1 Nutritional Epidemiology Branch, Division of Cancer Epidemiology and Genetics, National Cancer Institute, Bethesda, MD, USA.
2 Biometry Research Group, Division of Cancer Prevention, National Cancer Institute, Bethesda, MD, USA.
3 Department of Statistics, Texas A&M University, College Station, TX, USA.
4 Applied Research Program, Division of Cancer Control and Population Sciences, National Cancer Institute, Bethesda, MD, USA.
5 Medical Research Council, Dunn Human Nutrition Unit, Cambridge, UK.
6 University of Wisconsin, Madison, WI, USA.
7 Department of Mathematics, Statistics and Computer Science, Bar Ilan University, Ramat Gan, Israel, and Gertner Institute for Epidemiology and Health Policy Research, Tel Hashomer, Israel.
Correspondence: Dr Arthur Schatzkin, Nutritional Epidemiology Branch, National Cancer Institute, 6120 Executive BlvdEPS 3040, Bethesda,MD 208927232, USA. E-mail: schatzka{at}mail.nih.gov
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Abstract |
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Methods The OPEN study included 484 healthy volunteer participants (261 men, 223 women) from Montgomery County, Maryland, aged 4069. Each participant was asked to complete a FFQ and 24HR on two occasions 3 months apart, and a doubly labelled water (DLW) assessment and two 24-hour urine collections during the 2 weeks after the first FFQ and 24HR assessment. For both the FFQ and 24HR and for both men and women, we calculated attenuation factors for absolute energy, absolute protein, and protein density.
Results For absolute energy and protein, a single FFQs attenuation factor is 0.040.16. Repeat administrations lead to little improvement (0.080.19). Attenuation factors for a single 24HR are 0.100.20, but four repeats would yield attenuations of 0.200.37. For protein density a single FFQ has an attenuation of 0.30.4; for a single 24HR the attenuation factor is 0.150.25 but would increase to 0.350.50 with four repeats.
Conclusions Because of severe attenuation, the FFQ cannot be recommended as an instrument for evaluating relations between absolute intake of energy or protein and disease. Although this attenuation is lessened in analyses of energy-adjusted protein, it remains substantial for both FFQ and multiple 24HR. The utility of either of these instruments for detecting important but moderate relative risks (between 1.5 and 2.0), even for energy-adjusted dietary factors, is questionable.
Accepted 12 June 2003
Much of the current evidence on diet and disease has been gathered from prospective cohort studies in which large numbers of individuals report their dietary habits and are monitored for subsequent development of specific diseases. A consensus is emerging that such prospective studies give more reliable results than the retrospective case-control approach.1 Questions persist, however, regarding the most appropriate dietary report instrument to use in large cohort studies.24
Most large cohort studies have used a version of the food frequency questionnaire (FFQ), which has been shown to be sufficiently convenient and inexpensive to allow its use in tens or even hundreds of thousands of individuals. Day et al.2 and Bingham et al.4 have suggested use of a 7-day diet diary instead. Day et al.s argument rests on data from a study of 179 individuals who completed two FFQ and two 7-day diaries and also provided six 24-hour urines for analysis of nitrogen, potassium, and sodium. Assuming that these urinary biomarkers give unbiased measurements of the unobservable true intake, they showed that the diary was more closely correlated with the biomarker measurements for all three nutrients than was the FFQ. However, Day et al. could not study energy-adjusted nutrient intakes, because their study did not include a biomarker for energy intake.
In this paper, we describe the results of a study similar to that of Day et al., the Observing Protein and Energy Nutrition (OPEN) study.5 Two essential differences between the OPEN study and that of Day et al. were (1) the addition of doubly labelled water (DLW) measurements to estimate energy expenditure, a surrogate for energy intake,6 and (2) the use of two 24-hour recalls (24HR) instead of 7-day diaries. The design, therefore, allows us to investigate both absolute and energy-adjusted intakes, although unlike Day et al.,2 our comparison is between 24HR and FFQ.
