168 



Fishery Bulletin 89(1). 1991 



Table 1 



Probability levels associated with the randomization tests of 

 the individual hypotheses. Because these are a posteriori tests, 

 significance levels were adjusted according to the Bonferroni 

 inequality to 0.2 5cr (*P< 0.0125, "P<0.0025). 



Mean proportion 



Probability 



Sample 1 Sample 2 t- value level 



Tunicates 

 Crustaceans 

 Fish 

 "Others" 



-9.i <o.oor* 



5.0 0.011* 



3.7 0.046 



1.9 0.294 



combined, (2) the combined data are randomly sorted 

 into two new samples equal in size to the originals, and 

 (3) a value of T 2 is calculated for the two samples. 

 Steps 2 and 3 are repeated iteratively (the present 

 study uses 5000 iterations), and the probability distribu- 

 tion of the randomized values of T 2 is then calculated. 

 Next, the significance level of obtaining the original 

 T 2 is estimated by determining the proportion of the 

 randomized T 2 values that, ignoring signs, is equal to 

 or greater than the original. 



If between-sample equality of the diet is rejected, it 

 is then appropriate to test for the equality of individual 

 prey categories to determine which categories con- 

 tribute most to the difference in diet. In this case, the 

 measure of statistical difference used is the univariate 

 t- statistic. In matrix notation, the vector of t- statistics 

 (t) can be computed as 



t = D.-Mxi 



(4) 



where D." 1 is the inverse of a matrix formed from the 

 diagonal elements of S.. As before, the significance 

 of each t- statistic is determined from an empirical 

 probability distribution computed for each prey cate- 

 gory using randomization. Computation of these prob- 

 ability distributions is identical to that described for 

 the multivariate case, except that a matrix of univariate 

 t- statistics (Eq. 4) is used in the calculation instead 

 of the T 2 - statistic (Eq. 2). These individual tests are 

 a ■posteriori tests and require some adjustment of the 

 error rate considered to be significant. Using the 

 Bonferroni inequality (Morrison 1976, Miller 1981), an 

 appropriate adjustment is to assume significance at 



a- n -1 . 



The above procedures have been incorporated into 

 the computer program DIETTEST, which is designed 

 to run on IBM-compatible microcomputers. This pro- 

 gram is available from the author. 



As an example of the application of this method, it 

 has been used to test for dietary differences between 



two samples of pelagic armorhead: 55 fish collected in 

 June 1985, and 101 fish collected in August 1988 (only 

 fish with prey in their stomachs were used in the anal- 

 ysis). Both samples were obtained from the Southeast 

 Hancock Seamount (lat. 30°N, long. 180°W) on the 

 Hawaiian Ridge. Stomach contents were sorted to the 

 lowest taxonomic category possible, then blotted and 

 weighed to the nearest milligram. To simplify the 

 analysis, various prey items were pooled into four ma- 

 jor prey categories: tunicates, crustaceans, fish, and 

 "others." 



Results and discussion 



When the method was applied to the two armorhead 

 samples, the test of the simultaneous equality of all 

 dietary proportions between samples was highly sig- 

 nificant (P<0.001), indicating that the diet of pelagic 

 armorhead had changed between the two sampling 

 periods. Because of this, tests were also made for in- 

 dividual prey categories. Two of the four prey cate- 

 gories, tunicates and crustaceans, differed significantly 

 between samples (Table 1) and therefore appeared to 

 be responsible for the overall difference in diet. 



The measure of between-sample difference, T 2 , em- 

 ployed in the proposed test was selected primarily 

 because the absolute differences between samples are 

 scaled by the within-sample variances, a particularly 

 desirable feature when dealing with highly variable 

 quantities such as fish diets. This choice, however, im- 

 poses a constraint on the proposed statistical test; that 

 is, the method can only be applied to cases in which 

 the two diet samples lack mutually exclusive compo- 

 nents. This constraint arises because computation of 

 T 2 requires inversion of the variance-covariance 

 matrix (Eq. 2) which is singular and therefore not in- 

 vertible when a prey category is completely absent 

 from one of the samples. Although this constraint may 

 not be severe when the diet of a single predator is 

 being examined for spatial or temporal variation, espe- 

 cially if one is willing to accept the pooling of prey to 

 relatively high taxa, the proposed method is likely to 

 be of limited value for comparing the diets of different 

 predators. For such between-predator comparisons, a 

 more appropriate test could be developed by utilizing 

 some measure of diet overlap (Caillet and Barry 1979) 

 which is not affected by mutually exclusive prey cate- 

 gories, instead of T 2 as a measure of between-sample 

 difference. 



Acknowledgments 



I thank George Boehlert, Tim Gerrodette, John Hoenig, 

 Russell Kappenman, Jeff Polovina, and two anonymous 



