cIQR

cIQR stands for conditional interquartile range, a robust dispersion measure that describes the spread of a dependent variable Y conditional on one or more covariates X. It extends the idea of the interquartile range to the conditional distribution, allowing variability to vary with X rather than being constant across the population.

Definition and computation: Let Q1(x) and Q3(x) denote the 25th and 75th conditional quantiles of Y given X = x. The conditional IQR is cIQR(x) = Q3(x) − Q1(x). In practice, cIQR is estimated by estimating the two conditional quantile functions Q0.25(x) and Q0.75(x) and taking their difference: cIQR̂(x) = Q̂0.75(x) − Q̂0.25(x). Quantile regression provides a straightforward approach for this, and other methods include local (kernel) quantile regression or bootstrap procedures to quantify uncertainty.

Uses and interpretation: cIQR is used to study how dispersion of outcomes varies with covariates, to assess heteroscedasticity, and to characterize risk or variability across groups. It is more robust to outliers and heavy tails than a conditional standard deviation and does not assume normality of the conditional distribution.

Limitations and extensions: Estimation requires sufficient data across the range of X, and interpretation depends on which variables are conditioned on. Model choice and sample size can influence the reliability of cIQR estimates, especially in multivariate or high-dimensional settings. Extensions to multivariate conditioning or dependent data are discussed in the statistical literature.