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Methods |
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A complete description of the study can be found elsewhere.5 Briefly, each participant was asked to complete a FFQ and 24HR on two occasions. The FFQ was completed within 2 weeks of Visit 1 and approximately 3 months later, within a few weeks of Visit 3. The 24HR was completed at Visit 1 and approximately 3 months later at Visit 3. Participants received their dose of DLW at Visit 1 and returned 2 weeks later (Visit 2) to complete the DLW assessment. Participants provided two 24-hour urine collections, at least 9 days apart, during the 2-week period between Visit 1 and Visit 2, verified for completeness by the para amino benzoic acid (PABA) check method.7 Since approximately 81% of nitrogen intake is excreted through the urine,8 and nitrogen constitutes 16% of protein, the urinary nitrate (UN) values were adjusted, dividing by 0.81 and multiplying by 6.25, to estimate the individual protein intake.
In addition to the protocol for all study participants described above, we repeated the DLW procedure in 25 volunteers (14 men, 11 women). These participants received their second DLW dose at the end of Visit 2 and returned approximately 2 weeks later to complete the DLW assessment.
Dietary assessment methods
The food frequency questionnaire
In this study, we used the Diet History Questionnaire, an FFQ, developed and evaluated at NCI.913 This FFQ is a 36-page booklet which queries frequency of intake over the previous year for 124 individual food items and asks portion size for most of these by providing a choice of three ranges. For 44 of the 124 foods, the FFQ asks from one to seven additional embedded questions about related factors such as seasonal intake, food type, (e.g. low-fat, lean, diet, caffeine-free), and/or fat uses or additions. The FFQ also includes six additional questions about use of low-fat foods, four summary questions, and ten dietary supplement questions.
The 24-hour recall
The employed 24HR was a highly standardized version utilizing the five-pass method, developed by the US Department of Agriculture for use in national dietary surveillance.14 The recall data were collected in-person using a paper-and-pencil approach with standardized probes, food models, and coding. These data were linked to a nutrient database, the Food Intake Analysis System version 3.99, which obtains its database from updates to the 19941996 Continuing Survey of Food Intakes by Individuals.15
Biomarker measurements
Doubly labelled water
DLW, given orally at a dose of approximately 2 g 10 atom per cent H218O and 0.12 g 99.9 atom per cent 2H2O per kg of estimated total body water along with a subsequent 50 ml water rinse of the dose bottle, was used to assess total energy expenditure. Participants provided four spot urine samples, two shortly before and two shortly after the administration of the DLW dose. Participants 60 years of age also provided a blood specimen due to the possibility of delayed bladder emptying. At the follow-up visit, approximately 2 weeks later, participants provided two more spot urine samples. Investigators at the University of Wisconsin Stable Isotope Laboratory determined energy expenditure via mass spectroscopic analysis of urine and blood specimens for deuterium and oxygen-18.1618
Urine collections
In the 2-week period after Visit 1, participants collected their 24-hour urine on two separate occasions. To determine the completeness of urine collections, we asked study participants to take PABA tablets on each day they collected a 24-hour urine specimen. Investigators at the Dunn Nutrition Unit of the Medical Research Council in Cambridge, UK analysed UN and PABA. They analysed nitrogen by the Kjeldahl method and PABA by the colorimetric method. Collections with less than 70% PABA recovery were considered incomplete and removed from further analyses. Samples containing 7085% PABA were also considered incomplete, but the content of analytes were proportionally adjusted to 93% PABA recovery.5 To distinguish PABA from acetaminophen, taken by many participants, they used high protein liquid chromatography19,20 to re-analyse PABA values deemed high (>110% recovery) by the colorimetric method.
Statistical methods
Attenuation resulting from measurement error
The effects of dietary measurement error on the estimation of disease risks are well known.8 The most important concept is that of attenuation. Consider the disease model
![]() | (1) |
where R(D|T) denotes the risk of disease D on an appropriate scale (e.g. logistic) and T is the unobservable true long-term habitual intake of a given nutrient, also measured on an appropriate scale. The slope 1 represents an association between the nutrient intake and disease. In logistic regression, for example,
1 is the log relative risk (RR). Let
be the slope in the linear regression of habitual intake, T, on reported intake, Q, based on the dietary instrument. If the instrument-based values Q are used in place of habitual intake, then instead of estimating the risk parameter
1, one really estimates
1 =
1, the product of the slope
and the true risk parameter
1. Usually, in dietary studies, the value
of is between zero and one, and so the effect of error in the instrument is to cause an underestimate of the risk parameter. This underestimation is called attenuation, and typically
is called the attenuation factor. Values of
closer to zero lead to more serious underestimation of risk. For the logistic regression disease model (1), a true RR of 2 for a given change in dietary exposure would be observed as 20.4 = 1.27 if the attenuation factor were 0.4, and as 20.2 = 1.15 if the attenuation factor were 0.2.
