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APPENDIX E Analysis of Error in the Estimation of Nutrient Intake Using Three Sample Data Sets The impact of two different kinds of error on the preva- lence estimate is described in Chapter 7. There, the sub- cammittee examined in detail two potential sources of error that can affect the estimation of nutrient intake: errors in estimating the composition of the food item consumed and errors in estimating or recording the amount of each food item consumed. In this appendix, the committee ex ~ nes the potential impact of unmeasured errors of this kind on the probability approach. A distinction will be made between random errors (deviations moving in both directions around a true mean) and systematic errors or biases (consistent under- or over- estimat~on of the true value). A distinction will also be made between the impact of error in assessing a single serv- ing of a single food and in calculating intake from a ser- ies of servings of foods in one day. Emphasis is placed on the ef feet of these errors on the estimated distribution of usual intakes across people rather than on actual intakes of particular individuals. These constructs are first illus- trated using actual data, and then their theoretical implica- tions are developed. The initial assumption of this analyses is that the food composition analyses are correct (e.g., no systematic bias) but that there is variation in reported composition. 129

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130 VARIABILITY OF FOOD COMPOSITION The most recent reference tables on food composition developed by the U.S. Department of Agriculture (USDA, 1976- 1984) provide come information about the nether of samples analyzed and the standard error of the mean for some foods. For these foods, the standard deviation (SD) of the nutrient composition can be calculated, and the coefficient of variation (CV = 100 x SD/mean) can be derived. Although the standard error (SE) is dependent upon the nether of sam- ples analyzed and describes the reliability of the estimate of the mean, the SD is not dependent on the number of sam- ples per se (provided there are sufficient samples and anal- yses to supply a good description of the full range of foods) and furnishes a description of the range of values that can be taken by a specific sample of the food. The SE is a mea- sure of the variability of the mean of the population and in that sense is a measure of the error that might be encoun- tered in accepting the average composition of a particular food as the reference data. In the Chapter 6 analysis, therefore, the SE has been used to calculate confidence limits. For present purposes, however, the SD is more meaningful than the SE of the mean. The CV expresses this variability in relation to the mean, and it is useful in this exercise for comparing error in estimating nutrient content between several foods and for considering the Impact of the error on the estimate of the daily intake of a nutrient, as used in dietary evaluation. Because the SD cannot be estimated from the reference tables for all food items, the available SDs were examined and used to make a judgment about the possible CV or range of CVs that might apply for foods with missing data. The food composition tables indicate that the relative varia- bility of micronutrients is greater than the variability of protein; this difference seems biologically plausible. The USI)A provides no CV estimates for energy, because the ref erence data for energy concentration are computed rather than measured values. Two kinds of data analyses were used to examine the impact of variability on dietary evaluation. In the first analysis, hypothetical variance estimates are assigned to a food record for a vegetarian diet. The variability estimates used for this analysis are shown in Table E-1. The subcommittee assumed that the magnitude of the CV is different for various nutrients, but the level of nutrient was not taken into account. Subsequent analyses, based as much as possible on

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131 TABLE E-1. Assumed Variability in Food Composition Data Used in Estimating the Errora Class of Vector Range of CVs (%) Energy Protein Other nutrients - aData from G. H. Beaton, University of Toronto, per- sonal ca~unication, 1985. 10-30 10-20 10-45 reported variance estimates and complemented by imputed vari ances, are presented in a later section of this Appendix. These variance estimates were applied with a simulation procedure to the dietary intake record of a vegetarian sub- ject studied in Toronto. The food composition data reported by USDA (1976-1984) were used to estimate the average composition of each of the 21 foods included in the record. A variability was assigned to each food item by random selection within the ranges presented in Table E-1 by using the algorithm CV (food item X) = 10 + RNDt 1) x Y. where Y = 20 for energy, 10 for protein, and 35 for other - nutrients . Thus, for each food item and each nutrient, there was a mean composition and CV. his procedure was used to randomly assign a specific composition for each food item or nutrient combination. A random value from the normal distr~- bution, represented by the mean and CV for that food item, was chosen. Table E-2 presents the results that accrued from 1,000 repetitions of this exercise and computations of the SD and CV for the computed nutrient intake. The results show that the relative error is decreased for the total record of food intake in comparison to the individual food items. me exercise could be repeated by selecting new random values for the CVs of the food items and then obtaining composite error estimates, which would not be expected to differ mar- kedly from those shown in Table E-2. The table also presents the direct calculation of the variances and the SD and CV of the total intake as the sum of variances of the individual item by conventional statistical approaches. Given the assumptions of normality for the individual composition distributions, this is a much more rapid approach than the

