<s>
In	O
statistics	O
,	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
assesses	O
the	O
goodness	O
of	O
fit	O
of	O
two	O
competing	O
statistical	O
models	O
based	O
on	O
the	O
ratio	O
of	O
their	O
likelihoods	O
,	O
specifically	O
one	O
found	O
by	O
maximization	O
over	O
the	O
entire	O
parameter	O
space	O
and	O
another	O
found	O
after	O
imposing	O
some	O
constraint	B-Application
.	O
</s>
<s>
If	O
the	O
constraint	B-Application
(	O
i.e.	O
,	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
is	O
supported	O
by	O
the	O
observed	O
data	O
,	O
the	O
two	O
likelihoods	O
should	O
not	O
differ	O
by	O
more	O
than	O
sampling	O
error	O
.	O
</s>
<s>
Thus	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
tests	O
whether	O
this	O
ratio	O
is	O
significantly	B-General_Concept
different	I-General_Concept
from	O
one	O
,	O
or	O
equivalently	O
whether	O
its	O
natural	O
logarithm	O
is	O
significantly	B-General_Concept
different	I-General_Concept
from	O
zero	O
.	O
</s>
<s>
The	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
,	O
also	O
known	O
as	O
Wilks	B-General_Concept
test	I-General_Concept
,	O
is	O
the	O
oldest	O
of	O
the	O
three	O
classical	O
approaches	O
to	O
hypothesis	O
testing	O
,	O
together	O
with	O
the	O
Lagrange	B-General_Concept
multiplier	I-General_Concept
test	I-General_Concept
and	O
the	O
Wald	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
latter	O
two	O
can	O
be	O
conceptualized	O
as	O
approximations	O
to	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
,	O
and	O
are	O
asymptotically	O
equivalent	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
comparing	O
two	O
models	O
each	O
of	O
which	O
has	O
no	O
unknown	O
parameters	O
,	O
use	O
of	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
can	O
be	O
justified	O
by	O
the	O
Neyman	O
–	O
Pearson	O
lemma	O
.	O
</s>
<s>
The	O
lemma	O
demonstrates	O
that	O
the	O
test	O
has	O
the	O
highest	O
power	B-General_Concept
among	O
all	O
competitors	O
.	O
</s>
<s>
A	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
often	O
stated	O
by	O
saying	O
that	O
the	O
parameter	O
is	O
in	O
a	O
specified	O
subset	O
of	O
.	O
</s>
<s>
The	O
alternative	B-General_Concept
hypothesis	I-General_Concept
is	O
thus	O
that	O
is	O
in	O
the	O
complement	O
of	O
,	O
i.e.	O
</s>
<s>
The	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
statistic	I-General_Concept
for	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
given	O
by	O
:	O
</s>
<s>
As	O
all	O
likelihoods	O
are	O
positive	O
,	O
and	O
as	O
the	O
constrained	O
maximum	O
cannot	O
exceed	O
the	O
unconstrained	O
maximum	O
,	O
the	O
likelihood	O
ratio	O
is	O
bounded	B-Algorithm
between	O
zero	O
and	O
one	O
.	O
</s>
<s>
Multiplying	O
by	O
2	O
ensures	O
mathematically	O
that	O
(	O
by	O
Wilks	B-General_Concept
 '	I-General_Concept
theorem	I-General_Concept
)	O
converges	O
asymptotically	O
to	O
being	O
²-distributed	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
happens	O
to	O
be	O
true	O
.	O
</s>
<s>
The	O
finite	O
sample	O
distributions	O
of	O
likelihood-ratio	B-General_Concept
tests	I-General_Concept
are	O
generally	O
unknown	O
.	O
</s>
<s>
The	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
requires	O
that	O
the	O
models	O
be	O
nested	O
i.e.	O
</s>
<s>
Many	O
common	O
test	O
statistics	O
are	O
tests	O
for	O
nested	O
models	O
and	O
can	O
be	O
phrased	O
as	O
log-likelihood	B-General_Concept
ratios	I-General_Concept
or	O
approximations	O
thereof	O
:	O
e.g.	O
</s>
<s>
the	O
Z-test	B-General_Concept
,	O
the	O
