<s>
Mauchly	B-General_Concept
's	I-General_Concept
sphericity	I-General_Concept
test	I-General_Concept
or	O
Mauchly	O
's	O
W	O
is	O
a	O
statistical	O
test	O
used	O
to	O
validate	O
a	O
repeated	O
measures	O
analysis	O
of	O
variance	O
(	O
ANOVA	O
)	O
.	O
</s>
<s>
If	O
sphericity	O
is	O
violated	O
(	O
i.e.	O
,	O
if	O
the	O
variances	O
of	O
the	O
differences	O
between	O
all	O
combinations	O
of	O
the	O
conditions	O
are	O
not	O
equal	O
)	O
,	O
then	O
the	O
variance	O
calculations	O
may	O
be	O
distorted	O
,	O
which	O
would	O
result	O
in	O
an	O
inflated	O
F-ratio	B-General_Concept
.	O
</s>
<s>
If	O
sphericity	O
is	O
violated	O
,	O
a	O
decision	O
must	O
be	O
made	O
as	O
to	O
whether	O
a	O
univariate	B-General_Concept
or	O
multivariate	O
analysis	O
is	O
selected	O
.	O
</s>
<s>
If	O
a	O
univariate	B-General_Concept
method	O
is	O
selected	O
,	O
the	O
repeated-measures	O
ANOVA	O
must	O
be	O
appropriately	O
corrected	O
depending	O
on	O
the	O
degree	O
to	O
which	O
sphericity	O
has	O
been	O
violated	O
.	O
</s>
<s>
To	O
determine	O
whether	O
statistically	O
significant	O
differences	O
exist	O
between	O
the	O
variances	O
of	O
the	O
differences	O
,	O
Mauchly	B-General_Concept
's	I-General_Concept
test	I-General_Concept
of	I-General_Concept
sphericity	I-General_Concept
can	O
be	O
performed	O
.	O
</s>
<s>
Developed	O
in	O
1940	O
by	O
John	O
W	O
.	O
Mauchly	O
,	O
Mauchly	B-General_Concept
's	I-General_Concept
test	I-General_Concept
of	I-General_Concept
sphericity	I-General_Concept
is	O
a	O
popular	O
test	O
to	O
evaluate	O
whether	O
the	O
sphericity	O
assumption	O
has	O
been	O
violated	O
.	O
</s>
<s>
Statistical	O
software	O
should	O
not	O
provide	O
output	O
for	O
a	O
test	O
of	O
sphericity	O
for	O
two	O
levels	O
of	O
a	O
repeated	O
measure	O
factor	O
;	O
however	O
,	O
some	O
versions	O
of	O
SPSS	B-Algorithm
produce	O
an	O
output	O
table	O
with	O
degrees	O
of	O
freedom	O
equal	O
to	O
0	O
,	O
and	O
a	O
period	O
in	O
place	O
of	O
a	O
numeric	O
p	O
value	O
.	O
</s>
<s>
When	O
sphericity	O
has	O
been	O
established	O
,	O
the	O
F-ratio	B-General_Concept
is	O
valid	O
and	O
therefore	O
interpretable	O
.	O
</s>
<s>
In	O
instances	O
where	O
Mauchly	O
's	O
test	O
is	O
significant	O
,	O
modifications	O
need	O
to	O
be	O
made	O
to	O
the	O
degrees	O
of	O
freedom	O
so	O
that	O
a	O
valid	O
F-ratio	B-General_Concept
can	O
be	O
obtained	O
.	O
</s>
<s>
In	O
SPSS	B-Algorithm
,	O
three	O
corrections	O
are	O
generated	O
:	O
the	O
Greenhouse	O
–	O
Geisser	O
correction	O
(	O
1959	O
)	O
,	O
the	O
Huynh	O
–	O
Feldt	O
correction	O
(	O
1976	O
)	O
,	O
and	O
the	O
lower-bound	O
.	O
</s>
<s>
Each	O
of	O
these	O
corrections	O
have	O
been	O
developed	O
to	O
alter	O
the	O
degrees	O
of	O
freedom	O
and	O
produce	O
an	O
F-ratio	B-General_Concept
where	O
the	O
Type	O
I	O
error	O
rate	O
is	O
reduced	O
.	O
</s>
<s>
The	O
actual	O
F-ratio	B-General_Concept
does	O
not	O
change	O
as	O
a	O
result	O
of	O
applying	O
the	O
corrections	O
;	O
only	O
the	O
degrees	O