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132 TABLE E-2. Potential Error in a Person's Estimated Nutrient Intake Attributable to Variance in Ford Componition Data on Sample Vegetarian D4eta~b Food Composition Data With Variance Added to Food Compositions Nutrient No Variance, By Randomization Approach By Statististical Formula Vector Mean Mean SD CV (I) Mean SO CV (I) Energy (keel/day) 2,610.4 2,619.6 146.37 5.60 26,10.4 146.02 5.59 Protein (g/day) 68.8 68.7 3.96 5.76 68.8 4.04 5.87 Calcium (mg/day) 814.1 812.7 86.49 10.64 814.1 87.29 10.72 Iron (mg/day) 29.1 29.4 3.48 11.85 29.1 3.43 11.76 Vitamin A (It//day) 13,085.5 13,070.0 1,912.67 14.63 13,085.5 1,880.3 14.37 Thicken (mg/day) 2.3 2.3 0.3 12.69 2.3 0.29 12.73 Vitamin C (mg/day) 303.6 302.8 29.52 9.75 303.6 30.91 10.18 aMean and standard deviation. based on 1,000 iterations with normally randomized variables in randomization approach. Statistical formula represents addition of variances under the resumption that each variance is normally distributed with mean and CV as described. For the CV of ford composition randomly assigned to each nutrient, see Table E-~. There CVs we as high as 45% for individual foods. bData from G. H. Beaton, University of Toronto, personal communication, 1985. repeated calculations based on random selections. The com- parison of the two methods in Table E-2 shows that the results are practically identical. A member of the subcommittee (H. Cmiciklas-Wright, Penn- sylvania State University, personal communication, 1985) provided two food intake records for use in a second set of analyses. New USDA food composition data and variance estimates (reported standard errors and number of analy- ses) were available for most of the foods in these records (USDA, 1976-1984). The data provided by Smiciklas-Wright were used as more realistic examples for modeling the vari- ance in estimated intake attributable to variability in the food composition data. The first step was to impute variabilities for food com- position when they could not be derived directly from the USDA tables. An internalized empirical exercise was used: CVs were calculated f or all f gods, when data permitted, and were plotted in relation to the level of nutrient reported

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133 in the food. me plots suggested that the range of the CV increased markedly at low concentrations of nutrient. This increase may reflect limitations of methods for determining food composition, because an absolute contribution of method- ologic error may become a large relative error at the lowest levels of nutrient concentration. Alternatively, it may simply mean that at low levels, the biological variation is not proportional to the mean. Nevertheless, it appears that above a nutrient-specific break point, the variability seems to relate to the mean, and the range of CVs is diminished. This apparent relationship was used in imputing CVs in the two sample diets in the exercise. The stratification of CV ranges is shown in Table E-3. Using the ranges shown in Table E-3 and the randomized approach discussed earlier for the vegetarian diet, the sub- committee assigned estimates of variability to all foods for which a direct derivation could not be made from data pro- vided by the USDA. These data were then examined to deter- mine the error in the estimated 1-day intake (see Table E-4). TABLE E-3. Stratification of CV Ranges for Use in Assigning Variability of Food Composi- tion in Nonvegetarian Food Intake Recordsa Nutrient Cutoff Point CV Range Assumed (%) (per 100 g) Below Cutoff Above Cutoff Protein 2 g 5 - 50 5 - lS Calcium 20 mg 5 - 50 5 - 15 Iron 1 mg 5 - 65 10 - 30 Magnesium 10 mg 5 - 50 10 - 30 Sodium 100 mg 5 - 65 5 - 15 Zinc l mg 5 - 65 lo - 30 Thicken 0.05 mg 5 - 50 10 - 30 Riboflavin 0.05 mg 5 - 50 lo - 30 Niacin 0.5 mg 5 - 65 5 - 15 Vitamin C 7.5 mg 5 - 50 10 - 30 Vitami n B6 0.1 mg 5 - 50 10 - 30 Folacin 20 mg S - 65 lo - 30 Vitals n A 300 IU 5 - 65 10 - 30 aData frae H. Semi ciklas-Wright, Pennsylvania State University, personal communication, 1985.

