Inferring inequality: Testing for median-preserving spreads in ordinal data

Abstract.

The median-preserving spread (MPS) ordering for ordinal variables has become ubiquitous in the inequality literature. We devise statistical tests of the hypothesis that a distribution G is not an MPS of a distribution F. Rejecting this hypothesis enables the conclusion that G is more unequal than F according to the MPS criterion. Monte Carlo simulations and novel graphical techniques show that a simple, asymptotic Z test is sufficient for most applications. We illustrate our tests with two applications: happiness inequality in the US and self-assessed health in Europe.

Keywords:

• Hypothesis testing
• inequality measurement
• median-preserving spread
• ordinal data

JEL codes::

• C12
• D63
• I14
• I32

Acknowledgments

We would like to thank James MacKinnon, Ashraful Mahfuze, Andy Snell, seminar participants at Heriot-Watt and the 9th ECINEQ Meeting, and three anonymous referees and the editor for helpful comments and suggestions.

Disclosure statement

The authors report there are no competing interests to declare.

Notes

1. The MPS partial ordering is itself a special case of Mendelson (1987)’s “quantile-preserving spread”; see section 6.

2. See Davidson and Duclos (2013) for a similar framing of the null and alternative hypotheses.

3. ${\mathbb{N}}_0=\{0,1,2,...\}$.

4. Even though the sample size (n_x, n_y) is normally considered a parameter of the multinomial distribution, we do not consider it as such because in our applications it is fixed (e.g. by survey design).

5. For the case of median-preserving spreads with multiple median categories see Liu et al. (2024).

6. The set Θ₀ is rotationally symmetric, meaning that reversing the ordering of the categories does not alter the MPS partial ordering of the original distributions. Therefore, all the tests we propose are invariant to reverse ordering of the categories.

7. ${\hat{\sigma }}_i^L\left(\frac{n_x+n_y}{\sqrt{n_x{n}_y}}\right)$ is the pooled-sample formula for the standard errors of $Y_i/n_y-X_i/n_x$ and $X_i/n_x-Y_i/n_y$ under the null hypothesis that $(Y_i/n_y=X_i/n_x)$.

8. For more details on this application see section 5.

9. See lemma 2 in Appendix A.

10. Code implementing all four tests in R is available at https://cstapenhurst.academic.ws/projects/1551.

11. Code replicating these experiments in R is available at https://cstapenhurst.academic.ws/projects/1551.

12. We thank an anonymous referee for this suggestion.

13. With respect to any Euclidean metric.

Additional information

Funding

Christopher Stapenhurst gratefully acknowledges financial support from the Economic and Social Research Council through an ESRC 1+3 scholarship.