Sometimes, the RR is expressed for the standardized change of a certain amount of standard deviations of the distribution of dietary exposure, which is often interpreted as a comparison of quantiles.21 In this case, the observed RR between quantiles will be attenuated by the Pearson correlation coefficient, (Q,T), between the reported and true intakes.
Measurement error also leads to loss of statistical power for testing the significance of the diseaseexposure association. Approximately, the sample size required to reach the desired statistical power to detect a given risk is proportional to: , or equivalently,
, where
is the variance of the instrument-based reported intake and
is the variance of the true intake.22 In particular, for a given instrument, the required sample size is inversely proportional to the squared attenuation factor,
2. For example, if the true attenuation factor were 0.2, the sample size, calculated to achieve the nominal power under the assumption that
= 0.4, would be smaller by a factor of 0.42/0.22 = 4. On the other hand, the comparison of the necessary sample sizes for different dietary assessment instruments should be based on the squared correlation coefficients between the corresponding instruments and truth.
Note that discrepancies between the reported and the true group mean intake do not in themselves affect the performance of an instrument in a cohort study. For example, an instrument that leads to all individuals under-reporting intake by exactly 25% would be no less useful than an instrument that gives the true intake for each individual, mainly because the ranking of the individuals would be unchanged.
Statistical analysis
Estimation of the attenuation factor and correlation coefficient
(Q,T) requires collecting measurements on a second instrument, called the reference instrument, to compare with the main dietary instrument, in the same subset of individuals. Estimation of the attenuation factor requires that the adopted reference instrument have errors that are independent of both the true intake and errors in the instrument whose attenuation is being evaluated. Estimation of the correlation with true intake requires a more complex study design.21,23 The conventional design requires that the reference instrument be unbiased, and that at least two independent repeat reference measurements be collected. Commonly in nutritional epidemiology, investigators have used multiple day food diaries or 24HR as reference measurements to evaluate FFQ, assuming that these dietary-report instruments satisfy all the above conditions and produce unbiased estimates of both the attenuation factor and correlation with true intake. There is now increasing evidence of jointly correlated biases in all dietary-report instruments, suggesting that none of them satisfies the requirements for a valid reference measure.2,8,21,22,2426
In this paper we use a biomarker (M), either DLW, UN, or a combination of both, as the reference measurement. The evidence for both adjusted UN8 and DLW6 suggests that these are both valid, essentially unbiased reference instruments; that is, their errors have mean zero, and are unrelated to true intakes and errors in dietary-report instruments. We regard 24HR (F) as a second dietary instrument, on an equal footing with the FFQ (Q). Throughout, we applied the logarithmic transformation to energy and protein to make measurement error in the DLW and UN biomarkers additive and homoscedastic and to better approximate normality.
We use the same statistical model as in our previous work.8,25 Briefly, for individual i, let Ti denote usual nutrient intake, let Qij denote log nutrient intake as estimated from the jth repeat of the FFQ, j = 1, 2, let Fij denote log nutrient intake as estimated from the jth repeat of the 24-hour recall, j = 1, 2, and let Mij denote log nutrient intake as measured by the jth repeat of the biomarker, j = 1, 2. The statistical model specifies an error structure of the FFQ, 24HR, and biomarker, and is given by
![]() | (2) |
The model specifies that both the FFQ and 24HR values comprise (a) overall constant biases at the group level ßQ0 and ßF0, respectively; (b) intake-related biases (i.e. those correlated with an individuals true intake), reflected by the slopes ßQ1 and ßF1 of the regressions of FFQ and 24HR, respectively, on true intake; (c) person-specific biases (the difference between total within-person bias and its intake-related component), ri and si, that are independent of true intake Ti, have means zero, variances and
, respectively, and are correlated with the correlation coefficient
rs, and (d) within-person random errors
ij, uij (reflecting variation between repeat measurements due to a variety of physiological and behavioral factors) with means zero and variances
,
, respectively. The biomarker contains only within-person random error, uij, with mean zero and variance,
. (Note that, for purposes of statistical modelling, any instrument measuring short-term intakewhether 24HR or a biomarkerincludes deviations of short-term from longer-term intake as part of the error term.) Within-person random errors in all three instruments are assumed independent of each other and of other terms in the model, except that within-pair errors, (
ij, uij), (
ij, vij), and (uij, vij) are allowed to be correlated, if the corresponding measurements are taken contemporaneously. The model also includes time-specific group intercepts µQj, µFj, and µMj for the FFQ, 24HR, and biomarker, respectively, which reflect possible differences among mean reported intakes over time and which sum to zero over j.
