<s>
Analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
is	O
a	O
collection	O
of	O
statistical	O
models	O
and	O
their	O
associated	O
estimation	O
procedures	O
(	O
such	O
as	O
the	O
"	O
variation	O
"	O
among	O
and	O
between	O
groups	O
)	O
used	O
to	O
analyze	O
the	O
differences	O
among	O
means	O
.	O
</s>
<s>
ANOVA	B-General_Concept
was	O
developed	O
by	O
the	O
statistician	O
Ronald	O
Fisher	O
.	O
</s>
<s>
ANOVA	B-General_Concept
is	O
based	O
on	O
the	O
law	O
of	O
total	O
variance	O
,	O
where	O
the	O
observed	O
variance	O
in	O
a	O
particular	O
variable	O
is	O
partitioned	O
into	O
components	O
attributable	O
to	O
different	O
sources	O
of	O
variation	O
.	O
</s>
<s>
In	O
its	O
simplest	O
form	O
,	O
ANOVA	B-General_Concept
provides	O
a	O
statistical	O
test	O
of	O
whether	O
two	O
or	O
more	O
population	O
means	O
are	O
equal	O
,	O
and	O
therefore	O
generalizes	O
the	O
t-test	B-General_Concept
beyond	O
two	O
means	O
.	O
</s>
<s>
In	O
other	O
words	O
,	O
the	O
ANOVA	B-General_Concept
is	O
used	O
to	O
test	O
the	O
difference	O
between	O
two	O
or	O
more	O
means	O
.	O
</s>
<s>
While	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
reached	O
fruition	O
in	O
the	O
20th	O
century	O
,	O
antecedents	O
extend	O
centuries	O
into	O
the	O
past	O
according	O
to	O
Stigler	O
.	O
</s>
<s>
Around	O
1800	O
,	O
Laplace	O
and	O
Gauss	O
developed	O
the	O
least-squares	B-Algorithm
method	I-Algorithm
for	O
combining	O
observations	O
,	O
which	O
improved	O
upon	O
methods	O
then	O
used	O
in	O
astronomy	O
and	O
geodesy	O
.	O
</s>
<s>
Laplace	O
knew	O
how	O
to	O
estimate	O
a	O
variance	O
from	O
a	O
residual	O
(	O
rather	O
than	O
a	O
total	O
)	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
By	O
1827	O
,	O
Laplace	O
was	O
using	O
least	B-Algorithm
squares	I-Algorithm
methods	I-Algorithm
to	O
address	O
ANOVA	B-General_Concept
problems	O
regarding	O
measurements	O
of	O
atmospheric	O
tides	O
.	O
</s>
<s>
An	O
eloquent	O
non-mathematical	O
explanation	O
of	O
the	O
additive	O
effects	O
model	O
was	O
available	B-General_Concept
in	O
1885	O
.	O
</s>
<s>
His	O
first	O
application	O
of	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
was	O
published	O
in	O
1921	O
.	O
</s>
<s>
Analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
became	O
widely	O
known	O
after	O
being	O
included	O
in	O
Fisher	O
's	O
1925	O
book	O
Statistical	O
Methods	O
for	O
Research	O
Workers	O
.	O
</s>
<s>
The	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
can	O
be	O
used	O
to	O
describe	O
otherwise	O
complex	O
relations	O
among	O
variables	O
.	O
</s>
<s>
The	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
provides	O
the	O
formal	O
tools	O
to	O
justify	O
these	O
intuitive	O
judgments	O
.	O
</s>
<s>
There	O
are	O
three	O
classes	O
of	O
models	O
used	O
in	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
and	O
these	O
are	O
outlined	O
here	O
.	O
</s>
<s>
The	O
fixed-effects	O
model	O
(	O
class	O
I	O
)	O
of	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
