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Table 3. Food composition and variability estimates associated with food record HW2a

a. CV values are shown below average composition

A simulation operation was then conducted. For each food and nutrient item, a random composition value was obtained from the normal distribution described by the mean and CV developed as above. This was done independently for each food. The intake summed across all foods was then estimated. This exercise was repeated 1,000 times and then the mean and standard deviation of the 1,000 estimates of nutrient intake were calculated. This SD is a measure of the variability or error term associated with the one-day intake estimates that would be derived by using the average composition values presented in the food composition tables. The results of this exercise are shown in table 4. In comparison to table 1, there is a striking reduction in the relative variability of food composition in the mixed diet compared to that of the individual foods!

This can also be estimated by statistical formula. The variance of a sum is simply the sum of the variances. Applying this approach, the above results can be reproduced by statistical equation rather than by simulation analyses. Extending this approach, the relationship between the number of foods included in a diet and the variability of composition of individual foods can be presented in matrix form [table 5). It is to be noted that for the purposes of this table all foods are assumed to make an equal contribution to the total nutrient intake, which was not the case in the simulation analyses presented in table 4 or in other real diets. Table 5, then, must be seen as illustrating a principle.

The phenomenon portrayed is self-apparent: as the number of food items increases, the relative variability of composition decreases. That is, with a very diverse diet, variations or random errors in the composition of individual foods has Et much smaller impact than in a diet with only a few foods. Diets HW1 and HW2 contained 15 and 13 foods respectively.

Additional impact of a random error in intake estimation

Given the above construct of the impact of food composition variation, in order to offer informational perspective it is possible to extend the model to include the impact of another source of variation - an error in the estimation of portion size when obtaining the record of intake. In this case the two variances (composition and intake estimate) would be multiplicative. Assuming that they are not correlated, the following equation would serve to predict the variance of the estimated nutrient content of a single food item. The equation is based on one presented in the FAO/WHO/UNU report on Energy and Protein Requirements [10].

V_{(food 1)} = I²*V_(C) + C²*V_(I) + V_(c)*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 _{(food 1)} is the variance of content of food 1 where content is I*C;
- V_(I) is the variance of the intake measurement;
- V_(c) is the variance of the composition measurement.

(The equation presented assumes no correlation between values of I and C; approximations are available for situations in which correlation exists.)

This equation derives the combined variance for a single food. To obtain the variance (and then SD and CV) for a one-day intake, the variances of the individual foods must be summed. The exercise was carried out first in a simulation exercise with 1,000 iterations and then by the statistical formula. The results were in very close agreement. For simplicity, table 6 presents the results of application of the above equation only for diets HW1 and HW2. The assumed error term in food-intake estimation had a CV of 10 per cent of the reported intake of the food item.

As was done above, this model can be generalized to illustrate the principles involved (table 7). To extend this to a one-day intake, it was assumed that all foods made an equal contribution to the one-day intake and that the diet contained 15 foods. Table 8 then presents the estimate of the variability that would be associated with the estimate of one-day intake.

If the data in table 8 are compared with the line in table 5 portraying the impact of food composition variation in a diet containing 15 foods, the impact of the additional source of error in the estimation of food intake can be seen. Thus, for example, if the composition variability is taken as 20 per cent and the intake error as 0 (table 5), the variation term in the one-day intake is 5.2 per cent. If an error term of 10 per cent CV in the intake estimation is now added (table 8), the one-day intake estimate has a variability of 5.8 per cent. The addition of the second errot term has only relatively small impact!

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