Dietary variability and its impact on nutritional epidemiology

0021.968l/X3/030237-13$03.00~0 0 1983 Pergamon Press Ltd

J Chron Dis Vol. 36. No. 3, pp. 237-249, 1983 Printed ITIGrext Britain. All rights reserved

Copyright

DIETARY VARIABILITY AND ITS IMPACT NUTRITIONAL EPIDEMIOLOGY MOHAMED

Department

of Nutrition.

Harvard

(Ruceiced

ON

EL LOZY

School of Public Health, MA 02 I 15, U.S.A. in revised firm

665 Huntington

I6 September

Avenue,

Boston,

1982)

Abstract-Dietary intakes are subject to measurement errors and to day-to-day variation, which have contributed to obscuring the suspected relation between dietary lipids and ischemic heart disease. The effect of measurement error on the correlation between dietary intakes and serum cholesterol levels has been studied by others. In this paper we study the effects of errors on the categorization of subjects according to the quantiles of their intakes, and the effects of this misclassification on the observed relation between observed dietary intakes and disease. Our model is based on a bivariate normal joint distribution of true and observed intakes, from which various conditional probabilities can be calculated. Tables are given to simplify many of these computations. We conclude that the usual period of collection of dietary records, 1 week, is usually adequate. The model developed is applicable to any measurement recorded with error, and two examples of its application to the classification of subjects as normotensive or hypertensive are given. The model does depend on a large number of assumptions, some of which are clearly not met. Hence the actual numerical values obtained should be treated with some scepticism. If, however, the assumptions are approximately met, then the results should be reasonable approximations to the truth.

INTRODUCTION MUCH has been written about the effects of misclassification on measures of association [l-5]. Copeland et a/. [ 1) provide a good introduction to the earlier work. Less work has been done on the effect of errors in the estimation of continuous variables (other than blood pressure) which are then used to classify subjects into discrete categories. This is of great importance in studying dietary exposures, an area in which errors (both measurement and diurnal variation) are inevitable but can be reduced (at a price) by repeated measurements. Liu et al. [6] have discussed the problem of estimating “true mean” dietary intakes of individuals in the presence of intra-individual (day-to-day) variability. Their discussion of quantile classification, and of the probabilities of misclassification, appears to be incomplete, as they discuss the probabilities of misclassifying individuals with spec$ed true means, rather than the more general problem of misclassifying individuals falling into given rmges of values. Thus they ask how many days of dietary recording are required so that the probability that a subject whose true intake is in the lower two quintiles should be observed to be in the upper two quintiles is less than 0.05. Our interpretation of their question is that it asks for the probability of so misclassifying a subject whose true value is sornewlzere in the lower two quintiles. They calculate, however, the probability of misclassifying a subject whose true value is at the boundary between the second and third quintiles, and who is obviously at far greater risk of misclassification than a subject with a true value elsewhere in those two quintiles. Their calculation gives a very pessimistic upper bound

Supported Teaching,

in part by Grant HL25724 from the National Institutes of Health Department of Nutrition. Harvard School of Public Health. 237

and the Fund

for Research

and

238

FL Laze

MOHAMED

to the probability of misclassification; they compute that 11 days are needed. while we find that 4 days are sufficient. Knowing the extent of misclassification to expect at a given level of error (both measurement error and between days variation) can be of great help in planning studies on the relation of diet to, say, the subsequent development of coronary heart disease. Given a postulated relation between a categorized risk factor and an outcome, knowledge of the degree of misclassification to be expected from a proposed protocol allows one to predict the relationship between observed levels and the outcome [Z]. One can then decide what level of misclassification is acceptable, and hence how much accuracy is needed in recording food intakes. This, in turn, allows one to decide how many days of dietary records are needed for each subject, a decision for which there is otherwise little rational justification. This paper presents a genera1 formula for the misclassification problem as we interpret it, together with tables which cover many common situations. The model does depend on a large number of assumptions, some of which are clearly not met. Hence the actual numerical values obtained should be treated with some scepticism. If, however, the assumptions are approximately met, then the results should be reasonable approximations to the truth. THE

MODEL

Our model involves a normally distributed population of “true” values, with a known mean and variance. We measure the “observed” values. which differ from the true ones by “measurement error”, assumed to be independent of the true value and normally distributed with zero mean (unbiased) and known variance. This error may represent genuine measurement error, day-to-day variability, or a combination of the two. The population of observed values will have the same mean as that of the true values, but will have a variance equal to the sum of the variances of the true population and of the error; in symbols:

It is important to note that the variance of the observed values is greater than that of the true values. One consequence is that the quantiles of the distribution of the observed values differ from those of the true values, a point to which we shall return later. To simplify the algebra, we will express true and observed values as deviations from their common mean, in units of the true standard deviation. Errors will also be expressed in units of the true standard deviation. The standardized error standard deviation, S, will be the ratio of the error to the true standard deviations, and the observed variance will thus be 1 + s’. We can now ask two questions. which in their most general form are: (1) What is the probability that a subject, whose true value lies between LI and h, will have an observed value between c and d? (2) What is the probability that a subject, whose observed value is found to lie between c and d. has a true value between a and h? To calculate these two conditional probabilities we need the joint distribution of the true and observed values. Owen and Wiesen [7] have shown it to be bivariate normal, with zero means and the variances given above, and correlation 1 r =Jl Hence, if we denote

the joint

probability

+ .?’ by P(AB), we have: h

P(AB)

y(x,

ssa where g(x, y, Y) is the standard

dr

=

bivariate

I;

Y)

dx dr

c* normal

surface.