applies	O
to	O
situations	O
in	O
which	O
the	O
experimenter	O
applies	O
one	O
or	O
more	O
treatments	O
to	O
the	O
subjects	O
of	O
the	O
experiment	O
to	O
see	O
whether	O
the	O
response	O
variable	O
values	O
change	O
.	O
</s>
<s>
The	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
has	O
been	O
studied	O
from	O
several	O
approaches	O
,	O
the	O
most	O
common	O
of	O
which	O
uses	O
a	O
linear	B-Algorithm
model	I-Algorithm
that	O
relates	O
the	O
response	O
to	O
the	O
treatments	O
and	O
blocks	O
.	O
</s>
<s>
The	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
can	O
be	O
presented	O
in	O
terms	O
of	O
a	O
linear	B-Algorithm
model	I-Algorithm
,	O
which	O
makes	O
the	O
following	O
assumptions	O
about	O
the	O
probability	O
distribution	O
of	O
the	O
responses	O
:	O
</s>
<s>
Equality	O
(	O
or	O
"	O
homogeneity	O
"	O
)	O
of	O
variances	O
,	O
called	O
homoscedasticity	B-General_Concept
—	O
the	O
variance	O
of	O
data	O
in	O
groups	O
should	O
be	O
the	O
same	O
.	O
</s>
<s>
The	O
objective	O
random-assignment	O
is	O
used	O
to	O
test	O
the	O
significance	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
following	O
the	O
ideas	O
of	O
C	O
.	O
S	O
.	O
Peirce	O
and	O
Ronald	O
Fisher	O
.	O
</s>
<s>
Kempthorne	O
and	O
his	O
students	O
make	O
an	O
assumption	O
of	O
unit	O
treatment	O
additivity	O
,	O
which	O
is	O
discussed	O
in	O
the	O
books	O
of	O
Kempthorne	O
and	O
David	O
R	B-Language
.	O
Cox	O
.	O
</s>
<s>
Kempthorne	O
uses	O
the	O
randomization-distribution	O
and	O
the	O
assumption	O
of	O
unit	O
treatment	O
additivity	O
to	O
produce	O
a	O
derived	O
linear	B-Algorithm
model	I-Algorithm
,	O
very	O
similar	O
to	O
the	O
textbook	O
model	O
discussed	O
previously	O
.	O
</s>
<s>
The	O
test	O
statistics	O
of	O
this	O
derived	O
linear	B-Algorithm
model	I-Algorithm
are	O
closely	O
approximated	O
by	O
the	O
test	O
statistics	O
of	O
an	O
appropriate	O
normal	O
linear	B-Algorithm
model	I-Algorithm
,	O
according	O
to	O
approximation	O
theorems	O
and	O
simulation	O
studies	O
.	O
</s>
<s>
Since	O
the	O
randomization-based	O
analysis	O
is	O
complicated	O
and	O
is	O
closely	O
approximated	O
by	O
the	O
approach	O
using	O
a	O
normal	O
linear	B-Algorithm
model	I-Algorithm
,	O
most	O
teachers	O
emphasize	O
the	O
normal	O
linear	B-Algorithm
model	I-Algorithm
approach	O
.	O
</s>
<s>
The	O
normal-model	O
based	O
ANOVA	B-General_Concept
analysis	O
assumes	O
the	O
independence	O
,	O
normality	O
,	O
and	O
homogeneity	B-General_Concept
of	I-General_Concept
variances	I-General_Concept
of	O
the	O
residuals	O
.	O
</s>
<s>
Both	O
these	O
analyses	O
require	O
homoscedasticity	B-General_Concept
,	O
as	O
an	O
assumption	O
for	O
the	O
normal-model	O
analysis	O
and	O
as	O
a	O
consequence	O
of	O
randomization	O
and	O
additivity	O
for	O
the	O
randomization-based	O
analysis	O
.	O
</s>
<s>
However	O
,	O
studies	O
of	O
processes	O
that	O
change	O
variances	O
rather	O
than	O
means	O
(	O
called	O
dispersion	O
effects	O
)	O
have	O
been	O
successfully	O
conducted	O
