Computes the quantile ratio index (QRI) for measuring inequality from grouped frequency data using linear interpolation for quantile estimation. This function is intended to be used for administrative or tax data, which are very often in the form of grouped (binned) data. Therefore, sampling weights are not considered.
Arguments
- freq
Numeric vector of class frequencies (counts). Must be non-negative.
- lower_bounds
Numeric vector of lower class bounds.
- upper_bounds
Numeric vector of upper class bounds.
- M
Integer, number of quantile ratios to average (default: 100).
- midpoints
Optional numeric vector of class midpoints. Used only as fallback when a quantile class has zero frequency.
- na.rm
Logical, should missing values in frequencies be removed? (default: TRUE)
Value
A scalar numeric value representing the estimated inequality by the quantile ratio index (QRI) for grouped data.
Details
Consider grouped data divided into \(L\) classes with known boundaries and observed frequencies \(f_1, \ldots, f_L\). The QRI estimator for grouped data is approximated as:
$${QRI} \approx \frac{1}{M}\sum_{m=1}^{M}\left(1 - \frac{\widetilde{Q}(p_m/2)}{\widetilde{Q}(1 - p_m/2)}\right)$$
where:
\(p_m = (m - 0.5)/M\) for \(m=1, \ldots, M\)
\(\widetilde{Q}(p)\) denotes the \(p\)-th quantile computed from grouped data using linear interpolation (see
quantile_grouped)\(M\) is the number of quantile ratios to average (default: 100)
The quantiles \(\widetilde{Q}(p)\) are computed via quantile_grouped(),
which uses linear interpolation within classes and automatically handles
open-ended classes (with -Inf or Inf bounds).
The QRI ranges from 0 (perfect equality) to 1 (maximum inequality). The index measures inequality by averaging the relative differences between symmetric quantiles below and above the median, across the entire distribution.
Comparison with Microdata QRI
When individual-level (microdata) are available, use qri instead,
which provides more accurate estimates. The grouped data version
qri_grouped should be used when only frequency distributions are available,
such as in published statistical tables or administrative aggregates.
The grouped QRI will generally approximate the microdata QRI well when:
Classes are sufficiently narrow
The distribution within classes is approximately uniform
Sample sizes within classes are adequate
References
Prendergast LA, Staudte RG (2018). “A simple and effective inequality measure.” The American Statistician, 72, 328–343.
See also
qri for QRI estimation with microdata.
superpop_qri for QRI computation on parametric distributions
Other grouped data functions:
gini_grouped(),
quantile_grouped()
Examples
# Basic example with closed classes
income_freq <- c(120, 180, 150, 80, 40, 20, 10)
income_lower <- c(0, 15000, 30000, 45000, 60000, 80000, 100000)
income_upper <- c(15000, 30000, 45000, 60000, 80000, 100000, 150000)
qri_grouped(income_freq, income_lower, income_upper)
#> [1] 0.5888192
# Example with open-ended classes (Italian MEF-style data)
wage_freq <- c(150, 200, 180, 220, 180, 50, 15, 5)
wage_lower <- c(-Inf, 0, 10000, 15000, 26000, 55000, 75000, 120000)
wage_upper <- c(0, 10000, 15000, 26000, 55000, 75000, 120000, Inf)
# Compute QRI (automatically handles open classes)
qri_grouped(wage_freq, wage_lower, wage_upper)
#> [1] 0.7268536