Correlations between estimated and true dietary intakes

Correlations between Estimated and True Dietary Intakes GARY E. FRASER, PHD, MB, CHB, AND DAVID J. SHAVLIK, MPH

PURPOSE: It is unclear how well questionnaire or so-called reference methods of dietary assessment correlate with true dietary intake. We develop a method to estimate such correlations. METHODS: An error model is described that uses data from a food frequency questionnaire (Q), a reference method (R), and a biological marker (M). The model does not assume the classical error model for either R or M, or that the correlation between errors in the questionnaire and reference data is zero. Credible intervals can be placed about correlations between R, Q, M and true dietary data (T), also about the correlations between errors in reference and questionnaire data. RESULTS: Application of this model to a validation data set mainly found correlations in the range 0.4 to 0.8, and that correlations (R,T) generally exceeded correlations (Q,T), providing evidence that R is more valid than Q. Estimated correlations between errors in R and Q were often far from zero suggesting that regression calibration to imperfect reference data is problematic unless these error correlations can be estimated. CONCLUSION: A biological marker in addition to dietary data, allows calculation of correlations between estimated and true dietary intakes under reasonable assumptions about errors. However, sensitivity analyses are necessary on one variable. Ann Epidemiol 2004;14:287–295. 쑕 2004 Elsevier Inc. All rights reserved. KEY WORDS:

Measurement Error, Nutritional Epidemiology, Structural Equation Models, Validity.

INTRODUCTION A recurring problem in observational epidemiology is the bias in effect estimates caused by errors in exposure assessment (1). Nowhere is this more problematic than in nutritional epidemiology where reliance on subjects’ memories and perceptions about their diets inevitably lead to only approximate values being recorded. Recently it has become common to incorporate a validation or calibration substudy to a cohort study (1–3). One reason is to estimate the validity of dietary estimates that were developed from the main study food frequency questionnaire. Usually these substudies, conducted in a small group of subjects, collect food frequency and also so-called “reference” dietary data such as multi-day diet diaries or repeated 24-hour recalls. However, clear evidence that the reference methods provide more valid information than the simpler and less expensive food frequency questionnaire is difficult to find. Regression calibration has been proposed (2) to correct crude estimates of disease-exposure associations using the

From the Center for Health Research (G.E.F), and Department of Epidemiology and Biostatistics (D.J.S), School of Public Health, Loma Linda University, Loma Linda, CA 92350. Address correspondence to: Dr. Gary E. Fraser, Center for Health Research, School of Public Health, Loma Linda University, Loma Linda, CA 92313. Tel.: (909) 558-4753; Fax: (909) 558-0126. E-mail: gfraser@sph. llu.edu This work was partly supported by NIH grant number 1F33CA66287. Received November 20, 2002; accepted August 27, 2003. 쑕 2004 Elsevier Inc. All rights reserved. 360 Park Avenue South, New York, NY 10010

“reference” data (R) from the calibration study and its association with the crude food-frequency data (Q). R is inevitably an imperfect representation of some, perhaps long-term average, truth (T). A consequence is that naı¨ve regression calibration will only work (4) if ρ(εR,εQ), the correlation between errors in R and errors in Q, is zero, and if the errors in R conform to the classical error model, R = T ⫹ ε, E(ε|T) ⫽ 0. We use estimating equations that avoid these strong assumptions. The model however, requires measurement of a biological marker (M) for the dietary variable of interest. These equations provide: 1) estimates of ρ(Q,T), ρ(R,T) and ρ(M,T); and 2) a non zero estimate of ρ(εR,εQ) that does not require the classical error model for R. The method adds to previously described work in that the biomarker (M) does not need to be a so-called reference biomarker (5). A reference biomarker, which has errors that conform to the classical error model, simplifies the modeling (6), but at present has very limited applicability. A limitation of our approach is that the model is not fully identifiable and either requires sensitivity analyses on one variable or integrating this variable out using Bayesian methods. However, this latter method does provide an identifiable lower bound to estimates of ρ(Q,T) and ρ(R,T), which takes into account the random variation consequent on the calibration study sample size. MATERIALS AND METHODS The Model In this model, T is a latent, unobservable variable (3, 5–9). Then 1047-2797/04/$–see front matter doi:10.1016/j.annepidem.2003.08.008

