<s>
We	O
can	O
organize	O
this	O
in	O
a	O
contingency	B-Application
table	I-Application
:	O
</s>
<s>
Other	O
measures	O
of	O
effect	B-Application
size	I-Application
for	O
binary	O
data	O
such	O
as	O
the	O
relative	O
risk	O
do	O
not	O
have	O
this	O
symmetry	O
property	O
.	O
</s>
<s>
This	O
forms	O
the	O
contingency	B-Application
table	I-Application
:	O
</s>
<s>
If	O
we	O
wish	O
to	O
test	O
the	O
hypothesis	O
that	O
the	O
population	O
odds	O
ratio	O
equals	O
one	O
,	O
the	O
two-sided	O
p-value	B-General_Concept
is	O
,	O
where	O
P	O
denotes	O
a	O
probability	O
,	O
and	O
Z	O
denotes	O
a	O
standard	O
normal	O
random	O
variable	O
.	O
</s>
<s>
Note	O
however	O
that	O
the	O
standard	B-General_Concept
error	I-General_Concept
of	O
the	O
log	O
odds	O
ratio	O
does	O
depend	O
on	O
the	O
value	O
of	O
f	O
.	O
</s>
<s>
It	O
has	O
been	O
suggested	O
that	O
the	O
odds	O
ratio	O
should	O
only	O
be	O
presented	O
as	O
a	O
measure	O
of	O
effect	B-Application
size	I-Application
when	O
the	O
risk	O
ratio	O
cannot	O
be	O
estimated	O
directly	O
,	O
but	O
with	O
newly	O
available	O
methods	O
it	O
is	O
always	O
possible	O
to	O
estimate	O
the	O
risk	O
ratio	O
,	O
which	O
should	O
generally	O
be	O
used	O
instead	O
.	O
</s>
<s>
The	O
sample	B-General_Concept
odds	I-General_Concept
ratio	I-General_Concept
n11n00/n10n01	O
is	O
easy	O
to	O
calculate	O
,	O
and	O
for	O
moderate	O
and	O
large	O
samples	O
performs	O
well	O
as	O
an	O
estimator	O
of	O
the	O
population	O
odds	O
ratio	O
.	O
</s>
<s>
When	O
one	O
or	O
more	O
of	O
the	O
cells	O
in	O
the	O
contingency	B-Application
table	I-Application
can	O
have	O
a	O
small	O
value	O
,	O
the	O
sample	B-General_Concept
odds	I-General_Concept
ratio	I-General_Concept
can	O
be	O
biased	O
and	O
exhibit	O
high	O
variance	O
.	O
</s>
<s>
A	O
number	O
of	O
alternative	O
estimators	O
of	O
the	O
odds	O
ratio	O
have	O
been	O
proposed	O
to	O
address	O
limitations	O
of	O
the	O
sample	B-General_Concept
odds	I-General_Concept
ratio	I-General_Concept
.	O
</s>
<s>
One	O
alternative	O
estimator	O
is	O
the	O
conditional	O
maximum	O
likelihood	O
estimator	O
,	O
which	O
conditions	O
on	O
the	O
row	O
and	O
column	O
margins	O
when	O
forming	O
the	O
likelihood	O
to	O
maximize	O
(	O
as	O
in	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
)	O
.	O
</s>
<s>
Another	O
alternative	O
estimator	O
is	O
the	O
Mantel	B-General_Concept
–	I-General_Concept
Haenszel	I-General_Concept
estimator	I-General_Concept
.	O
</s>
<s>
The	O
following	O
four	O
contingency	B-Application
tables	I-Application
contain	O
observed	O
cell	O
counts	O
,	O
along	O
with	O
the	O
corresponding	O
sample	B-General_Concept
odds	I-General_Concept
ratio	I-General_Concept
(	O
OR	O
)	O
and	O
sample	O
log	O
odds	O
ratio	O
(	O
LOR	O
)	O
:	O
</s>
<s>
There	O
are	O
various	O
other	O
summary	O
statistics	O
for	O
contingency	B-Application
tables	I-Application
that	O
measure	O
association	O
between	O
two	O
events	O
,	O
such	O
as	O
Yule	B-General_Concept
's	I-General_Concept
Y	I-General_Concept
,	O
Yule	B-General_Concept
's	I-General_Concept
Q	I-General_Concept
;	O
these	O
two	O
are	O
normalized	O
so	O
they	O
are	O
0	O
for	O
independent	O
events	O
,	O
1	O
for	O
perfectly	O
correlated	O
,	O
1	O
for	O
perfectly	O
negatively	O
correlated	O
.	O
</s>
