The term quantile is synonymous with percentile; the median is the best-known example of a quantile, being the 50th in a ranked distribution of numbers. If we say that an infant's height is at the
rth quantile, then they are taller than the proportion
r of the reference group of infants and shorter than the proportion 1 −
r. The median (50th), lower (25th) and upper (75th) quartiles are specific quantiles. The 95% reference interval commonly used for measurements in medicine captures the data between the 2.5 and the 97.5 quantiles. Often a measurement varies with age, such as in infant growth, and the challenge is to estimate the 95% reference intervals at different specific ages. The problem then becomes one of regression where limits around the average (mean) relationship between, say, the infant's height and age are captured. However, just as a mean often gives an incomplete picture of a single distribution of numbers (it only indicates the center of the distribution), so the single OLS regression line gives only an incomplete picture of the relationship between two variables. The alternative is quantile regression: Just as classic linear regression methods based on minimizing sums of squared residuals enables one to estimate models for conditional mean functions, regression quantile methods offer a mechanism for estimating models for the conditional median function, and the full range of conditional quantile functions. Estimation here is based on a weighted sum (with weights depending on the order of the quantiles) of absolute values of residuals. In short, we can generate a regression line at any point in the distribution of values, and this is what we sought to do with rim area against optic disc size in this work. There are variants on how to fit the lines but we have adopted the standard method used in a package quantreg
31,32 from the open-source statistical programming environment R.
33,34 For more technical detail, the statistically minded reader is directed toward Koenker.
27 Those who are curious but require a more accessible descriptor of the methods are referred to a well-written paper by Cade and Noon.
29