F-test	B-General_Concept
,	O
the	O
G-test	B-General_Concept
,	O
and	O
Pearson	B-General_Concept
's	I-General_Concept
chi-squared	I-General_Concept
test	I-General_Concept
;	O
for	O
an	O
illustration	O
with	O
the	O
one-sample	O
t-test	O
,	O
see	O
below	O
.	O
</s>
<s>
If	O
the	O
models	O
are	O
not	O
nested	O
,	O
then	O
instead	O
of	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
,	O
there	O
is	O
a	O
generalization	O
of	O
the	O
test	O
that	O
can	O
usually	O
be	O
used	O
:	O
for	O
details	O
,	O
see	O
relative	O
likelihood	O
.	O
</s>
<s>
A	O
simple-vs.-simple	O
hypothesis	O
test	O
has	O
completely	O
specified	O
models	O
under	O
both	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
and	O
the	O
alternative	B-General_Concept
hypothesis	I-General_Concept
,	O
which	O
for	O
convenience	O
are	O
written	O
in	O
terms	O
of	O
fixed	O
values	O
of	O
a	O
notional	O
parameter	O
:	O
</s>
<s>
For	O
this	O
case	O
,	O
a	O
variant	O
of	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
is	O
available	O
:	O
</s>
<s>
The	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
provides	O
the	O
decision	O
rule	O
as	O
follows	O
:	O
</s>
<s>
The	O
Neyman	O
–	O
Pearson	O
lemma	O
states	O
that	O
this	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
is	O
the	O
most	B-General_Concept
powerful	I-General_Concept
among	O
all	O
level	O
tests	O
for	O
this	O
case	O
.	O
</s>
<s>
The	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
rejects	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
if	O
the	O
value	O
of	O
this	O
statistic	O
is	O
too	O
small	O
.	O
</s>
<s>
How	O
small	O
is	O
too	O
small	O
depends	O
on	O
the	O
significance	B-General_Concept
level	I-General_Concept
of	O
the	O
test	O
,	O
i.e.	O
</s>
<s>
on	O
what	O
probability	O
of	O
Type	O
I	O
error	O
is	O
considered	O
tolerable	O
(	O
TypeI	O
errors	O
consist	O
of	O
the	O
rejection	O
of	O
a	O
null	B-General_Concept
hypothesis	I-General_Concept
that	O
is	O
true	O
)	O
.	O
</s>
<s>
The	O
numerator	O
corresponds	O
to	O
the	O
likelihood	O
of	O
an	O
observed	O
outcome	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
Low	O
values	O
of	O
the	O
likelihood	O
ratio	O
mean	O
that	O
the	O
observed	O
result	O
was	O
much	O
less	O
likely	O
to	O
occur	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
as	O
compared	O
to	O
the	O
alternative	O
.	O
</s>
<s>
High	O
values	O
of	O
the	O
statistic	O
mean	O
that	O
the	O
observed	O
outcome	O
was	O
nearly	O
as	O
likely	O
to	O
occur	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
as	O
the	O
alternative	O
,	O
and	O
so	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
cannot	O
be	O
rejected	O
.	O
</s>
<s>
Thus	O
,	O
our	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
and	O
our	O
alternative	B-General_Concept
hypothesis	I-General_Concept
is	O
.	O
</s>
<s>
If	O
the	O
distribution	O
of	O
the	O
likelihood	O
ratio	O
corresponding	O
to	O
a	O
particular	O
null	O
and	O
alternative	B-General_Concept
hypothesis	I-General_Concept
can	O
be	O
explicitly	O
determined	O
then	O
it	O
can	O
directly	O
be	O
used	O
to	O
form	O
decision	O
regions	O
(	O
to	O
sustain	O
or	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
.	O
</s>
<s>
This	O
implies	O
that	O
for	O
a	O
great	O
variety	O
of	O
hypotheses	O
,	O
we	O
can	O
calculate	O
the	O
likelihood	O
ratio	O
for	O
the	O
data	O
and	O
then	O
compare	O
the	O
observed	O
to	O
the	O
value	O
corresponding	O
to	O
a	O
desired	O
statistical	B-General_Concept
significance	I-General_Concept
as	O
an	O
approximate	O
statistical	O
test	O
.	O
</s>