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134 TABLE E-4. Comparison of Potential Error Due to Variability of Food Composition Associated with Estimated 1-Day Intakes, Non- vegetarian Dietsa, b Espy mated 1-Day Intake _ _- Diet HW1 Diet HW2 Nutrient Mean SD CV ( ~ ) Mean SD CV ( ~ ) Protein 104.6 6.20 5.93 97. 5 2.21 2.27 Calcium 1,540.2 80.77 5.24 1,135.2 61.31 5.40 Iron 8.03 1.19 14.85 10.4 1.66 16.00 Magnesium 250.1 15.70 6.28 222.4 13 .04 5.86 Sodium 4,129.5 157.36 3.81 2,589.8 121.73 4.70 Zinc 11.6 0.909 7.85 13.3 1.64 12.33 Miami n 2.10 0. 37S 17.92 0.715 0. 076 10. 59 Ribof lavin 2 . 60 0 .2 05 7 . 90 2 .13 0 . 154 7. 22 Niacin 15.9 0.908 5.72 13.5 0.879 6.53 Vitamin Be 1.45 0.136 9. 37 1.43 0. 210 14.62 Vitam; n C 153.1 11. 91 7. 77 11. 8 1. 54 13 . 00 Folacin 184.3 19.80 10.74 97.1 ~ 2. 02 12. 38 Vitals n A 3,798.4 281.24 7.40 5,142.0 603.61 11.74 =sylvania State University, per- sonal communication, 1985. bSee Tables E-11 and E-12 for diet composition. Here the variance of 1-day intake was computed by statisti- cal algorithm rather than by simulation. For most of the foods reported in the first diet (HW1), there were standard errors from which variance estimates could be derived (see Tables E-9 through E-12 at the end of this appendix). The results are realistic estimates of the potential error of the estimated 1-day intake. For the second diet (HW2), a higher proportion of the variability for individual foods had to be imputed ( see Table E-12). Differences in the CV of the intake estimate for the two diets can be attributed to differences in variability asso- ciated with individual foods. The CV of the diet is also affected by the relative contributions to intake f rem indi- vidual foods with particularly high or low variabilities.

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135 Effect of Increasing the Number of Foods in the Diet Although it may not be apparent from a comparison of the three diets, it can be demonstrated by statistical theory that increasing the number of foods included in the record will decrease the relative variance of the total intake estimate. This effect is illustrated in Table E-5. In this model it is assumed that all foods make an equal con- tribution to total intake and thus exert the same impact upon variance of the sum. The table displays the impact of the number of foods in the record by using several hypo- thetical CVs for the food composition data. RANDOM ERROR IN THE MEaSUREMENT OF FOOD INTAKE - If the measurement or recording of actual intake of individual food items includes an implicit error because some items are underestimated and some are overestimated, then there measurements will lead to error in estimation of the 1-day intake of nutrients. TABLE E-5. impact of the Number of Food Items in a Record on the Error Term for Computed Nutrient Intakea Number of CV (I) of Nutrient Content of Individual Foods in Food Serving Record 10 20 30 40 50 2 7.1 14.1 21.2 28.3 35.4 3 5.8 11.6 17.3 23.1 28.9 4 5.0 10.0 15.0 20.0 25.0 5 4.5 8.9 13.4 17.9 22.4 10 3.2 6.3 9.5 12.7 15.8 15 2.6 5.2 7.8 10.3 12.9 20 2.2 4.5 6.7 8.9 11.2 25 2.0 4.0 6.0 8.0 10.0 30 1.8 3.7 5.5 7.3 9.1 aThese calculations assume that all foods make an equal contribution to the total intake and that all food servings have the same error terms. The values are based on a simulated distribution.

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136 For analysis of measurement error when no variance in the food composition is taken into account, the considerations are identical to those discussed in the preceding section. The solution can be obtained by adding the variances, and the effects will be exactly as calculated for the variability of food composition tables. When the model includes error both from measurement and from variation in food composition, the variance of a prod- uct must be computed. Statistical equations for the approx- imation of this variance have been developed by FAO/WHO/UNU (in press). If it is accepted that there is no correlation between the two variations, the following equation can be used to estimate the variance of the product of intake and food composition: V = I2 x v(c) + c2 x V(I) + v(c) x V(I)' where I2 is the square of reported mean intake of units of food; c2 is the square of reported mean concentration of nutrient per unit of food; V is the variance of content of a food whose content is I x C; V(I) is the variance of the intake measurement; and V(C) is the variance of the com- position measurement. Thus the equation assumes no corre- lation between values of I and C, although approximations are available for situations in which there is a correla- tion. The result is a variance for each it-m that is then summed for the total intake. To illustrate the impact of variation on estimations of the actual amount of the food items consumed, a hypothetical 10% CV for measurement will be assumed (see Table E-6). This illustration is based on the vegetarian diet described ear- lier. In the simulation, values were selected at random from two normal distributions (one for the intake estimate and one for the composition estimate) for each food item, and 1,000 iterations were performed. Using statistical calculations rather than the simulated approach, a member of the subcom- mittee performed a similar exercise for the data sets for diets HW1 and HW2. Comparison of these variance estimates with those devel- oped earlier for food composition alone reveal that the effect of adding a second source of variation, although real, is less than might have been anticipated. Unless the random error is very large , there will be a limited additional effect on the error term generated by food composition varia-