(1)

Dietary

Variability

and its Impact

on Nutritional

Epidemiology

239

In Appendix 1 various approaches to computing the required volumes under the bivariate normal surface are described. Tables are given below for important special cases. SPECIAL

CASES

The simplest special case (of little practical importance) is when we want the probability that a subject with a true value below the mean will have an observed value above the mean. It is simply tan’s where s is considered to be expressed in radians. This formula is readily derived from Zelen and Sever0 [g], equation 26.3.19. A second simple (but very important) case deals with the probabilities of misclassification when a continuous variable is dichotomized around an arbitrary cutoff point, say h units of the true standard deviation. Table 1 gives the joint probability of having a true value below b and an observed one above it for different values of b and s. The other three cells of the two by two table are obtained by noting that the (marginal) probabilities of the true and observed values being below b are areas under the normal curve to the left of b and

respectively.

A simple

approximation

to the probability

tabulated

in Table

1 is given

in

c91. A final very important special case deals with questions of the type: what is the probability that a subject with a true value in the lower two quintiles of the true distributiorz will have an observed value in the upper two quintiles of’ rhe obserced distribution. The italicized phrases are there to remind us that while the true and observed distributions have the same means they have different variances. As a result, the values (in original units or in units of the true standard deviation) of the cut-off points for the quantiles of the observed distribution will differ from those of the true distribution.

TABLE

I. JOIST

PROBABILITY

OF HAVING

A TRUE

VALIJE

LESS THAN

FOR DIFFEREXT

VALUES

h ALD

AU OBSERVED

VALUt

GRFATER

THAX

OF 5

S-value h -

2.0 I.8 1.6 1.4 1.2 1.0

-0.X

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.X 1.o 1.2 1.4 1.6 I .8 2.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

I .6

I.8

2.0

0.0034 0.005 1 0.0073 0.0100 0.0133 0.0 169 0.0207 0.0244 0.0276 0.0301 0.03 14 0.0316 0.0305 0.0283 0.0253 0.0217 0.0179 0.0142 0.0 108 0.0079 0.0056

0.0055 0.0083 0.0121 0.0169 0.0228 0.0295 0.036X 0.0442 0.0510 0.0567 0.0606 0.0623 0.0616 0.0587 0.0538 0.0475 0.0403 0.0330 0.0260 0.0197 0.0144

0.0068 0.0 104 0.0153 0.0216 0.0295 0.0388 0.0490 0.0597 0.0700 0.0790 0.0860 0.0902 0.0912 0.0889 0.0835 0.0757 0.0662 0.0559 0.0455 0.0358 0.0272

0.0077 0.0118 0.0175 0.0250 0.0344 0.0455 0.058 I 0.0716 0.0850 0.0972 0.1074 0.1145 0.1178 0. I I70 0.1 124 0.1043 0.0937 0.0814 0.0685 0.0558 0.044 I

0.0083 0.0128 0.0191 0.0274 0.0379 0.0506 0.065 1 0.0808 0.0968 0.1119 0.1250 0.1349 0.1408 0.1422 0.1390 0.1317 0.1209 0.1077 0.0932 0.0784 0.0642

0.00x7 0.0136 0.0203 0.0292 0.0406 0.0544 0.0704 0.0880 0.1061 0.1237 0.1394 0.1520 0.1605 0.1642 0.1629 0.1568 0.1467 0.1335 0.1183 0.1022 0.0862

0.009 1 0.0141 0.02 12 0.0306 0.0427 0.0575 0.0747 0.0937 0.1 136 0.1333 0.1513 0.1663 0.1771 0.1831 0.1838 0.1793 0. I 704 0.1577 0.1426 0.1259 0.1088

0.0093 0.0146 0.02 19 0.03 17 0.0443 0.0599 0.07x0 0.0983 0.1198 0.141 I 0.1611 0.1782 0.1912 0.1993 0.2020 0.1993 0.1917 0.1800 0.1653 0.1487 0.1312

0.0096 0.0149 0.0224 0.0326 0.0457 O.OhlU 0.0808 0.1021 0. I248 0.1477 0.1693 0.1882 0.2032 0.2132 0.2177 0.7168 0.2106 0.2001 0. I862 0. I700 0.1525

0.0097 0.0152 0.0229 0.0333 0.0468 0.0634 0.083 1 0.1052 0.1290 0.1531 0.1762 0.1968 0.2134 0.2252 0.23 IS 0.2321 0.2274 0.2182 0.2052 0.1897 0.1725

