<s>
An	O
F-test	B-General_Concept
is	O
any	O
statistical	O
test	O
in	O
which	O
the	O
test	B-General_Concept
statistic	I-General_Concept
has	O
an	O
F-distribution	B-General_Concept
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Exact	O
"	O
F-tests	B-General_Concept
"	O
mainly	O
arise	O
when	O
the	O
models	O
have	O
been	O
fitted	O
to	O
the	O
data	O
using	O
least	B-Algorithm
squares	I-Algorithm
.	O
</s>
<s>
Fisher	O
initially	O
developed	O
the	O
statistic	B-General_Concept
as	O
the	O
variance	O
ratio	O
in	O
the	O
1920s	O
.	O
</s>
<s>
Common	O
examples	O
of	O
the	O
use	O
of	O
F-tests	B-General_Concept
include	O
the	O
study	O
of	O
the	O
following	O
cases	O
:	O
</s>
<s>
The	O
hypothesis	O
that	O
the	O
means	O
of	O
a	O
given	O
set	O
of	O
normally	O
distributed	O
populations	O
,	O
all	O
having	O
the	O
same	O
standard	B-General_Concept
deviation	I-General_Concept
,	O
are	O
equal	O
.	O
</s>
<s>
This	O
is	O
perhaps	O
the	O
best-known	O
F-test	B-General_Concept
,	O
and	O
plays	O
an	O
important	O
role	O
in	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
.	O
</s>
<s>
See	O
Lack-of-fit	B-General_Concept
sum	I-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
In	O
addition	O
,	O
some	O
statistical	O
procedures	O
,	O
such	O
as	O
Scheffé	O
's	O
method	O
for	O
multiple	B-General_Concept
comparisons	I-General_Concept
adjustment	O
in	O
linear	O
models	O
,	O
also	O
use	O
F-tests	B-General_Concept
.	O
</s>
<s>
The	O
F-test	B-General_Concept
is	O
sensitive	O
to	O
non-normality	O
.	O
</s>
<s>
In	O
the	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
,	O
alternative	O
tests	O
include	O
Levene	B-General_Concept
's	I-General_Concept
test	I-General_Concept
,	O
Bartlett	B-General_Concept
's	I-General_Concept
test	I-General_Concept
,	O
and	O
the	O
Brown	B-General_Concept
–	I-General_Concept
Forsythe	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
However	O
,	O
when	O
any	O
of	O
these	O
tests	O
are	O
conducted	O
to	O
test	O
the	O
underlying	O
assumption	O
of	O
homoscedasticity	B-General_Concept
(	O
i.e.	O
</s>
<s>
homogeneity	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
)	O
,	O
as	O
a	O
preliminary	O
step	O
to	O
testing	O
for	O
mean	O
effects	O
,	O
there	O
is	O
an	O
increase	O
in	O
the	O
experiment-wise	O
Type	O
I	O
error	O
rate	O
.	O
</s>
<s>
Most	O
F-tests	B-General_Concept
arise	O
by	O
considering	O
a	O
decomposition	O
of	O
the	O
variability	O
in	O
a	O
collection	O
of	O
data	O
in	O
terms	O
of	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
.	O
</s>
<s>
The	O
test	B-General_Concept
statistic	I-General_Concept
in	O
an	O
F-test	B-General_Concept
is	O
the	O
ratio	O
of	O
two	O
scaled	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
reflecting	O
different	O
sources	O
of	O
variability	O
.	O
</s>
<s>
These	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
are	O
constructed	O
so	O
that	O
the	O
statistic	B-General_Concept
tends	O
to	O
be	O
greater	O
when	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
true	O
.	O
</s>
<s>
In	O
order	O
for	O
the	O
statistic	B-General_Concept
to	O
follow	O
the	O
F-distribution	B-General_Concept
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
the	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
should	O
be	O
statistically	O
independent	O
,	O
and	O
each	O
should	O
follow	O
a	O
scaled	O
χ²-distribution	O
.	O
</s>
<s>
The	O
F-test	B-General_Concept
in	O
one-way	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
is	O
used	O