using	O
ANOVA	B-General_Concept
.	O
</s>
<s>
Problems	O
which	O
do	O
not	O
satisfy	O
the	O
assumptions	O
of	O
ANOVA	B-General_Concept
can	O
often	O
be	O
transformed	O
to	O
satisfy	O
the	O
assumptions	O
.	O
</s>
<s>
ANOVA	B-General_Concept
is	O
used	O
in	O
the	O
analysis	O
of	O
comparative	O
experiments	O
,	O
those	O
in	O
which	O
only	O
the	O
difference	O
in	O
outcomes	O
is	O
of	O
interest	O
.	O
</s>
<s>
So	O
ANOVA	B-General_Concept
statistical	O
significance	O
result	O
is	O
independent	O
of	O
constant	O
bias	O
and	O
scaling	O
errors	O
as	O
well	O
as	O
the	O
units	O
used	O
in	O
expressing	O
observations	O
.	O
</s>
<s>
The	O
calculations	O
of	O
ANOVA	B-General_Concept
can	O
be	O
characterized	O
as	O
computing	O
a	O
number	O
of	O
means	O
and	O
variances	O
,	O
dividing	O
two	O
variances	O
and	O
comparing	O
the	O
ratio	O
to	O
a	O
handbook	O
value	O
to	O
determine	O
statistical	O
significance	O
.	O
</s>
<s>
ANOVA	B-General_Concept
uses	O
traditional	O
standardized	O
terminology	O
.	O
</s>
<s>
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
(	O
SS	O
)	O
,	O
the	O
result	O
is	O
called	O
the	O
mean	O
square	O
(	O
MS	O
)	O
and	O
the	O
squared	O
terms	O
are	O
deviations	O
from	O
the	O
sample	O
mean	O
.	O
</s>
<s>
ANOVA	B-General_Concept
estimates	O
3	O
sample	O
variances	O
:	O
a	O
total	O
variance	O
based	O
on	O
all	O
the	O
observation	O
deviations	O
from	O
the	O
grand	O
mean	O
,	O
an	O
error	O
variance	O
based	O
on	O
all	O
the	O
observation	O
deviations	O
from	O
their	O
appropriate	O
treatment	O
means	O
,	O
and	O
a	O
treatment	O
variance	O
.	O
</s>
<s>
The	O
fundamental	O
technique	O
is	O
a	O
partitioning	O
of	O
the	O
total	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
SS	O
into	O
components	O
related	O
to	O
the	O
effects	O
used	O
in	O
the	O
model	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
model	O
for	O
a	O
simplified	O
ANOVA	B-General_Concept
with	O
one	O
type	O
of	O
treatment	O
at	O
different	O
levels	O
.	O
</s>
<s>
The	O
number	O
of	O
degrees	O
of	O
freedom	O
DF	O
can	O
be	O
partitioned	O
in	O
a	O
similar	O
way	O
:	O
one	O
of	O
these	O
components	O
(	O
that	O
for	O
error	O
)	O
specifies	O
a	O
chi-squared	O
distribution	O
which	O
describes	O
the	O
associated	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
,	O
while	O
the	O
same	O
is	O
true	O
for	O
"	O
treatments	O
"	O
if	O
there	O
is	O
no	O
treatment	O
effect	O
.	O
</s>
<s>
The	O
F-test	B-General_Concept
is	O
used	O
for	O
comparing	O
the	O
factors	O
of	O
the	O
total	O
deviation	O
.	O
</s>
<s>
to	O
the	O
F-distribution	B-General_Concept
with	O
,	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
Using	O
the	O
F-distribution	B-General_Concept
is	O
a	O
natural	O
candidate	O
because	O
the	O
test	O
statistic	O
is	O
the	O
ratio	O
of	O
two	O
scaled	O
sums	O
of	O
squares	O
each	O
of	O
which	O
follows	O
a	O
scaled	O
chi-squared	O
distribution	O
.	O
</s>
<s>
As	O
values	O
of	O
F	O
increase	O
above	O
1	O
,	O
the	O
evidence	O
is	O
increasingly	O
inconsistent	O