b

240

MOHAMED EL LOZY

TABLE 2. JOINT PROBABILITY OF HAVING A TRUE VALUE IN TERTILE 1 AND AU OBSERVFD VALUE IK TFRTlLE J PI 1

P21

P22

0.3 189

0.0 I45

0.3047 0.29 I I 0.2782 0.2662 0.255 1 0.2450 0.2358 0.2275 0.2200 0.2 I32 0.207 1 0.2015 0.1965 0.1919

0.0286 0.0422 0.0545 0.0649 0.0733 0.0799 0.0850 0.089 I 0.0924 0.0950 0.097 I 0.0989 0.1004 0.1016

0.3044 0.276 I 0.2489 0.2243 0.2035 0.1868 0.1736 0. I633 0.1551 0.14X6 0.1433 0.1390 0. I355 0.1326 0.1302

5

0. IO 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 I .oo 1.10 1.20 1.30 1.40 I .50 Note

1. P(J, I) = P(I. J): 1~3 2. 1 P(l. J) = 0.3333 I I

TABLE 3. JOINT PROBARII.ITYOF IIAVIUG A IRI:F VALLE IN QUAKHLF AUD Ai\ OBSERVED VALUE IN QUARTILE J s

PI 1

P2l

P22

P3l

P32

P33

0.10

0.2374 0.2250 0.2131 0.2019 0.1915 0.1819 0.1731 0.1652 0.15X1 0.1517 0.1459 0.1407 0.1360 0.1317 0. I279

0.0126 0.0250 0.0365 0.0460 0.0530 0.0580 0.0616 0.0641 0.0658 0.067 I 0.0680 0.0686 0.0690 0.0693 0.0695

0.2215 0. I936 0.1675 0.1456 0.1288 0.1 161 0. IO65 0.0993 0.0936 0.0892 0.0X56 0.0827 0.0x04 0.07X4 0.076X

0.0000 0.0000

0.0159 0.0314 0.0456 0.0563 0 062X 0.0662 0.0677 0.068 I 0 06X0 0.0676 0.067 I 0.0666 0.066 I 0.0656 0.0652

0.22 I5 0.1936 0.1675 0.1456 0.128X 0.1 161 0. I Oh5 0.0993 0.0936 0.0892 0.0X56 0.0827 0.0X04 0.0784 0.0768

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 I .oo I.10 I .20 1.30 I .40 1.50 Note

I. P(J. I) = P(1, J): 2.

0.0004 0.002 I 0.0055 0.0097 0.0142 0.01 X6 0.0226 0.0262 0.0293 0.0321 0.0345 0.0367 0.0385

C ~(1. J) = 0.25. I=,

TABLE ~.JOINT PROBABILITY OF HAVING A TRUE VALUE IU QLKTILE s

PI 1

0.10

0.1889 0.1780 0.1675 0.1577 0.14X5 0.1402 0.1326 0.1257 0.1195 0.1140 0.1090 0.1045 0.1005 0.0969 0.0936

0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 I .oo 1.10 1.20 I .30 1.40 1.50

Note

P2l 0.0111 0.0220 0.0317 0.03X9 0.0439 0.047 I 0.049 1 0.0504 0.05 I2 0.05 I6 0.05 18 0.05 IX 0.05 17 0.05 I6 0.05 14

1. P(1, J) = P(J. I); 2.

P22

P31

0.1735 0.1476 0.1242 0.105X 0.0923 0.0824 0.075 I 0.0696 0.0653 0.0620 0.0593 0.0571 0.0553 0.0538 0.0526

0.0000 0.0000 0.0008 0.0033 0.007 1 0.0112 0.01.51 0.0185 0.0214 0.0239 0.0260 0.0277 0.029 I 0.0304 0.03 I4

1 ~(1, J) = 02, I-1

I

P?Z 0.0 I54 0.0303 0.0422 0.0489 0.05 16 0.0520 0.05 15 0.0505 0.0494 0.04x4 0.0475 0.0467 0.0459 0.0453 0.044x

P3.3 0.1693 0.1394 0. I I40 0.0955 0.0826 0.0735 0.0669 0.0620 0.0583 0.0554 0.053 1 0.0513 0.0498 0.0486 0.0476

I AND AS OBSERVPD VALUE IY QUINIIL~ J P4l

P42

P4.1

P44

0.0000 0.0000 0.0000 0.0001

0.0000

0.0154 0.0303 0.0422 0.0489 0.05 I6 0.0520 0.05 I5 0.0505 0.0494 0.0484 0.0475 0.0467 0.0459 0.0453 0.044X

0.1735 0.1476 0.1242 0.105x 0.0923 0.0824 (3.075 1 0.0696 0.0653 0.0620 0.0593 0.057 I 0.0553 0.053x 0.0526

0.0005 0.001 5 0.0030 0.0049 0.0069 0.0089 00109 0.0 l2X 0.0145 00160 0.0175

0.000 1 0.0019 0.0062 0.0117 0.0169 0.02 13 0.0246 0.027 I 0.029 I 0.0305 0.03 16 0.0325 0.0332 0.033x

Dietary

Variability

and its Impact

on Nutritional

Epidemiology

2‘41

(Their values in units standardized for both true and observed values are the same.) Tables 2 to 4 give, for tertiles, quartiles and quintiles respectively, P(i, j), the joint probability of having a true value in the ith quantile and observed value in thejth one for different values of s. The tables have been reduced in size by noting that P(i, j) = P(,i, i) and that the sum of the probabilities in each row (or column) is one third, fourth or fifth respectively. (Further condensation would have been possible exploiting other symmetries, but it was felt that this would make the tables less easy to use.)

EXAMPLES

While the above discussion of the effects of errors is perfectly general, we will largely confine our examples to applications to the problems of estimating dietary intakes, where the error is composed of both measurement error and day-to-day variability. The most useful methods of estimating dietary intakes involve keeping records for several days, so the error variance is easily estimated. Alternatively, it can be estimated from two estimates obtained by any method, by computing the correlation between the first and the second measurement [6. Appendix C]. Given these two variances, the true variance can readily be obtained.

Table 1 of Liu rt rrl. [6] gives the quintile boundaries for the Keys dietary score for l-day food records, which are 49.4, 55.0, 61.2 and 67.1. From these numbers we can estimate the mean to be 58.2 and the standard deviation 11.4. The authors state that the within subject standard deviation is 12.5 units, hence, as they used the average of seven the error variance is 12.52/7 = 22.32. The true variance is thus measurements, 11.42 - 22.32 = 107.64, and the true standard deviation is 10.4 units. For one day s = 12.5!:10.4 = 1.20, while for II days s = 1.2/y’ri (= 0.4536 for 7 days). We are assuming. as we will in all the examples, that intake on successive days are independent. This assumption is probably not valid, but the magnitude of the effects of non-independence appears to be small (unpublished observations). Let us now answer some of the questions they pose in their example 1. What is the probability, on the basis of a 7 day record, of misclassifying an individual from the first to the fifth quintile? The result can be obtained from Table 4. We want P( 1, 5) which equals i=4

P(5, 1) = 0.2 -

c

i=l

P(5, i),

which is zero to four decimals for s = 0.5. A more exact computation using equation shows it to be 6.23 x lo-‘, rather than 1.08 x 10m4, which is the value they give.

