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OCR for page 183

The Target Nutrient Density
of a Single Fooc]
As discussed in Chapter 4, planning for groups that include indi-
vicluals with different nutrient requirements as well as different
energy requirements is complicated. This is because inclivicluals vary
not only with respect to the amount of food they consume, but also
in their choice of foods. However, if all individuals in the group
consume a cliet consisting of a single, nutritionally complete food
(e.g., in an emergency fouling situation), then planners neeci to
account only for the variability across inclivicluals in the amount of
food they consume. In this simplified scenario, the target nutrient
density in a food can be directly obtained from the distribution of
requirements expressed as a density, as clescribeci below.
The first step in determining intake of a diet composed of a single
food (or of a mix of foocis with similar nutrient density is to obtain
a target nutrient density of the food for each subgroup in the heter-
ogeneous group.
Given a distribution of usual energy intakes in the subgroup, what
is the target density of the nutrient in the food so that the preva-
lence of nutrient inacloquacy in the subgroup is low? Calculation of
the target nutrient density in a single (nutritionally complete) food
to achieve a certain acceptable prevalence of inacloquate intakes is
simple if the distribution of density requirements is available. The
concept of a distribution of requirements of a nutrient expressed as
a density is now introduced, because it makes the planning of
intakes of a diet consisting of a single food a relatively simple task
even for a heterogeneous group.
183

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184
DIETARY REFERENCE INTAKES
THE DISTRIBUTION OF REQUIREMENTS FOR A
NUTRIENT EXPRESSED AS A DENSITY
To obtain the distribution of requirements expressed as a nutrient
density, it is necessary to know the distributions of nutrient require-
ments and the distributions of usual energy intakes in the various
subgroups that comprise the target group. For most nutrients for
which an Estimated Average Requirement (EAR) has been estab-
lisheci, the distributions of requirements have been implicitly
assumed to be normal, with mean (anci meclian) equal to the EAR,
and the coefficient of variation (CV) of 10 percent (except for nia-
cin, copper, and molybdenum, which have a CV of 15 percent, and
vitamin A and iodine, which have a CV of 20 percent tIOM 1997,
1998a, 2000b, 20014~. Even if a nutrient has a skewoci requirement
distribution, as in the case of iron and protein, the method intro-
cluceci in this section can still be applied. Following the discussion
presented in Chapter 4, it is assumed that estimates of the clistribu-
tions of usual energy intakes are available for each of the subgroups
that comprise the heterogeneous group of interest.
The approach clescribeci below to derive the distribution of require-
ments of a nutrient expressed as a density is flexible. It can be used
for any nutrient Including iron, for which the requirement clistri-
bution is known to be nonnormal). Because reliable information to
derive the distribution of nutrient density requirements when nutri-
ent requirements and energy intakes are not inclepenclent is not
available, this approach assumes independence.
To derive the requirement distribution of a nutrient expressed as
a density, proceed as follows:
1. Simulate a large number n of requirements from the clistribu-
tion of nutrient requirements in the group. For most nutrients, this
implies drawing n random values from a normal distribution with a
mean equal to the EAR of the nutrient in the subgroup and a CV
equal to 10 percent of the EAR (15 or 20 percent for some nutrients).
2. Simulate a large number n of usual energy intakes from the
distribution of usual energy intakes in the subgroup, or in a group
that is believed to be reasonably similar in energy intakes to the
subgroup of interest.
3. For each pair of simulated nutrient requirements and usual
energy intakes, construct the ratio nutrient requirement/usual energy
intake. The distribution of these n ratios is an estimate of the require-
ment distribution of the nutrient expressed as a density.

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APPENDIX C
185
As an example, the distribution of vitamin C requirements for
nonsmoking women age ci 19 to 50 years is assumed to be normal
with an EAR of 60 mg/ciay (IOM, 2000b) and a stanciarci deviation
of 10 percent of the EAR, or 6 mg/ciay. For boys age ci 14 to 18
years, the distribution of vitamin C requirements is normal with an
EAR of 63 mg/ciay and stanciarci deviation of 6.3 mg/ciay. For energy,
this example uses normal distributions with means equal to 1,900
kcal/day and 2,300 kcal/day for women and boys, respectively, and
a CV of 20 percent to represent the distributions of usual energy
intakes in each of the two subgroups. (In practice, the actual usual
energy intake distributions would be used to construct the clistribu-
tion of nutrient requirements expressed as densities. However, the
mean energy intakes and CV of energy intake used in this example
closely correspond to those that would be obtained from an analysis
of the 1994-1996 Continuing Survey of Food Intakes by Inclivicluals
PARS, 19981.)
The Statistical Analysis System program used to derive the clistri-
bution of vitamin C requirements expressed as a density in each of
the two subgroups is given at the end of this appendix. A sample
size of n = 10,000 values of vitamin C requirements and of usual
energy intakes for each of the two groups was simulated and the
ratio was constructed as clescribeci in step 3 above. The resulting
two density requirement distributions are shown in Figure C-1.
Notice that the two density requirement distributions shown in
the figure are skewoci, even though the distributions of vitamin C
requirements and of usual energy intakes were assumed to be
normal. Notice too that it is possible to compute the mean, meclian,
or any percentile of the cleriveci requirement distributions for the
nutrient densities because through the simulation, there are many
observations (in this example, 10,000) from each of the clistribu-
tions.
THE PERCENTILE METHOD TO DERIVE THE TARGET
NUTRIENT DENSITY OF A SINGLE FOOD
The target nutrient density of a single food can be directly estab-
lished from the distribution of nutrient requirements expressed as
density that was derived in the preceding section.
In the following illustrations, 3 percent is used as the desired prev-
alence of inadequate intake. Continuing with the example used
earlier, consider the problem of estimating the target vitamin C
density in a single food so that the prevalence of inadequate vita-
min ~ intakes in nonsmoking women aged 19 to 50 years and boys

