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APPENDIX A
Adjustment of Intake Distributions
Used in This Report
All original analyses in this report have been based
on data from the 19771978 Nationwide Food Consumption
Survey (NFCS), which were provided by the U.S. Department
of Agriculture (USDA) for this purpose. Data were avail
able for approximately 2,400 women and 1,750 men between
the ages of 23 and 34 years.
AS described in Chapter 4, food intakes estimated on
each of 3 consecutive days were not collected by the
same technique each day. The first method of observa
tion consisted of an interview and recall of foods eaten
on the day prior to the interview. The respondent was
then instructed to keep a record of food intake for the
remainder of the day of the interview and the following
day. Subsequent statistical analyses have suggested
that either the method or the sequence of observation
days has an effect on reported intake; however, this
effect has not been considered in the analysis presented
herein. The resulting variance has been pooled with
intraindividual variance. Because the data refer to
adjacent days rather than to independent estimates of
intake, there is a potential for loss of statistical
power as a result of the design of data collection,
because of possible correlation of food intake between
days for a given person.

Note: The data analyzed in this report are for nutri
ents ingested in foods. Infonmation about dietary sup
plements was not included in the 19771978 NFCS. As a
result, all analyses presented in the report underesti
mate intake and overestimate the prevalence of inade
quate intake. The magnitude of this bias is not known.
110
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111
The USDA provided data in the fore of fixed fre
quency interval distributions. The data for individual
subjects were ranked and the mean intake computed for
each interval. Altogether there were 200 intervals,
each consisting of 0.5% of the subject days. me data
were presented in three ways: (1) data for 1 day with
out grouping data for each person (i.e., as if all data
were independent), (2) mean values for 3 days of intake
data for each person, and (3) mean values of the loga
rithm of intakes for each of 3 days for each person.
These were the basic working data sets for the analyses
presented in this report.
The USDA also conducted and reported to the subcom
mittee its analysis of variance (ANOVA) results for the
NFCS data. For this analysis, the 1day data were loga
rithmically transformed, and the subcommittee performed
an ANOVA by standard techniques, assigning variance to
model (subjects), to day (sequence), and to residual.
Subsequently, variance was assigned to on' y two
componentsmodel and residual. A typical ANOVA is
displayed in Table A1, together with an illustration of
the derivation of interindividual and intraindividual
variance estimates. From the data transformation shown
in the table, the variance attributable to subjects is
computed as:
V(subject) = (0.40930366  0.16502502)/3 a 0.081427866,
and the standard deviations (SDs) attributable to subjects
(interindividual) and to daytoday variation within subjects
TABLE A1. ANOVA: Protein Intake by Adult teen, Shown by
Logarithmically Transformed Data
Degrees
of
Source Freedom Sum of_Squares MeSE 05 aid
Modela 1 ~ 751 716e 69947157 0 ~ 40930866
Errorb 3 ~ 498 577e 257S1426 0 ~ 16502502
TotalC 5'249 1'293 ~ 95698583

a3 V( subj ect) ~ V(error).
bV(error).
CCorrected.
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112
( intrainaividua1 ) are computed as the square roots of
V(subject) and V(error). Thus, SD(~nter) = 0.2853556 and
SD(intra) = 0.4062326. The adjusted SD of 3day data may be
estimated as the square root of the sum of variances
[V(subject) + V(error)]/3 to yield a 3day SD of 0.369372.
The results of ANOVAs carried out for the NFCS data sets
are presented in Table A2. For comparison, the observed
SDs in the original logarithmically transformed data sets
are presented as well as the 3day SD derived as described
above.
In estimating the distribution of usual intakes, the
obj ective was to remove the effects of the daytoday
variation in intake, the error term in the ANOVA. This
component of variation includes both real daytoaay
variation in intake and any random error in methodology
(e.g., daytoday variation in under and overreporting of
actual intake attributable to method). Of course, it does
not adjust for any systematic bias in the data sets (consis
tent under or overreporting for individual subjects).
TABLE A2. Estimate" of Interindividual and Intra
individual Variation in Logarithmically
Transformed Datsa
Nutrient
Estimates of Variation
Nether of SD(inter SD(intra
Sub~ects individual) individual)
Males:
Protein 1,752 0.2853 0.4062
Iron 1,752 0.2909 0.3825
Vitamin A 1,752 0.5119 0.8547
Vitamin B 1,752 0.6493 0.8441
Thiamin/day 1, 752 0.3497 0.4415
Thiamin/koal 1, 752 0.1898 0.3421
Females:
Protein 2,394 0.3370 0.4468
Iron 2,394 0.3518 0.3987
Vitamin A 2,394 0.6092 0.8834
Vitamin C 2,394 0.7090 0.8843
. . .