The model allows all its parameters to be estimated and tested. For example, the absence of overall group-level bias (type (a)) would be indicated by ßQ0 = ßF0 = 0. The absence of intake-related biases (type (b)) would be indicated by ßQ1 = ßF1 = 1. The absences of person-specific biases (type (c)) would be indicated by . The absence of a relationship between the person-specific biases on the two instruments would be indicated by
rs = 0.
Model (2) specifies a much more parsimonious parameterization of dietary measurement error structure than the model considered by Plummer and Clayton,21 but fits the data equally well.8 The model used by Day et al.2 is mathematically equivalent to model (2), but instead of introducing correlated person-specific biases in dietary-report instruments it allows within-person errors to be correlated both between instruments and between repeats within the same instrument.
In addition to absolute intakes, the OPEN study also allows us to investigate energy-adjusted intakes. We used two energy adjustment methods: nutrient density and nutrient residual.1 Protein density was calculated as the percentage of energy coming from protein sources and then log transformed. The protein residual was calculated from the linear regression of protein on energy intake on the log scale. Both protein density and residual were calculated for each instrument using the protein and energy intakes as measured by this instrument. The convention used for dealing with biomarker-based derived measures is explained in the Appendix.
For all dietary variables, we excluded extreme outlying values that fell outside the interval given by 25th percentile minus twice the inter-quartile range to 75th percentile plus twice the inter-quartile range. For each variable and each instrument, no more than six outlying values for men and four for women were excluded from the analyses.
The estimates of the model parameters and their standard errors were obtained using the method of maximum likelihood under the assumption of normality of the random terms in the model. Standard errors were checked for accuracy by the bootstrap method. This method is similar to the method of moments used by Day et al.,2 but is more efficient when the numbers of measurements per individual are not equal due to missing data and the normality assumptions are approximately correct.
Using the model parameters, the attenuation factor for the FFQ is expressed as
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and the correlation of the FFQ and true intake is given by
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Both are estimated by replacing the parameters by their estimates based upon model (2). This is essentially equivalent to adjusting for random within-person measurement error in the reference biomarker. Similarly, the attenuation factor for the 24HR and the correlation coefficient between the 24HR and true intake are estimated by plugging in the estimated parameters in the expressions
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and
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When the main dietary-assessment instrument in the study is based on the average of a series of k repeat measurements, then, for the FFQ, is replaced by
and, for the 24HR,
is replaced by
.
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Results |
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In Table 1, we present sample sizes, medians, and quartiles for absolute energy, protein, and protein density, respectively. We note the 3040% underreporting of median energy intake by the FFQ, as compared with the 1020% underreporting for the 24HR. Median absolute protein intakes are underestimated by approximately 30% when using the FFQ, and by approximately 10% using the 24HR. In contrast, there is a slight overestimation of protein density by the FFQ and the 24HR, especially among women. Note, however that differences in group-mean reported nutrient intake do not necessarily invalidate an instrument for use in a cohort study. As explained in Methods, attenuation factor and the correlation with true intake are more important. We therefore examine the nature of the individual biases, and the attenuation and correlation with truth of the two instruments.
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For absolute intakes, within-person random variation in the FFQ is of the same magnitude as between-person variation
of true intake. Similar to person-specific bias, it is considerably reduced by energy adjustment. As could be expected due to day-to-day variation in intake, within-person random variation
in the 24HR is substantially greater. Interestingly, relative to variation of true intake, it is only moderately reduced by energy adjustment. In contrast to person-specific bias, within-person random error variance is higher for the 24HR than for the FFQ by 1.5- to 3-fold when measuring energy or protein and by 5-fold for protein density. Comparing variances of the different sources of error, it appears that for the FFQ the person-specific bias dominates, whereas for the 24HR the within-person variation dominates.