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Ri ⫽ αR ⫹ βRTi ⫹ εR,i Qi ⫽ αQ ⫹ βQTi ⫹ εQ,i Mi ⫽ αM ⫹ βMTi ⫹ εM,i It is assumed that errors in M are independent of those in Q and R, but that errors in Q and R may not be independent. None of βR, βQ, or βM need to equal 1.0, as has often been necessary in other related work, and so the classical error model is not assumed. It is not necessary in this model to distinguish subject-specific (4, 6, 9) and random withinperson errors, as the estimates of ρ(Q,T) and ρ(R,T) do not change. Writing expressions for all variances and covariances between the different methods of dietary assessment and equating these to the variances and covariances actually observed, leads to the estimating equations below, where we assume Cov(εQ,εM) ⫽ Cov(ε R,εM) ⫽ 0; also Cov(εk,T) ⫽ 0 as usual; and where βkσT ⫽ θk, k ⫽ R,Q,M. R as used below refers to the mean of two estimates of the reference data. σ2R s2Q

⫽

β2R s2T

⫽

2 2 βQ σT

⫹

s2ε R

θ2R⫹σ2ε R

⫽

⫹

σ2ε Q

⫽ θ2Q⫹σ2ε Q

s2M ⫽ β2Mσ2T ⫹ σ2ε M ⫽ θ2M ⫹ σ2ε M Cov(R,Q) ⫽ βRβQσT2 ⫹ Cov(εR,εQ) ⫽ θRθQ ⫹ Cov(εR,εQ) Cov(R,M) ⫽ βRβMσ2T ⫽ qRθM Cov(Q,M) ⫽ βQβMσ2T ⫽ θQθM Thus, there are 6 equations, and 7 unknowns. Solutions can be found only if one variable, ψ, is arbitrarily fixed at different values and sensitivity analyses are performed to determine how solutions vary with different values of ψ. The assumptions in the above model are all relatively weak, although possible violations are discussed below. Note that θk/σk ⫽ ρ(k,T), the correlation between the dietary estimator and the true intake T (where k ⫽ R, Q, or M). These correlations are variables of great interest. One consequence of this model is that a βk never occurs without an accompanying σT. Thus, neither σT nor any of the βk can be independently estimated without further assumptions that we prefer not to make. The new variables, θk ⫽ βkσT, reflect this situation. A resulting advantage is that the number of unknowns is reduced by one, and that assumptions of the classical error model are no longer required. Estimating Correlations and Other Parameters We chose ψ ⫽ ρ(M,T) as the unknown variable for most analyses, but estimates of ρ(εR,εQ) were more stable when ψ ⫽ σ2r/Cov(R1,R2) (see Appendix for definition of r, R1, and R2). As demonstrated by Kaaks (5) also, a fundamental result is that under the above model,

ρ(R,M)⫽ρ(R,T)·ρ(M,T) and ρ(Q,M)⫽ρ(Q,T)·ρ(M,T). Hence ρ(R,M)/ρ(Q,M)⫽ρ(R,T)/ρ(Q,T), a useful indicator of the relative validity of R and Q that does not involve ψ. The equivalent result for ρ(Q,R) is more complicated as in Cov(εR,εQ) general ρ(Q,R) ⫽ ρ(Q,T)·ρ(R,T)⫹ . σR σQ When ψ ⫽ ρ(M,T), there are the following solutions for parameters. Note that each solution is in terms of observed quantities only, plus ψ. ρˆ (R,T) ⫽ ρˆ (R,M)/ψ ρˆ (Q,T) ⫽ ρˆ (Q,M)/ψ Coˆv(eR,eQ) ⫽ [ρˆ (Q,R)⫺ρˆ (Q,M)·ρˆ (R,M)/ψ2]SRSQ S2eR ⫽ S2R(1⫺ρˆ (R,M)2/ψ2) 2 (1⫺ρˆ (Q,M)2/ψ2) S2eQ ⫽ SQ

