<s>
In	O
statistics	O
,	O
a	O
contingency	B-Application
table	I-Application
(	O
also	O
known	O
as	O
a	O
cross	B-Application
tabulation	I-Application
or	O
crosstab	B-Application
)	O
is	O
a	O
type	O
of	O
table	B-Application
in	O
a	O
matrix	B-Architecture
format	O
that	O
displays	O
the	O
(	O
multivariate	O
)	O
frequency	O
distribution	O
of	O
the	O
variables	O
.	O
</s>
<s>
The	O
term	O
contingency	B-Application
table	I-Application
was	O
first	O
used	O
by	O
Karl	O
Pearson	O
in	O
"	O
On	O
the	O
Theory	O
of	O
Contingency	O
and	O
Its	O
Relation	O
to	O
Association	O
and	O
Normal	O
Correlation	O
"	O
,	O
part	O
of	O
the	O
Drapers	O
 '	O
Company	O
Research	O
Memoirs	O
Biometric	O
Series	O
I	O
published	O
in	O
1904	O
.	O
</s>
<s>
A	O
crucial	O
problem	O
of	O
multivariate	B-General_Concept
statistics	I-General_Concept
is	O
finding	O
the	O
(	O
direct	O
-	O
)	O
dependence	O
structure	O
underlying	O
the	O
variables	O
contained	O
in	O
high-dimensional	O
contingency	B-Application
tables	I-Application
.	O
</s>
<s>
In	O
order	O
to	O
do	O
this	O
one	O
can	O
use	O
information	O
theory	O
concepts	O
,	O
which	O
gain	O
the	O
information	O
only	O
from	O
the	O
distribution	O
of	O
probability	O
,	O
which	O
can	O
be	O
expressed	O
easily	O
from	O
the	O
contingency	B-Application
table	I-Application
by	O
the	O
relative	O
frequencies	O
.	O
</s>
<s>
A	O
pivot	B-Application
table	I-Application
is	O
a	O
way	O
to	O
create	O
contingency	B-Application
tables	I-Application
using	O
spreadsheet	O
software	O
.	O
</s>
<s>
A	O
contingency	B-Application
table	I-Application
can	O
be	O
created	O
to	O
display	O
the	O
numbers	O
of	O
individuals	O
who	O
are	O
male	O
right-handed	O
and	O
left-handed	O
,	O
female	O
right-handed	O
and	O
left-handed	O
.	O
</s>
<s>
Such	O
a	O
contingency	B-Application
table	I-Application
is	O
shown	O
below	O
.	O
</s>
<s>
The	O
grand	O
total	O
(	O
the	O
total	O
number	O
of	O
individuals	O
represented	O
in	O
the	O
contingency	B-Application
table	I-Application
)	O
is	O
the	O
number	O
in	O
the	O
bottom	O
right	O
corner	O
.	O
</s>
<s>
The	O
table	B-Application
allows	O
users	O
to	O
see	O
at	O
a	O
glance	O
that	O
the	O
proportion	O
of	O
men	O
who	O
are	O
right-handed	O
is	O
about	O
the	O
same	O
as	O
the	O
proportion	O
of	O
women	O
who	O
are	O
right-handed	O
although	O
the	O
proportions	O
are	O
not	O
identical	O
.	O
</s>
<s>
The	O
strength	O
of	O
the	O
association	O
can	O
be	O
measured	O
by	O
the	O
odds	O
ratio	O
,	O
and	O
the	O
population	O
odds	O
ratio	O
estimated	O
by	O
the	O
sample	B-General_Concept
odds	I-General_Concept
ratio	I-General_Concept
.	O
</s>
<s>
The	O
significance	B-General_Concept
of	O
the	O
difference	O
between	O
the	O
two	O
proportions	O
can	O
be	O
assessed	O
with	O
a	O
variety	O
of	O
statistical	O
tests	O
including	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
,	O
the	O
G-test	B-General_Concept
,	O
Fisher	B-General_Concept
's	I-General_Concept
exact	I-General_Concept
test	I-General_Concept
,	O
Boschloo	B-General_Concept
's	I-General_Concept
test	I-General_Concept
,	O
and	O
Barnard	B-General_Concept
's	I-General_Concept
test	I-General_Concept
,	O
provided	O
the	O
entries	O
in	O
the	O
table	B-Application
represent	O
individuals	O
randomly	O
sampled	O
from	O
the	O
population	O
about	O
which	O
conclusions	O
are	O
to	O
be	O
drawn	O
.	O
</s>
<s>
The	O
example	O
above	O
is	O
the	O
simplest	O
kind	O
of	O
contingency	B-Application
table	I-Application
,	O
a	O
table	B-Application
in	O
which	O
each	O
variable	O
has	O
