<s>
In	O
statistics	O
Wilks	B-General_Concept
 '	I-General_Concept
theorem	I-General_Concept
offers	O
an	O
asymptotic	O
distribution	O
of	O
the	O
log-likelihood	B-General_Concept
ratio	I-General_Concept
statistic	I-General_Concept
,	O
which	O
can	O
be	O
used	O
to	O
produce	O
confidence	O
intervals	O
for	O
maximum-likelihood	O
estimates	O
or	O
as	O
a	O
test	B-General_Concept
statistic	I-General_Concept
for	O
performing	O
the	O
likelihood-ratio	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Statistical	O
tests	O
(	O
such	O
as	O
hypothesis	O
testing	O
)	O
generally	O
require	O
knowledge	O
of	O
the	O
probability	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
.	O
</s>
<s>
A	O
convenient	O
result	O
by	O
Samuel	O
S	O
.	O
Wilks	O
says	O
that	O
as	O
the	O
sample	O
size	O
approaches	O
,	O
the	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
asymptotically	O
approaches	O
the	O
chi-squared	O
(	O
)	O
distribution	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
.	O
</s>
<s>
This	O
result	O
means	O
that	O
for	O
large	O
samples	O
and	O
a	O
great	O
variety	O
of	O
hypotheses	O
,	O
a	O
practitioner	O
can	O
compute	O
the	O
likelihood	O
ratio	O
for	O
the	O
data	O
and	O
compare	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>
<s>
In	O
that	O
event	O
,	O
the	O
likelihood	O
test	O
is	O
still	O
a	O
sensible	O
test	B-General_Concept
statistic	I-General_Concept
and	O
even	O
possess	O
some	O
asymptotic	O
optimality	O
properties	O
,	O
but	O
the	O
significance	O
(	O
the	O
-value	O
)	O
can	O
not	O
be	O
reliably	O
estimated	O
using	O
the	O
chi-squared	O
distribution	O
with	O
the	O
number	O
of	O
degrees	O
of	O
freedom	O
prescribed	O
by	O
Wilks	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
the	O
asymptotic	O
null-hypothesis	B-General_Concept
distribution	O
of	O
the	O
statistic	O
is	O
a	O
mixture	O
of	O
chi-square	O
distributions	O
with	O
different	O
numbers	O
of	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
The	O
test	B-General_Concept
statistic	I-General_Concept
(	O
often	O
denoted	O
by	O
)	O
is	O
twice	O
the	O
log	O
of	O
the	O
likelihoods	O
ratio	O
,	O
i.e.	O
,	O
it	O
is	O
twice	O
the	O
difference	O
in	O
the	O
log-likelihoods	O
:	O
</s>
<s>
Where	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
represents	O
a	O
special	O
case	O
of	O
the	O
alternative	O
hypothesis	O
,	O
the	O
probability	O
distribution	O
of	O
the	O
test	B-General_Concept
statistic	I-General_Concept
is	O
approximately	O
a	O
chi-squared	O
distribution	O
with	O
degrees	O
of	O
freedom	O
equal	O
to	O
,	O
respectively	O
the	O
number	O
of	O
free	O
parameters	O
of	O
models	O
alternative	O
and	O
null	O
.	O
</s>
<s>
The	O
observations	O
can	O
be	O
put	O
into	O
a	O
contingency	B-Application
table	I-Application
with	O
rows	O
corresponding	O
to	O
the	O
coin	O
and	O
columns	O
corresponding	O
to	O
heads	O
or	O
tails	O
.	O
</s>
<s>
The	O
elements	O
of	O
the	O
contingency	B-Application
table	I-Application
will	O
be	O
the	O
number	O
of	O
times	O
each	O
coin	O
came	O
up	O
heads	O
or	O
tails	O
.	O
</s>
<s>
The	O
space	O
of	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
the	O
subspace	O
where	O
.	O
</s>
<s>
The	O
dimensionality	O
of	O
the	O
full	O
parameter	O
space	O
is	O
2	O
(	O
either	O
of	O
the	O
and	O
either	O
of	O
the	O
may	O
be	O
treated	O
as	O
free	O
parameters	O
under	O
the	O
hypothesis	O
)	O
,	O
and	O
the	O
dimensionality	O
of	O
is	O
1	O
(	O
only	O
one	O
of	O
the	O
may	O
be	O
considered	O
a	O
free	O
parameter	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
)	O
.	O
</s>
<s>
The	O
hypothesis	O
and	O
null	B-General_Concept
hypothesis	I-General_Concept
can	O
be	O
rewritten	O
slightly	O
so	O
that	O
they	O
satisfy	O
the	O
constraints	O
for	O
the	O
logarithm	O
of	O
the	O
likelihood	O
ratio	O
to	O
have	O
the	O
desired	O
distribution	O
.	O
</s>
<s>
This	O
is	O
commonly	O
violated	O
in	O
random	B-General_Concept
or	O
mixed	B-General_Concept
effects	I-General_Concept
models	I-General_Concept
,	O
for	O
example	O
,	O
when	O
one	O
of	O
the	O
variance	B-General_Concept
components	I-General_Concept
is	O
negligible	O
relative	O
to	O
the	O
others	O
.	O
</s>
<s>
In	O
some	O
such	O
cases	O
,	O
one	O
variance	B-General_Concept
component	I-General_Concept
can	O
be	O
effectively	O
zero	O
,	O
relative	O
to	O
the	O
others	O
,	O
or	O
in	O
other	O
cases	O
the	O
models	O
can	O
be	O
improperly	O
nested	O
.	O
</s>
<s>
To	O
be	O
clear	O
:	O
These	O
limitations	O
on	O
Wilks’	O
theorem	O
do	O
not	O
negate	O
any	O
power	B-General_Concept
properties	O
of	O
a	O
particular	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
only	O
issue	O
is	O
that	O
a	O
distribution	O
is	O
sometimes	O
a	O
poor	O
choice	O
for	O
estimating	O
the	O
statistical	B-General_Concept
significance	I-General_Concept
of	O
the	O
result	O
.	O
</s>
<s>
In	O
general	O
,	O
to	O
test	O
random	B-General_Concept
effects	I-General_Concept
,	O
they	O
recommend	O
using	O
Restricted	O
maximum	O
likelihood	O
(	O
REML	O
)	O
.	O
</s>
<s>
For	O
fixed-effects	O
testing	O
,	O
they	O
say	O
,	O
“	O
a	O
likelihood	B-General_Concept
ratio	I-General_Concept
test	I-General_Concept
for	O
REML	O
fits	O
is	O
not	O
feasible	O
”	O
,	O
because	O
changing	O
the	O
fixed	O
effects	O
specification	O
changes	O
the	O
meaning	O
of	O
the	O
mixed	O
effects	O
,	O
and	O
the	O
restricted	O
model	O
is	O
therefore	O
not	O
nested	O
within	O
the	O
larger	O
model	O
.	O
</s>
<s>
As	O
a	O
demonstration	O
,	O
they	O
set	O
either	O
one	O
or	O
two	O
random	B-General_Concept
effects	I-General_Concept
variances	O
to	O
zero	O
in	O
simulated	O
tests	O
.	O
</s>
