Detecting Differences in Fish Diets 



David A. Somerton 



Honolulu Laboratory, Southwest Fisheries Science Center 

 National Marine Fisheries Service, NOAA, 2570 Dole Street 

 Honolulu, Hawaii 96822-2396 



Statistical comparison of the diet of 

 a predator between areas or time- 

 periods allows one to distinguish 

 true dietary differences from sam- 

 pling variability and may lead to a 

 better understanding of a species' 

 feeding habits. Despite the utility of 

 statistical testing, few procedures 

 appropriate for dietary comparisons 

 have been developed. Perhaps one 

 impediment to the development of 

 a general approach to dietary com- 

 parisons is the wide variety of ways 

 in which diets have been expressed 

 and the lack of consensus about 

 which is best. For example, diets ex- 

 pressed as the numeric or gravi- 

 metric proportions of the total food 

 consumed will require different ap- 

 proaches to statistical comparison 

 than those expressed either as the 

 proportion of the samples contain- 

 ing each of the various prey types, 

 or as the index of relative impor- 

 tance of each prey type (Pinkas et 

 al. 1971). 



For cases in which diets are ex- 

 pressed in terms of gravimetric pro- 

 portions, Crow (1979) and Ellison 

 (1979) have recommended statis- 

 tical tests of between-sample differ- 

 ences based on multivariate analysis 

 of variance (MANOVA). Validity of 

 such tests, however, requires that 

 the prey proportions have a multi- 

 variate normal distribution and that 

 the variance-covariance structure of 

 the prey proportions is identical 

 among samples (Morrison 1976). 

 Recognizing that dietary data are 

 unlikely to have these properties, 

 Crow (1979) further recommended 

 using MANOVA that incorporates 

 non-parametric procedures. Herein, 

 this recommendation is followed, 



and a new approach for testing dif- 

 ferences in diets using non-para- 

 metric MANOVA is examined. This 

 approach combines the usual mea- 

 sure of between-sample differences 

 employed in parametric MANOVA 

 (i.e., Hotelling's T 2 statistic; Mor- 

 rison 1976) and a non-parametric 

 procedure (i.e., a randomization 

 test; Edgington 1987) to determine 

 the significance of T 2 . The method 

 is then applied to determine wheth- 

 er the diet of pelagic armorhead 

 Pseudopentaceros wheeleri from 

 the Southeast Hancock Seamount 

 changed between two sampling 

 periods. 



Materials and methods 



Testing for between-sample differ- 

 ences is accomplished in three steps: 

 (1) calculating for each sample the 

 gravimetric dietary proportions and 

 their variances and covariances, (2) 

 calculating a measure of the statis- 

 tical difference between samples, 

 and (3) determining the statistical 

 significance of the measure. The 

 gravimetric proportion of the diet 

 contributed by prey category j(pj) 

 can be estimated as the total weight 

 of prey category j in all stomach 

 samples divided by the total weight 

 of all prey categories combined 

 (Hyslop 1980). Algebraically this is 

 expressed as 



W; 



Pj = 



2jSkW ]k w. 



(1) 



where Wjk is the weight of prey 

 category j for individual k, wj is 

 the total weight of prey category j 



summed across all individuals, and 

 w is the total weight of all prey. 

 Each Pj is transformed to Xj, where 

 xj = arcsin ^Pj , so that it conforms 

 more closely to a normal random 

 variable (Sokol and Rohlf 1969). Be- 

 cause Xj is estimated as a pooled 

 proportion rather than the average 

 of the proportions for individual 

 fish, the variance of Xj and the co- 

 variance between Xj and x, cannot 

 be calculated directly and instead 

 are approximated by using the delta 

 method (Seber 1973). In the follow- 

 ing, X; indicates the vector of Xj for 

 sample i, and S ; indicates the ma- 

 trix of variance and covariance esti- 

 mates for X;. 



The measure of statistical differ- 

 ence used is the Hotelling's T 2 sta- 

 tistic, a multivariate extension of 

 the t- statistic (Morrison 1976). In 

 matrix notation, this statistic is ex- 

 pressed as 



T 2 = 



(x 1 -x 2 )'S.- 1 (xi-x 2 ), (2) 



where S. _1 is the inverse of the 

 pooled estimate of the variance- 

 covariance matrix (Morrison 1976). 

 S. is approximated, assuming rea- 

 sonably large sample sizes (>50 

 stomachs with prey per sample), as 



2(N 1 S 1 + N 2 S 2 ) 



o. = - , (o) 



Nj+No 



where Nj and N 2 are the sizes of 

 the two samples. 



Once a value of T 2 is computed, 

 its significance is determined from 

 an empirical probability distribution 

 of T 2 computed by using a tech- 

 nique known as randomization (Ed- 

 gington 1987). Computation of the 

 empirical probability distribution 

 using this technique proceeds as 

 follows: (1) stomach content data 

 from both time or area samples are 



Manuscript accepted 12 October 1990. 

 Fishery Bulletin, U.S. 89:167-169 (1991). 



167 