(1)

A second question they ask is: how many measurements are needed so that the probability that a subject with a true value in the upper two quintiles will be misclassified into the lower two quintiles is less than 0.05? There is no simple formula to answer this question, but the probability of so misclassifying a subject on the basis of one, two,. . measurements can be computed, using our Table 4 or direct calculation of the bivariate integral, and noting that the error standard deviation is 1.2/,/s. The joint probability of having a true value in the lower two quintiles and an observed one in the upper two is P(1, 4) + P(1, 5) + P(2, 4) + P(2, 5). For four days s = 0.6 and the required probability is 0.02 which, divided by the marginal probability of the true value being in the lower two quintiles (0.4) gives us a conditional probability of 0.05. For 1 week (the most common period of collection) the probability is 0.024. Hence, by this criterion, 1 week of dietary records is more than enough to give an adequately accurate picture of the subject’s nutrient intake.

MOHAM~D LI. Lorr

242

So far we have only considered the effects of measurement error on the probability of misclassification (and hence on the proportion of subjects likely to be misclassified). Avoidance of misclassification should not be considered an end in itself, but rather a means towards obtaining accurate measures of association between postulated risk factors and outcomes. Denote by aij the probability that a subject with a true value in the ith quantile is classified as being in the jth quantile, and by A the matrix with elements a,j in the ith row and,jth column. Let us refer to the probability of a subject whose true value falls in the ith quintile having the outcome of interest (assumed to be exactly determined) as 71i and let I7 to be the matrix with zi as its ith row. Finally, let pi be the probability that a person with an ohsrrwd value in the ith quantile should have the outcome, with [FDbeing the matrix with elements pi. Barron [2] has shown that P = A’lZ and, since we can calculate A if we know the error variance. we can calculate the effect of a given level of error on the apparent relation between the risk factor and the outcome. There are two distinct ways in which this can be used. In the first place, it can be used in planning studies. Here we start with a postulated quantitative relation (which is essential in any case for rational study design) between risk factor and outcome, and use the formula to obtain the expected observed relation in the presence of a given level of measurement error. This will give us a rational basis for determining how many days of observation are needed for each subject, and to weigh the trade-offs between increased accuracy and an increased number of days of observation (with the accompanying increased cost and decreased compliance). On the other hand, we may use the formula after the study to estimate what the observed risks would have been in the absence of misclassification. Since P’ = 2i’rl we have I7 = (A’)-’ P. (In practice, of course. we would not invert the matrix but simply solve the simultaneous equations 4’11 = P.) This will allow us to compute an estimate (with no known statistical properties) of the relation between the true level of risk factor and outcome. Unlike the first approach, the legitimacy of this approach is not clear. An example will clarify this. Let us assume that the subjects used in the examples above were followed up for a period, and that the expected risk of an outcome, say death from ischemic heart disease, was believed to be related to the (true) Keys dietary score, with the (properly adjusted) probabilities of that outcome being 0.07, 0.10 and 0.13 fol subjects in the lowest, middle and upper tertiles of score respectively. (This hypothetical example is inspired by the results of Shekelle et m/.[lO]). The (true) relative risks in the upper and middle tertiles (relative to the lowest tertile) are 1.86 and 1.43 respectively. If we classify them into tertiles on the basis of a single day’s intake, the error standard deviation will be 1.2. We obtain the required conditional probabilities by dividing the joint probabilities given in Table 2 by the marginal probability, one third. (i.e. multiplying by three) and get the following values for the /J matrix:

A =

0.621 0.291 [ 0.087

0.291 0.418 0.29 1

The probabilities of an outcome will be 0.084, 0.1 lower, middle and upper tertile on the basis of that will have been appreciably lowdered, to 1.38 and Should we use 4 days of data, the error standard A matrix will be: /Y =

0.765 0.220 I 0.015

0.220 0.560 0.220

0.087 0.291 0.62 1 I and 0.1 16 for those classified into the single day. The observed relative risks 1.19 for the upper and middle tertile. deviation will go down to 0.6, and the

0.015 0.220 0.765

1

The probabilities of an outcome are now 0.07X. 0.1 and 0.122 for subjects classified into the lower, middle and upper tertile, and the relative risks are now I .56 and 1.28 respect-

Dietary

Variability

and its Impact

1.8-

Y 0 ti Q .? 2 5

on Nutritional

Epidemiology

143

upper Yew”* lower teelIe

1.8l.41.2-

Middle veraua lower tertlre

a

‘bo+

21 Days

FIG. I. Relation

of observed

relative

of

Recording

risks to period

ively. One week would give an error standard deviation would be 1.63 and 1.32. Further increases in the number of days of observation ishing returns. Figure 1 gives the relative risks observed collection, showing a rapidly increasing accuracy as we days, and a slower subsequent improvement. Example

of observation

of 0.454, and the relative

risks

would bring in rapidly diminwith different periods of data go from one to three or four

4

As an example of the converse process we will apply the inverse transformation to the data of Shekelle et al. [lo] on the relation of diet to subsequent ischemic heart disease. Table 1 of their paper gives a correlation of 0.589 between Keys dietary score obtained at two examinations a year apart. From equation (7) of [6] we can estimate s to be 0.7. The observed risk of death from coronary heart disease (P’) in subjects in the first, second and third tertiles of Keys dietary score was reported to be 9.3, 10.6 and 13.4”,, respectively. We can then obtain A from Table 2, and solve the equations A’ll = P to obtain the risks related to true dietary score, which are 8.6, 10.9 and 14.4y0. The observed relative risk in the third tertile relative to the first was 1.44, correction for misclassification increases it to 1.67. Unfortunately there is little we can say about the statistical properties of this corrected estimate. E.xatnplc