Hence, ρˆ (εR,εQ) ⫽

ρˆ (R,Q)·ψ2⫺ρˆ (Q,M)·ρˆ (R,M) 冪(ψ2⫺ρˆ (R,M)2)(ψ2⫺ρˆ (Q,M)2)

Confidence Intervals For fixed values of ψ, confidence intervals for the parameters of interest were estimated by BCa resampling (10). Small values of ψ resulted in failure of the BCa method when estimating confidence intervals for ρ(εR,εQ), as estimates in the parent sample required by the method, reached the boundary for a correlation coefficient (⫾1.0). This does not preclude the possibility of smaller values of ψ, but our data cannot then provide confidence intervals using this method. Further details of the confidence interval calculation are found in the Appendix. By using similar information, and a non-informative Bayesian prior for a part of the range of ψ, a credible region can be constructed for the various other quantities of interest, across values of ψ (see Appendix). In particular this produces an identifiable lower bound for the correlations.

The Validation Study In preparation for a new cohort of Seventh-day Adventists, a validation study of 159 randomly selected non-Hispanic white subjects from California was formed using church directories to find subjects. Blood and subcutaneous fat aspirates were obtained from only those study subjects living in the San Diego area, the Los Angeles basin, and the Oakland Bay area. Thus we collected 96 blood specimens, and 96 subcutaneous fat aspirates using the squeeze technique (11). Of the fat aspirates, 24 were either contaminated with blood or of inadequate size and finally 72 were available for use.

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Once enrolled, subjects gave a first block of four telephone 24-hour recalls. Contact was unannounced, and recalls were obtained on 2 weekdays, a Saturday, and Sunday. No recall was closer than 10 days from the previous recall. Six to 8 weeks later a second block of four recalls began. Between the two blocks subjects attended a clinic and completed a food frequency instrument containing 210 foods, each with a nominated standard portion size and the opportunity to indicate whether the usual portion was smaller (⬍50% standard) or larger (⭓150 percent standard). The 24-hour recalls were obtained using Nutritional Data System (NDS) (12) software in an interactive fashion. Two synthetic weeks of reference data were averaged, each being constructed by summing the 2 weekend days and 5/2 times the 2 weekdays for each block of recalls. The food frequency data were also analyzed for nutrient content using NDS software after being double-entered into a computer file and verified. Nutrient indices were constructed from 130 items of the food frequency questionnaire. These were foods and food groups selected by considering the mass that a particular food contributes to this nutrient in the study population’s diet, and also whether the addition of that item to foods already selected, improved the correlation with reference (24-hour recall) data using cross-validation (resampling) methods (10). Dietary fatty acids were expressed as a proportion of total fat to correspond to the subcutaneous fat data. Dietary vitamin intakes were log-transformed and adjusted for total energy by the residual method (13).

RESULTS The age-sex distributions of the study populations are shown in Table 1. In keeping with the structure of the proposed cohort, there is a modest excess of women. Mean estimated intakes of those vitamins and fatty acids for which we had an independent biological estimator, and values of corresponding biological (blood and subcutaneous fat) data, are shown in Table 2. All dietary vitamin estimates included supplements. Correlations between energy-adjusted vitamins, fatty acids (% of total fat), and biological estimators are shown in

TABLE 1. Age and sex distribution of the study populations Serum vitamins Number Mean age (SD) in years Subcutaneous fatty acids Number Mean age (SD) in years

Men

Women

40 55.1 (16.8)

56 50.9 (17.2)

27 54.4 (16.3)

45 53.1 (17.7)

