<s>
In	O
statistics	O
,	O
(	O
between	O
-	O
)	O
study	B-General_Concept
heterogeneity	I-General_Concept
is	O
a	O
phenomenon	O
that	O
commonly	O
occurs	O
when	O
attempting	O
to	O
undertake	O
a	O
meta-analysis	O
.	O
</s>
<s>
Differences	O
between	O
outcomes	O
would	O
only	O
be	O
due	O
to	O
measurement	O
error	O
(	O
and	O
studies	O
would	O
hence	O
be	O
homogeneous	B-General_Concept
)	O
.	O
</s>
<s>
Study	B-General_Concept
heterogeneity	I-General_Concept
denotes	O
the	O
variability	O
in	O
outcomes	O
that	O
goes	O
beyond	O
what	O
would	O
be	O
expected	O
(	O
or	O
could	O
be	O
explained	O
)	O
due	O
to	O
measurement	O
error	O
alone	O
.	O
</s>
<s>
Meta-analysis	O
increases	O
the	O
power	B-General_Concept
of	O
statistical	O
analyses	O
by	O
pooling	O
the	O
results	O
of	O
all	O
available	O
trials	O
.	O
</s>
<s>
Any	O
excess	O
variation	O
(	O
whether	O
it	O
is	O
apparent	O
or	O
detectable	O
or	O
not	O
)	O
is	O
called	O
(	O
statistical	O
)	O
heterogeneity	B-General_Concept
.	O
</s>
<s>
The	O
presence	O
of	O
some	O
heterogeneity	B-General_Concept
is	O
not	O
unusual	O
,	O
e.g.	O
,	O
analogous	O
effects	O
are	O
also	O
commonly	O
encountered	O
even	O
within	O
studies	O
,	O
in	O
multicenter	O
trials	O
(	O
between-center	O
heterogeneity	B-General_Concept
)	O
.	O
</s>
<s>
Different	O
types	O
of	O
effect	B-Application
measures	I-Application
(	O
e.g.	O
,	O
odds	O
ratio	O
vs.	O
relative	O
risk	O
)	O
may	O
also	O
be	O
more	O
or	O
less	O
susceptible	O
to	O
heterogeneity	B-General_Concept
.	O
</s>
<s>
In	O
case	O
the	O
origin	O
of	O
heterogeneity	B-General_Concept
can	O
be	O
identified	O
and	O
may	O
be	O
attributed	O
to	O
certain	O
study	O
features	O
,	O
the	O
analysis	O
may	O
be	O
stratified	O
(	O
by	O
considering	O
subgroups	O
of	O
studies	O
,	O
which	O
would	O
then	O
hopefully	O
be	O
more	O
homogeneous	B-General_Concept
)	O
,	O
or	O
by	O
extending	O
the	O
analysis	O
to	O
a	O
meta-regression	B-General_Concept
,	O
accounting	O
for	O
(	O
continuous	O
or	O
categorical	O
)	O
moderator	O
variables	O
.	O
</s>
<s>
In	O
addition	O
,	O
heterogeneity	B-General_Concept
is	O
usually	O
accommodated	O
by	O
using	O
a	O
random	B-General_Concept
effects	I-General_Concept
model	I-General_Concept
,	O
in	O
which	O
the	O
heterogeneity	B-General_Concept
then	O
constitutes	O
a	O
variance	B-General_Concept
component	I-General_Concept
.	O
</s>
<s>
The	O
centre	O
of	O
this	O
symmetric	O
distribution	O
describes	O
the	O
average	O
of	O
the	O
effects	O
,	O
while	O
its	O
width	O
describes	O
the	O
degree	O
of	O
heterogeneity	B-General_Concept
.	O
</s>
<s>
It	O
is	O
difficult	O
to	O
establish	O
the	O
validity	O
of	O
any	O
distributional	O
assumption	O
,	O
and	O
this	O
is	O
a	O
common	O
criticism	O
of	O
random	B-General_Concept
effects	I-General_Concept
meta-analyses	O
.	O
</s>
<s>
However	O
,	O
variations	O
of	O
the	O
exact	O
distributional	O
form	O
may	O
not	O
make	O
much	O
of	O
a	O
difference	O
,	O
and	O
simulations	O
have	O
shown	O
that	O
methods	O
are	O
relatively	O
robust	O
even	O
under	O
extreme	O
distributional	O
assumptions	O
,	O
both	O
in	O
estimating	O
heterogeneity	B-General_Concept
,	O
and	O
calculating	O
an	O
overall	O
effect	B-Application
size	I-Application
.	O
</s>
<s>
Inclusion	O
of	O
a	O
random	B-General_Concept
effect	I-General_Concept
to	O
the	O
model	O
has	O
the	O
effect	O
of	O
making	O
the	O
inferences	O
(	O
in	O
a	O
sense	O
)	O
more	O
conservative	O
or	O
cautious	O
,	O
as	O
a	O
(	O
non-zero	O
)	O
heterogeneity	B-General_Concept
will	O
lead	O
to	O
greater	O
uncertainty	O
(	O
and	O
avoid	O
overconfidence	O
)	O
in	O
the	O
estimation	O
of	O
overall	O
effects	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
of	O
a	O
zero	O
heterogeneity	B-General_Concept
variance	O
,	O
