Generalized lambda distribution for flexibly testing differences beyond the mean in the distribution of a dependent...

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.

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.

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.

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.