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137 TA=E E-6. Error Term in 1-Day Intakes Associated with Variability of Food Composition and E rror in Intake Estimate in Non~regetarian Diets a - Diet - 1 Diet BW2 ~ SD CV ( ~ ) Mean SD CV ( ~ Nutrient Protein 109.6 7.56 7.23 97.5 5.81 5.96 Calcium 1,540.2 103.7 6.74 1,135.2 82.52 7.26 Iron 8.03 1.23 15.35 10.40 1.73 16.62 Magnesium 250. 0 17. 72 7. 08 222 .4 15. 51 6. 97 Sodium 4,129.5 239.3 5.80 2,589.8 180.3 6.95 Zinc 11.58 1.00 8.67 13.32 1.76 13.22 miamin 2.10 0.395 18.85 0.716 0. 080 11.13 Riboflavin 2.60 0.226 8.71 2.13 0.175 8.21 Niacin 15.89 1.18 7.43 13.46 1.29 9.49 Vitamin Be 1.45 0.149 10.26 1.43 0.227 15.83 Vitamin C 153.1 14.78 9.65 11.85 1.61 13.56 Folacin 184.3 21.12 11.46 97. 07 12. 72 13.10 Vitamin A 3,198.4 313.2 8.25 5,142.0 683.0 13.28 ~ . . aData from H. &iciklas-Wright, Pennsylvania State University, personal communication, 1985. For composition of diets and food composition variability estimates, see Tables E-ll and E-12. (CV is based on the ass~mptlon that measurement error in 10. normally distributed.) bility. The estimates of protein intake in the HW1 data lead to a 5.9% CV of the estimate of total protein intake when only food composition variability is considered (see Table E-7). However, when measurement error is added, the CV increases to 7.2% (see Table E-6). are 14.9% and 15.4%. For iron, the two CVs The magnitude of the effect depends on many factors, including the relative contributions of various food items to the final intake (weighting of the relative variances); the nether of food items as discussed in the preceding sec- tion for food composition variation; and, importantly, the magnitude of the two variances. Table E-8 illustrates the effect of the estimated variability (error teem) for an individual food item when there is variability both in food composition and in estimation of food quantity. As shown in Table E-5, the relative variance of the total intake for many individual foods would decrease as the number of foods increases.

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138 TAME E-7. Error in 1-Day Intakes Attributable to Varia- bility in Food Composition and Intake Esti- matea Sample Diets HW1 HW2 ~ . Nutrient Mean SD CV (96) Mean SD Cal (9e ) . Protein 109.6 7.56 7.23 97.5 5.81 5.96 Calcium 1,540.2 103.7 6.74 1,135.2 82.52 7.26 Iron 8.03 1.23 15.35 10.40 1.73 16.62 Magnesium 2SO.0 17.72 7.08 222.4 15.51 6.97 Sodium 4,129.5 239.3 5.80 2,589.8 10.3 6.95 aNormally distributed with CV measurement error assumed to be 10%. TABLE E-8. Impact of ~ Random Error in Intake and Food Composition Data on the CV Calculated for Nutrient Content of an Individual Serving of Fooda~b CV 2 0 10 20 30 40 0 0 10 20 30 40 10 10 14.2 22.4 31.8 41.4 20 20 22.4 28.6 36.6 45.4 30 30 31.8 36.6 43.4 51.4 40 40 41.4 45.4 51.4 58.8 l aData from NFCS. Values are relative. ball values expressed as CV = 100 x SD/mean. not important to know which variable is 1 or 2. m e error term for a diet comprising several individual servings of foods would necessitate a summation of variances (see Table E-5).

