Technical Report

Generalized lambda distribution for flexibly testing differences beyond the mean in the distribution of a dependent variable such as body mass index

Received:
Revised:
Accepted:
Published online:

Abstract

Background/Objectives:

Conventional statistical methods often test for group differences in a single parameter of a distribution, usually the conditional mean (for example, differences in mean body mass index (BMI; kg m−2) by education category) under specific distributional assumptions. However, parameters other than the mean may of be interest, and the distributional assumptions of conventional statistical methods may be violated in some situations.

Subjects/Methods:

We describe an application of the generalized lambda distribution (GLD), a flexible distribution that can be used to model continuous outcomes, and simultaneously describe a likelihood ratio test for differences in multiple distribution parameters, including measures of central tendency, dispersion, asymmetry and steepness. We demonstrate the value of our approach by testing for differences in multiple parameters of the BMI distribution by education category using the Health and Retirement Study data set.

Results:

Our proposed method indicated that at least one parameter of the BMI distribution differed by education category in both the complete data set (N=13 571) (P<0.001) and a randomly resampled data set (N=300 from each category) to assess the method under circumstances of lesser power (P=0.044). Similar method using normal distribution alternative to GLD indicated the significant difference among the complete data set (P<0.001) but not in the smaller randomly resampled data set (P=0.968). Moreover, the proposed method allowed us to specify which parameters of the BMI distribution significantly differed by education category for both the complete and the random subsample, respectively.

Conclusions:

Our method provides a flexible statistical approach to compare the entire distribution of variables of interest, which can be a supplement to conventional approaches that frequently require unmet assumptions and focus only on a single parameter of distribution.

  • Subscribe to International Journal of Obesity for full access:

    $652

    Subscribe

Additional access options:

Already a subscriber?  Log in  now or  Register  for online access.

References

  1. 1.

    , , , . Trends in the skewness of the body mass index distribution among urban Australian adults, 1980 to 2007. Ann Epidemiol 2015; 25: 26–33.

  2. 2.

    , . The trend of BMI values of US adults by deciles, birth cohorts 1882-1986 stratified by gender and ethnicity. Econ Hum Biol 2011; 9: 234–250.

  3. 3.

    . Quantile regression-opportunities and challenges from a user's perspective. Am J Epidemiol 2014; 180: 330–331.

  4. 4.

    , , . A simple significance test for quantile regression. Stat Med 2004; 23: 2587–2597.

  5. 5.

    , , . Breastfeeding and childhood obesity: shift of the entire BMI distribution or only the upper parts? Obesity (Silver Spring) 2008; 16: 2730–2733.

  6. 6.

    , , . Educational level, fatness, and fatness differences between husbands and wives. Am J Clin Nutr 1989; 50: 740–745.

  7. 7.

    , , . Determinants of body mass index: a study from northern Italy. Int J Obes Relat Metab Disord 1994; 18: 497–502.

  8. 8.

    , , , , , et al. The association of education with body mass index and waist circumference in the EPIC-PANACEA study. BMC Public Health 2011; 11: 169.

  9. 9.

    , . Trends in the association between obesity and socioeconomic status in US adults: 1971 to 2000. Obes Res 2004; 12: 1622–1632.

  10. 10.

    , , , , . Educational level, relative body weight, and changes in their association over 10 years: an international perspective from the WHO MONICA Project. Am J Public Health 2000; 90: 1260–1268.

  11. 11.

    , , , . Cross-sectional and longitudinal associations of BMI with socioeconomic characteristics. Obes Res 2005; 13: 1412–1421.

  12. 12.

    . Overweight and poor? On the relationship between income and the body mass index. Econ Hum Biol 2011; 9: 342–355.

  13. 13.

    RAND. RAND HRS Data, Version M. RAND Center for the Study of Aging, with funding from the National Institute on Aging and the Social Security Administration: Santa Monica, CA, USA 2013.

  14. 14.

    , , . Flexible distribution modeling with the generalized lambda distribution. MPRA 2012; 43333 Available at: .

  15. 15.

    , . A study of fitting the generalized lambda distribution to solar-radiation data. J Appl Meteorol 1982; 21: 10.

  16. 16.

    . Multiple imputation under the generalized lambda distribution. J Biopharm Stat 2009; 19: 77–89.

  17. 17.

    , . On the problem of the most efficient tests of statistical hypotheses. Philos Trans R Soc Lond A 1933; 231: 289–337.

  18. 18.

    . Method of moments and method of maximum likelihood. Biometrika 1936; 28: 34–59.

  19. 19.

    . Fitting flexible parametric regression models with GLDreg in R. J Mod Appl Stat Methods 2016; 15: 46.

  20. 20.

    Information theory and an extension of the maximum likelihood principle. In: , , eds. Breakthroughs in Statistics: Foundations and Basic Theory. Springer: New York, NY, USA, 1992; 610–624.

  21. 21.

    , , , . A novel method for estimating distributions of body mass index. Popul Health Metr 2016; 14: 6.

Download references

Acknowledgements

We thank Dr Jasmin Divers for reviewing our initial manuscript and providing constructive comments. We also thank Jennifer Holmes for editing our manuscript. This study was supported in part by NIH grants P30DK056336 and UL1TR001417, Japan Society for Promotion of Science (JSPS) grant KAKENHI 15J00009 and Grant-in-Aid for Epidemiological Research from St Luke’s International University. The opinions expressed are those of the authors and do not necessarily represent those of the NIH or any other organization.

Author information

Affiliations

  1. Office of Energetics, School of Health Professions, University of Alabama at Birmingham, Birmingham, AL, USA

    • K Ejima
    •  & D B Allison
  2. Institute of Industrial Science, The University of Tokyo, Tokyo, Japan

    • K Ejima
  3. Nutrition Obesity Research Center, University of Alabama at Birmingham, Birmingham, AL, USA

    • K Ejima
    • , G Pavela
    •  & D B Allison
  4. Department of Health Behavior, University of Alabama at Birmingham, Birmingham, AL, USA

    • G Pavela
  5. Department of Biostatistics, University of Alabama at Birmingham, Birmingham, AL, USA

    • P Li
    •  & D B Allison
  6. Department of Nutrition Sciences, University of Alabama at Birmingham, Birmingham, AL, USA

    • D B Allison

Authors

  1. Search for K Ejima in:

  2. Search for G Pavela in:

  3. Search for P Li in:

  4. Search for D B Allison in:

Competing interests

The authors declare no conflict of interest.

Corresponding author

Correspondence to D B Allison.

Supplementary information

Word documents

  1. 1.

    Supplementary Material

Author contributions

KE, GP and PL prepared the original draft. KE and PL led the data analysis. GP conducted the literature search and managed the data. DBA conceived and supervised the project.

Supplementary Information accompanies this paper on International Journal of Obesity website (http://www.nature.com/ijo)