Table 3 shows the estimated attenuation factors and correlations with truth for the two instruments, for a single administration, averages of 2, 4, or 14 repeats, and the estimated theoretical maximum that can be attained as the number of repeats becomes very large (
). When considering energy or protein, the FFQs attenuation factors are very low (below 0.2), and repeated administrations of the instrument do not lead to much improvement. Attenuation factors for 24HR are somewhat better, and are improved by repeat administrations. With four repeats the attenuations approach or exceed 0.3 except for energy intake among women.
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The overall pattern of results for estimated correlations of reported intakes with truth generally follows that of the attenuation factors. For the FFQ, correlations are very low (below 0.2) for energy, and are only slightly higher (around 0.3) for protein with not much improvement with repeat administration. Correlations improve with energy adjustment, although not as substantially as attenuation factors. For the 24HR, correlations are considerably better for energy, even without repeats, and become substantially better for protein and protein density with averaging over increasing number of repeats. Interestingly, although energy adjustment improves correlations for men, for women it leads to slightly lower correlations.
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Discussion |
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In a previous analysis of a study of 160 women conducted by the Medical Research Council Dunn Nutrition Unit in Cambridge, UK, we reported8 an estimated attenuation factor of 0.19 for protein intake assessed by a FFQ similar to the one used in the Day et al. study. Again, this is a little higher than the 0.14 found in our OPEN study, but well within the margin of sampling error.
Willett3 criticized Day et al.2 for not adjusting for heterogeneity in their population. We have addressed this issue in our study by analysing males and females separately. We also performed the analyses not reported earlier that included as covariates age in 5-year groups and the logarithm of body mass index. These analyses did not change materially the results reported in this paper.
In his commentary on Kipnis et al.,25 Willett27 criticizes the OPEN study for underestimating within-person variation in our biomarkers by not including repeat measurements of DLW and UN at the informative interval of 6 or 12 months. Underestimated within-person variation in the reference biomarker would not affect the estimated attenuation factor, which is essentially equal to the regression slope of biomarker measurement on reported intake. But it would lead to overestimated between-person variation of true intake and therefore to underestimated correlation with true intake. It is important to note that, with a valid reference instrument that is unbiased and has errors independent of those in dietary-report measurements, such as DLW or UN, within-person variation will be correctly estimated with two independent repeat administrations. We have performed an analysis of studies with repeat UN and DLW measurements separated by different time intervals.28 The results demonstrate that the OPEN design with two consecutive averages of DLW over 2-week periods each, as well as two UN repeats separated by at least 9 days, indeed produces independent biomarker replicates and unbiased estimates of within-person variation. The criticism may be justified, though, for estimating within-person variation in protein density. As explained in the Appendix, due to the convention used to derive biomarker-based reference measure for protein density, the correlation between reported and true intake may have been underestimated by at most approximately 4%.
While our results basically confirm the observations of Day et al., they also provide a partial answer to a challenge laid down by Willett,3 who contended that the results of Day et al. were not convincing because they did not examine the energy-adjusted nutrient intakes that are used by many epidemiologists. When investigating the measurement of protein density, we have found that the performance of both the FFQ and the 24HR are considerably improved. A single administration of the FFQ has an estimated attenuation factor of 0.300.40, which may be improved slightly to 0.400.50 by administering the instrument twice. A single 24HR has an attenuation factor of 0.150.25, but this can be substantially improved by repeat administrations. Four repeats lead to attenuations of 0.400.50, and further improvements can be achieved by extra administrations. (The models that are used for these predictions do account for the tendency to report lower amounts on repeated administrations of a 24HR. See description of model (2) in Statistical Methods.)
Our results indicate that the FFQ cannot be recommended as an instrument for evaluating the absolute intakes of energy and protein in relation to disease. Even with two administrations of an FFQ, the attenuation factors for energy and protein in men are 0.089 and 0.173 (Table 3). A true RR of 2.0 for would be observed as 2.00.077 = 1.05 and 2.00.173 = 1.13 for absolute intakes of energy and protein in men. The attenuation would be even a little greater for women (Table 3
). It seems unlikely that the exact form of the FFQ would change this conclusion. The FFQ used in this and the Day et al. study carried substantial differences, yet yielded similarly poor results. The attenuation factors are somewhat greater if multiple 24HR, rather than FFQ, are used for assessing absolute intakes of energy and protein (Table 3
), but the attenuation (and consequent RR dilution) remains considerable.
