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Random errors, from any source, will attenuate epidemiological risk estimates. Before we launched the Shanghai Men's Health Study (SMHS), a large population-based cohort study investigating the diet–cancer association among Chinese men, a dietary calibration study was conducted among 96 men aged 40–75 years (mean age 56.5 years), with biweekly 24-hour dietary recalls (24HDRs) implemented over a 1-year period. Data from this study were analysed to evaluate the nature and magnitude of variances for intake of 26 nutrients among SMHS participants, to compare variance ratios of 26 nutrients among Chinese men and women and individuals in other studies, and to estimate the number of 24HDRs required for future dietary calibration studies in similar populations.
Design
Ninety-six healthy, free-living men in Shanghai were administered biweekly 24HDR interviews 24 times over a 1-year period. To assess between-individual and within-individual contributions to variance, a mixed effects model was fitted and ratios of within-individual to between-individual dietary intake variances were computed.
Setting
Shanghai, China.
Results
In agreement with reports from studies conducted in the USA and many other countries, we found that within-individual variances were usually larger than between-individual variances in dietary intake for all nutrients. The sum of all other variation (e.g. weekday and weekend, seasonal, interviewer) accounted for less than 5% of total variation. Ratios of within- to between-individual variances (for log-transformed data) ranged from 1.25 for carbohydrate intake to near 8 for δ-tocopherol intake.
Conclusions
The results of this study suggest that among middle-aged and elderly Chinese men in Shanghai, within- and between-individual variation account for more than 95% of the total variation for 26 nutrients. Further dietary validation studies in the same population could be adequately carried out with only 12 days of dietary recalls, if 100 participants were enrolled.
Formulae are given for the variances and covariances for mean squares in anova under the broadest possible assumptions. The results of ther authors are obtained by specializing appropriately: these include ones concerning randomization and/or random sampling models, as well as additive (linear) models consisting of mutually independent sets of exchangeable effects. Although the illustrations given refer only to doubly and triply-indexed arrays, the approach is quite general. Particular attention is drawn to the generalized cumulants (and their natural unbiased estimators) which vanish when additive models are assumed.
Earlier work of the author exploiting the role of partition lattices and their Mbius functions in the theory of cumulants, k-statistics and their generalisations is extended to multiply-indexed arrays of random variables. The natural generalisations of cumulants and k-statistics to this context are shown to include components of variance and the associated linear combinations of mean-squares which are used to estimate them. Expressions for the generalised cumulants of arrays built up as sums of independent arrays of effects as in anova models are derived in terms of the generalized cumulants of the effects. The special case of degree two, covering the unbiased estimation of components of variance, is discussed in some detail.
The standard ANOVA models with random effects for multi-indexed arrays of random variables with an arbitrary nesting structure on the indices are considered from the viewpoint of symmetry. It is found that the covariance matrix of such an array has sufficient symmetry to permit viewing the usual components of variance as a generalised spectrum and the linear models of random effects as a generalised spectral decomposition.
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