5

As an example of the use of Table 1, let us estimate the probability that a person with a true diastolic blood pressure below 90 mmHg will have an observed pressure above that value. The National Center for Health Statistics reports that white men aged 35-44 yr have a mean diastolic pressure of 84.2 mmHg with a standard deviation of 1 1.3 [ 1 1, Table 251. Hebel et a/.[121 have reported a within person standard deviation of 7.8 mmHg for diastolic blood pressure. Hence we may compute the true population variance as 11.3’-7.8’, which equals 66.85, with a population standard deviation of 8.18. Hence s (the standardized error standard deviation) equals 0.95, and the cutoff point h is (90-84.2)/8.18 standard units above the mean (= 0.7 l), 0.5 1 units of the observed distribution. The probability of a true and observed value being below the cutoff point are then the areas to the left of 0.71 and 0.51 respectively. Assuming normality (an assumption which is false in this case, see [ll], Table 24), these areas are 0.761 and 0.697 respectively. The assumption of normality does not lead us too far off in this case, as from [ 1 I]. Table 14, 32.8% of the subjects had observed diastolic pressures above 90 mmHg, compared to our calculated value of 30.3% Interpolating in Table 1 gives us a joint probability of 0.135 of having a true value below 90 mmHg and an observed value above that level. Using the previously calculated marginal probabilities we can easily fill in the whole two by two table (Table 5), from which the various conditional probabilities can readily be calculated. It is disturbing to

244

MOHAMED TV LOZY

Observed True blood prfxure Below 00 YO or more Total

blood

Below 90

pressure

90 o,- more

0.676 0.07 I 0.6Y7

0 13.5 0. I68 0.303

Total 0.761 0.239

I .ooo

note that almost half of the 30.30/, of subjects with an observed diastolic pressure above 90 mmHg have a true pressure below that level. Conversely. about one third of the X9”,, of subjects with a true blood pressure above that level have an observed pressure below it. Example

6

All the examples given above could be solved using the tables given in this paper. This final example will require the computation of bivariate normal volumes. Rather than dichotomizing the diastolic blood pressure. as we did in example 5. let us divide it into three ranges: below 90 mmHg (normal). 90 to below 95 (borderline hypertensive) and 95 or more (definite hypertensive). What are the joint probabilities of having a true value in category i and an observed one in category ,i? As before, we first compute the cutoff points for each marginal distribution in units of its standard deviation. For the true values, the cutoff points are 0.71 and 1.32, for the observed values they are 0.51 and 0.96. From these we readily compute the marginal probabilities of having true or observed values in each category (Table 6). Next. we compute r = 0.725. We then start to fill out the three by three table, noting that as we have already calculated the marginal probabilities we will only need to fill in four cells. The first cell. representing the joint probability of both true and observed values being below 90 is given by BvN (0.71, 0.51, 0.725), using the notation established in Appendix I. This is found (using tables or by computer) to be 0.626, the same value we obtained in example 5. We next calculate the joint probability of a true value below 90 and an observed value below 95, BvN (0.71, 0.96, 0.725), and find it to be 0.707. Subtracting 0.626 from it. we get 0.081 as the probability of having a true value below 90 and an observed one between 90 and 95. Subtracting 0.707 from 0.761 (the marginal probability of having a true value below 90), we get 0.054 as the probability of a true value below 90 and an observed value above 95. Proceeding in this way, the whole table can readily be filled in, and from it all conditional probabilities can be calculated.

