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nature of the data in the examples given, namely, the reversal or the cessation of
an association in the overall population; they are in no way ad hoc. Third, the
paradox is “structural” in character, in the sense that the reasoning that leads
to it is deductive. Consider our examples, which involve simple arithmetic. The
overall rates of acceptance for both females and males follow from their rates of
acceptance in two departments taken separately. Note that both conditions of
the paradox can be defined in terms of the probability theory, which is purely
deductive [3]. Fourth, unless someone uses the notion of causation trivially, for
example, believes that 2+2 “causes” 4, there is no reason to assume that there
are causal intuitions lurking in the background. We will return to the last point
in greater detail in the following sections.
2.2 Why is SP “Paradoxical”?
To answer question (i), “why is SP a paradox?” we now provide an explanation of
how the paradox arises in people's minds and why it is found perplexing. In other
words, what is the reasoning that the “average person” follows that leads him/her
to a paradoxical conclusion? For our purposes, we have reconstructed our type
I version of SP in terms of its premises and conclusion to show how the paradox
arises. However, the point of the reconstruction will be adequately general to
be applicable to all types of SP. We introduce a numerical principle called the
collapsibility principle (CP) which plays a crucial role in the reconstruction. CP
says that relationships between variables that hold in the sub-populations (e.g.,
the rate of acceptance of females being higher than the rate of acceptance of
males in both sub-populations) must hold in the overall population as well (i.e.,
the rate of acceptance of females must be higher than the rate of acceptance of
males in the population). There are two versions of CP corresponding to the two
types of SP represented by Tables 1 and 2. The first version of CP (CP1) says
that a dataset is collapsible if and only if [(
A
1
>B
1
)&(
A
2
>B
2
)
(
ʱ>ʲ
)].
The second version of CP (CP2) states that a dataset is collapsible if and only if
[(
A
1
=
B
1
)&(
A
1
=
B
2
)
→
(
ʱ
=
ʲ
)]. That CP1 and CP2 can lead to paradoxical
results demonstrates that both versions of the principle are not, in all their
applications, true. That is, CP
→
→∼
→
”
is to be construed as the implication sign. If
f
1
,
F
2
,
m
1
,
M
2
,
A
1
,
A
2
,
B
1
,
B
2
,
ʱ
,and
ʲ
have the same meanings as given in section 2.1, then CP1 takes the
following form.
f
1
SP, whether it is CP1 or CP2, where “
&
f
2
f
1
+
f
2
F
1
>
m
1
F
2
>
m
2
F
1
+
F
2
>
m
1
+
m
2
→
M
1
M
2
M
1
+
M
2
Likewise, CP2 says
f
1
&
f
2
f
1
+
f
2
F
1
=
m
1
F
2
=
m
2
F
1
+
F
2
=
m
1
+
m
2
→
M
1
M
2
M
1
+
M
2
