<s>
Multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
covariance	I-General_Concept
(	O
MANCOVA	B-General_Concept
)	O
is	O
an	O
extension	O
of	O
analysis	B-General_Concept
of	I-General_Concept
covariance	I-General_Concept
(	O
ANCOVA	B-General_Concept
)	O
methods	O
to	O
cover	O
cases	O
where	O
there	O
is	O
more	O
than	O
one	O
dependent	O
variable	O
and	O
where	O
the	O
control	O
of	O
concomitant	O
continuous	O
independent	O
variables	O
–	O
covariates	O
–	O
is	O
required	O
.	O
</s>
<s>
The	O
most	O
prominent	O
benefit	O
of	O
the	O
MANCOVA	B-General_Concept
design	O
over	O
the	O
simple	O
MANOVA	B-General_Concept
is	O
the	O
'	O
factoring	O
out	O
 '	O
of	O
noise	B-General_Concept
or	O
error	O
that	O
has	O
been	O
introduced	O
by	O
the	O
covariant	O
.	O
</s>
<s>
A	O
commonly	O
used	O
multivariate	O
version	O
of	O
the	O
ANOVA	B-General_Concept
F-statistic	B-General_Concept
is	O
Wilks	O
 '	O
Lambda	O
( Λ	O
)	O
,	O
which	O
represents	O
the	O
ratio	O
between	O
the	O
error	O
variance	O
(	O
or	O
covariance	O
)	O
and	O
the	O
effect	O
variance	O
(	O
or	O
covariance	O
)	O
.	O
</s>
<s>
Similarly	O
to	O
all	O
tests	O
in	O
the	O
ANOVA	B-General_Concept
family	O
,	O
the	O
primary	O
aim	O
of	O
the	O
MANCOVA	B-General_Concept
is	O
to	O
test	O
for	O
significant	O
differences	O
between	O
group	O
means	O
.	O
</s>
<s>
The	O
process	O
of	O
characterising	O
a	O
covariate	O
in	O
a	O
data	O
source	O
allows	O
the	O
reduction	O
of	O
the	O
magnitude	O
of	O
the	O
error	O
term	O
,	O
represented	O
in	O
the	O
MANCOVA	B-General_Concept
design	O
as	O
MSerror	O
.	O
</s>
<s>
This	O
grants	O
the	O
researcher	O
more	O
statistical	B-General_Concept
power	I-General_Concept
to	O
detect	O
differences	O
within	O
the	O
data	O
.	O
</s>
<s>
The	O
multivariate	O
aspect	O
of	O
the	O
MANCOVA	B-General_Concept
allows	O
the	O
characterisation	O
of	O
differences	O
in	O
group	O
means	O
in	O
regards	O
to	O
a	O
linear	O
combination	O
of	O
multiple	O
dependent	O
variables	O
,	O
while	O
simultaneously	O
controlling	O
for	O
covariates	O
.	O
</s>
<s>
Example	O
situation	O
where	O
MANCOVA	B-General_Concept
is	O
appropriate	O
:	O
</s>
<s>
Certain	O
assumptions	O
must	O
be	O
met	O
for	O
the	O
MANCOVA	B-General_Concept
to	O
be	O
used	O
appropriately	O
:	O
</s>
<s>
Homogeneity	B-General_Concept
of	I-General_Concept
variances	I-General_Concept
:	O
Each	O
dependent	O
variable	O
must	O
demonstrate	O
similar	O
levels	O
of	O
variance	O
across	O
each	O
independent	O
variable	O
.	O
</s>
<s>
This	O
violation	O
is	O
often	O
called	O
'	O
heteroscedasticity	B-General_Concept
 '	O
and	O
can	O
be	O
tested	O
for	O
using	O
Levene	B-General_Concept
's	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Violation	O
of	O
this	O
assumption	O
may	O
lead	O
to	O
an	O
increase	O
in	O
Type	O
I	O
error	O
rates	O
as	O
well	O
as	O
decreased	O
statistical	B-General_Concept
power	I-General_Concept
.	O
</s>
<s>
Invariance	O
considerations	O
imply	O
the	O
MANOVA	B-General_Concept
statistic	O
should	O
be	O
a	O
measure	O
of	O
magnitude	O
of	O
the	O
singular	O
value	O
decomposition	O
of	O
this	O
matrix	O
product	O
,	O
but	O
there	O
is	O
no	O
unique	O
choice	O
owing	O
to	O
the	O
multi-dimensional	O
nature	O
of	O
the	O
alternative	O
hypothesis	O
.	O
</s>
<s>
The	O
aim	O
of	O
such	O
techniques	O
as	O
ANCOVA	B-General_Concept
is	O
to	O
remove	O
the	O
effects	O
of	O
such	O
uncontrolled	O
variation	O
,	O
in	O
order	O
to	O
increase	O
statistical	B-General_Concept
power	I-General_Concept
and	O
to	O
ensure	O
an	O
accurate	O
measurement	O
of	O
the	O
true	O
relationship	O
between	O
independent	O
and	O
dependent	O
variables	O
.	O
</s>