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Table 3. Note that in these calculations ρˆ (R,M)c is corrected for the attenuation that would otherwise be present due to random within-person errors in R when estimating the variable of interest, total energy, and total fat (14–15). Application of the methods described above, and the required sensitivity analyses, produced a range of estimates (according to level of ψ) for ρ(R,T), ρ(Q,T), and ρ(εR,εQ). Values for ρˆ (R,T) and ρˆ (Q,T) at different values of ψ are shown in Figure 1. For most dietary variables, values of ρˆ (R,T) were in the range 0.5 to 0.8. However, values for ρˆ (Q,T) were generally rather lower, but in the range 0.4 to 1.0. Two exceptions to this were the very high values of ρˆ (Q,T) for vitamin E and the very low values of vitamin C. Values of ψ, that produce estimates of ρˆ (Q,T) approaching 1.0 by chance, are implausible as they imply σ2ε Q⫽0. Values of ρˆ (εR,εQ) and BCa 95% confidence at different values of ψ are found in Table 4. Confidence intervals are frequentlyverywide forthisparameter,indicatingtheneedfor larger calibration studies. Even point estimates can often not be estimated at lower values of ψ, as boundary conditions are violated. Nevertheless, those that could be estimated were usually strongly positive, and far from the value of zero which has often been assumed in naı¨ve regression calibration. The distributions of values of these same variables when ψ is removed by integration were also calculated, and Table 5 shows median values and the 95 percent credible intervals using an uninformative prior for a portion of the distribution of ψ (see Appendix). Again values of ρˆ (Q,T) are lower (except for vitamin E) than those of ρˆ (R,T). Except for vitamin C, the lower bounds of the confidence intervals for ρˆ (R,T) are relatively far from zero, adding further evidence that the reference data contains much valid information about T. This is also so for the questionnaire data, although values of ρˆ (Q,T) are generally lower. With the exception of vitamin C confidence intervals do not include zero. A larger sample would have provided more precise information. The second column of results in Table 5 contains values of ρˆ (R,T) corrected for the attenuation due to within-person random error in R, which moves estimates of the correlations substantially higher. As the within-person error is of little interest, it is corrected values that provide the best information about ρˆ (R,T), and to which ρˆ (Q,T) should be compared when assessing the relative validity of Q and R. With the exception of vitamin C the lower bound of the interval is far from zero. As expected, in most cases the biological variable, M, correlated well with T, and correlations are in the same range as those for the corresponding reference dietary data. Values of ρˆ (εR,εQ) calculated after ψ is removed by integration are also found in Table 5. In agreement with the ψ-specific point estimates of Table 4, median values of ρˆ (εR,εQ) were again usually strikingly positive. Saturated

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TABLE 2. Mean Estimated Daily Dietary Intakes (Including Supplements) and Blood or Subcutaneous Fat Levels of Variables (Standard Deviations in Parentheses) Vitamins β-carotene Vitamin E* Vitamin C Folate Fatty Acids Saturated (%) Polyunsaturated (%)

Diet (R) 6.17 60.0 335.1 485.0

(6.97) mg (83.2) mg (337.8) mg (252.2) µg

Diet (Q) 9.07 92.9 662.6 590.6

(6.36) mg (148.1) mg (587.9) mg (376.6) µg

Diet (R)

Diet (Q)

32.75 (5.82) 23.02 (4.55)

28.01 (5.99) 26.21 (5.22)

Blood (M) 164.2 16.3 28.8 452.6

(63.5) µg/ml (5.0) µg/ml (12.2) µg/108 WBC (177.7) ng/ml

Subcutaneous Fat (M) 22.9 (3.30) 23.4 (3.30)

*Total α-tocopherol equivalents.

fatty acids, folate, and perhaps vitamin C gave the strongest evidence of positive values for this quantity as the credible intervals either excluded, or nearly excluded, zero.

