Enigmatic relationship between chlorophyll a concentrations and photosynthetic rates at Station ALOHA

An ordinary least squares (OLS) analysis of the relationship between chlorophyll a (chl a) concentrations and photosynthetic rates at depths of 5 and 25 m at Station ALOHA produced a slope that was only 28% of the mean productivity index at those depths and an intercept at zero chl a that equaled 70% of the mean photosynthetic rate. OLS regression lines are known to produce a slope and intercept that are biased estimates of the true slope and intercept when the explanatory variable, X, is uncontrolled, but in this case the measurement errors and natural variability of the chl a concentrations were much too small to explain the apparent bias. The bias was traceable to the fact that the photosynthetic rates were determined by more than one explanatory variable, a source of variability that is typically overlooked in discussions of OLS bias. Modeling the photosynthetic rates as a function of the product of chl a and surface irradiance produced a much more accurate and realistic description of the data, but the OLS continued to be biased, presumably because the photosynthetic rates were functions of factors in addition to chl a and surface irradiance (e.g., temperature, macronutrients, trace metals, and vitamins). The results underscore the need to recognize that the absence of bias in an OLS when X is not controlled implies that all scatter in the data about the OLS is due to errors in the dependent variable, an unlikely scenario. In most cases, resolution of the bias problem will require identification of the explanatory variables in addition to X that determine the dependent variable.