Mesozooplankton grazing and primary production: An alternative assessment

In a recent paper in the journal, Calbet (2001) describes the analysis of data on primary production and mesozoo-plankton grazing from a wide variety of marine ecosystems. The somewhat surprising result of Calbet's analysis is that, ''The slope of the log-log relationship between ingestion rates and PP was significantly Ͻ1, indicating a decline of relative importance of mesozooplankton grazing with increasing PP'' (Calbet 2001, p. 1824). It is a well-known fact, however, that model I regression analysis leads to a biased estimate of the slope between two variables X and Y when both variables are subject to error (Ricker 1973; Laws and Archie 1981; Sokal and Rohlf 1981). The nature of the bias causes the magnitude of the expectation value of the model I slope to underestimate the magnitude of the true slope of the underlying relationship between X and Y. Calbet's model I analysis, treating the logarithm of primary production (PP) as the independent variable (X) and the logarithm of meso-zooplankton ingestion rate (I) as the dependent variable (Y), indicates that I varies as PP is raised to the 0.64 power. However, had Calbet performed a similar analysis treating the logarithm of I as the independent variable and the logarithm of PP as the dependent variable, he would have concluded that I varied as PP raised to the 0.64r Ϫ2 power (Laws 1997), where r is the product-moment correlation coefficient between log(I) and log(PP). In this case, r ϭ 0.45, and (0.64)(0.45) Ϫ2 ϭ 3.2. Calbet's application of a t-test to determine whether the slope of the regression line is significantly different from 1.0 is inappropriate, because both the X and Y variables are subject to error, and neither was under the control of the investigator (Laws 1997). With respect to error, it is important to realize that for variables such as I and PP there are two sources of error. One is the measurement error. The other, as noted by Ricker (1973, p. 410), is the error ''inherent in the material being measured.'' In the case of I and PP, it is reasonable to assume that factors other than PP control I, and vice versa. Thus one would expect to see scatter in a plot of PP and I, even if both variables were measured with great accuracy. Model I regression methods will provide an un-biased estimate of the slope between X and Y even when both X …