TARLF URES

6. JOIN-I PROBABILITIES "Y

THF

OF TRUE

AWII OBSEKV~II

BASIS OF A SIUGLL: VlSlT AUD

Observed True blood pressure Below 90 90 to below 95 95 oi- more Total

Below 90

A SINGLE

blood

90 to below 95

I

0.626

0.08

0.086 0.014 0.696

0.036 0.017 0.134

DIASTOLIC

PRFSS-

MEASUKI.MlXT

pressure 95 or more

Total

0.054

0.761

0.053 0.063 0. I 70

0.145 0.094

Dietary

Variability

and its Impact

on Nutritional

Epidemiology

245

DISCUSSION

Errors, due to both measurement error and between occasion variability, are a fact of life. While most extensively documented in the areas of blood pressure [12-171 and dietary intakes [l&35], they are found whenever they are sought [36-391. We will not discuss the subject of blood pressure now; the examples in the text were merely used to illustrate the use of the formulae. The diagnosis of hypertension is based on the use of both the systolic and diastolic blood pressures, while the examples given considered the diastolic pressure alone. By analogy with the development given above, the joint distribution of the true and observed systolic blood pressures is quadrivariate normal. While computing such probabilities without a computer is impossible, there exist programs which can compute quadrivariate normal probabilities in a reasonable time [40,41]. From that joint distribution we can compute conditional probabilities, such as that, for example, of a subject being observed to be borderline hypertensive when in fact he is normotensive. The recent suggestion [42] that the systolic and diastolic pressures be combined to give ;I single figure, the “effective blood pressure”, would return us to the simpler bivariate situation. Alternatively, the systolic and diastolic blood pressures could be combined to give a pair of orthogonal indices [43]. If we assume that the errors in observing the two indices are independent we can use the present argument for each index separately. Tables 5 and 6 should warn us that a considerably a considerable degree of misclassification is to be expected. Wide day-to-day fluctuations in food intakes have been extensively documented [I g-351. While most of the published work is descriptive, two analytical approaches have been used. The first deals with the effect of day-to-day variation on the estimated prevalence of extreme dietary intakes [30,31,33-351. The variance of the population of observed intakes will be the sum of the variances of the population of “true mean” dietary intakes and the between day variance. It is therefore possible to deflate the observed variance if we have an estimate of the between day variance, and use this estimate (together with the observed mean) to compute the proportion of the population falling into given ranges of intake. Parker [35] has shown how this inflated variance can lead to a gross overestimate of the proportion of subjects with intakes below a given level. She reports that in a group of 209 elderly women 13% had an observed intake of protein that was below two thirds of the Recommended Dietary Allowances. After adjusting for day-to-day variation she estimated that almost none (0.06’/$ had an intake below that level. The other approach has been to demonstrate the effects of this variability on the relationship between diet and serum lipids [6,27,33,34], showing why the expected relationship between them has so rarely been observed in field studies. All three papers reach essentially the same conclusion, pointing out the attenuation of the regression or correlation coefficients that arises from the measurement error, and giving formulae allowing the user to recover the lost information. The question of what confidence we can place in these corrected estimates is not touched on by any of these authors. As noted in the introduction, Liu et a/.[61 have developed a model for the probability of misclassifying subjects that differs from the one presented here, and that leads to substantially different conclusions. Specifically, their model suggests that the usual period of collection of dietary records, namely 1 week, is insufficient to separate the upper two quintiles from the lowest two with a probability of misclassification of less than 5%. Indeed, according to their model, 11 days are required. Our model suggests that 4 days are adequate, and that 1 week of observation will lead to a probability of misclassification of less than 2..5”/,. The reader will have to decide which model he prefers. Neither Liu et ~1.. nor any other authors, seem to have discussed the effects of this misclassification on the question that is of ultimate interest: namely the relationship of diet to outcomes. We have shown how, given a postulated relation of outcomes to true diet levels and an estimate of the measurement error, we can calculate the expected degree of attenuation of the relation of outcomes to observed diet levels. Using Liu et

246

MOHAMED

EL Lozr

~I/.‘s data we have shown that one week is again an adequate period for the collection of diet records, and that longer periods of observation will give a fairly small improvement in our ability to estimate the relation between diet and risk factors. Note that we assumed that no errors were made in assigning outcomes, a clearly overoptimistic assumption. How to deal with errors in both risk factors and outcomes is treated in Barron’s paper [2]. Barron has also pointed out that this process is mathematically invertible, and has shown how, starting from a relationship between an outcome and the observed level of a risk factor (together with an estimate of the probabilities of misclassification), one can obtain a corrected estimate of the relation of the outcome to the true levels of the risk factors. Unfortunately, he gives no method of assessing the statistical significance of the corrected estimates. The model used here, based on the bivariate normal distribution of true and observed values has been extensively used in quality control [7 and references cited therein]. It has rarely been used in medical work; the only paper that was found to mention it explicitly is that of Gardner and Heady [36]. One possible reason for the neglect of this model is its computational difficulty. and models such as Liu et a/.‘~ [6], based on the univariate normal. are computationally far simpler. Perhaps the tables included in this paper will take some of the computational pain out of the bivariate model. The above development depends on a large number of assumptions. Hence the actual numeric values it leads to should be treated with some scepticism. The four decimal places given in Tables l-4 are certainly more than seems reasonable in view of the uncertainties in the model. The first is the assumption of normality which pervades most biometric work. We rarely have any evidence that it is correct, and indeed often have good evidence to the contrary (e.g. for blood pressure). However, as long as we consider our results as being merely general indications, and do not consider extreme values, we will not be too far off the truth. In example 5 above, we obtained a close correspondence between estimates based on the normal distribution and the observed distribution. In cases of extreme skewness. a logarithmic transformation will often produce a distribution that is far closer to normal than that of the raw data [35]. we rarely know much about the distribution of errors at the Furthermore. start of a study. During the planning stages published results and guesses may be our only sources of information. This can be refined by a pilot study, and further information on the error distribution can be obtained by replicate examination of a sub-sample. A special difficulty arises in studying food records on successive days. There is a common-sense presumption of within-week patterns, with different patterns of food consumption on weekends than on week days. This has been observed in some studies [1X. 22,24,34] but not in others [23,26]. Furthermore, homeostatic reasoning might lead one to predict corrective patterns for total food intakes, with periods of above average intake being followed by periods of below average intake. Both the weekend and homeostatic effects counter the fundamental assumption that successive errors be independent. in this case giving us greater accuracy than expected under independence. It should be noted that Sukhatme and Margen [44] have recently proposed a different model that predicts larger between-week variances than under independence. As an added complication. Dalvit [45] has recently shown that food intakes in young women vary during the menstrual cycle, being higher in the ten days following ovulation than in the ten days preceding it. This increased accuracy is less marked than one might have hoped; under the most favorable reasonable combination of assumptions a week gives us about as much information as 10 independent days would (unpublished observations). Even this modest result seems to be applicable only to total food intake, as there is no reason to expect negative feedback for intakes of individual nutrients, and empirically the weekend effect is more marked for total intake than for individual nutrients.

Dietary

Variability

and its Impact

on Nutritional

Epidemiology

247

In addition to these short range deviations from the model of random variation, there are longer range ones. It is very probable that there are seasonal effects, as noted by McHenry er al. [ 191. Not only does food availability have a seasonal pattern, but so does a person’s lifestyle, and it is not reasonable to assume that this has no effect on food intake. Furthermore, it is almost certain that food patterns of individuals change over long periods of time. This leads to considerable scepticism about studies in which diet is ascertained at the start of a long follow up and related to outcomes 10 or 20 yr later. Thus, while we feel that our computations strongly suggest that a week is indeed an adequate period for determining the current consumption of major nutrients by a subject, we do not believe that it is adequate for follow-up epidemiological studies. Unless subjects are re-examined during different seasons over the years, we will be unable to assess the effects of long term and seasonal variations.