to	O
assess	O
whether	O
the	O
expected	O
values	O
of	O
a	O
quantitative	O
variable	O
within	O
several	O
pre-defined	O
groups	O
differ	O
from	O
each	O
other	O
.	O
</s>
<s>
The	O
ANOVA	B-General_Concept
F-test	B-General_Concept
can	O
be	O
used	O
to	O
assess	O
whether	O
any	O
of	O
the	O
treatments	O
are	O
on	O
average	O
superior	O
,	O
or	O
inferior	O
,	O
to	O
the	O
others	O
versus	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
all	O
four	O
treatments	O
yield	O
the	O
same	O
mean	O
response	O
.	O
</s>
<s>
The	O
advantage	O
of	O
the	O
ANOVA	B-General_Concept
F-test	B-General_Concept
is	O
that	O
we	O
do	O
not	O
need	O
to	O
pre-specify	O
which	O
treatments	O
are	O
to	O
be	O
compared	O
,	O
and	O
we	O
do	O
not	O
need	O
to	O
adjust	O
for	O
making	O
multiple	B-General_Concept
comparisons	I-General_Concept
.	O
</s>
<s>
The	O
disadvantage	O
of	O
the	O
ANOVA	B-General_Concept
F-test	B-General_Concept
is	O
that	O
if	O
we	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
,	O
we	O
do	O
not	O
know	O
which	O
treatments	O
can	O
be	O
said	O
to	O
be	O
significantly	O
different	O
from	O
the	O
others	O
,	O
nor	O
,	O
if	O
the	O
F-test	B-General_Concept
is	O
performed	O
at	O
level	O
α	O
,	O
can	O
we	O
state	O
that	O
the	O
treatment	O
pair	O
with	O
the	O
greatest	O
mean	O
difference	O
is	O
significantly	O
different	O
at	O
level	O
α	O
.	O
</s>
<s>
This	O
F-statistic	O
follows	O
the	O
F-distribution	B-General_Concept
with	O
degrees	O
of	O
freedom	O
and	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
The	O
statistic	B-General_Concept
will	O
be	O
large	O
if	O
the	O
between-group	O
variability	O
is	O
large	O
relative	O
to	O
the	O
within-group	O
variability	O
,	O
which	O
is	O
unlikely	O
to	O
happen	O
if	O
the	O
population	O
means	O
of	O
the	O
groups	O
all	O
have	O
the	O
same	O
value	O
.	O
</s>
<s>
Note	O
that	O
when	O
there	O
are	O
only	O
two	O
groups	O
for	O
the	O
one-way	O
ANOVA	B-General_Concept
F-test	B-General_Concept
,	O
where	O
t	O
is	O
the	O
Student	O
's	O
statistic	B-General_Concept
.	O
</s>
<s>
This	O
use	O
of	O
the	O
F-test	B-General_Concept
is	O
known	O
as	O
the	O
Chow	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
One	O
approach	O
to	O
this	O
problem	O
is	O
to	O
use	O
an	O
F-test	B-General_Concept
.	O
</s>
<s>
where	O
RSSi	O
is	O
the	O
residual	B-Algorithm
sum	I-Algorithm
of	I-Algorithm
squares	I-Algorithm
of	O
model	O
i	O
.	O
</s>
<s>
If	O
the	O
regression	O
model	O
has	O
been	O
calculated	O
with	O
weights	O
,	O
then	O
replace	O
RSSi	O
with	O
χ2	O
,	O
the	O
weighted	O
sum	B-Algorithm
of	I-Algorithm
squared	I-Algorithm
residuals	I-Algorithm
.	O
</s>
<s>
Under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
model	O
2	O
does	O
not	O
provide	O
a	O
significantly	O
better	O
fit	O
than	O
model	O
1	O
,	O
F	O
will	O
have	O
an	O
F	B-General_Concept
distribution	I-General_Concept
,	O
with	O
(	O
p2−p1	O
,	O
n−p2	O
)	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
The	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
rejected	O
if	O
the	O
F	O
calculated	O
from	O
the	O
data	O
is	O
greater	O
than	O
the	O
critical	O
value	O
of	O
the	O
F-distribution	B-General_Concept
for	O
some	O
desired	O
false-rejection	O
probability	O
(	O
e.g.	O
</s>
<s>
Since	O
F	O
is	O
a	O
monotone	O
function	O
of	O
the	O
likelihood	B-General_Concept
ratio	I-General_Concept
statistic	I-General_Concept
,	O
the	O
F-test	B-General_Concept
is	O
a	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
.	O
</s>