DISCUSSION The main result is that the validity of both repeated 24-hour recalls and questionnaire data is generally quite good for the variables available and in this population (Figure 1 and Table 5). Second, estimates from the reference method are usually more closely correlated with the underlying truth, than similar data from a food frequency instrument. The third important result is that ρ(εR,εQ) appears to be decidedly nonzero, in some instances at least. The estimated median correlations (excluding vitamin C) in Table 5 between the repeated recalls and the truth, corrected for attenuation, are relatively high (0.65–0.80, and mean of medians of 0.71), but those between the questionnaire and the truth are lower (0.50–0.71, and mean of medians is 0.61). This provides evidence that the mean of eight 24-hour recalls is usually more valid than the food frequency estimate. Results for vitamin C are much lower than those for other variables and confidence intervals are wider. This is apparently a consequence of the poor validity of the biomarker as suggested by the low correlations (R, M) and (Q, M). The reason for this poor validity is uncertain, but may be due to a very high fruit intake in this California population. Levels may often be over the threshold where it is largely urinary excretion that determines the blood levels (16). As M is a concentration biomarker it is unlikely that ψ(⫽ρ(M,T)) would exceed 0.80. Thus, Figure 1 suggests that for vitamin E and polyunsaturated fat the values of ρ(Q,T) are quite high, and this is also so for saturated fat and folate also when ρ(R,T) is the variable of interest. Our small sample size introduces a good deal of uncertainty, as shown by the confidence interval. This could be overcome by a larger calibration study.

In practical regression calibration one must usually rely on imperfect reference data, and ρ(εR,εQ) must be either zero, or if non-zero, estimated, for the procedure to be valid (4). It appears from our analyses that an assumption of the zero value may often be seriously in error. This is in agreement with the findings of Kipnis et al. (3, 6) and Day et al. (17). Our structural equation model to explore nutritional data is broadly similar to that described by several others (3, 5– 9, 17–19), but there are the advantages of not requiring the classical error model assumption for any variable (hence βR, βQ, and βM are free to vary), and not requiring a reference-type biomarker. The requirement for sensitivity analysis on one variable is a limitation, but despite this, useful estimates of some important quantities can be found, partly because natural constraints on ψ are used to advantage. There is no conceptual difference between R and M, except for the inconvenience that as distinct from ρ(εM,εQ), ρ(εR,εQ) cannot be assumed to equal zero. The requirement that ρ(εQ,εM)⫽0 needs careful consideration in each analysis. However, if M is an objectively estimated biological variable it is not easy to envisage error mechanisms for M that would produce associations with errors in the subjective food frequency dietary estimates (Q). The assumption of zero correlations between such errors has also been proposed by others (4, 5, 18–20). It is important to note that just because Q is correlated with age, gender, smoking or some other factor (say C), this does not imply that errors in Q are associated with C. Individuals endeavor to report on their true diets and naturally take account of these other characteristics. Associations between C and M may simply be caused by dietary differences associated with C, and their effects on M. However, in some cases C may be associated with systematic metabolic differences independent of diet. Only then, and if C was also associated with systematic errors in Q, would errors in M and Q be correlated. Such error correlations, if they exist, will often be very small, being less than the product of partial correlations between Q and C, M and C, both conditional on T (see Appendix). The population that we worked with was non-smoking

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FIGURE 1. Correlations (R, T) and (Q, T) for Selected Nutrients and Vitamins, Conditional on Values of ψ [⫽ρ(M,T)], with 95% confidence intervals. * The vertical line truncating the curves for ρ(R,T) indicates the point at which a boundary condition is reached, for all lower values of ψ.

291

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TABLE 3. Correlations between estimated energy-adjusted† vitamins, and fatty acids§: dietary (R ⫽ reference; Q ⫽ food frequency) and biological levels (M) Variable

ρˆ (R,M)

ρˆ (R,M)c‡

ρˆ (Q,M)

β-carotene Carotene (adj. for serum cholesterol) Vitamin E Vitamin E (adj. for serum cholesterol) Vitamin C Folate Saturated fatty acids Polyunsaturated fatty acids

