<s>
In	O
statistics	O
,	O
the	O
two-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
is	O
an	O
extension	O
of	O
the	O
one-way	B-General_Concept
ANOVA	I-General_Concept
that	O
examines	O
the	O
influence	O
of	O
two	O
different	O
categorical	O
independent	O
variables	O
on	O
one	O
continuous	O
dependent	O
variable	O
.	O
</s>
<s>
The	O
two-way	B-General_Concept
ANOVA	I-General_Concept
not	O
only	O
aims	O
at	O
assessing	O
the	O
main	B-General_Concept
effect	I-General_Concept
of	O
each	O
independent	O
variable	O
but	O
also	O
if	O
there	O
is	O
any	O
interaction	O
between	O
them	O
.	O
</s>
<s>
In	O
1925	O
,	O
Ronald	O
Fisher	O
mentions	O
the	O
two-way	B-General_Concept
ANOVA	I-General_Concept
in	O
his	O
celebrated	O
book	O
,	O
Statistical	O
Methods	O
for	O
Research	O
Workers	O
(	O
chapters	O
7	O
and	O
8	O
)	O
.	O
</s>
<s>
In	O
2005	O
,	O
Andrew	O
Gelman	O
proposed	O
a	O
different	O
approach	O
of	O
ANOVA	B-General_Concept
,	O
viewed	O
as	O
a	O
multilevel	B-General_Concept
model	I-General_Concept
.	O
</s>
<s>
Let	O
us	O
imagine	O
a	O
data	B-General_Concept
set	I-General_Concept
for	O
which	O
a	O
dependent	O
variable	O
may	O
be	O
influenced	O
by	O
two	O
factors	O
which	O
are	O
potential	O
sources	O
of	O
variation	O
.	O
</s>
<s>
From	O
these	O
data	O
,	O
we	O
can	O
build	O
a	O
contingency	B-Application
table	I-Application
,	O
where	O
and	O
,	O
and	O
the	O
total	O
number	O
of	O
replicates	O
is	O
equal	O
to	O
.	O
</s>
<s>
Upon	O
observing	O
variation	O
among	O
all	O
data	O
points	O
,	O
for	O
instance	O
via	O
a	O
histogram	B-Algorithm
,	O
"	O
probability	O
may	O
be	O
used	O
to	O
describe	O
such	O
variation	O
"	O
.	O
</s>
<s>
The	O
two-way	B-General_Concept
ANOVA	I-General_Concept
models	O
all	O
these	O
variables	O
as	O
varying	O
independently	O
and	O
normally	O
around	O
a	O
mean	O
,	O
,	O
with	O
a	O
constant	O
variance	O
,	O
(	O
homoscedasticity	B-General_Concept
)	O
:	O
</s>
<s>
where	O
is	O
the	O
grand	O
mean	O
,	O
is	O
the	O
additive	O
main	B-General_Concept
effect	I-General_Concept
of	O
level	O
from	O
the	O
first	O
factor	O
(	O
i-th	O
row	O
in	O
the	O
contingency	B-Application
table	I-Application
)	O
,	O
is	O
the	O
additive	O
main	B-General_Concept
effect	I-General_Concept
of	O
level	O
from	O
the	O
second	O
factor	O
(	O
j-th	O
column	O
in	O
the	O
contingency	B-Application
table	I-Application
)	O
and	O
is	O
the	O
non-additive	O
interaction	O
effect	O
of	O
treatment	O
for	O
samples	O
from	O
both	O
factors	O
(	O
cell	O
at	O
row	O
i	O
and	O
column	O
j	O
in	O
the	O
contingency	B-Application
table	I-Application
)	O
.	O
</s>
<s>
Another	O
equivalent	O
way	O
of	O
describing	O
the	O
two-way	B-General_Concept
ANOVA	I-General_Concept
is	O
by	O
mentioning	O
that	O
,	O
besides	O
the	O
variation	O
explained	O
by	O
the	O
factors	O
,	O
there	O
remains	O
some	O
statistical	B-General_Concept
noise	I-General_Concept
.	O
</s>
<s>
Following	O
Gelman	O
and	O
Hill	O
,	O
the	O
assumptions	O
of	O
the	O
ANOVA	B-General_Concept
,	O
and	O
more	O
generally	O
the	O
general	O
linear	O
model	O
,	O
are	O
,	O
in	O
decreasing	O
order	O
of	O
importance	O
:	O
</s>
<s>
In	O
the	O
classical	O
approach	O
,	O
testing	O
null	O
hypotheses	O
(	O
that	O
the	O
factors	O
have	O
no	O
effect	O
)	O
is	O
achieved	O
via	O
their	O
significance	B-General_Concept
which	O
requires	O
calculating	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
Five	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
are	O
calculated	O
:	O
</s>
<s>
Finally	O
,	O
the	O
sums	O
of	O
squared	O
deviations	O
required	O
for	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
can	O
be	O
calculated	O
.	O
</s>