with	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
There	O
are	O
two	O
methods	O
of	O
concluding	O
the	O
ANOVA	B-General_Concept
hypothesis	O
test	O
,	O
both	O
of	O
which	O
produce	O
the	O
same	O
result	O
:	O
</s>
<s>
If	O
F	O
≥	O
FCritical	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
.	O
</s>
<s>
The	O
computer	O
method	O
calculates	O
the	O
probability	O
(	O
p-value	B-General_Concept
)	O
of	O
a	O
value	O
of	O
F	O
greater	O
than	O
or	O
equal	O
to	O
the	O
observed	O
value	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
if	O
this	O
probability	O
is	O
less	O
than	O
or	O
equal	O
to	O
the	O
significance	O
level	O
( α	O
)	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
F-test	B-General_Concept
is	O
known	O
to	O
be	O
nearly	O
optimal	O
in	O
the	O
sense	O
of	O
minimizing	O
false	O
negative	O
errors	O
for	O
a	O
fixed	O
rate	O
of	O
false	O
positive	O
errors	O
(	O
i.e.	O
</s>
<s>
For	O
example	O
,	O
to	O
test	O
the	O
hypothesis	O
that	O
various	O
medical	O
treatments	O
have	O
exactly	O
the	O
same	O
effect	O
,	O
the	O
F-test	B-General_Concept
'	O
s	O
p-values	B-General_Concept
closely	O
approximate	O
the	O
permutation	B-General_Concept
test	I-General_Concept
's	O
p-values	B-General_Concept
:	O
The	O
approximation	O
is	O
particularly	O
close	O
when	O
the	O
design	O
is	O
balanced	O
.	O
</s>
<s>
Such	O
permutation	B-General_Concept
tests	I-General_Concept
characterize	O
tests	B-General_Concept
with	I-General_Concept
maximum	I-General_Concept
power	I-General_Concept
against	O
all	O
alternative	B-General_Concept
hypotheses	I-General_Concept
,	O
as	O
observed	O
by	O
Rosenbaum	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
F-test	B-General_Concept
(	O
of	O
the	O
null-hypothesis	B-General_Concept
that	O
all	O
treatments	O
have	O
exactly	O
the	O
same	O
effect	O
)	O
is	O
recommended	O
as	O
a	O
practical	O
test	O
,	O
because	O
of	O
its	O
robustness	O
against	O
many	O
alternative	O
distributions	O
.	O
</s>
<s>
ANOVA	B-General_Concept
consists	O
of	O
separable	O
parts	O
;	O
partitioning	O
sources	O
of	O
variance	O
and	O
hypothesis	O
testing	O
can	O
be	O
used	O
individually	O
.	O
</s>
<s>
ANOVA	B-General_Concept
is	O
used	O
to	O
support	O
other	O
statistical	O
tools	O
.	O
</s>
<s>
Regression	O
is	O
first	O
used	O
to	O
fit	O
more	O
complex	O
models	O
to	O
data	O
,	O
then	O
ANOVA	B-General_Concept
is	O
used	O
to	O
compare	O
models	O
with	O
the	O
objective	O
of	O
selecting	O
simple(r )	O
models	O
that	O
adequately	O
describe	O
the	O
data	O
.	O
</s>
<s>
"	O
Such	O
models	O
could	O
be	O
fit	O
without	O
any	O
reference	O
to	O
ANOVA	B-General_Concept
,	O
but	O
ANOVA	B-General_Concept
tools	O
could	O
then	O
be	O
used	O
to	O
make	O
some	O
sense	O
of	O
the	O
fitted	O
models	O
,	O
and	O
to	O
test	O
hypotheses	O
about	O
batches	O
of	O
coefficients.	O
"	O
</s>
<s>
"[W]e	O
think	O
of	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
as	O
a	O
way	O
of	O
understanding	O
and	O
structuring	O
multilevel	O
models	O
—	O
not	O