0.38*** 0.43*** 0.41*** 0.47*** 0.17 0.50*** 0.48*** 0.57***

0.49 0.55 0.44 0.50 0.17 0.55 0.56 0.70

0.28* 0.29* 0.40** 0.56*** 0.03 0.32** 0.31** 0.53***

*p ⬍ 0.05. **p ⬍ 0.01. ***p ⬍ 0.001, test that ρ ⫽ 0. † Residual method of energy-adjustment. ‡ Corrected for attenuation due to random within-person error in both the variables of interest and energy estimates contributing to energy-adjusted values of R. Statistical testing was not performed due to the complexity of the deattenuated function. § Fatty acids are a proportion of total fat.

and for practical purposes did not drink alcohol, both possible candidates for C in some circumstances. Obesity may need special consideration, as it is thought that more obese subjects tend to under-report energy consumption (21). If the biological factor, M conditional on T, is also dependent on obesity, this could cause correlations between the errors in M and those in the dietary estimates. There are at least two possible solutions: 1) Use BMI-adjusted residuals of M in place of M; and/or 2) It is reasonable to assume that the energy-adjustment procedure used for R and Q will often remove much of the bias that is characteristic of obese subjects, as the under-reporting probably occurs in the dietary variable of interest, as well as estimated energy. The dependence of a few known biomarkers [e.g., certain fatty acids (16)] on BMI is quite weak, and probably would only induce much weaker correlations between errors. Moreover it is not clear whether positive associations between subcutaneous saturated fatty acids and BMI simply reflect greater proportionate dietary intake of these acids in the obese, or is

an effect independent of T. It would not be necessary to avoid nonzero correlations between BMI and the errors in both Q and M, but only doing so for either Q or M, would resolve any difficulty. Values of M were not energy-adjusted because they are concentration variables or proportions. As the energy-adjustment for Q and R is imperfect, errors in Q and R could become correlated with those of M, if M (conditional on T) was dependent on total energy intake. As far as we know there is no evidence of such associations for the biomarkers that we used. However, this should be evaluated in each situation. The requirement that ρ(εR,εM) ⫽ 0 also needs careful thought, and many of the considerations discussed above apply here. If R is a relatively short-term measure (e.g., a 1week diet diary), and temporally related to M, a short-term distortion in dietary habits could cause similar-signed errors in both R and M and hence a positive correlation between these errors. In our calibration study this should not be a problem as R was the average of eight 24-hour recalls spread over 6 months or more, and M is estimated in the middle of this period. When eliminating ψ to find credibility intervals, use of a constant density for part of the prior distribution was proposed for illustrative purposes. This will probably result in conservative interval calculations. There are of course other options, such as decreasing the density for ψ (a correlation coefficient) at unlikely values, above 0.90 for instance. In more informative data sets the form of the prior will have less influence on the posterior distribution (22). In summary, despite the difficulties of a non-identifiable latent variable model used in the analyses above, its relative simplicity otherwise, and the lack of strong assumptions has led to useful conclusions. These are that for most variables the questionnaire data had reasonably good validity, but “reference” dietary data was usually more valid, and that naı¨ve calibration to a reference method may be incorrect unless ρ(εR,εQ) can be estimated.

TABLE 4. Point estimates and 95% confidence intervals for ρ(εR,εQ) at selected values of ψ (ψ ⫽ ρ(M,T)) ψ Fat or vitamin variable

0.40

Saturated fat Polyunsaturated fat β-carotene Vitamin E Vitamin C Folate

— — — — 0.65 —

95% CI

0.35, 1.00

–Indicates that boundary conditions apply.