"Derived from the subcommittee's analysis of the 1977
1978 NFCS.
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113
If all data sets fit perfectly to the normal distribution,
it would be possible to use the mean and interindividual SD
to completely describe the new distribution. However, ex~m;
nation of the distributions revealed a number of departures
from normality. An approach that was adopted might preserve
some of the uniqueness of the original distribution while
removing the effect of intraindividual variation. This
approach is described by the following algorithm, which was
applied to each interval of intake in the original trans
f ormed data set:
Adjusted intake = (observed intake  mean intake)
x SD(interindividual) + mean intake.
SD(observed)
This adjustment created a new distribution with 200 inter
vals, still in logarithmically transformed form. By com
puting the exponential of the values, the distribution was
converted back to the original units and could then be used
in subsequent computations as an estimate of the distribu
tion of usual intakes.
Descriptive information on some of the distributions used
in this report is presented in Table A3. The 1day intake
distribution consists of all singleday measurements analyzed
as if they were independent observations. The 3day intake
distribution represents the means, calculated at the level
of individuals, for three replicates of intake. The loga
rithmically transformed 3day distribution represents the
mean log of each day calculated at the level of the individ
ual. The transformed distribution, in original units, is as
described above. The most critical measure in the data pre
sented is the degree to which the transformed data fit the
normal assumption. It would have been preferable to develop
a transformation algorithm appropriate to the individual data
set before conducting the ANOVA (Box and Cox, 1964). How
ever, this exercise was not conducted for the present report
REFERENCES
Box, G. E. P., and D. R. Cox. 1964. An analysis of
transformations. J. R. Stat. Soc. B26: 211252.
Cochran, W. G., and G. W. Snedecor. 1980. Statistical
Methods, Seventh edition. Iowa State University Press,
Ames.
.
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114
TABLE A3. Characteristics of the Distributions of Nutrient Intake in
This Report
Nutrient and Observed i, b
Data Set Mean Median SD Skew Kurtosis
MALES:
Protein ( g/day )
1day data 97.8 91.4 45.0 1.185 3.021
3day data 97.8 93.9 33.9 1.038 2.639
Tran~fonned data 4.4744 4.503 0.3695 0. S71 1.273
Adjusted data 91.2 89.6 25.0 0.578 1.308
Iron ( mg/day )
1day data 15.9 14.7 7.4 1.369 3.820
3day data 15.9 15.} S.7 1.302 3.880
Transformed data 2.6570 2.676 0.3655 0.335 0.849
Adjusted data 14.9 14.5 4.3 0.849 2. 046
Vitamin A ( IU/day )
lordly data 5,S70 3,37S 9,12S 7.939 82.743
3day data 5,600 4,155 5,800 4.645 30.160
Transformed data 8.1160 8.150 0.7194 0. S2S 1.1S2
Ad jutted data 3,780 3.420 1.890 1.411 3.594
Vitami n C (m/day)
1day data 85.2 57.0 84.6 2.279 7.869
3day data 85.3 66.7 67.5 2.072 6.630
Transformed data 3.939' 3.980 0.8770 0.372 0.118
Adjusted data 62.4 52.8 39.4 1.476 3.297
Thiamin ( mg/day )
1day data 1. 53 1.36 0.87 1. 596 4.012
3day data 1.54 1.44 0.67 1.494 4.422
Transformed data 0.2197 0.310 0.4334 0.234 0.516
Adjusted data 1.40 1.35 0.49 1.135 2.903
l~hiamin (mg/1, 000 kcal )
1day data 0. 64 0 . 59 0 . 2 7 1. 570 3 . 862
3day data 0.64 0.62 0.18 1.095 2.837
Transformed data 0. 5175 0. 521 0.2738 0.179 1.232
Adjusted data 0.61 0.60 0.12 0.697 1.914
FEMALES:
Protein ( g/day )
1day data 65.5 61.6 31.1 0. 918 1.605
3day data 65.6 63.0 24.1 0.78? 1.264
Transformed data 4.0527 4.097 0.4377 1.277 4.469
Adjusted data 61.3 S9.5 18.3 0.346 0.682
Iron (mg/day)
1day data 10.8 10.0 5.3 1.367 3.800
3day data 10.8 10.2 4.2 1.318 3.962
Transformed data 2.2567 2.290 0.4195 0.848 2.796
Ad3 usted data 10. 2 9. 8 3 .3 0. 812 2. 0 88
Vitamin A ( IU/day)
1day data
3day data
Transf orbed data
Adjusted data
Vitamin C ( mg/dav )
1day data
3day data
Transf orbed data
Adjusted data
4,620
4, 690
7.8647
3, 160
73.1
72.6
3.7219
52.8
2,740
3, 340
7.916
2, 700
48.0
57.3
3.785
43.1 _
7, 360
5, 065
0.8488
1,800
72.9
56.8
0.9789
34.8
6.60S 58.298
3.911 20.675
G . 853 2.472
1.519 3.891
1.933 S. 440
1.509 2.872
0.528 0.438
1.316 2.312
71~79 .
aAlgorithms for skew calculations f ran Cochran and Snedecor, 1980, pp.
bEran Cochran and Snedecor, 1980, pp. 7981.