as	O
an	O
alternative	O
to	O
regression	O
but	O
as	O
a	O
tool	O
for	O
summarizing	O
complex	O
high-dimensional	O
inferences	O
...	O
"	O
</s>
<s>
The	O
simplest	O
experiment	O
suitable	O
for	O
ANOVA	B-General_Concept
analysis	O
is	O
the	O
completely	O
randomized	O
experiment	O
with	O
a	O
single	O
factor	O
.	O
</s>
<s>
A	O
relatively	O
complete	O
discussion	O
of	O
the	O
analysis	O
(	O
models	O
,	O
data	O
summaries	O
,	O
ANOVA	B-General_Concept
table	O
)	O
of	O
the	O
completely	O
randomized	O
experiment	O
is	O
available	B-General_Concept
.	O
</s>
<s>
There	O
are	O
some	O
alternatives	O
to	O
conventional	O
one-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
e.g.	O
</s>
<s>
ANOVA	B-General_Concept
generalizes	O
to	O
the	O
study	O
of	O
the	O
effects	O
of	O
multiple	O
factors	O
.	O
</s>
<s>
The	O
use	O
of	O
ANOVA	B-General_Concept
to	O
study	O
the	O
effects	O
of	O
multiple	O
factors	O
has	O
a	O
complication	O
.	O
</s>
<s>
In	O
a	O
3-way	O
ANOVA	B-General_Concept
with	O
factors	O
x	O
,	O
y	O
and	O
z	O
,	O
the	O
ANOVA	B-General_Concept
model	O
includes	O
terms	O
for	O
the	O
main	O
effects	O
(	O
x	O
,	O
y	O
,	O
z	O
)	O
and	O
terms	O
for	O
interactions	O
(	O
xy	O
,	O
xz	O
,	O
yz	O
,	O
xyz	O
)	O
.	O
</s>
<s>
The	O
ability	O
to	O
detect	O
interactions	O
is	O
a	O
major	O
advantage	O
of	O
multiple	O
factor	O
ANOVA	B-General_Concept
.	O
</s>
<s>
Caution	O
is	O
advised	O
when	O
encountering	O
interactions	O
;	O
Test	O
interaction	O
terms	O
first	O
and	O
expand	O
the	O
analysis	O
beyond	O
ANOVA	B-General_Concept
if	O
interactions	O
are	O
found	O
.	O
</s>
<s>
Texts	O
vary	O
in	O
their	O
recommendations	O
regarding	O
the	O
continuation	O
of	O
the	O
ANOVA	B-General_Concept
procedure	O
after	O
encountering	O
an	O
interaction	O
.	O
</s>
<s>
A	O
lengthy	O
discussion	O
of	O
interactions	O
is	O
available	B-General_Concept
in	O
Cox	O
(	O
1958	O
)	O
.	O
</s>
<s>
A	O
variety	O
of	O
techniques	O
are	O
used	O
with	O
multiple	O
factor	O
ANOVA	B-General_Concept
to	O
reduce	O
expense	O
.	O
</s>
<s>
One	O
technique	O
used	O
in	O
factorial	O
designs	O
is	O
to	O
minimize	O
replication	O
(	O
possibly	O
no	O
replication	O
with	O
support	O
of	O
analytical	B-General_Concept
trickery	I-General_Concept
)	O
and	O
to	O
combine	O
groups	O
when	O
effects	O
are	O
found	O
to	O
be	O
statistically	O
(	O
or	O
practically	O
)	O
insignificant	O
.	O
</s>
<s>
Besides	O
the	O
power	B-General_Concept
analysis	I-General_Concept
,	O
there	O
are	O
less	O
formal	O
methods	O
for	O
selecting	O
the	O
number	O
of	O
experimental	O
units	O
.	O
</s>
<s>
Power	B-General_Concept
analysis	I-General_Concept
is	O
often	O
applied	O
in	O
the	O
context	O
of	O
ANOVA	B-General_Concept
in	O
order	O
to	O
assess	O
the	O
probability	O
of	O
successfully	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
we	O
assume	O
a	O
certain	O
ANOVA	B-General_Concept
design	O
,	O
effect	O
size	O
in	O
the	O
population	O
,	O
sample	O
size	O
and	O
significance	O
level	O
.	O
</s>
<s>
Power	B-General_Concept
