(1)

For a given estimate of the joint distribution ƒ_Y,R, obtaining equation 1 is trivial. The problem is not the actual probability calculation but rather the estimation of the joint distribution of intakes and requirements in the population.

To reduce the data burden for estimating ƒ_Y,R, approaches such as the probability approach proposed by the National Research Council (NRC, 1986) and the Estimated Average Requirement (EAR) cut-point method proposed by Beaton (1994), make an implicit assumption that intakes and requirements are independent random variables —that what an individual consumes of a nutrient is not correlated with that individual's requirement for the nutrient. If the assumption of independence holds, then the joint distribution of intakes and requirements can be factorized into the product of the two marginal densities as follows:

ƒ_Y,R(r, y) = ƒ_R(r)ƒ_Y(y) (2)

where ƒ_Y(y) and ƒ_R(r) are the marginal densities of usual intakes of the nutrient, and of requirements respectively, in the population of interest.

Note that under the formulation in equation 2, the problem of assessing prevalence of nutrient inadequacy becomes tractable. Indeed, methods for reliable estimation of ƒ_Y(y) have been proposed (e.g., Guenther et al., 1997; Nusser et al., 1996) and data are abundant. Estimating ƒ_R(r) is still problematic because requirement data are scarce for most nutrients, but the mean (or perhaps the median) and the variance of ƒ_R(r) can often be computed with some degree of reliability (Beaton, 1999; Beaton and Chery, 1988; Dewey et al., 1996; FAO/WHO, 1988; FAO/WHO/UNU, 1985). Approaches for combining ƒ_R(r) and ƒ_Y(y) for prevalence assessments that require different amounts of information (and assumptions) about the unknown requirement density ƒ_R(r) and the joint distribution F_Y,R (y, r) are discussed next.