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equate intakes as the proportion of the population with usual intakes below the median requirement (EAR).

To understand how the cut-point method works, the reader is referred to Chapter 4, where the joint distribution of intakes and requirements is defined. Figure 4-8 shows a simulated joint distribution of intakes and requirements. To generate the joint distribution, usual intakes and requirements for 3,000 individuals were simulated from a χ2 distribution with 7 degrees of freedom and a normal distribution, respectively. Intakes and requirements were generated as independent random variables. The usual intake distribution was rescaled to have a mean of 1,600 and standard deviation of 400. The normal distribution used to represent requirements had a mean of 1,200 and standard deviation of 200. Note that intakes and requirements are uncorrelated (and in this example, independent) and that the usual intake distribution is skewed. An individual whose intake is below the mean requirement does not necessarily have an inadequate intake.

Because inferences are based on joint rather than the univariate distributions, an individual consuming a nutrient at a level below the mean of the population requirement may be satisfying the individual 's own requirements. That is the case for all the individuals represented in Figure 4-8 by points that appear below the 45° line and to the left of the vertical EAR reference line, in triangular area B.

To estimate prevalence, proceed as in equation 1, or equivalently, count the points that appear above the 45° line (the shaded area), because for them y < r. This is not a practical method because typically information needed for estimating the joint distribution is not available. Can this proportion be approximated in some other way? The probability approach in the previous section is one such approximation. The EAR cut-point method is a shortcut to the probability approach and provides another approximation to the true prevalence of inadequacy.

When certain assumptions hold, the number of individuals with intakes to the left of the vertical intake = EAR line is more or less the same as the number of individuals over the 45° line. That is,

or equivalently,

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