of	O
freedom	O
.	O
</s>
<s>
The	O
test	O
statistic	O
for	O
these	O
estimates	O
is	O
denoted	O
by	O
epsilon	B-Language
( ε	O
)	O
and	O
can	O
be	O
found	O
on	O
Mauchly	O
's	O
test	O
output	O
in	O
SPSS	B-Algorithm
.	O
</s>
<s>
Epsilon	B-Language
provides	O
a	O
measure	O
of	O
departure	O
from	O
sphericity	O
.	O
</s>
<s>
By	O
evaluating	O
epsilon	B-Language
,	O
we	O
can	O
determine	O
the	O
degree	O
to	O
which	O
sphericity	O
has	O
been	O
violated	O
.	O
</s>
<s>
If	O
the	O
variances	O
of	O
differences	O
between	O
all	O
possible	O
pairs	O
of	O
groups	O
are	O
equal	O
and	O
sphericity	O
is	O
exactly	O
met	O
,	O
then	O
epsilon	B-Language
will	O
be	O
exactly	O
1	O
,	O
indicating	O
no	O
departure	O
from	O
sphericity	O
.	O
</s>
<s>
If	O
the	O
variances	O
of	O
differences	O
between	O
all	O
possible	O
pairs	O
of	O
groups	O
are	O
unequal	O
and	O
sphericity	O
is	O
violated	O
,	O
epsilon	B-Language
will	O
be	O
below	O
1	O
.	O
</s>
<s>
The	O
further	O
epsilon	B-Language
is	O
from	O
1	O
,	O
the	O
worse	O
the	O
violation	O
.	O
</s>
<s>
When	O
epsilon	B-Language
is	O
>.75	O
,	O
the	O
Greenhouse	O
–	O
Geisser	O
correction	O
is	O
believed	O
to	O
be	O
too	O
conservative	O
,	O
and	O
would	O
result	O
in	O
incorrectly	O
rejecting	O
the	O
null	O
hypothesis	O
that	O
sphericity	O
holds	O
.	O
</s>
<s>
Collier	O
and	O
colleagues	O
showed	O
this	O
was	O
true	O
when	O
epsilon	B-Language
was	O
extended	O
to	O
as	O
high	O
as	O
.90	O
.	O
</s>
<s>
Girden	O
recommended	O
a	O
solution	O
to	O
this	O
problem	O
:	O
when	O
epsilon	B-Language
is	O
>.75	O
,	O
the	O
Huynh	O
–	O
Feldt	O
correction	O
should	O
be	O
applied	O
and	O
when	O
epsilon	B-Language
is	O
<	O
.75	O
or	O
nothing	O
is	O
known	O
about	O
sphericity	O
,	O
the	O
Greenhouse	O
–	O
Geisser	O
correction	O
should	O
be	O
applied	O
.	O
</s>
<s>
Another	O
alternative	O
procedure	O
is	O
using	O
the	O
multivariate	O
test	O
statistics	O
(	O
MANOVA	B-General_Concept
)	O
since	O
they	O
do	O
not	O
require	O
the	O
assumption	O
of	O
sphericity	O
.	O
</s>
<s>
O’Brien	O
and	O
Kaiser	O
suggested	O
that	O
when	O
you	O
have	O
a	O
large	O
violation	O
of	O
sphericity	O
(	O
i.e.	O
,	O
epsilon	B-Language
<	O
.70	O
)	O
and	O
your	O
sample	O
size	O
is	O
greater	O
than	O
k+10	O
(	O
i.e.	O
,	O
the	O
number	O
of	O
levels	O
of	O
the	O
repeated	O
measures	O
factor	O
+10	O
)	O
,	O
then	O
a	O
MANOVA	B-General_Concept
is	O
more	O
powerful	O
;	O
in	O
other	O
cases	O
,	O
repeated	O
measures	O
design	O
should	O
be	O
selected	O
.	O
</s>
<s>
Additionally	O
,	O
the	O
power	O
of	O
MANOVA	B-General_Concept
is	O
contingent	O
upon	O
the	O
correlations	O
between	O
the	O
dependent	O
variables	O
,	O
so	O
the	O
relationship	O
between	O
the	O
different	O
conditions	O
must	O
also	O
be	O
considered	O
.	O
</s>
<s>
SPSS	B-Algorithm
provides	O
an	O
F-ratio	B-General_Concept
from	O
four	O
different	O
methods	O
:	O
Pillai	O
's	O
trace	O
,	O
Wilks’	O
lambda	O
,	O
Hotelling	O
's	O
trace	O
,	O
and	O
Roy	O
's	O
largest	O
root	O
.	O
</s>
