<s>
A	O
mixed	B-General_Concept
model	I-General_Concept
,	O
mixed-effects	O
model	O
or	O
mixed	B-General_Concept
error-component	I-General_Concept
model	I-General_Concept
is	O
a	O
statistical	O
model	O
containing	O
both	O
fixed	B-General_Concept
effects	I-General_Concept
and	O
random	B-General_Concept
effects	I-General_Concept
.	O
</s>
<s>
Because	O
of	O
their	O
advantage	O
in	O
dealing	O
with	O
missing	O
values	O
,	O
mixed	O
effects	O
models	O
are	O
often	O
preferred	O
over	O
more	O
traditional	O
approaches	O
such	O
as	O
repeated	O
measures	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
This	O
page	O
will	O
discuss	O
mainly	O
linear	B-General_Concept
mixed-effects	I-General_Concept
models	I-General_Concept
(	O
LMEM	O
)	O
rather	O
than	O
generalized	B-General_Concept
linear	I-General_Concept
mixed	I-General_Concept
models	I-General_Concept
or	O
nonlinear	O
mixed-effects	B-General_Concept
models	I-General_Concept
.	O
</s>
<s>
Ronald	O
Fisher	O
introduced	O
random	B-General_Concept
effects	I-General_Concept
models	I-General_Concept
to	O
study	O
the	O
correlations	O
of	O
trait	O
values	O
between	O
relatives	O
.	O
</s>
<s>
provided	O
best	O
linear	O
unbiased	O
estimates	O
of	O
fixed	B-General_Concept
effects	I-General_Concept
and	O
best	O
linear	O
unbiased	O
predictions	O
of	O
random	B-General_Concept
effects	I-General_Concept
.	O
</s>
<s>
Mixed	B-General_Concept
models	I-General_Concept
are	O
applied	O
in	O
many	O
disciplines	O
where	O
multiple	O
correlated	O
measurements	O
are	O
made	O
on	O
each	O
unit	O
of	O
interest	O
.	O
</s>
<s>
is	O
an	O
unknown	O
vector	O
of	O
fixed	B-General_Concept
effects	I-General_Concept
;	O
</s>
<s>
is	O
an	O
unknown	O
vector	O
of	O
random	B-General_Concept
effects	I-General_Concept
,	O
with	O
mean	O
and	O
variance	O
–	O
covariance	O
matrix	O
;	O
</s>
<s>
and	O
are	O
known	O
design	B-Algorithm
matrices	I-Algorithm
relating	O
the	O
observations	O
to	O
and	O
,	O
respectively	O
.	O
</s>
<s>
Assuming	O
normality	O
,	O
,	O
and	O
,	O
and	O
maximizing	O
the	O
joint	O
density	O
over	O
and	O
,	O
gives	O
Henderson	O
's	O
"	O
mixed	B-General_Concept
model	I-General_Concept
equations	O
"	O
(	O
MME	O
)	O
for	O
linear	O
mixed	B-General_Concept
models	I-General_Concept
:	O
</s>
<s>
One	O
method	O
used	O
to	O
fit	O
such	O
mixed	B-General_Concept
models	I-General_Concept
is	O
that	O
of	O
the	O
expectation	B-Algorithm
–	I-Algorithm
maximization	I-Algorithm
algorithm	I-Algorithm
(	O
EM	O
)	O
where	O
the	O
variance	B-General_Concept
components	I-General_Concept
are	O
treated	O
as	O
unobserved	O
nuisance	O
parameters	O
in	O
the	O
joint	O
likelihood	O
.	O
</s>
<s>
Currently	O
,	O
this	O
is	O
the	O
method	O
implemented	O
in	O
statistical	O
software	O
such	O
as	O
Python	B-Language
(	O
statsmodels	B-Application
package	O
)	O
and	O
SAS	B-Language
(	O
proc	O
mixed	O
)	O
,	O
and	O
as	O
initial	O
step	O
only	O
in	O
R	B-Language
's	O
nlme	O
package	O
lme( )	O
.	O
</s>
<s>
The	O
solution	O
to	O
the	O
mixed	B-General_Concept
model	I-General_Concept
equations	O
is	O
a	O
maximum	O
likelihood	O
estimate	O
when	O
the	O
distribution	O
of	O
the	O
errors	O
is	O
normal	O
.	O
</s>
<s>
There	O
are	O
several	O
other	O
methods	O
to	O
fit	O
mixed	B-General_Concept
models	I-General_Concept
,	O
including	O
using	O
an	O
EM	O
initially	O
,	O
and	O
then	O
Newton-Raphson	O
(	O
used	O
by	O
R	B-Language
package	O
nlme	O
's	O
lme( )	O
)	O
,	O
penalized	O
least	O
squares	O
to	O
get	O
a	O
profiled	O
log	O
likelihood	O
only	O
depending	O
on	O
the	O
(	O
low-dimensional	O
)	O
variance-covariance	O
parameters	O
of	O
,	O
i.e.	O
,	O
its	O
cov	O
matrix	O
,	O
and	O
then	O
modern	O
direct	O
optimization	O
for	O
that	O
reduced	O
objective	O
function	O
(	O
used	O
by	O
R	B-Language
's	O
lme4	O
package	O
lmer( )	O
and	O
the	O
Julia	B-Application
package	O
MixedModels.jl	O
)	O
and	O
direct	O
optimization	O
of	O
the	O
likelihood	O
(	O
used	O
by	O
e.g.	O
</s>
<s>
R	B-Language
's	O
glmmTMB	O
)	O
.	O
</s>
