Is Logarithmic Transformation Really the Best Procedure for Estimating Stock-Recruitment Relationships?

Abstract

The conventional wisdom favoring the use of log-transformation in fitting stock–recruitment relationships is based on the assumed lognormality of the error structure as well as the assumed homogeneity of variance under log-transformation. Although rarely recognized, the latter is an important requirement underlying the conventional bias correction for back-transformation (i.e., exp(s ²/2), where s ² is the error variance). The homogeneity assumption may be violated much more frequently than is commonly recognized. Stock and recruitment data sets for two of eight West Coast groundfish stocks showed significant heterogeneity of variance under log-transformation, despite the low statistical power of the test (circa 25%). We simulated stock and recruitment data sets of various sample sizes and known amounts of variance heterogeneity and subjected them to three methods of regression analysis: regression under conventional log-transformation, with bias correction for the back-transformation; untransformed (i.e., arithmetic) least-squares regression; and iteratively reweighted arithmetic least-squares regression with weights proportional to the square of expected recruitment. The latter method duplicates many of the properties of regression under log-transformation. Log-transformed regression produced strongly biased estimates when variance heterogeneity was present, whereas the arithmetic approaches produced very little bias. Simple arithmetic regression tended to be unbiased but was less precise than the alternatives. Based on the root mean squared errors of the estimates of five conventional management reference points, iteratively reweighted regression generally performed as well as log-transformed regression, and it appears to be superior for small sample sizes (n ≤ 15) and in cases of suspected variance heterogeneity under log-transformation.