If, however, our objective is to evaluate relations between protein density and disease, what is the optimum dietary assessment strategy? The OPEN data indicate that, for a single administration of an FFQ, a true RR of 2.0 would be observed as 1.33 in men and 1.24 in women (derived from Table 3). These attenuated RR certainly approach the limits of detection for observational epidemiological research. Studies would have to be very large to have adequate power to detect these associations. Moreover, uncontrolled confounding could easily account for a substantial portion of such modest excess risks. If we were interested in detecting a smaller, but entirely plausible and potentially important, RR of 1.6 for a nutritional factor and disease, that RR would reduce to 1.21 in men, 1.16 in women.
What about using two FFQ in a cohort study? This would increase costs substantially but might still be a feasible approach in future nutritional epidemiological research. Even with two FFQ, though, a true RR of 2.0 for protein density would be reduced to 1.40 in men and 1.29 in women; a RR of 1.6 would become 1.26 in men, 1.19 in women (derived from Table 3). Thus, for moderate RR, the administration of two FFQ would still leave us with substantial attenuation that challenges the capabilities of even our best epidemiological studies.
Even the use of multiple 24HRa very expensive undertaking in a large cohort study due to the up front administrative costswould not provide a solution to this attenuation dilemma. For protein density, the attenuation factors associated with the administration of four 24HR recalls are comparable to those for administration of two FFQ (Table 3).
The OPEN data speak only to the 24HR in comparison to the FFQ. It is plausible that for protein density the multiple-day diary techniquewhich is considerably less expensive than the multiple 24HR approach because the diary data can be entered and analysed on a nested case-control basis at the end of a studyyields qualitatively less RR attenuation than that produced by either multiply repeated 24HR or the FFQ. Data in Table 3 show that use of 14 24HR would yield attenuation factors qualitatively greater than those for 2 FFQ or 4 24HR. Table 3
data also show that the correlation coefficients for 14 24HR are substantially greater than those for 2 FFQ or 4 24HR; this translates into greater power for detecting true RR (and reduced sample size requirements). To the extent that these results for 14 24HR can be extrapolated to two 7-day diaries, then it follows that use of the multiple-day diary might allow us to detect the modest RR that neither multiple 24HR nor FFQ could detect. We need new biomarker-based dietary assessment data, obtained from diverse populations, to evaluate this possibility.
Finally, we make two cautionary comments: (1) These results are based on a univariate analysis that relates disease risk to a single dietary variable. Whether deviations from these results are substantial in a multivariate analysis is yet to be determined. (2) Our results are based on only two nutritional factors, energy and protein (and the ratio, protein density). Although we have no direct evidence, it is reasonable that our findings and conclusions with respect to both absolute and energy-adjusted intake can be extended to other dietary factors, especially other energy-containing macronutrients. Nevertheless, we need to develop new unbiased biomarkers of dietary factors other than energy and protein if we are to evaluate more comprehensively the strengths and limitations of our dietary assessment instruments.
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Appendix |
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However, a technical difficulty arose in the analysis of energy-adjusted protein. The error in the biomarker-based derived reference measure was almost entirely influenced by the error in the UN measurements where the coefficient of variation was 17.6%. As a result, attempting to estimate the within-person variance of the derived reference measure as a parameter in the model led to relatively large standard errors in the main analysis and to instability in the procedure for bootstrap calculations.
Based on these facts, in dealing with the derived reference measurements for energy-adjusted protein, we used the following convention. When defining biomarker-based reference measures for nutrient density and nutrient residual, we used the first DLW observation with both the first and second repeat UN observations. In theory, this induced some correlation between repeat biomarker-based reference observations and therefore lead to somewhat underestimated within-person variation, but the measurement error in DLW was so small that this underestimation could be ignored in practice. For example, because the logarithm of the ratio of UN to DLW is equal to the difference between their corresponding logarithms, using the first DLW measurement underestimates within-person variation in the derived reference measure by within-person variance of DLW. From the components-of-variance analysis of the 25 subjects with two DLW observations, this variation was estimated as 0.0025. Subtracting this value from the estimates of between-person variation of true protein density intake (Table 2) would decrease them by approximately 8%. This would have very little effect on estimated model parameters that depend on between-person variation of true intake. For example, it would increase the estimated correlation coefficient between reported and true protein density by 4%.
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Acknowledgments |
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References |
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