Comparison of Ferguson’s δ and the Gini coefficient used for measuring the inequality of data related to health quality of life outcomes

BACKGROUND: Ferguson’s δ and Gini coefficient (GC) are defined as contrasting statistical measures of inequality among members within populations. However, the association and cutting points for these two statistics are still unclear; a visual display is required to inspect their similarities and differences. METHODS: A simulation study was conducted to illustrate the pertinent properties of these statistics, along with Cronbach’s α and dimension coefficient (DC) to assess inequality. We manipulated datasets containing four item lengths with two number combinations (0 and 33%) in item length if two domains exist. Each item difficulty with five-point polytomous responses was uniformly distributed across a ± 2 logit range. A simulated response questionnaire was designed along with known different structures of true person scores under Rasch model conditions. This was done for 20 normally distributed sample sizes. A total of 320 scenarios were administered. Four coefficients (Ferguson’s δ, GC, test reliability Cronbach’s α, and DC) were simultaneously calculated for each simulation dataset. Box plots were drawn to examine which of these presented the correct property of inequality on data. Two examples were illustrated to present the index on Google Maps for securing the discriminatory power of individuals. RESULTS: We found that 1-Ferguson’s δ coefficient has a high correlation (0.95) with GC. The cutting points of Ferguson’s δ, GC, test reliability Cronbach’s α, and the DC are 0.15, 0.50, 0.70, and 0.67, respectively. Two applications are shown on Google Maps with GCs of 0.14 and 0.42, respectively. Histogram legends and Lorenz curves are used to display the results. CONCLUSION: The GC is recommended to readers as an index for measuring the extent of inequality (or lower discrimination power) in a given dataset. It can also show the study results of person measures to determine the inequality in the health-related quality of life outcomes.