analysis	I-General_Concept
can	O
assist	O
in	O
study	O
design	O
by	O
determining	O
what	O
sample	O
size	O
would	O
be	O
required	O
in	O
order	O
to	O
have	O
a	O
reasonable	O
chance	O
of	O
rejecting	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
when	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
is	O
true	O
.	O
</s>
<s>
Several	O
standardized	O
measures	O
of	O
effect	O
have	O
been	O
proposed	O
for	O
ANOVA	B-General_Concept
to	O
summarize	O
the	O
strength	O
of	O
the	O
association	O
between	O
a	O
predictor(s )	O
and	O
the	O
dependent	O
variable	O
or	O
the	O
overall	O
standardized	O
difference	O
of	O
the	O
complete	O
model	O
.	O
</s>
<s>
Sometimes	O
tests	O
are	O
conducted	O
to	O
determine	O
whether	O
the	O
assumptions	O
of	O
ANOVA	B-General_Concept
appear	O
to	O
be	O
violated	O
.	O
</s>
<s>
Residuals	O
are	O
examined	O
or	O
analyzed	O
to	O
confirm	O
homoscedasticity	B-General_Concept
and	O
gross	O
normality	O
.	O
</s>
<s>
A	O
statistically	O
significant	O
effect	O
in	O
ANOVA	B-General_Concept
is	O
often	O
followed	O
by	O
additional	O
tests	O
.	O
</s>
<s>
Often	O
the	O
follow-up	O
tests	O
incorporate	O
a	O
method	O
of	O
adjusting	O
for	O
the	O
multiple	B-General_Concept
comparisons	I-General_Concept
problem	I-General_Concept
.	O
</s>
<s>
Follow-up	O
tests	O
to	O
identify	O
which	O
specific	O
groups	O
,	O
variables	O
,	O
or	O
factors	O
have	O
statistically	O
different	O
means	O
include	O
the	O
Tukey	B-General_Concept
's	I-General_Concept
range	I-General_Concept
test	I-General_Concept
,	O
and	O
Duncan	B-General_Concept
's	I-General_Concept
new	I-General_Concept
multiple	I-General_Concept
range	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
In	O
turn	O
,	O
these	O
tests	O
are	O
often	O
followed	O
with	O
a	O
Compact	B-Application
Letter	I-Application
Display	I-Application
(	O
CLD	O
)	O
methodology	O
in	O
order	O
to	O
render	O
the	O
output	O
of	O
the	O
mentioned	O
tests	O
more	O
transparent	O
to	O
a	O
non-statistician	O
audience	O
.	O
</s>
<s>
There	O
are	O
several	O
types	O
of	O
ANOVA	B-General_Concept
.	O
</s>
<s>
Many	O
statisticians	O
base	O
ANOVA	B-General_Concept
on	O
the	O
design	O
of	O
the	O
experiment	O
,	O
especially	O
on	O
the	O
protocol	O
that	O
specifies	O
the	O
random	O
assignment	O
of	O
treatments	O
to	O
subjects	O
;	O
the	O
protocol	O
's	O
description	O
of	O
the	O
assignment	O
mechanism	O
should	O
include	O
a	O
specification	O
of	O
the	O
structure	O
of	O
the	O
treatments	O
and	O
of	O
any	O
blocking	O
.	O
</s>
<s>
It	O
is	O
also	O
common	O
to	O
apply	O
ANOVA	B-General_Concept
to	O
observational	O
data	O
using	O
an	O
appropriate	O
statistical	O
model	O
.	O
</s>
<s>
Some	O
popular	O
designs	O
use	O
the	O
following	O
types	O
of	O
ANOVA	B-General_Concept
:	O
</s>
<s>
One-way	B-General_Concept
ANOVA	I-General_Concept
is	O
used	O
to	O
test	O
for	O
differences	O
among	O
two	O
or	O
more	O
independent	O
groups	O
(	O
means	O
)	O
,	O
e.g.	O
</s>
<s>
Typically	O
,	O
however	O
,	O
the	O
one-way	B-General_Concept