REFERENCES 1. 2. 3. 4. 5. 6.

Copeland KT. Checkoway H. McMichael AJ ef al: Bias due to misclassification in the estimation of relative risk. Am J Epid 105: 488-495, 1977 Barron BA: The effects of misclassification on the estimation of relative risks. Biometrics 33: 414-418. 1977 Quade D, Lachenbruch PA, Whaley FS et ul: Effects of misclassification on statistical inferences In epidemiology. Am J Epid I I I : 503-515. 1980 Whaley FS. Quade D. Haley RW: Effects of method error on the power of a statistical test: implications of imperfect sensitivity and specificity in retrospective chart review. Am J Epid 111 : 534-542. 1980 Greenland S: The effect of misclassification in the presence of covariates. Am J Epid 112: 564569, 1980 Liu K. Stamler J. Dyer A er trl: Statistical methods to assess and minimize the role of intra-individual variability in obscuring the relationship between dietary lipids and serum cholesterol. J Chron Dis 3 I : 39Y-41X.

7.

Owen 553-572.

8. 9.

IO. II 12 13

197x

DB. Wiesen

JM:

A method

of computing

bivariate

normal

probabilities.

Bell Syst Tech J 3X:

I959

Zclcn M. Sever0 NC: Probability functions. In Handbook of Mathematical Functions. Abramowitr M, Stegun IA (Eds) Washington: US Government Printing Office. 1964 el Lozv M : Simple computation of a bivariate normal integral arising from a problem of misclassification with applicatlo& to hypertension. Commun Stat (in press)Shekelle RB. Shrvock AM, Paul 0. er ul: Diet. serum cholesterol. and death from coronary heart disease. The Western Electric study. N Engl J Med 304: 65-70, 1981 Natlonal Center for Health Statistics: Blood pressure levels of persons 6- 74 years, United States 1971-1974. Vital and Health Statistics Series 1 I, no. 203, 197X Hebel JR. Apostolides AY. Dlschinger P. rr al: Within-person variability in diastolic blood pressure for a cohort of normotensiver J Chron Dis 33: 745-750, 1980 Armltage P. Rose GA: The variability of measurements of casual blood pressure. 1. A laboratory study. Clin Sci 30: 325 335. I966

14 15 16

Armitage P, Fox W. Rose GA: The variability of measurements of casual blood pressure. II. Survey experience. Clin Sci 30: 337 344, 1966 Souchek J. Stamler J, Dyer AR. et trl: The value of two or three versus a single reading of blood pressure at a first visit. J Chron Dis 32: 197-210, 1979 Rosner B. Polk BF: The implications of blood pressure variability for clinical and screening purposes. J Chron Dis 32: 451~461.

17 IX

Home

19

1979

Shepard DS: Reliability of blood pressure measurements: implications grams to control hypertension. J Chron Dis 34: 191-209, 1981 Leverton RM, Marsh AG: Comparison of food intakes for weekdays Econ 31: 111-114.

McHcnr)

EW, Ferguson

for designing

and evaluating

and for Saturdays

pro-

and Sundays.

J

1939

HP. Gurland

J: Sources

of error in dietary

surveys.

Can J Puhl Hlth 36: 355-361.

1945

25

Widdowson EM: A study of individual children’s diets. Med Res Count GB Spec Rep Ser No 257. 1937 Woolf B: StatIstical aspects of dietary surveys. Proc Nutr Sot 13: X2-94, 1954 Carry RC. Passmore R. Warnock GM. et ul: Studies on expenditure of energy and consumption of food in miners and clerks. Fife. Scotland, 1952. Med Res Count GB Spec Rep Ser No 2X9 1955 Head) JA: Diets of bank clerks: development of a method of classifying the diets of individuals for use in cpidemiological studies. J Roy Stat Sot 124: 336-361, 1961 Cellier KM. Hankin ME: Studies on nutrition in pregnancy. I. Some considerations in collecting dietary information. Am J Clin Nutr 13: 55-62, 1963 Fry PG. Fox HM. Linkswiller H: Nutrient intakes of healthy older women. J Am Diet Assoc 42: 218 272,

26

I963 Hankin

20 ‘I 72 23 24

27 2x

JH, Reynolds WE, Margen S: measured nutrient intakes. Am J Clin Kels A: Dietary epidemiology. Am J Easty DL: The relatlonship of diet to x)7-~?

12. I Y70

A short dietary

method

for epidemiological

studies

II. Variability

of

Br J Nutr

24:

Nutr 20: 935 945. 1967 Clin Nutr 20: I l51- 1157, 1967

serum cholesterol

levels in young

men in Antarctica.

MOHA~ED

24X

29.

Balough

M, Kahn

HA, Medalie

JH: Randon

repeat

EL LOZY

24.hr

dietary

recalls.

Am J Clin Nutr

24: 304 310.

1971 30.

Hegsted

DM: Problems

in the use and interpretation of recommended dietary allo~anccs. Ecol Food Nutr 1972 Hegsted DM: Dietary standards. J Am Diet Assoc 66: 13-21, 1975 Garn SM. Larkin FA. Cole PE: The problem with one-day dietary intakes. Ecol Food Uutr 5: 215-747. 1976 Jacobs DR Jr, Anderson JT. Blackburn H: Diet and serum cholesterol: do zero correlations negate the relationships’? Am J Epid 110: 77-87. 1979

I : 255-265.

31. 32. 33. 34. 35. 36. 31. 3x. 39. 40. 41.

