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Fishery Bulletin 91 12). 1993 



Appendix 



We encountered many statistical difficulties and un- 

 certainties in our biostatistical approach to detecting 

 contaminant influences. Anyone attempting to inter- 

 pret our results or to conduct a similar analysis should 

 be cognizant of the following issues concerning data, 

 biological understanding, and statistical methodology. 



Difficulties relating to the data 



• imprecision in data on stock and recruitment; 



• imprecision in measurements of physical (climatic) 

 conditions; 



• imprecision in measurements and reconstructions of 

 contaminant loadings; 



• lack of residence-time models for contaminants in 

 the water column and sediments, necessitating the 

 use of loading (influx) values, rather than estimated 

 in situ concentrations; 



• substantial environmental variability that cannot be 

 modeled effectively, and that would thus become 

 "noise"; 



• strong trends in some resource and contaminant his- 

 tories that can produce ambiguous correlations be- 

 tween these quantities; 



• time-series that are short by statistical standards, 

 although long by comparison with others available; 



• multicollinearity: similar patterns over time in dif- 

 ferent explanatory variables. 



Shortcomings in biological understanding 



• only rudimentary knowledge of biological response 

 to environmental variability and contaminants; 



• poor models of environmental influences on larval 

 production and recruitment; 



• poor knowledge of spawning-at-age schedules, espe- 

 cially in light of increased evidence of multiple 

 spawnings within seasons; 



• little knowledge of natural mortality rates for most 

 species; 



• little or no knowledge of the true form of stock- 

 recruitment relationships for most species. 



Problems with statistical methodology 



• lack of methods to address the problem of errors-in- 

 variables, except in the simplest cases; 



• lack of methods to determine the statistical power of 

 hypothesis tests; and 



• lack of methods to identify conclusively a model with 

 the optimum number of parameters, or, for that mat- 

 ter, the correct non-zero parameters; 



• possible presence of nonlinear effects, but insuffi- 

 cient data to identify and estimate a nonlinear model. 



Similar problems arise in diverse areas of biostatisti- 

 cal analysis of populations and ecosystems, including 

 most studies conducted for fishery management. 



We attempted to reduce the effects of collinearity 

 among the explanatory variables by using principal- 

 components analysis. Unfortunately, stock size was at 

 times correlated to other variables. The inability to 

 identify effects of individual climate or contaminant 

 variates is an unavoidable result of collinearity. This 

 problem was accentuated by autocorrelation in the ex- 

 planatory data and in the stock and recruitment se- 

 ries. Time trends in several explanatory variables, in- 

 cluding stock size, compounded the problem of 

 multi-collinearity, as they can lead to spurious 

 (noncausal) correlations. 



Even after the principal-component analysis, the 

 suite of possible explanatory variables was large. For 

 parsimony, we chose subset models for each stock. As 

 stated, it is nonetheless probable that our models con- 

 tain specification error. There are three main forms of 

 specification error: ( 1 ) omission of relevant variables 

 or inclusion of irrelevant ones, (2) a wrong functional 

 form for the model (e.g., quadratic response when lin- 

 ear is specified), and (3) changes in the true param- 

 eters over time. Consequences of specification error 

 can include biased parameter estimates and biased 

 estimates of variance (Kennedy 1979), which in turn 

 can lead to reduced statistical power. Using principal 

 components, which are biologically arbitrary combina- 

 tions of variables, may conceivably contribute to speci- 

 fication error (including errors in variables, as defined 

 below). 



Most of the recruitment time-series derived from 

 VPA were short and did not coincide exactly with the 

 contaminant data series, themselves not exceptionally 

 long. A major consequence of modeling with a small 

 sample size is low statistical power. 



Observation error in explanatory variables causes 

 difficulties in OLS parameter estimates, including in- 

 consistency and, in the bivariate linear case, bias to- 

 wards zero (Theil 1971, Kennedy 1979). This is known 

 in the field of econometrics as the "errors-in-variables" 

 or "errors-in-predictors" problem, and is sometimes con- 

 sidered a special case of specification error (e.g., Theil 

 1971). It has been demonstrated that recruitment mod- 

 els used in fisheries are subject to this problem (Walters 

 & Ludwig 1981, Goodyear & Christensen 1984). If one 

 is interested only in a predictive model, the bias is not 

 important. If the parameter estimates are the object, 

 however — as in the present case — the bias is especially 

 problematic because its direction and magnitude are 



