<s>
In	O
statistics	O
,	O
a	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
is	O
a	O
type	O
of	O
location	B-General_Concept
test	I-General_Concept
that	O
is	O
used	O
when	O
comparing	O
two	O
sets	O
of	O
paired	O
measurements	O
to	O
assess	O
whether	O
their	O
population	O
means	O
differ	O
.	O
</s>
<s>
A	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
uses	O
additional	O
information	O
about	O
the	O
sample	O
that	O
is	O
not	O
present	O
in	O
an	O
ordinary	O
unpaired	O
testing	O
situation	O
,	O
either	O
to	O
increase	O
the	O
statistical	B-General_Concept
power	I-General_Concept
,	O
or	O
to	O
reduce	O
the	O
effects	O
of	O
confounders	O
.	O
</s>
<s>
Specific	O
methods	O
for	O
carrying	O
out	O
paired	B-General_Concept
difference	I-General_Concept
tests	I-General_Concept
are	O
,	O
for	O
normally	O
distributed	O
difference	O
t-test	B-General_Concept
(	O
where	O
the	O
population	O
standard	O
deviation	O
of	O
difference	O
is	O
not	O
known	O
)	O
and	O
the	O
paired	O
Z-test	B-General_Concept
(	O
where	O
the	O
population	O
standard	O
deviation	O
of	O
the	O
difference	O
is	O
known	O
)	O
,	O
and	O
for	O
differences	O
that	O
may	O
not	O
be	O
normally	O
distributed	O
the	O
Wilcoxon	B-General_Concept
signed-rank	I-General_Concept
test	I-General_Concept
as	O
well	O
as	O
the	O
paired	O
permutation	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
most	O
familiar	O
example	O
of	O
a	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
occurs	O
when	O
subjects	O
are	O
measured	O
before	O
and	O
after	O
a	O
treatment	O
.	O
</s>
<s>
Paired	B-General_Concept
difference	I-General_Concept
tests	I-General_Concept
for	O
reducing	O
variance	O
are	O
a	O
specific	O
type	O
of	O
blocking	O
.	O
</s>
<s>
The	O
key	O
issue	O
that	O
motivates	O
the	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
is	O
that	O
unless	O
the	O
study	O
has	O
very	O
strict	O
entry	O
criteria	O
,	O
it	O
is	O
likely	O
that	O
the	O
subjects	O
will	O
differ	O
substantially	O
from	O
each	O
other	O
before	O
the	O
treatment	O
begins	O
.	O
</s>
<s>
Although	O
the	O
mean	O
difference	O
is	O
the	O
same	O
for	O
the	O
paired	O
and	O
unpaired	O
statistics	O
,	O
their	O
statistical	B-General_Concept
significance	I-General_Concept
levels	O
can	O
be	O
very	O
different	O
,	O
because	O
it	O
is	O
easy	O
to	O
overstate	O
the	O
variance	O
of	O
the	O
unpaired	O
statistic	O
.	O
</s>
<s>
Suppose	O
we	O
are	O
using	O
a	O
Z-test	B-General_Concept
to	O
analyze	O
the	O
data	O
,	O
where	O
the	O
variances	O
of	O
the	O
pre-treatment	O
and	O
post-treatment	O
data	O
and	O
are	O
known	O
(	O
the	O
situation	O
with	O
a	O
t-test	B-General_Concept
is	O
similar	O
)	O
.	O
</s>
<s>
The	O
power	O
of	O
the	O
unpaired	O
,	O
one-sided	B-General_Concept
test	I-General_Concept
carried	O
out	O
at	O
level	O
can	O
be	O
calculated	O
as	O
follows	O
:	O
</s>
<s>
where	O
is	O
a	O
random	B-General_Concept
effect	I-General_Concept
that	O
is	O
shared	O
between	O
the	O
two	O
values	O
in	O
the	O
pair	O
,	O
and	O
is	O
a	O
random	O
noise	O
term	O
that	O
is	O
independent	O
across	O
all	O
data	O
points	O
.	O
</s>
<s>
This	O
is	O
non-negative	O
,	O
so	O
it	O
leads	O
to	O
better	O
performance	O
for	O
the	O
paired	B-General_Concept
difference	I-General_Concept
test	I-General_Concept
compared	O
to	O
the	O
unpaired	O
test	O
,	O
unless	O
the	O
are	O
constant	O
over	O
,	O
in	O
which	O
case	O
the	O
paired	O
and	O
unpaired	O
tests	O
are	O
equivalent	O
.	O
</s>
<s>
If	O
the	O
two	O
measurements	O
within	O
a	O
subject	O
are	O
positively	O
correlated	O
,	O
the	O
unpaired	O
test	O
overstates	O
the	O
variance	O
of	O
,	O
making	O
it	O
a	O
conservative	O
test	O
in	O
the	O
sense	O
that	O
its	O
actual	O
type	O
I	O
error	O
probability	O
will	O
be	O
lower	O
than	O
the	O
nominal	O
level	O
,	O
with	O
a	O
corresponding	O
loss	O
of	O
statistical	B-General_Concept
power	I-General_Concept
.	O
</s>
<s>
The	O
paired	B-General_Concept
test	I-General_Concept
is	O
generally	O
used	O
when	O
repeated	O
measurements	O
are	O
made	O
on	O
the	O
same	O
subjects	O
,	O
since	O
it	O
has	O
the	O
correct	O
level	O
regardless	O
of	O
the	O
correlation	O
of	O
the	O
measurements	O
within	O
pairs	O
.	O
</s>
<s>
A	O
reasonable	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
that	O
there	O
is	O
no	O
effect	O
of	O
the	O
treatment	O
within	O
either	O
the	O
"	O
high	O
"	O
or	O
"	O
low	O
"	O
student	O
groups	O
,	O
so	O
that	O
.	O
</s>
<s>
If	O
the	O
expected	O
value	O
of	O
under	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
is	O
not	O
equal	O
to	O
zero	O
,	O
then	O
a	O
situation	O
where	O
we	O
reject	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
could	O
either	O
be	O
due	O
to	O
an	O
actual	O
differential	O
effect	O
between	O
teaching	O
strategies	O
and	O
,	O
or	O
it	O
could	O
be	O
due	O
to	O
non-independence	O
in	O
the	O
assignment	O
of	O
students	O
to	O
the	O
and	O
groups	O
(	O
even	O
in	O
the	O
complete	O
absence	O
of	O
an	O
effect	O
due	O
to	O
the	O
teaching	O
strategy	O
)	O
.	O
</s>
<s>
Consequently	O
,	O
if	O
the	O
null	B-General_Concept
hypothesis	I-General_Concept
holds	O
,	O
the	O
expected	O
value	O
of	O
will	O
equal	O
zero	O
,	O
and	O
statistical	B-General_Concept
significance	I-General_Concept
levels	O
have	O
their	O
intended	O
interpretation	O
.	O
</s>
