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Fishery Bulletin 91(2). 1993 



coupled to human population size, which generates 

 them. This multicollinearity causes problems in fitting 

 and interpreting statistical models such as ours. In 

 extreme cases of multicollinearity, numerical algorithms 

 used for fitting can fail. A more likely problem is that 

 interpretation is not straightforward, because param- 

 eter values depend upon the other variables in the 

 model. When sea-surface temperature is included in a 

 model, for example, it may explain a large amount of 

 the variance associated with sea level, so there is little 

 to be gained from including sea level itself in the model. 

 Yet an effect may truly be due to sea level (or to the 

 strength of the California Current), and this relation- 

 ship may thus be overlooked. Because of this difficulty, 

 any biostatistical study of this nature is inherently 

 unable to isolate the effects of individual causes. An 

 important related practical problem is that a large 

 number of potential explanatory variables makes any 

 fitting procedure unwieldy. 



We addressed some of these statistical problems by 

 combining the explanatory data into new composite 

 variables that did not contain duplicate informa- 

 tion. These new variables, constructed by principal- 



component analysis, are linear combinations of the 

 original variables. To reduce the number of variables 

 (initially -216) to a number more easily analyzed by 

 standard statistical software, data on sea level, sea- 

 surface temperature, and salinity were converted to 

 bimonthly means; also, surface and bottom salinities 

 at Scripps Pier were averaged. Monthly rainfall val- 

 ues were transformed into an annual value of total 

 rainfall (preceding 1 July to current 30 June) and a 

 value representing the median date of the season's 

 rainfall. The explanatory dataset with these changes 

 contained 94 variables before reduction by principal- 

 component analysis. 



We constructed two separate sets of principal com- 

 ponents. The first set was constructed from physical 

 and climate data (Table 3) but not data on contami- 

 nants or general stressors (Table 2). A recruitment 

 model based on this analysis would show how much of 

 the variability in spawning success could be attributed 

 to stock size and climatic variation alone. The second 

 principal-component analysis included all explanatory 

 variables, and was used to reveal how much more vari- 

 ability could be explained by adding contaminant in- 

 formation to the analysis. 



Weighting was used in the principal-component 

 analyses to avoid giving undue emphasis to variables 

 (e.g., SST) measured at many locations. To determine 

 weights, each variable was assigned to one of six 

 groups, as shown in Tables 2 and 3. Each group re- 

 ceived 1/6 of the total weighting, which was divided 

 equally among the variables within the group. 



The results of principal-component analyses are fre- 

 quently difficult to interpret, as the components are 

 formed on purely statistical grounds. Extensive graphi- 

 cal analysis (presented in Prager & MacCall 1987c, 

 1990) allowed attaching an interpretation to some, but 

 not all, of the components used in these analyses 

 (Table 4). In interpreting results of the recruitment 

 models, we used a different approach, that of measur- 

 ing the correlation of the model's explanatory effect 

 with the individual variables, as explained below. 



Application of model to three 

 fish stocks 



Overview 



For each stock, we developed two alternative models: one 

 using the principal components of climate data only, 

 and the second using the principal components of cli- 

 mate and contaminant data. We then examined corre- 

 lations of each model's estimated explanatory time- 

 series (i.e., the summation on the right side of Eq. 3) 

 to the original climate and contaminant variables. Be- 



