STATISTICAL-MODELS FOR LIMITING NUTRIENT RELATIONS IN INLAND WATERS

The ecological theory of limiting factors holds that the observed leve l of response in a biological process will be governed by the input fa ctor in least supply-the limiting factor. This theory has formed the b asis for numerous attempts by aquatic ecologists to describe the relat ion between the biological productivity of inland waters and the avail ability of plant nutrients required for algal growth. Regression analy sis has been the primary statistical tool used in the development of s uch relations, yet any statistical model that represents the limiting effect of some explanatory factor as an expectation contradicts the su bstantive theory of limiting factors. Limnological data not resulting in an adequate regression of chlorophyll on phosphorus have been viewe d as failing to support the limiting effect of this nutrient on algal biomass in lakes. But when represented by a more appropriate model, su ch data may be seen to provide similar evidence for the relation of ch lorophyll to phosphorus as does data resulting in a strong regression. Data from limnological studies often exhibit a scatter of points dist ributed in the shape of a triangle lying beneath an upper boundary. Ap propriate models for such data are introduced to describe the upper bo undary or potential limit, the distribution of points falling below th e limit, and the degree of random error. An application of the EM algo rithm provides marginal maximum likelihood estimates of the parameters in the more complex models considered. Several results are given for the models, including a goodness-of-fit diagnostic and estimation of t he Large-sample parameter covariance matrix. Application of the models is illustrated by fitting empirical relationships between chlorophyll and the plant nutrient phosphorus in temperate lakes.