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Fishery Bulletin 91(2). 1993 



the final model structure and set of explanatory vari- 

 ables are correctly chosen before fitting. Here, we use 

 elements of hypothesis testing, but acknowledge the 

 impossibility of testing the hypotheses rigorously. 



Detecting the influence of contaminants 



We started with the following null hypothesis: "any 

 apparent influence of contaminants can be explained 

 by random variability alone, including chance correla- 

 tions of contaminant loadings with climate variabil- 

 ity." We imposed three statistical criteria to be met 

 before we would consider rejecting this null hypoth- 

 esis on the basis of a model including contaminant- 

 loadings data. The first criterion was qualitative: the 

 correlations of the model's explanatory effect with most 

 contaminant variables had to be negative. The other 

 criteria were quantitative: the regression coefficients 

 of the model had to be statistically significant at oc=0.05, 

 and the inclusion of contaminant information had to 

 provide more explanatory power than use of climate 

 data alone. If all three criteria were met, we would 

 admit the possibility of contaminant influence on 

 spawning success. The three criteria are described in 

 more detail immediately below. 



Sign test Of the variables in Table 3, the metals and 

 organochlorines have been found by bioassay to have 

 deleterious effects on fish (Weis & Weis 1989). (We 

 omitted nickel because of an incomplete time-series 

 and, more importantly, its relatively low toxicity.) 

 Power-plant cooling flow also is generally considered 

 deleterious, but the remaining variables in the indica- 

 tors group could be favorable as well as unfavorable. 

 The signs of the correlations with the 16 "deleterious" 

 variables were subjected to a (one-tailed) statistical 

 test of the following null hypothesis: "The observed 

 number of negative correlations could arise by chance 

 alone." Under this null hypothesis, the probability of a 

 negative correlation between a contaminant variable 

 and the model's estimated explanatory time-series is 

 0.5. Assuming independence among the 16 contami- 

 nant variables (an assumption that may be violated; 

 see below), the probability of obtaining x negative cor- 

 relations by chance is given by the binomial probabil- 

 ity fix; « = 16, p=0.5). The probability of observing 11 or 

 more negative correlations is 0.105, and of observing 

 >12 is 0.038. Thus, the conventional error rate of a<0.05 

 requires >12 negative correlations between the 16 del- 

 eterious contaminant variables and the explanatory 

 time-series from a model that uses the combined ex- 

 planatory data. 



Such a test concludes that apparent contaminant 

 effects are qualitatively significant if 12 or more nega- 

 tive correlations are found. However, the true prob- 



ability of Type-I error is >0.038, because many of the 

 contaminant variables are positively correlated with 

 one another, violating the assumption of independence 

 in the binomial probability model and increasing the 

 tail probabilities. While we cannot calculate an exact 

 critical value of x in view of lack of independence, the 

 value would be larger than the nominal value of 12 

 required for a model to pass this test. 



Test of coefficients This test was used to assess 

 whether the coefficients of the chosen model were sig- 

 nificantly different from zero. Although we always chose 

 the model with lowest C p , that criterion does not de- 

 pend directly on how precisely the regression coeffi- 

 cients are estimated. To test the significance of regres- 

 sion coefficients, we used the standard <-tests (at 

 a=0.05) provided by the statistical software, and re- 

 quired all coefficients to be significant for the model to 

 pass the test. 



Improvement-in-fit test Our third criterion required 

 that the model that incorporated contaminant infor- 

 mation (i.e., which used the combined explanatory data) 

 had to provide a substantially better fit to the spawn- 

 ing success data than the model using the climate data 

 alone. To judge this, we used the Schwarz criterion 

 (Smith 1988), a statistic for comparing non-nested mod- 

 els. The Schwarz criterion requires an estimate of o, 

 the true model variance; we used the MSE of the bet- 

 ter-fitting model. To pass the test, the model with com- 

 bined data was required to have a higher value of this 

 statistic than the model with climate data. 



Results 



Northern anchovy 



Not unexpectedly for a short-lived species, the spawn- 

 ing biomass of this northern anchovy stock appears to 

 depend strongly upon the preceding few years' recruit- 

 ments. A peak in spawning biomass generally follows 

 a corresponding peak in recruitment by 1 or 2yr (Fig. 

 la). The stock-recruitment relationship, although noisy, 

 appears density-dependent, in that spawning success 

 improves at lower stock sizes (Fig. lc), and each model 

 includes a significant compensatory term (the nega- 

 tive stock-size parameter; Table 5). Ricker (1954) and 

 MacCall (1980) have pointed out that cannibalism of 

 eggs or larvae by the adults could cause such compen- 

 sation. 



Neither the climate model nor the combined model 

 explained even half of the variability in log spawning 

 success of northern anchovy (Table 5). Although the 

 model incorporating contaminants had somewhat 



