Minimum detectable effect size for main outcomes (accounting for sample
design and clustering)

To conduct the power analysis, we rely on the study conducted by Schmid et al. (2021) which tests the adhesion to different principles of distributive justice in a context of public infrastructure planning. Although our context differs from the one studied by Schmid et al. (2021) we argue that it may be relevant for the following reason. First, they study fairness in a water-related context (ageing of wastewater systems). Second, their analysis is conducted in Switzerland, a country sharing similarities with France. Third, Schmid et al. (2021) focuses on three principles of distributive justice –equity, equality, and need– which are also at the core of our analysis.
Our central outcomes of interest are the subjective perception of fairness associated to the different water sharing rules which are measured using a scale going from 0 (Totally unfair) to 10 (Totally fair). In Schmid et al. (2021), perception of fairness associated to the three principles of distributive justice is measured using a 6-points scale (Not at all fair, Not fair, Rather not fair, Rather fair,Fair, Totally Fair). In order to match our way of measuring fairness (quantitative scale from 0 to 10), we have converted this 6-points qualitative scale into a quantitative scale going from 0 (Not at all fair) to 10 (Totally Fair). We have then computed a fairness score for each principle of distributive justice with its associated standard deviation.
Table: Descriptive results on adhesion to fairness principles
Mean score Standard deviation
Equality 5,01 2,71
Equity 5,70 2,59
Need 7,07 2,68
Source: Author’s computation based on Schmid et al. (2021)
We conduct a power analysis to detect significant differences between fairness scores associated to the three principles of distributive justice. We compute sample sizes, power and minimal detectable effects using two-sample means tests. All tests are implemented using the power function in Stata version 15.1.
First, we determine the sample sizes needed to obtain a power of 0.8 given an alpha of 0.05 (5%-level two-sided test). For Equality and Need (resp. Equity and Need ; Equality and Equity), the power analysis indicates that we require a sample size of at least n = 28 (resp n=60 ; n=233) in each treatment.
Based on our sample size (n=222) we get a power for Equality and Need (resp. Equity and Need ; Equality and Equity) equal to 100% (resp. 99.9% ; 78.1%). This translates into effect size for Equality and Need (resp. Equity and Need ; Equality and Equity) equal to 0,76 (resp. 0,52 ; -0.26).
Schmid, S., Vetschera, R., & Lienert, J. (2021). Testing Fairness Principles for Public Environmental Infrastructure Decisions. Group Decision and Negotiation, 30(3), 611-640.
Appendix: STATA code for power analyzes
* Sample size for a two-sample means test
power twomeans 5.01 7.07, sd1(2.71) sd2(2.68)
power twomeans 5.01 5.70, sd1(2.71) sd2(2.59)
power twomeans 5.70 7.07, sd1(2.59) sd2(2.68)
* Computing power
power twomeans 5.01 7.07 , n(222) sd1(2.71) sd2(2.68)
power twomeans 5.70 7.07 , n(222) sd1(2.59) sd2(2.68)
power twomeans 5.01 5.70 , n(222) sd1(2.71) sd2(2.59)
* Effect size
power twomeans 5.01, n(222) power(0.8) direction(lower) sd1(2.71) sd2(2.68)
power twomeans 5.70, n(222) power(0.8) direction(lower) sd1(2.59) sd2(2.68)
power twomeans 5.01, n(222) power(0.8) direction(lower) sd1(2.71) sd2(2.59)