Beaton GH. Milner J, Corey P, et ul: Sources of variance in 24-hour dietarq recall data: implications for nutrition study design and interpretation. Am J Clin Nutr 32: 2546 2559. 1979 Parker LM: Evaluation of twenty-four recall dietary survey data. D SC Thesis, Harvard School of Public Health 1980 Gardner MJ. Heady JA: Some effects of within-person variability in epidemiological studies. J Chron Dis 26: 781-795. 1973 Statland BE, Winkel P: Variations in serum cholesterol and total lipid concentrations in sera of healthy young men. Am J Clin Pathol 66: 935 943. 1976 el Lory M: Computer simulation of the effects of errors in birth registration on age-depcndcnt anthropometric methods. Am J Clin Nutr 29: 585- 590. 1976 Foster TA. Webber LS. Srinivasan SR. et 01: Measurement error of risk factor variables in an epidemiological study of childrenthe Bogalusn heart study. J Chron Dis 33: 661-672. 19X0 Milton RC: Computer evaluation of the multivariate normal integral. Technometrics 14: XX1~ X89, 1973 Bohrer R. Schervish MJ: An error-bounded algorithm for normal probabilities of rectangular regions. Technometrics

42. 43. 44.

: 297-300.

19X 1

GI, Lundstrom A: Hypertension and mortality. J Chron Dis 35: I65 172. 19X2 Dyer AR. Stamler. J. Shekelle RB (of al: Pulse pressure I. Level and associated factors in four Chicago epidemiologic studies. J Chron Dis 35: 259. 1982 Sukhatme PV, Margen S: Autoregulatory homeostatic nature of energy balance. Am J Clin \iutr 35: 355 -365,

45.

23

Svensson

Dalvit

1982

SP: The effect of the menstrual

cycle on pattern

of food intake.

Am J Clin Yutr

3-1:

1YI I I SIS.

19x1

53.

Owen DB: Tables for computing bivariate normal probabilities. Ann Math Statist 27: IO75 1090. 1956 Pearson K : Tables for Statisticians and Biometricians, Vol. 2. Cambridge L’niversitv Press. I93 I National Bureau of Standards: Tables of the Bivariate Normal Distribution Function and Related Functions. Washington US Government Printing Office. 1959 Nicholson C: The probability Integral for two variables. Biometrika 33: 59-72. 1943 Daley DJ: Computation of bi- and tri-variate normal integrals. Appl Statist 23: 435-438. 1974 Donnell! TG: Algorithm 462. Bivariate normal distribution. Commun ACM 16: 63X 1973 International Mathematical and Statistical Libraries, Inc.: Library I Reference Manual, 5 edn Houston: InternatIonal Mathematical and Statistical Libraries. Inc., 1975 Dagennis MC, Loner&m G: Accurate estimation of bivariate normal integrals. Cah Cent Etud Rech Operat

16: I53

54.

Gideon

RA. Gurland

46. 47. 4X. 49. 50. 51. 52.

Math 55. 56.

Drezner Divgi I979

160. 1974

34: 6X1-684,

J: A polynormial

2: Computation Calculation

DR:

for bivariate

normal

variates.

SIAM

J Appl

of the bivariate normal integral. Math Comp 32: 277-279. 197X of umvariate and bivariate normal probability functions. Ann Stat 7: 903 910.

APPENDIX Bivarlate that do not. some effort) Following

type approximation

1978

I

: PRACTICAL

COMPUTATION

normal volumes are at the boundary separating computations that need a computer from those While these computations are very much easier with a computer, they can be carried out (with without it. We will therefore discuss both approaches. Owen [46], let us define

where g(.~, I‘. r) is the standard

blvariate

normal

surface. We then have our fundamental

formula.

* d isa

c

~(u. y. r) = BvN(h. tl.r) ~ BvN(u,d.

r) - BvN(h. (‘.r) + B\,N(tr. c.r)

(A21

All practical methods of computation primarily compute BvN. from which we can compute volumes over any rectangular area. When the boundaries of the area are all finite this is the simplest way to compute the \oIumc\. When some are infinite they may simply be replaced by a large number (five or more) with the approprtate slgt,. This is inelticient. but acceptable when using a computer. When doing the calculations by hand. formulae 26.3.4 to 26.26.3.9 of Zelen and Sever0 [X] can be used to diminish the needed computations. Note that the function they use, L(I1. k. r). equals BvN (-h, -k, r). When computing a large crosstabulation. merhods Gmilar to Ihose employed in example 6 will save considerable computing time. Tables giving volumes under the bivariate normal surface 147. 4X] exist. but both sets al-e out of print. Furthermore. they are bulky. involve three arguments and hence usually require Interpolation 1n three dlmen-

Dietary

Variability

and its Impact

on Nutritional

Epidemiology

249

sions, and are generally inconvenient to use. Nicholson [49] has derived a function of two arguments that can be used to obtain bivariate normal volumes more easily, but while it is a definite improvement it is still inconvenient to interpolate. Owen [46] has developed a modification of Nicholson’s function and produced a set of tables, specially designed for easy interpolation, that seems to be the preferred method of obtaining bivarlate normal volumes to high accuracy (six decimal places) without a computer. An alternatlve to using these tables is to compute Owen’s function using an approximation due to Daley [50]: 3 T(x, a) = 2?r exp(-fx’a/Y)(l

+ 0.00868~~~~)

(A3)

where 3 = tan-’ a. This formula has a maximal error of slightly over five units in the fourth decimal piace for u d 1, for larger values of a, use the relation given by Owen. Owen’s paper should be consulted whether his tables or Daley’s approximation is used. For those with ready access to a computer there is a published FORTRAN subroutine to compute bivariate volumes [Sl], as well as a similar one in the commercially available IMSL Library 1521. Many other approaches to computing bivariate normal volumes exist [SO and references cited therein, for more recent methods see 53-561, but none are in the form of published, fully debugged routines, and 1 do not feel that the effort needed to implement them is justified. While general-purpose methods are needed to answer the most general questions. the tables given here will solve many specific problems.