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Classification rule A , B C
benefit ( C )
group denied granted
protected ( A )
a
b
n 1
unprotected ( ¬ A )
c
d
n 2
m 1
m 2
n (total of B )
p 1 = a / n 1
p 2 = c / n 2
p = m 1 / n
p 1
p 2
1 p 1
1 p 2
RR
RC =
a / b
c / d
RD = p 1 p 2
RR =
RC =
OR =
p 1
p
1
p 1
1 p
ED = p 1 pER =
EC =
Fig. 5.1 Contingency table and discrimination measures
Definition 3. Let A
,
B
C be a PD classification rule with con f
(
B
C
) >
0. The
extended lift of the rule is:
(
,
)
con f
A
B
C
elift
(
A
,
B
C
)=
.
con f
(
B
C
)
Arule SEX = FEMALE , CAR = OWN
CREDIT = NO with an extended lift of 3 means
that being a female increases 3 times the probability of being refused credit with
respect to the average confidence of people owning a car. While this means that
women are discriminated among car owners, notice that we cannot conclude that
being a woman is the actual reason of discrimination (see Sect. 5.5 for a discussion).
An alternative way, yet equivalent, of defining the extend lift is as the ratio between
the proportion of the disadvantaged group A in context B obtaining the benefit C
over the overall proportion of A in B :
(
,
)
con f
B
C
A
.
con f
(
B
A
)
This makes it clear how extended lift relates to the principle of group over-represen-
tation in benefit denying, or, equivalently, of group under-representation in benefit
granting. In addition to extended lift, other measures can be formalized starting
from different definitions of discrimination provided by laws. They can be defined
over the 2
2 contingency table shown in Figure 5.1, showing the absolute num-
ber of transactions in the underlying dataset
×
satisfying the itemsets in the X-Y
coordinates and the context B .Let p 1 (resp., p 2 ) be the proportion of people in the
protected group (resp., not in the protected group) that were not granted a bene-
fit, and let p be the proportion of all people (both protected and not) that were not
granted the benefit. The following discrimination measures can be defined:
D
risk difference (RD
=
p 1
p 2 ), also known as absolute risk reduction ,
risk ratio or relative risk (RR
=
p 1 /
p 2 ),
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