only	O
two	O
levels	O
;	O
this	O
is	O
called	O
a	O
2×2	O
contingency	B-Application
table	I-Application
.	O
</s>
<s>
There	O
may	O
also	O
be	O
more	O
than	O
two	O
variables	O
,	O
but	O
higher	O
order	O
contingency	B-Application
tables	I-Application
are	O
difficult	O
to	O
represent	O
visually	O
.	O
</s>
<s>
The	O
relation	O
between	O
ordinal	O
variables	O
,	O
or	O
between	O
ordinal	O
and	O
categorical	O
variables	O
,	O
may	O
also	O
be	O
represented	O
in	O
contingency	B-Application
tables	I-Application
,	O
although	O
such	O
a	O
practice	O
is	O
rare	O
.	O
</s>
<s>
For	O
more	O
on	O
the	O
use	O
of	O
a	O
contingency	B-Application
table	I-Application
for	O
the	O
relation	O
between	O
two	O
ordinal	O
variables	O
,	O
see	O
Goodman	B-General_Concept
and	I-General_Concept
Kruskal	I-General_Concept
's	I-General_Concept
gamma	I-General_Concept
.	O
</s>
<s>
Significance	B-General_Concept
tests	O
.	O
</s>
<s>
Typically	O
,	O
either	O
column	O
comparisons	O
,	O
which	O
test	O
for	O
differences	O
between	O
columns	O
and	O
display	O
these	O
results	O
using	O
letters	O
,	O
or	O
,	O
cell	O
comparisons	O
,	O
which	O
use	O
color	O
or	O
arrows	O
to	O
identify	O
a	O
cell	O
in	O
a	O
table	B-Application
that	O
stands	O
out	O
in	O
some	O
way	O
.	O
</s>
<s>
The	O
simplest	O
measure	O
of	O
association	O
for	O
a	O
2×2	O
contingency	B-Application
table	I-Application
is	O
the	O
odds	O
ratio	O
.	O
</s>
<s>
where	O
is	O
computed	O
as	O
in	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
,	O
and	O
N	O
is	O
the	O
grand	O
total	O
of	O
observations	O
.	O
</s>
<s>
Then	O
its	O
sign	O
equals	O
the	O
sign	O
of	O
the	O
product	O
of	O
the	O
main	B-Algorithm
diagonal	I-Algorithm
elements	O
of	O
the	O
table	B-Application
minus	O
the	O
product	O
of	O
the	O
off	O
–	O
diagonal	O
elements	O
.	O
</s>
<s>
Two	O
alternatives	O
are	O
the	O
contingency	B-Application
coefficient	I-Application
C	O
,	O
and	O
Cramér	O
's	O
V	O
.	O
</s>
<s>
C	O
suffers	O
from	O
the	O
disadvantage	O
that	O
it	O
does	O
not	O
reach	O
a	O
maximum	O
of	O
1.0	O
,	O
notably	O
the	O
highest	O
it	O
can	O
reach	O
in	O
a	O
2×2	O
table	B-Application
is	O
0.707	O
.	O
</s>
<s>
It	O
can	O
reach	O
values	O
closer	O
to	O
1.0	O
in	O
contingency	B-Application
tables	I-Application
with	O
more	O
categories	O
;	O
for	O
example	O
,	O
it	O
can	O
reach	O
a	O
maximum	O
of	O
0.870	O
in	O
a	O
4×4	O
table	B-Application
.	O
</s>
<s>
C	O
can	O
be	O
adjusted	O
so	O
it	O
reaches	O
a	O
maximum	O
of	O
1.0	O
when	O
there	O
is	O
complete	O
association	O
in	O
a	O
table	B-Application
of	O
any	O
number	O
of	O
rows	O
and	O
columns	O
by	O
dividing	O
C	O
by	O
where	O
k	O
is	O
the	O
number	O
of	O
rows	O
or	O
columns	O
,	O
when	O
the	O
table	B-Application
is	O
square	O
,	O
or	O
by	O
where	O
r	O
is	O
the	O
number	O
of	O
rows	O
and	O
c	O
is	O
the	O
number	O
of	O
columns	O
.	O
</s>
<s>
The	O
lambda	B-General_Concept
coefficient	I-General_Concept
is	O
a	O
measure	O
of	O
the	O
strength	O
of	O
association	O
of	O
the	O
cross	B-Application
tabulations	I-Application
when	O
the	O
variables	O
are	O
measured	O
at	O
the	O
nominal	O
level	O
.	O
</s>
<s>
Gamma	B-General_Concept
test	I-General_Concept
:	O
No	O
adjustment	O
for	O
either	O
table	B-Application
size	O
or	O
ties	O
.	O
</s>
<s>
Kendall	B-General_Concept
's	I-General_Concept
tau	I-General_Concept
:	O
Adjustment	O
for	O
ties	O
.	O
</s>
<s>
Tau-b	B-General_Concept
:	O
Used	O
for	O
square	O
tables	O
.	O
</s>
<s>
Tau-c	B-General_Concept
:	O
Used	O
for	O
rectangular	O
tables	O
.	O
</s>
