Statistical Methods & Applications

, Volume 27, Issue 4, pp 715–734 | Cite as

Box–Cox t random intercept model for estimating usual nutrient intake distributions

  • Giovana Fumes-GhantousEmail author
  • Silvia L. P. Ferrari
  • José Eduardo Corrente
Original Paper


The issue of estimating usual nutrient intake distributions and prevalence of inadequate nutrient intakes is of interest in nutrition studies. Box–Cox transformations coupled with the normal distribution are usually employed for modeling nutrient intake data. When the data present highly asymmetric distribution or include outliers, this approach may lead to implausible estimates. Additionally, it does not allow interpretation of the parameters in terms of characteristics of the original data and requires back transformation of the transformed data to the original scale. This paper proposes an alternative approach for estimating usual nutrient intake distributions and prevalence of inadequate nutrient intakes through a Box–Cox t model with random intercept. The proposed model is flexible enough for modeling highly asymmetric data even when outliers are present. Unlike the usual approach, the proposed model does not require a transformation of the data. A simulation study suggests that the Box–Cox t model with random intercept estimates the usual intake distribution satisfactorily, and that it should be preferable to the usual approach particularly in cases of highly asymmetric heavy-tailed data. In applications to data sets on intake of 19 micronutrients, the Box–Cox t models provided better fit than its competitors in most of the cases.


Box–Cox Cole–Green distribution Box–Cox t distribution Box–Cox transformation NCI method Nutrient intake 



We thank the reviewers for their valuable comments and suggestions on an earlier version of the paper. We gratefully acknowledge the financial support of the Brazilian agencies FAPESP (grants 2008/10261-8 and 2012/21788-2) and CNPq (grant 304388/20149).


  1. Block G (1982) A review of validations of dietary assessment methods. Am J Epidemiol 115:492–505CrossRefGoogle Scholar
  2. Borrelli R, Simonetti MS, Fidanza F (1992) Inter and intra individual variability in food intake of elderly people in Perugia (Italy). Br J Nutr 68:3–10CrossRefGoogle Scholar
  3. Box GEP, Cox DR (1964) An analysis of transformations. J R Stat Soc Ser B 26:211–252zbMATHGoogle Scholar
  4. Carriquiry AL (1998) Assessing the prevalence of nutrient inadequacy. Public Health Nutr 2:23–33Google Scholar
  5. Cole TJ, Green PJ (1992) Smoothing reference centile curves: the LMS method and penalized likelihood. Stat Med 10:1305–1319CrossRefGoogle Scholar
  6. Dodd KW, Guenther PM, Freedman LS, Subar AF, Kipnis V, Midthune D, Tooze JA, Smith SMK (2006) Statistical methods for estimating usual intake of nutrients and foods: a review of the theory. J Am Diet Assoc 106:1640–1650CrossRefGoogle Scholar
  7. Ferrari SLP, Fumes G (2017) Box–Cox symmetric distributions and applications to nutritional data. Adv. Stat Anal 101:321–344MathSciNetCrossRefzbMATHGoogle Scholar
  8. Hubert M, Vandervieren E (2008) An adjusted boxplot for skewed distributions. Comput Stat Data Anal 52:5186–5201MathSciNetCrossRefzbMATHGoogle Scholar
  9. Institute of Medicine, Food and Nutrition Board (2003) Dietary reference intakes: applications in dietary planning. National Academies Press, Washington. [Acessed in 31 May, 2017], Subcommittee on interpretation and uses of dietary reference intakes and the standing committee on the scientific evaluation of dietary reference intakes. Available at
  10. Nutrition Coordenating Center (2012) Nutrition data system for research. NDS-R. Features. Accessed 31 May 2017
  11. Pinheiro JC, Bates DM (1995) Approximations to the log-likelihood function in the nonlinear mixed-effects model. J Comput Graph Stat 4:12–35Google Scholar
  12. R Core Team (2008) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN: 3-900051-07-0Google Scholar
  13. Rigby RA, Stasinopoulos DM (2006) Using the Box–Cox t distribution in GAMLSS to model skewness and kurtosis. Stat Model 6:209–229MathSciNetCrossRefGoogle Scholar
  14. SAS Institute Inc. (2012) SAS/STAT 12.1 user’s guide. SAS Institute Inc., CaryGoogle Scholar
  15. Slater B, Marchioni DL, Fisberg RM (2004) Estimating prevalence of inadequate nutrient intake. Rev. Saúde Pública 38:599–605CrossRefGoogle Scholar
  16. Souverein OW, Dekkers AL, Geelen A, Haubrock J, Vries JH, Ocke MC, Harttig U, Boeing H, Veer P (2011) Comparing four methods to estimate usual intake distribution. Eur J Clin Nut 65:S92–S101CrossRefGoogle Scholar
  17. Stasinopoulos DM, Rigby RA (2007) Generalized additive models for location, scale and shape (GAMLSS) in R. J Stat Softw 23:1–46CrossRefGoogle Scholar
  18. Tooze JA, Kipnis V, Buckman DW, Carroll RJ, Freedman LS, Guenther PM, Krebs-Smith SM, Subar AF, Dodd KW (2010) A mixed-effects model approach for estimating the distribution of usual intake of nutrients: the NCI method. Stat Med 29:2857–2868MathSciNetCrossRefGoogle Scholar
  19. Voudouris V, Gilchrist R, Rigby RA, Sedgwick J, Stasinopoulos DM (2012) Modelling skewness and kurtosis with BCPE density in GAMLSS. J Appl Stat 39:1279–1293MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Basic Sciences, FZEAUniversity of São PauloPirassunungaBrazil
  2. 2.Department of StatisticsUniversity of São PauloSão PauloBrazil
  3. 3.Department of BiostatisticsUniversity of São Paulo StateBotucatuBrazil

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