Equality of Outcome metrics

The idea of equality of outcome metrics, is to compare the rate of success in the privileged group with the rate of success in the unprivileged group. We define the success rate $SR_g$of a particular group $g$ as:

$$SR_g =\frac{\text{Number of individuals in g with successful outcome}}{\text{Number of individuals in g}}$$

The idea is that an unbiased system would present roughly similar success rates across groups. There are two main ways of quantifying this across-groups comparison: one is by taking the ratio (disparate impact) and the other one is to take the difference (statistical parity). We will refer to the success rate for the unprivileged group as $SR_{min}$, and to the success rate for the privileged group as $SR_{maj}$. Typically, these metrics are used in recruitment or in an academic context.

We list here the mathematical definitions for a few common metrics:

• Disparate Impact: Measures the ratio of success rates. The ideal value is 1. The system is considered not biased if this quantity falls between 0.8 and 1.2.

$$Disparate Impact =\frac{SR_{min}}{SR_{maj}}$$

• Statistical Parity: Measures the difference in success rates. The ideal value is 0. A negative value means that the unprivileged group is unfavoured.

$$Statistical Parity=SR_{min}-SR_{maj}$$

• Cohen-D: this is essentially a standardised statistical parity. This value should ideally be small or negligible.

$$CohenD=\frac{SR_{min}-SR_{maj}}{poolSTD} \quad \text{with} \quad poolSTD=\frac{(n_{maj}-1)STD_{maj}^2 + (n_{min}-1)STD_{min}^2}{n_{min}+n_{maj}-2}$$

with $n_{min} \text{ and } n_{maj}$ being the number of individuals in the minority and majority groups respectively, and:

$$STD_{maj}=\sqrt{SR_{maj} \cdot (1-SR_{maj})} \quad \text{and} \quad STD_{min}=\sqrt{SR_{min} \cdot (1-SR_{min})}$$

• 2-SD Rule: Another metric that normalises statistical parity. The ideal value is 0.

$$2SD = \frac{SR_{min}-SR_{maj}}{\sqrt{\frac{SR_{tot}(1-SR_{tot})}{N\cdot P_{min}(1-P_{min})}} }$$

where $SR_{tot}$ is the total success rate (across all groups), and $P_{min}=\frac{n_{min}}{N}$, with $N$being the total number of individuals.

An example of how to measure bias in a binary classification problem in recruitment can be found in our notebook, which can be accessed here or downloaded as the following file:

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Measuring_Bias.ipynb