<s>
In	O
statistics	O
,	O
multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
MANOVA	B-General_Concept
)	O
is	O
a	O
procedure	O
for	O
comparing	O
multivariate	B-General_Concept
sample	O
means	O
.	O
</s>
<s>
As	O
a	O
multivariate	B-General_Concept
procedure	O
,	O
it	O
is	O
used	O
when	O
there	O
are	O
two	O
or	O
more	O
dependent	O
variables	O
,	O
and	O
is	O
often	O
followed	O
by	O
significance	O
tests	O
involving	O
individual	O
dependent	O
variables	O
separately	O
.	O
</s>
<s>
In	O
this	O
case	O
there	O
are	O
k+p	O
dependent	O
variables	O
whose	O
linear	O
combination	O
follows	O
a	O
multivariate	B-General_Concept
normal	O
distribution	O
,	O
multivariate	B-General_Concept
variance-covariance	O
matrix	O
homogeneity	O
,	O
and	O
linear	O
relationship	O
,	O
no	O
multicollinearity	O
,	O
and	O
each	O
without	O
outliers	O
.	O
</s>
<s>
MANOVA	B-General_Concept
is	O
a	O
generalized	O
form	O
of	O
univariate	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	B-General_Concept
)	O
,	O
although	O
,	O
unlike	O
univariate	B-General_Concept
ANOVA	I-General_Concept
,	O
it	O
uses	O
the	O
covariance	O
between	O
outcome	O
variables	O
in	O
testing	O
the	O
statistical	O
significance	O
of	O
the	O
mean	O
differences	O
.	O
</s>
<s>
Where	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
appear	O
in	O
univariate	O
analysis	B-General_Concept
of	I-General_Concept
variance	I-General_Concept
,	O
in	O
multivariate	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
certain	O
positive-definite	B-Algorithm
matrices	I-Algorithm
appear	O
.	O
</s>
<s>
The	O
diagonal	O
entries	O
are	O
the	O
same	O
kinds	O
of	O
sums	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
that	O
appear	O
in	O
univariate	B-General_Concept
ANOVA	I-General_Concept
.	O
</s>
<s>
Under	O
normality	O
assumptions	O
about	O
error	O
distributions	O
,	O
the	O
counterpart	O
of	O
the	O
sum	B-General_Concept
of	I-General_Concept
squares	I-General_Concept
due	O
to	O
error	O
has	O
a	O
Wishart	O
distribution	O
.	O
</s>
<s>
MANOVA	B-General_Concept
is	O
based	O
on	O
the	O
product	O
of	O
model	O
variance	O
matrix	O
,	O
and	O
inverse	O
of	O
the	O
error	O
variance	O
matrix	O
,	O
,	O
or	O
.	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>
A	O
further	O
complication	O
is	O
that	O
,	O
except	O
for	O
the	O
Roy	O
's	O
greatest	O
root	O
,	O
the	O
distribution	O
of	O
these	O
statistics	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
straightforward	O
and	O
can	O
only	O
be	O
approximated	O
except	O
in	O
a	O
few	O
low-dimensional	O
cases	O
.	O
</s>
<s>
An	O
algorithm	O
for	O
the	O
distribution	O
of	O
the	O
Roy	O
's	O
largest	O
root	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
was	O
derived	O
in	O
while	O
the	O
distribution	O
under	O
the	O
alternative	O
is	O
studied	O
in	O
.	O
</s>
<s>
MANOVA	B-General_Concept
's	O
power	O
is	O
affected	O
by	O
the	O
correlations	O
of	O
the	O
dependent	O
variables	O
and	O
by	O
the	O
effect	O
sizes	O
associated	O
with	O
those	O
variables	O
.	O
</s>
<s>
For	O
example	O
,	O
when	O
there	O
are	O
two	O
groups	O
and	O
two	O
dependent	O
variables	O
,	O
MANOVA	B-General_Concept
's	O
power	O
is	O
lowest	O
when	O
the	O
correlation	O
equals	O
the	O
ratio	O
of	O
the	O
smaller	O
to	O
the	O
larger	O
standardized	O
effect	O
size	O
.	O
</s>