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186
DIETARY REFERENCE INTAKES
1 900 -
1 800 -
1 700 -
1 600 -
1 500 -
1 400 -
1 300 -
1 200 -
1100 -
1 000 -
900 -
800 -
700 -
600 -
500 -
400 -
300 -
200 -
100 -
o -
1 500 -
1 400 -
1 300 -
1 200 -
1100 -
1 000 -
900 -
800-
~ 700-
IL 600-
500 -
400 -
300 -
200 -
100 -
Panel A
12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75
mg of Vitamin C/1,000 kcal
Panel B
12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58
mg of Vitamin C/1,000 kcal
FIGURE C-1 Simulated requirement distributions of vitamin C expressed as densi-
ties for nonsmoking women aged 19 to 50 years (Panel A) and for boys aged 14 to
18 years (Panel B). The distributions were constructed using the SAS program
presented at the end of this appendix and using information on requirements of
vitamin C for the two subgroups (IOM, 2000b). The usual energy intake distribu-
tions used in the example are hypothetical.

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APPENDIX C
187
age ci 14 to 18 years floes not exceed 3 percent. To obtain the appro-
priate density, it is necessary to estimate the 97th percentiles of
each of the density distributions so only 3 percent would have require-
ments above this density. In this example, the values obtained are
63.6 mg/1,000 kcal and 42.9 mg/1,000 kcal for women and boys,
respectively (see Figure C-1~. That is, to ensure that the prevalence
of inacloquate vitamin C intakes among nonsmoking women age ci
19 to 50 years floes not exceed 3 percent, the planner must provide
a food with a vitamin C density equal to 63.6 mg/1,000 kcal. In the
case of boys age ci 14 to 18 years, the target vitamin C density in the
food is 42.9 mg/1,000 kcal.
To plan intakes of a single food in a heterogeneous group consist-
ing of these two subgroups, the planner would provide a food with
vitamin C of density at least 63.6 mg/1,000 kcal, the higher of the
two target densities computed above. This is called the reference
nutrient density, and is a key tool for planning diets for heteroge-
neous groups.
The reference nutrient density is defined as the highest target nutrient
density among the subgroups in the group being plannedfor. It is designed
to lead to an acceptable prevalence of nutrient inadequacy in the subgroup
with the highest target nutrient density. For the entire group, the preva-
lence of inadequacy would be even lower.
By basing planning on the highest target nutrient density, the
planner guarantees that the group with the highest density require-
ments will have its neecis met. In the group with the lowest density
requirements, in this case boys 14 tol8 years of age, the prevalence
of inacloquate nutrient intakes will very likely be lower than the
target. In fact, the target nutrient density of 63.6 mg of vitamin C/
1,000 kcal is approximately equal to the 99.5 percentile of the clen-
sity requirement distribution computed for the boys. Therefore, if
the food proviclecT has a vitamin C density of 63.6 mg/1,000 kcal,
only about 0.5 percent of the boys in the group will have inacle-
quate vitamin C intakes. This target nutrient density would also
need to be evaluated to ensure an acceptably low prevalence of
intakes above the Tolerable Upper Intake Level (UL) in the boys.
The actual densities derived in this example are for illustration pur-
poses only. In practice, the planner would use a better estimate of
the distribution of energy intakes in the subgroups of interest.
The percentile method to obtain the reference nutrient density is
very general in that there are essentially no underlying assumptions
that must hold for the method to work well. In fact, in principle this

