Minireview: On Sample Sizes and Reliable Information

Abstract

drawn regarding the validity of the results and discussions from such studies? Given that inadequate sample sizes will bias the results of even the most carefully designed and conducted study, am afraid that the answer to this question will be a sad commentary on the state of most of our investigations. Data, of course, provide the foundation for all ornithological knowledge. Data can range from the handwritten field notes of a competent observer to values obtained through the use of complex apparatus. Whether data are in a qualitative form such as I saw species A foraging consistently higher than species B, or a quantitative form such as We found a significant (P < 0.05, t-test) difference between the foraging heights of species A (X = 6.3, SD = 2.1 m) and species B (X = 20.2, SD = 1.4 m), we have been provided data on a biological phenomenon. But upon what are we to base our trust in the biological reality of these results? Using my simple two-species example, which result would you tend to trust? suspect most people would initially rely on the quantitative data analyzed by the t-test (or other appropriate analysis). Unfortunately, there is absolutely no reason to place any more trust in the quantitative assessment over the more qualitative one. If had added, On 132 occasions saw to the qualitative description, and n = 10 to the quantitative analysis, would certainly put my trust in the former data set. This example may seem somewhat trivial, it is meant to make a simple point: without both the presentation and justification of one's sample size, no data set has much, if any, validity. Of course, my comments here assume a proper sampling design. The sampling design is an early, critical step in any study and, along with the proper statistical analyses, must be of a proper form to answer the question at hand. Regardless of the size of the sample, improperly collected data may be of little use. The question of study design, is, of course, a complex topic that will not be developed in this short review. Many texts on the subject are available (e.g., Cochran 1977, Williams 1978, Green 1979, Kerlinger 1986, Kish 1987). have often heard the comment: but my sample size must have been adequate. ... found a significant difference .. . to justify the associated sample size in a study. Erroneous and often contradictory conclusions may be reached with variations in sample size, which may result in fluctuating alpha levels. The probability of committing a Type error (a; rejection of a null hypothesis when it is actually true) is inversely related to the probability of committing a Type II error (p; failing to reject the null hypothesis when it is in fact false), for a given n. Lower probabilities of committing a Type error are associated with higher probabilities of committing a Type II error, and the only way to minimize both types of error is to increase n. Thus, for a given a, larger sample sizes will result in statistical tests with greater power (1 #) (see Zar 1984:43-45). In theory, then, we have a strong basis for increasing our sample sizes. Further, the biological interpretation of the results can vary as sample sizes increase and the resulting responses of an animal change. For example, take the use of terminal buds by Chestnut-backed (Parus rufescens) and Mountain (P. gambeli) chickadees shown in Figure 1: different conclusions could be reached regarding the percent use of terminal buds by each species, and the overlap in use between the species, based on the sample size used. Results can differ dramatically with even small changes in sample sizes (compare results for the Mountain Chickadee for samples of 30, 40, and 50 individuals). Many formulas are available in basic statistics texts for determining proper sample sizes (e.g., Cochran 1977, Williams 1978, Green 1979, Sokal and Rohlf 1981). Many of these formulas require, however, an estimate of the population variance. Of course, the population variance is seldom known. Alternatively, one can collect data sequentially, with data being evaluated by these formulas at each step, and a decision being made on the adequacy of the samples and possible need for more data collection. Such sequential sampling procedures have been discussed by several workers (e.g., Kuno 1969, 1972; see also Green 1979:126-136). Sequential sampling provides a valuable, although seldom used, method of determining proper sample sizes without gross over-sampling. (Sequential techniques for estimating necessary sample sizes, as used in this paper, should not be confused with sequential sampling for classifying populations into categories; see Waters 1955.) 'Received 2 December 1987. Final acceptance 16 December 1987.