NOTE Hayes A statistical method for evaluating age-length keys for Melanogrammus aeglefmus 



555 



for length class j is 



Ho: p, lX = p :j2 for all age classes i from source 1 and 



source 2 

 Ha: p«i * p, j2 for all age classes i from source 1 and 



source 2. 



Consider as an example haddock (Melanogrammus 

 aeglefinus) in the 54-cm size class sampled from the 

 commercial catch from the first and second quarters of 

 1983. Numbers at age from each of these samples are 



The question asked is whether the two samples are 

 likely to be drawn from the same population or whether 

 they differ sufficiently to indicate that the sampled 

 populations are different. Assuming fixed marginal to- 

 tals, an appropriate test of this hypothesis is Fisher's 

 exact test (Siegel, 1956). Previously this test was im- 

 practical for contingency tables greater than 2x2 

 because of the amount of computational power required 

 by the algorithms available for its solution. Recent 

 improvements (Pagano and Halvorsen, 1981; 

 Mehta and Patel, 1983; implemented in Ver- 

 sion 6 of SAS [SAS Institute, 1990]) allow 

 problems of this size to be readily solved. 



Alternative tests exist in the chi-square 

 test of homogeneity (Hennemuth, 1965) and 

 the G 1 test (Bishop et al., 1975). These tests 

 have the advantage that age-length keys 

 can be compared in their entirety. Some 

 studies indicate that these tests may also 

 have greater power than Fisher's exact test 

 (DAgostino et al, 1988; Storer and Kim, 

 1990), but others have disputed the validity 

 of this these assertions (Little, 19894. Also, 

 the chi-square and G 2 tests are often viewed 

 as inappropriate when some of the expected 

 values are less than 5 (e.g., Sokal and Rohlf, 

 1981; Haberman, 1988); a situation that 

 commonly occurs in comparisons of age- 

 length keys. Although grouping data across 

 age or length classes is a way of increasing 

 the expected values for each cell in the con- 

 tingency table, such a procedure results in 

 a loss of statistical power (Cochran, 1952). 



With Fisher's exact test, there are no re- 

 strictions on the expected values for any 

 cell within the contingency table (Siegel, 

 1956). In practice, however, each source (i.e., 

 time period) should contain at least six ob- 



servations because smaller sample sizes do not have 

 sufficient power to resolve even major discrepancies 

 between sources (Bennett and Hsu, 1960). 



In summary, Fisher's exact test provides a means of 

 testing differences between age-length keys derived 

 from different sources or from different time periods. 

 When age-length keys are pooled across time periods 

 or sources that do not differ significantly, the resulting 

 estimates of catch at age are more precise and impor- 

 tantly do not appear to be biased by the pooling proce- 

 dure. For Georges Bank haddock, age-length keys from 

 commercially harvested haddock from the first and sec- 

 ond quarters can be combined, as well as keys from 

 haddock sampled in the NEFSC research survey. In 

 the future, allocation of sampling effort should con- 

 sider the benefits of pooling age-length keys. 



Acknowledgments 



I thank S. Clark, J. Forrester, S. Gavaris, and F. 

 Serchuk for their constructive comments on earlier 

 drafts of this manuscript. Thanks are also expressed 

 to J. Brodziak for his helpful discussions on this 

 subject. 