ANOVA	I-General_Concept
is	O
used	O
to	O
test	O
for	O
differences	O
among	O
at	O
least	O
three	O
groups	O
,	O
since	O
the	O
two-group	O
case	O
can	O
be	O
covered	O
by	O
a	O
t-test	B-General_Concept
.	O
</s>
<s>
When	O
there	O
are	O
only	O
two	O
means	O
to	O
compare	O
,	O
the	O
t-test	B-General_Concept
and	O
the	O
ANOVA	B-General_Concept
F-test	B-General_Concept
are	O
equivalent	O
;	O
the	O
relation	O
between	O
ANOVA	B-General_Concept
and	O
t	O
is	O
given	O
by	O
.	O
</s>
<s>
Factorial	B-General_Concept
ANOVA	I-General_Concept
is	O
used	O
when	O
there	O
is	O
more	O
than	O
one	O
factor	O
.	O
</s>
<s>
Repeated	O
measures	O
ANOVA	B-General_Concept
is	O
used	O
when	O
the	O
same	O
subjects	O
are	O
used	O
for	O
each	O
factor	O
(	O
e.g.	O
,	O
in	O
a	O
longitudinal	O
study	O
)	O
.	O
</s>
<s>
Multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
MANOVA	B-General_Concept
)	O
is	O
used	O
when	O
there	O
is	O
more	O
than	O
one	O
response	O
variable	O
.	O
</s>
<s>
For	O
single-factor	O
(	O
one-way	O
)	O
ANOVA	B-General_Concept
,	O
the	O
adjustment	O
for	O
unbalanced	O
data	O
is	O
easy	O
,	O
but	O
the	O
unbalanced	O
analysis	O
lacks	O
both	O
robustness	O
and	O
power	O
.	O
</s>
<s>
This	O
means	O
that	O
the	O
usual	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
techniques	O
do	O
not	O
apply	O
.	O
</s>
<s>
In	O
the	O
general	O
case	O
,	O
"	O
The	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
can	O
also	O
be	O
applied	O
to	O
unbalanced	O
data	O
,	O
but	O
then	O
the	O
sums	O
of	O
squares	O
,	O
mean	O
squares	O
,	O
and	O
F-ratios	B-General_Concept
will	O
depend	O
on	O
the	O
order	O
in	O
which	O
the	O
sources	O
of	O
variation	O
are	O
considered.	O
"	O
</s>
<s>
ANOVA	B-General_Concept
is	O
(	O
in	O
part	O
)	O
a	O
test	O
of	O
statistical	O
significance	O
.	O
</s>
<s>
ANOVA	B-General_Concept
is	O
considered	O
to	O
be	O
a	O
special	O
case	O
of	O
linear	B-General_Concept
regression	I-General_Concept
which	O
in	O
turn	O
is	O
a	O
special	O
case	O
of	O
the	O
general	O
linear	B-Algorithm
model	I-Algorithm
.	O
</s>
<s>
The	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
and	O
the	O
Friedman	B-General_Concept
test	I-General_Concept
are	O
nonparametric	B-General_Concept
tests	I-General_Concept
which	O
do	O
not	O
rely	O
on	O
an	O
assumption	O
of	O
normality	O
.	O
</s>
<s>
Below	O
we	O
make	O
clear	O
the	O
connection	O
between	O
multi-way	O
ANOVA	B-General_Concept
and	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
<s>
In	O
one-way	B-General_Concept
ANOVA	I-General_Concept
and	O
in	O
two-way	O
ANOVA	B-General_Concept
.	O
</s>
<s>
In	O
order	O
to	O
obtain	O
a	O
fully	O
general	O
-way	O
interaction	O
ANOVA	B-General_Concept
we	O
must	O
also	O
concatenate	O
every	O
additional	O
interaction	O
term	O
in	O
the	O
vector	O
and	O
then	O
add	O
an	O
intercept	O
term	O
.	O
</s>
<s>
With	O
this	O
notation	O
in	O
place	O
,	O
we	O
now	O
have	O
the	O
exact	O
connection	O
with	O
linear	B-General_Concept
regression	I-General_Concept
.	O
</s>