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139 These analyses demonstrate that the true intake of nutri- ents by a person on a particular day differs from the esti- mated intake and suggests that the standard deviation of this error for mixed diets containing 15 to 20 different foods is likely to fall in the range of 5% to 15%, depending on a nether of factors. Thus it can be assumed that 95% of the time the estimated intake will fall within 10% to 30% of the actual intake of a nutrient. The error in the estimate of a particular person's intake on a certain day is appreciable. CONCLUSIONS These analyses demonstrate that random variation in food composition (including random errors in analysis) and in the estimation of food intake introduces an element of variation in computed nutrient intake across days for 1-day records and that the relative impact, although not as large as might have been expected, is nevertheless real. These considera- tions suggest that part of the reported difference between calculated intake and chemically determined intake for duplicate meals or composite diets may arise from random error and that perfect agreement should not be expected. In considering the distributions of nutrient intake in population data, the data on variability of food com- position discussed in this appendix are not normally included. That is, the true variability of 1-day intake is greater than would be estimated with conventional techniques based on average composition data from the food composition table. More import ant in the context of the present report is the impact of random variation on estimation of the prevalence of inadequate intake. Part of the unmeasured variation associated with the 1-day intake estimate would clearly be factored out by the analysis of variance (ANOVA) procedure used to estimate the distributions of usual intake in the population. This part of the variation would have no final impact on the estimate of prevalence. Thus, there is no need to measure or estimate its magnitude. To determine if the entire effect is factored out in the ANOVA, a statistical model was developed (see Chapter 8). For this model, SEs were estimated from the food composition table for the diet HWl presented in this appendix.

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140 A similar approach for deriving the SE of a 1-day intake was used to estimate the SD and CV, but SEs rather than SDs of composition of individual foods were used as the starting point. The results demonstrate that random variation as dis- cussed in this appendix influences the confidence 1 ~ ts of the estimate of usual intake and may also influence the esti- mate of prevalence. If the prevalence estimate is below 50%, the effects will lead to a slight underestimation of the prevalence, and if the prevalence is above 50%, the effects will somewhat overestimate it. Fortunately, as demonstrated in Chapter 8, the under- or overestimations and the impact of confidence limits are not so great as to invalidate the approach to assessment. Nevertheless, it is clear that Improvement of food composition data bases can improve the estimate of the prevalence of inadequate intake. True bio- logical variation between individual samples of food will limit the improvement that can be gained. Modeling ap- proaches such as those presented in this appendix together with those presented in Chapter 8 can be used to ascertain which types of improvements in the food composition data base would have the greatest impact on estimations of the prevalence of inadequate intakes. Analyses of this kind can provide the basis for establishing priorities for future analytical work. True systematic biases in either food composition or food intake data are not considered in the analyses presented herein, but are discussed in Chapter 7. As was shown, these effects, if present, will influence the prevalence estimates. Elimination of systematic biases due to errors in methods should receive a high priority for this reason. REFERENCES FAO/WHO/UNU (Food and Agriculture Organization/World Health Organization/United Nations University). In press. Energy and Protein Requirements. Report of a Joint FAO/WHO/UNU meeting. World Health Organization, Geneva. USDA (U.S. Depart ment of Agriculture) . 1976-1984. Com- position of Foods: Raw, Processed, Prepared. Agri- culture Handbook No. 8. Sections 1-12. Agricultural Research See vice, U. S . Department of Agriculture, Washington, D.C.

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142 TABLE E-10. Example of the tion Eat: mates Application of Random Selection of Food Compo~i- for Calciuma . . Tabulated Hypothetical Calcium Content Content (mg in food consumed) Food Item ( mg ) 1 2 3 4 5 6 7 8 9 10 Watermelon 56 16 28 63 51 70 53 63 87 89 31 Cherries 113 185 178 184 134 90 67 191 190 140 77 Soy mi lk concentrate 38 40 46 53 52 16 37 42 32 52 25 Cereal mix 67 77 64 101 40 62 109 87 54 43 61 Figs 11 11 7 14 6 20 8 15 13 18 11 Lettuce 8 7 12 8 7 9 9 8 8 5 9 Cucumber 18 14 20 22 14 13 16 17 12 17 24 Tomato 10 2 13 12 6 14 10 19 13 15 9 Cabbage 54 75 48 40 62 35 51 74 64 57 44 Green peppers } 1 1 2 2 1 1 1 1 1 1 Avocado 4 6 3 5 3 4 5 6 5 2 3 Olives 29 31 27 35 28 32 28 33 26 27 33 Green onions 8 8 9 8 5 5 11 9 7 10 8 Bread, white 49 49 90 25 35 62 42 81 24 36 49 ( nonmilk) Mayonnaise 14 16 11 10 11 9 13 9 20 14 11 Corn on the cob 4 5 2 5 6 6 4 4 5 3 3 Peanut butter 126 134 143 136 131 142 102 148 166 157 104 Kidney beans 110 73 167 178 127 183 95 87 230 100 274 Celery 2 2 2 2 3 3 2 1 1 3 2 Cantaloupe 24 24 30 34 17 20 38 32 16 24 33 Black currants 6 6 8 6 6 4 5 6 7 8 7 Total 814 784 907 942 746 801 706 933 980 822 820 abased on vegetarian diet described in Table E-9. Overall mean = 844. 5; SD ~ 91.45; and Cal = 10.83~.

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