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•
relative chance
(RC
=(
1
−
p
1
)
/
(
1
−
p
2
)
), also known as
selection rate
,
•
),
and the versions of RD, RR, and RC when the protected group is compared to the
average proportion
p
, rather than to the proportion of the unprotected group:
•
odds ratio
(OR
=
p
1
(
1
−
p
2
)
/
(
p
2
(
1
−
p
1
))
extended difference
(ED
=
p
1
−
p
);
•
extended ratio
or
extended lift
(ER
=
p
1
/
p
);
•
).
Since one is interested in contexts of higher benefit denial (resp., lower benefit grant-
ing) for the protected group compared to the unprotected group or to the average,
the values of interest for RR, OR, and ER are those greater than 1; for RD and ED
are those greater than 0; and for RC and EC are those lower than 1. Confidence
intervals and tests of statistical significance of the above measures are discussed in
(Pedreschi et al., 2009; Ruggieri et al., 2010c). Here, we only mention that statistical
tests will rank the rules according to how unlikely it is that they would be observed
if there was equal treatment, not according to the severity of discrimination. As an
example, a mild discrimination among a large population will be ranked higher than
a much more severe discrimination in a small community.
From the legal side, different measures are adopted worldwide. UK law
(U.K. Legislation, 2011, (a)) mentions risk difference, EU Court of Justice has given
more emphasis to the risk ratio (see Schiek et al., 2007, Section 3.5), and US laws
and courts mainly refer to the selection rate
2
. Notice that the risk ratio is the ratio
of the proportions of
benefit denial
between the protected and unprotected groups,
while selection rate is the ratio of the proportions of
benefit granting
.TheEUis
more concerned about the ratio of denials, while the US is more concerned about
the ratio of grants; unfortunately, they do not lead to the same conclusions in dis-
crimination discovery.
Once we are provided with a quantitative measure of discrimination and a thresh-
old between “legal” and “illegal” degree, we are in the position to isolate classifi-
cation rules whose measure is below/above the threshold (for simplicity, we limit
ourselves to the extended lift measure).
extended chance
(EC
=(
1
−
p
1
)
/
(
1
−
p
)
Definition 4 (
a
-protection).
We say that a PD classification rule
A
,
B
→
C
is
a
-
protective if
elift
(
A
,
B
→
C
)
<
a
. Otherwise, we say that it is
a
-discriminatory.
Intuitively,
a
is a fixed threshold stating an acceptable level of discrimination accord-
ing to laws, regulations, and jurisprudence. Classification rules denying a benefit and
with a measure below such a level are considered safe, whilst rules whose measure
is greater or equal than such a level can then be considered a
prima facie
3
evidence
2
(U.S. Federal Legislation, 2011, (d)) goes further by stating that “a selection rate for any
race, sex, or ethnic group which is less than four-fifths (or eighty percent) of the rate for
the group with the highest rate will generally be regarded as evidence of adverse impact”.
This is called the
four-fifths rule
. It turns out to fix a minimum threshold value for
RC
of
4
/
5
=
0
.
8.
3
Prima facie
is a Latin term meaning “at first look,” or “on its face,” and refers to evidence
that, unless rebutted, would be sufficient to prove a particular proposition or fact.
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