the	O
random-effects	O
model	O
again	O
reduces	O
to	O
the	O
special	O
case	O
of	O
the	O
common-effect	B-General_Concept
model	O
.	O
</s>
<s>
Statistical	O
testing	O
for	O
a	O
non-zero	O
heterogeneity	B-General_Concept
variance	O
is	O
often	O
done	O
based	O
on	O
Cochran	O
's	O
Q	O
or	O
related	O
test	O
procedures	O
.	O
</s>
<s>
This	O
common	O
procedure	O
however	O
is	O
questionable	O
for	O
several	O
reasons	O
,	O
namely	O
,	O
the	O
low	O
power	B-General_Concept
of	O
such	O
tests	O
especially	O
in	O
the	O
very	O
common	O
case	O
of	O
only	O
few	O
estimates	O
being	O
combined	O
in	O
the	O
analysis	O
,	O
as	O
well	O
as	O
the	O
specification	O
of	O
homogeneity	B-General_Concept
as	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
which	O
is	O
then	O
only	O
rejected	O
in	O
the	O
presence	O
of	O
sufficient	O
evidence	O
against	O
it	O
.	O
</s>
<s>
While	O
the	O
main	O
purpose	O
of	O
a	O
meta-analysis	O
usually	O
is	O
estimation	O
of	O
the	O
main	O
effect	O
,	O
investigation	O
of	O
the	O
heterogeneity	B-General_Concept
is	O
also	O
crucial	O
for	O
its	O
interpretation	O
.	O
</s>
<s>
A	O
large	O
number	O
of	O
(	O
frequentist	B-General_Concept
and	O
Bayesian	O
)	O
estimators	O
is	O
available	O
.	O
</s>
<s>
Bayesian	O
estimation	O
of	O
the	O
heterogeneity	B-General_Concept
usually	O
requires	O
the	O
specification	O
of	O
an	O
appropriate	O
prior	O
distribution	O
.	O
</s>
<s>
An	O
incorrect	O
zero	O
between-study	O
variance	O
estimate	O
is	O
frequently	O
obtained	O
,	O
leading	O
to	O
a	O
false	O
homogeneity	B-General_Concept
assumption	O
.	O
</s>
<s>
Overall	O
,	O
it	O
appears	O
that	O
heterogeneity	B-General_Concept
is	O
being	O
consistently	O
underestimated	O
in	O
meta-analyses	O
.	O
</s>
<s>
The	O
heterogeneity	B-General_Concept
variance	O
is	O
commonly	O
denoted	O
by	O
τ²	O
,	O
or	O
the	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
its	O
square	O
root	O
)	O
by	O
τ	O
.	O
</s>
<s>
Heterogeneity	B-General_Concept
is	O
probably	O
most	O
readily	O
interpretable	O
in	O
terms	O
of	O
τ	O
,	O
as	O
this	O
is	O
the	O
heterogeneity	B-General_Concept
distribution	O
's	O
scale	O
parameter	O
,	O
which	O
is	O
measured	O
in	O
the	O
same	O
units	O
as	O
the	O
overall	O
effect	O
itself	O
.	O
</s>
<s>
Another	O
common	O
measure	O
of	O
heterogeneity	B-General_Concept
is	O
I²	O
,	O
a	O
statistic	O
that	O
indicates	O
the	O
percentage	O
of	O
variance	O
in	O
a	O
meta-analysis	O
that	O
is	O
attributable	O
to	O
study	B-General_Concept
heterogeneity	I-General_Concept
(	O
somewhat	O
similarly	O
to	O
a	O
coefficient	O
of	O
determination	O
)	O
.	O
</s>
<s>
I²	O
relates	O
the	O
heterogeneity	B-General_Concept
variance	O
's	O
magnitude	O
to	O
the	O
size	O
of	O
the	O
individual	O
estimates	O
 '	O
variances	O
(	O
squared	O
standard	B-General_Concept
errors	I-General_Concept
)	O
;	O
with	O
this	O
normalisation	O
however	O
,	O
it	O
is	O
not	O
quite	O
obvious	O
what	O
exactly	O
would	O
constitute	O
"	O
small	O
"	O
or	O
"	O
large	O
"	O
amounts	O
of	O
heterogeneity	B-General_Concept
.	O
</s>
<s>
For	O
a	O
constant	O
heterogeneity	B-General_Concept
( τ	O
)	O
,	O
the	O
availability	O
of	O
smaller	O
or	O
larger	O
studies	O
(	O
with	O
correspondingly	O
differing	O
standard	B-General_Concept
errors	I-General_Concept
associated	O
)	O
would	O
affect	O
the	O
I²	O
measure	O
;	O
so	O
the	O
actual	O
interpretation	O
of	O
an	O
I²	O
value	O
is	O
not	O
straightforward	O
.	O
</s>
<s>
The	O
joint	O
consideration	O
of	O
a	O
prediction	O
interval	O
along	O
with	O
a	O
confidence	O
interval	O
for	O
the	O
main	O
effect	O
may	O
help	O
getting	O
a	O
better	O
sense	O
of	O
the	O
contribution	O
of	O
heterogeneity	B-General_Concept
to	O
the	O
uncertainty	O
around	O
the	O
effect	O
estimate	O
.	O
</s>
