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

## 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; kgâ€‰mâˆ’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=13â€‰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.

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## 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 Lukeâ€™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

Authors

### Corresponding author

Correspondence to D B Allison.

## Ethics declarations

### Competing interests

The authors declare no conflict of interest.

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

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Ejima, K., Pavela, G., Li, P. et al. Generalized lambda distribution for flexibly testing differences beyond the mean in the distribution of a dependent variable such as body mass index. Int J Obes 42, 930â€“933 (2018). https://doi.org/10.1038/ijo.2017.262

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• DOI: https://doi.org/10.1038/ijo.2017.262