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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 1977-1978 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 1977-1978 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 1-day 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 components--model and residual. A typical ANOVA is displayed in Table A-1, 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 day-to-day variation within subjects TABLE A-1. 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 3-day data may be estimated as the square root of the sum of variances [V(subject) + V(error)]/3 to yield a 3-day SD of 0.369372. The results of ANOVAs carried out for the NFCS data sets are presented in Table A-2. For comparison, the observed SDs in the original logarithmically transformed data sets are presented as well as the 3-day SD derived as described above. In estimating the distribution of usual intakes, the obj ective was to remove the effects of the day-to-day variation in intake, the error term in the ANOVA. This component of variation includes both real day-to-aay variation in intake and any random error in methodology (e.g., day-to-day 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 A-2. 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 A-3. The 1-day intake distribution consists of all single-day measurements analyzed as if they were independent observations. The 3-day intake distribution represents the means, calculated at the level of individuals, for three replicates of intake. The loga- rithmically transformed 3-day 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: 211-252. Cochran, W. G., and G. W. Snedecor. 1980. Statistical Methods, Seventh edition. Iowa State University Press, Ames. .

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114 TABLE A-3. 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 ) 1-day data 97.8 91.4 45.0 1.185 3.021 3-day 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 ) 1-day data 15.9 14.7 7.4 1.369 3.820 3-day 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 3-day 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) 1-day data 85.2 57.0 84.6 2.279 7.869 3-day 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 ) 1-day data 1. 53 1.36 0.87 1. 596 4.012 3-day 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 ) 1-day data 0. 64 0 . 59 0 . 2 7 1. 570 3 . 862 3-day 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 ) 1-day data 65.5 61.6 31.1 0. 918 1.605 3-day 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) 1-day data 10.8 10.0 5.3 1.367 3.800 3-day 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) 1-day data 3-day data Transf orbed data Adjusted data Vitamin C ( mg/dav ) 1-day data 3-day 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. 79-81.