The statistical analysis of gill net catches

The validity of parametric versus nonparametric statistical analysis of gill net catch data is briefly discussed. It is concluded that the nonparametric ranking tests of Mann-Whitney and Kruskal -Wall is are the best methods available for analyzing such data. The theory and conduct of these tests are explained with the aid of simple examples. Also described are computational checks, various ways of handling tied observations, limitations on the use of the methods, and some peculiarities not associated with the more common parametric methods. Numerous examples are provided in order to illustrate pertinent points. Introduction Gill net sampling is widely used by fisheries biologists to obtain a variety of information about fish populations. One standard use is to gather information about the relative size of a fish population. Although such data cannot be manipulated so as to produce an estimate of the actual number of fish in a population, it nevertheless has seemed reasonable that, under the appropriate conditions, some kind of positive relationship should exist between the density of fish in a body of water and the number of fish caught in a series Hhis study was part of Montana Federal Aid in Fish Restoration Project F-4-R. Burwell Gooch Page 2 of gill net sets that have been fished in that body of water. Furthermore, even though more refined methods of population size estimation do exist, physical and/or economic limitations often conspire to make gill netting the only practical means available for such work. Since gill net catches do provide only an index to population size, the primary value of such figures is in reference to similar figures for other populations of fish (where "other populations" may represent fish in different bodies of water, in the same body of water at different locations, or in the same location at different times). That is, can we infer that population A is greater or smaller (or more or less dense) than population B, based on the gill net catches available from each? Clearly, this is the kind of situation in which statistical analysis was designed to play an important role, although a review of recent literature indicates that, with the exception of Moyle's pioneering work (Moyle, 1950; Moyle and Lound, 1960), this role has not received the attention that it should. Statistical methods of analysis are conventionally divided into two broad categories. First, and far more popular, are the parametric methods. These are related by the fact that they all assume some kind of model and/or frequency distribution associated in some way with the population of interest. As a result, they are intimately concerned with estimating the Burwell Gooch Page 3 parameters that describe the assumed models and distributions. Usually, they are also constrained by several additional assumptions, some of which concern the population, others the sample. Second are the nonparametric methods. These do not specify a particular kind of model or parent frequency distribution, nor do they normally require so many other restrictions. Parametric Methods Most of the conventional methods of parametric statistical analysis are based on so-called "normal theory." This theory is predicated partly on the following three assumptions: (1) the attribute of interest (e.g. , length, speed, temperature) occurs on a continuous scale of measure, rather than in discrete categories (e.g. , as with counts); (2) the essentially infinite population of such attributes has a "normal" frequency distribution (Figure 1); and (3) the standard deviation of the distribution is independent of the mean. Under these conditions, and several others, estimates of population parameters and tests of hypotheses have been developed that are the best possible. There are several reasons why statistical analysis in terms of normal theory enjoys such great popularity. Foremost among these is the fact that, when all assumptions are satisfied, no nonnormal method is as good. Second, the methods are "robust". This means that the power and efficiency of tests of hypotheses are not significantly reduced by reasonably small departures from the required assumptions. Third, a wide variety of parent Burwell Gooch Page 4 distributions from many different fields of study can be reasonably well approximated by the normal. Fourth, even if a parent distribution is not normal, the distribution of the sample mean approaches the normal with increasing sample size, thus preserving the validity of much of the theory. The gill netting problems that Moyle attempted to resolve concern assumptions (2) and (3) listed previously. First, in the majority of series of catches, the frequency distribution of catch data is highly skewed, i . e . , nonnormal (Figure 2). Second, the standard deviation is positively related to the mean (Figure 3). Moyle did not mention that assumption (1) also is not satisfied by gill net data. Based on his comprehensive review of the situation, Moyle concluded that statistical analysis of gill net catches in terms of an assumed normal distribution is not legitimate. This means that it would not be valid to use the t_ test to compare the mean catches of two gill net series, or to use tabulated t values to impose confidence limits on these mean catches. I concur in his evaluation. Moyle and Lound (1960) proposed two solutions to the problems stated above: one parametric, the other nonparametric. The parametric approach consists of assuming that gill net catches follow a negative binomial distribution. Under this assumption, a suitable transformation may be applied to the individual catches so that the resulting data are distributed more nearly normally. Thus, conventional confidence limits and ;t tests may be calculated. Burwell Gooch Page 5 There are two drawbacks to this proposal. First, even though many series of gill net catches do satisfactorily approximate a negative binomial distribution, the fact remains that a significant number do not. In other words, the negative binomial distribution is considerably more applicable than the normal distribution, but far from universally so. Second, the estimator of the dispersion index (1/k), which plays a central role in the theory of the negative binomial function, is simple to calculate, but does not provide the best possible estimate. The best estimate can only be obtained as the root of an explicitly insoluble equation. This can, of course, be found iteratively, but without a computer it is not practical to do so. As a result of the foregoing, I feel that we may justifiably conclude that parametric statistical analysis is inappropriate in general for use on gill net catch data. Rather than use methods that, at best, fit the situation only part of the time, or spend more time searching for other theoretical distributions that might conceivably fit all the data, it seems more logical to investigate the possibility that some standard nonparametric method might be suitable for our needs. Nonparametric Methods Nonparametric methods possess a number of features that make them very attractive alternatives to parametric methods (Siegel, 1956). Four of these that are particularly relevant to our problem Burwell Gooch Page 6 of gill net catches are: (1) no particular form of the parent distribution is assumed; (2) both continuous and discrete data are equally well handled; (3) several tests are far superior to normal theory tests under conditions where the latter are inappropriate, whereas they are only slightly inferior where normal tests are appropriate; and (4) exact tests exist for any sample size, no matter how small. The nonparametric method proposed for gill net catch analysis by Moyle and Lound (1960) is based on use of the median (rather than the mean) catch of a series. In addition to the advantages of a nonparametric method, the median method possesses three useful features: (1) confidence limits may be constructed about the median, although without appropriate tables this may be quite tedious; (2) the test of a difference between sample medians is simple to calculate and understand; and (3) the test may be applied to any number of samples. The primary disadvantage of the method is that it does not provide the most efficient analysis of the data possible. That is, other methods exist that extract more information from the samples, and thus discriminate more effectively between them. The purpose of this paper is to promote the understanding and use of these methods.