0.60 — — 0.09 ⫺0.09 0.62 —

95% CI

⫺0.21, 1.00 ⫺1.0, 0 0.44, 0.77

0.70

95% CI

0.91 — 0.30 0.58 0.62 0.58

0.54, 1.00 ⫺0.35, 1.0 ⫺0.80, 0.95 0.74, 1.24 0, 1.0

0.80

95% CI

0.95

95% CI

0.72 ⫺0.27 0.37 0.70 0.61 0.58

0, 1.00 ⫺1.0, 0.29 ⫺0.18, 0.86 0.19, 0.92 0.43, 0.76 0, 0.85

0.69 0.13 0.43 0.76 0.61 0.59

0, 1.00 ⫺0.27, 0.52 0, 0.77 0.48, 0.92 0.42, 0.75 0.27, 0.80

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TABLE 5. Estimated quantiles of dietary parameters of interest across values of ψ using a partial non-informative prior distribution for ψ‡ Parameter of interest Vitamin or Fatty Acid

Quantile

ρ(R,T)

ρ(R,T)c

ρ(Q,T)

ρ(M,T)

ρ(εR,εQ)

Saturated fatty acids*

2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5% 2.5% 50% 97.5%

0.32 0.63 1.00 0.47 0.71 1.00 0.30 0.50 0.88 0.34 0.58 1.00 ⫺0.05 0.21 0.46 0.38 0.61 1.00

0.37 0.76 1.00 0.49 0.80 1.00 0.38 0.65 1.00 0.38 0.65 1.00 ⫺0.06 0.23 0.51 0.43 0.70 1.00

0.11 0.41 0.91 0.41 0.63 1.00 0.13 0.38 0.69 0.44 0.65 1.00 ⫺0.22 0.03 0.24 0.09 0.35 0.70

0.19 0.69 1.00 0.47 0.72 1.00 0.32 0.64 1.00 0.33 0.63 1.00 ⫺0.04 0.27 1.00 0.31 0.67 1.00

0.13 0.68 1.00 ⫺1.00 0.05 0.54 ⫺0.83 0.39 0.83 ⫺1.00 0.54 0.92 ⫺0.04 0.71 1.00 0.08 0.58 1.00

Polyunsaturated fatty acids*

β-carotene† Vitamin E†

Vitamin C§

Folate§

*Proportion of total fat. † Energy and serum cholesterol adjusted. § Energy-adjusted. ‡ ψ ⫽ ρ(M,T), except that when estimating ρ(εR, εQ) (for greater stability), and when the variable of interest was ρ(M,T), ψ ⫽ σr/Cov(R1,R2).

APPENDIX

a) When it is necessary to apply the condition ρ(R,M)c ⫽ ψ, this implies that σ2r ⫽ 0 as

A. Further Details of ψ-specific Confidence Interval calculation

ρ(R,M)c ⫽ ψ

Some information can be found about the lower boundary of ψ. Note that

i)

ψ ⫽ ρ(M,T)⫽

ρ(R,M) ρ(R,M)·σR ⫽ ρ(R,T) θR

√

⭓ρ(R,M)

σ2R ⫽ ρ(R,M)c, Cov(R1,R2)

where Cov(R1,R2) ⫽ θ2R ⫹ σr2, and r is the subject-specific portion of εR after the random within-subject error is subtracted, and R1, R2 are the two independent estimates of R. The quantity that follows the above inequality can be estimated from the data independent of ψ. As also noted above ψ ⫽ ρ(M,T) ⫽ ρ(Q,M)/ρ(Q,T) ⭓ ρ(Q,M). Then a lower bound for ψ is Max[ρ(R,M)c, ρ(Q,M)]. The lower limit of a one sided 95% confidence interval for ψ when estimated by resampling, gives the following values: saturated fats 0.38; polyunsaturated fats 0.54; beta carotene 0.40; vitamin E 0.40; vitamin C 0.02; folate 0.43. When values of ρ(Q,M) or ρ(R,M)c in a particular bootstrap sample were outside the boundary represented by an imposed value of ψ, parameters were then reassigned to boundary values as follow:

θR

√

θ2R

⫹ σ2r

ρˆ (R,T) ⫽

⫽ ψ/√1 ⫹ σ2r /θ2R, and then

√

Cov(R1,R2) i.e., ρˆ (R,T)c ⫽ 1.0 S2R

ii) ρˆ (Q,T) is unchanged iii) ρˆ (εR,εQ) ⫽ 0 Let εR ⫽ r ⫹ w, where w is the random withinperson component. Then ρˆ (εR,εQ) ⫽ Cov(r ⫹ w, εQ)/σεRσεQ ⫽ 0 as r is constant when σ2r ⫽ 0 and Cov(w,εQ) ⫽ 0 by definition of w. b) When it is necessary to impose the condition ρ(Q,M) ⫽ ψ this implies that σ2ε Q ⫽ 0, and then i) ρˆ (R,T) calculated as usual ii) ρˆ (Q,T) ⫽ 1.0 iii) ρˆ (εR,εQ) ⫽ 0, using similar logic to part iii) in a) above c) When it is necessary to impose both the above boundary conditions, then Cov(R1,R2) i. ρˆ (R,T) ⫽ S2R

√

ii. ρˆ (Q,T) ⫽ 1.0

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iii. ρˆ (εR,εQ) ⫽ 0

Thus from Equation (2) above ρ(εQ,εM) ⬍ ρp(Q,C|T). ρp(M,C|T)

B. A Bayesian Probability Distribution and Credibility Interval for The Quantities of Interest, Eliminating ψ. Let γ be one such quantity of interest (e.g., ρ(R,T),ρ(Q,T),⫺⫺⫺). pr(g) ⫽

兰10 pr(γ|ψ).pr(ψ).dψ.⫺⫺⫺⫺⫺⫺⫺⫺

(1)

Then, if it were possible to resample pr(γ|ψ) such that each bootstrap sample was associated with a random value of ψ, a BCa distribution function and credibility interval for γ could be estimated. This requires knowledge of pr(ψ) which we estimated as follows: pr(ψ) ⫽ 兰0 pr(ψ|B,ψ ⬎ B)·pr(B)dB ψ

where B ⫽ Max(ρ(M,R)c, ρ(M,Q)), the value of a lower bound for ψ which corresponds to either σ2r ⫽ 0 or σ2ε Q ⫽ 0. The distribution function for B was constructed using the BCa resampling technique (N ⫽ 10,000), and is not dependent on ψ. A 1 So, pr(ψ ⫽ A) ⫽ 兰0 .pr(B)·dB, using a non-infor1⫺B mative prior over the range ψ ⭓ B. On randomly selecting a large number of values of Ai (i ⫽ 1,…,10,000) on the range 0 to 1.0, the pr(ψ ⫽ Ai) was repeatedly estimated, and then an approximate distribution function for ψ was formed. Next, random values of ψ were found by sampling from the distribution function for use in equation (1) above. When ψ ⫽ σ2r /Cov(R1,R2) a similar approach was used and ψ is restricted to the range (0,1). A non-informative prior distribution was assumed for ψ on this range, resulting in a rather simpler procedure than that above. Except when estimating credible intervals for ρ(M,T) or ρ(εR,εQ) however, we preferred ψ ⫽ ρ(M,T) as a more intuitive variable.

Using the same algebra as in the Methods section of this manuscript but based on equation εQ ⫽ αεQ ⫹ βεQC ⫹ εεQ and a similar equation for εM as a function of C, these lead to the relationship (2)

so long as ρ(εεQ,εεM) ⫽ 0. (If this is not true some factor other than C is also causing the correlation between εQ and εM). ρ(εM,C) ⫽ Cov(εM,αc ⫹ βcT ⫹ εc)/σεMσc⫽ Cov(εM,εc)/σεMσc ⫽ ρp(M,C|T)σεc/σc ⬍ ρp(M,C|T), where subscript p refers to partial correlation. Similarly ρ(εQ,C) ⬍ ρp(Q,C|T)

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C. The Effect of a Common Correlate (C) of Q and M on r(eQ,eM).

ρ(εQ,εM) ⫽ ρ(εQ,C).ρ(εM,C)⫺⫺⫺⫺⫺

The authors gratefully acknowledge the helpful advice of Dr. Daniel Stram, University of Southern California, Los Angeles, CA.

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