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188
DIETARY REFERENCE INTAKES
approach floes not even require that nutrient requirements and
usual energy intakes be inclepenclent; however, in practice, the incle-
penclence assumption is macle as there is no reliable information
that would allow statistical estimation of the joint distribution of
nutrient requirement and usual energy intake. Because the cleriva-
tion of the density requirement distribution and its desirable per-
centile is clone by simulation, it is not even necessary to assume that
the distribution of nutrient requirements or of usual energy intakes
is normal. Therefore, this approach can be used for iron even
though the distribution of requirements is known to be skewoci
(IOM, 2001~.
This percentile approach applies only to planning scenarios where
the target group consumes a single food item or mix of foocis with
very similar nutrient densities. In these scenarios, the variability in
intakes across inclivicluals in the heterogeneous group is clue only to
variability in the amounts of the food (or mix of foocis) consumed.
In most planning situations, however, inclivicluals vary both in the
amount of food consumed and in the choice of the foocis they con-
sume. If they choose from a selection of foocis with different nutri-
ent densities, then even if the average nutrient density is set as
above, it is possible that some inclivicluals will consume the lower-
clensity food items, while others may consume the higher-clensity
food items. When there is heterogeneity in food choices among
inclivicluals in a group, one cannot use this simple percentile
approach to estimate the necessary food density that will guarantee
a low risk of inacloquacy for almost all inclivicluals in the group.
MATHEMATICAL PROOF
~ 1
A simple mathematical proof for the result is presented here. The
symbol oc is used to denote the nutrient density, or units of the
nutrient per 1,000 kcal.
The percentile method attempts to provide an answer to the
following question: Given a certain distribution of usual energy
intakes, what is the target density, or, of the nutrient so that the
prevalence of nutrient inacloquacy in the group is low, for example,
2.5 percent?
The result proved below establishes that if the target prevalence
of inacloquacy is set at p%, then oc is the (1 - pith percentile (the
upper t1 - pith point) of the distribution of the random variable
nutrient requirement/usual energy intake.

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APPENDIX C
189
Proof of Result
To prove that the result presented above is correct, some notation
is introcluceci:
· The symbol x denotes requirement of the nutrient, and is a
random variable with some known distribution.
· The symbol y denotes the usual energy intake in the group, and
is also a random variable with some distribution.
· The symbol oc is the target density or concentration of the nutri-
ent in 1,000 kcal of the food uncler consideration. Given a usual
energy intake equal to y, the target usual intake of the nutrient is
equal to oh.
An incliviclual floes not have an acloquate target intake of the nutri-
ent if ocy < x, that is, if his or her target usual nutrient intake is less
than his or her requirement.
Suppose one wan teci to plan a nutrient density so that p% of the
group consumes an acloquate amount of the nutrient, given a cer-
tain distribution of energy intakes in the group.
Finci oc ~ (0,1) such that
Pr (ocy> a) =p
If x is cleleteci from both sicles of the
· 1e
Implies
and, therefore
Then
(1)
it
Pr (ocy- x> 0) =p
Pr (oc - x/y > 0) = p
Pr(x/y

190
DIETARY REFERENCE INTAKES
Assumptions
The result is true for just about any case. The proof above requires
only that x and y be positive. There are no conditions on the clistri-
butions of requirements and usual intakes; neither normality nor
symmetry of the two distributions is required for the result to holci.
In fact, it is not even necessary to assume that intakes and require-
ments are inclepenclent.
. .
However, In orcter to obtain a numerical value for or, specific clis-
tributions for requirements of the nutrient and for energy intakes
neeci to be chosen. Note that the result above holds even if the
distribution of requirements happens to be skewoci. Thus, the per-
centile method works for iron in menstruating women.
In the special case in which both the nutrient requirement and
the energy intake distributions are normal, it is possible to derive
an analytical expression for oc.
SAS PROGRAM TO COMPUTE THE REQUIREMENT
DISTRIBUTIONS EXPRESSED AS DENSITIES
The program below was used to obtain the two density require-
ment distributions shown in Figure C-1. Comments are given
between /* and */ symbols. The integer numbers given in paren-
theses after the rannor statements are semis to initialize the random
number generators. Any value between 1 and 99999 can be used as
a semi. The requirement distribution of a nutrient expressed as a
density is neecleci to plan intakes of a single food or of a cliet com-
poseci of various foocis with similar nutrient density.
ciata one;
do i = 1 to 10000; /* Start simulation of 10,000 vit C requirements
and energy intakes */
vcreq_w = rannor(675~6 + 60; /* women: vit C req ~ N(60, 62) */
vcreq_b = rannor(903~6.3 + 63; /*boys: vit C req ~ N(63, 6.32) */
ereq_w = rannor(432~380 + 1900; /* women: energy intake ~
N ~ 1900, 3802) */
ereq_b = rannor(500~460 + 2300; /* boys: energy intake ~ N(2300,
4602, */
ratio_w = (vcreq_w/ ereq_w)*l000; /* women: vit C requirements /
1000 kcal */
ratio_b = (vcreq_b/ ereq_b)*l000; /* boys: vit C requirements /
1000 kcal */

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APPENDIX C
output;
end;
run;
proc "chart ciata = one; /* Obtain the charts in Figure C-1 */
vbar ratio_w ratio_b/ levels = 50 space = 0;
run;
191
proc sort ciata = one; by ratio_w; /* women: obtain target density
for single food */
run;
data temp; set one; if
requirements a/
run;
proc print data = temp; run
single food */
n_ = 9700; /~ women: 97th percentile of density
/* women: print target density for
proc sort ciata = one; by ratio_b; /* boys: obtain target density for
single food */
run;
run;
data temp; set one; if _n_ = 9700;,
requirements a/
/~ boys: 97th percentile of density
proc print data = temp; run; /* boys: print target density for single
food */