<s>
In	O
statistics	O
,	O
the	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
,	O
also	O
called	O
the	O
Holm	B-General_Concept
method	I-General_Concept
or	O
Bonferroni	O
–	O
Holm	B-General_Concept
method	I-General_Concept
,	O
is	O
used	O
to	O
counteract	O
the	O
problem	B-General_Concept
of	I-General_Concept
multiple	I-General_Concept
comparisons	I-General_Concept
.	O
</s>
<s>
It	O
is	O
intended	O
to	O
control	O
the	O
family-wise	O
error	O
rate	O
(	O
FWER	O
)	O
and	O
offers	O
a	O
simple	O
test	O
uniformly	B-General_Concept
more	I-General_Concept
powerful	I-General_Concept
than	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
.	O
</s>
<s>
The	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
is	O
one	O
of	O
many	O
approaches	O
for	O
controlling	O
the	O
FWER	O
,	O
i.e.	O
,	O
the	O
probability	O
that	O
one	O
or	O
more	O
Type	O
I	O
errors	O
will	O
occur	O
,	O
by	O
adjusting	O
the	O
rejection	O
criteria	O
for	O
each	O
of	O
the	O
individual	O
hypotheses	O
.	O
</s>
<s>
Suppose	O
you	O
have	O
p-values	B-General_Concept
,	O
sorted	O
into	O
order	O
lowest-to-highest	O
,	O
and	O
their	O
corresponding	O
hypotheses	O
（null	O
hypotheses	O
)	O
.	O
</s>
<s>
You	O
want	O
the	O
FWER	O
to	O
be	O
no	O
higher	O
than	O
a	O
certain	O
pre-specified	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
And	O
so	O
on	O
:	O
for	O
each	O
P	B-General_Concept
value	I-General_Concept
,	O
test	O
whether	O
.	O
</s>
<s>
If	O
so	O
,	O
reject	O
and	O
continue	O
to	O
examine	O
the	O
larger	O
P	B-General_Concept
values	I-General_Concept
,	O
otherwise	O
EXIT	O
.	O
</s>
<s>
The	O
simple	O
Bonferroni	B-General_Concept
correction	I-General_Concept
rejects	O
only	O
null	O
hypotheses	O
with	O
p-value	B-General_Concept
less	O
than	O
,	O
in	O
order	O
to	O
ensure	O
that	O
the	O
FWER	O
,	O
i.e.	O
,	O
the	O
risk	O
of	O
rejecting	O
one	O
or	O
more	O
true	O
null	O
hypotheses	O
(	O
i.e.	O
,	O
of	O
committing	O
one	O
or	O
more	O
type	O
I	O
errors	O
)	O
is	O
at	O
most	O
.	O
</s>
<s>
The	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
also	O
controls	O
the	O
FWER	O
at	O
,	O
but	O
with	O
a	O
lower	O
increase	O
of	O
type	O
II	O
error	O
risk	O
than	O
the	O
classical	O
Bonferroni	B-General_Concept
method	I-General_Concept
.	O
</s>
<s>
The	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
sorts	O
the	O
p-values	B-General_Concept
from	O
lowest	O
to	O
highest	O
and	O
compares	O
them	O
to	O
nominal	O
alpha	O
levels	O
of	O
to	O
(	O
respectively	O
)	O
,	O
namely	O
the	O
values	O
.	O
</s>
<s>
The	O
index	O
identifies	O
the	O
first	O
p-value	B-General_Concept
that	O
is	O
not	O
low	O
enough	O
to	O
validate	O
rejection	O
.	O
</s>
<s>
If	O
then	O
no	O
p-values	B-General_Concept
were	O
low	O
enough	O
for	O
rejection	O
,	O
therefore	O
no	O
null	O
hypotheses	O
are	O
rejected	O
.	O
</s>
<s>
If	O
no	O
such	O
index	O
could	O
be	O
found	O
then	O
all	O
p-values	B-General_Concept
were	O
low	O
enough	O
for	O
rejection	O
,	O
therefore	O
all	O
null	O
hypotheses	O
are	O
rejected	O
(	O
none	O
are	O
accepted	O
)	O
.	O
</s>
<s>
Let	O
be	O
the	O
family	O
of	O
hypotheses	O
sorted	O
by	O
their	O
p-values	B-General_Concept
.	O
</s>
<s>
The	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
can	O
be	O
viewed	O
as	O
a	O
closed	B-General_Concept
testing	I-General_Concept
procedure	I-General_Concept
,	O
with	O
the	O
Bonferroni	B-General_Concept
correction	I-General_Concept
applied	O
locally	O
on	O
each	O
of	O
the	O
intersections	O
of	O
null	O
hypotheses	O
.	O
</s>
<s>
The	O
Holm-Bonferroni	B-General_Concept
method	I-General_Concept
is	O
a	O
shortcut	O
procedure	O
,	O
since	O
it	O
makes	O
or	O
less	O
comparisons	O
,	O
while	O
the	O
number	O
of	O
all	O
intersections	O
of	O
null	O
hypotheses	O
to	O
be	O
tested	O
is	O
of	O
order	O
.	O
</s>
<s>
Consider	O
four	O
null	O
hypotheses	O
with	O
unadjusted	O
p-values	B-General_Concept
,	O
,	O
and	O
,	O
to	O
be	O
tested	O
at	O
significance	B-General_Concept
level	I-General_Concept
.	O
</s>
<s>
Since	O
the	O
procedure	O
is	O
step-down	O
,	O
we	O
first	O
test	O
,	O
which	O
has	O
the	O
smallest	O
p-value	B-General_Concept
.	O
</s>
<s>
The	O
p-value	B-General_Concept
is	O
compared	O
to	O
,	O
the	O
null	O
hypothesis	O
is	O
rejected	O
and	O
we	O
continue	O
to	O
the	O
next	O
one	O
.	O
</s>
<s>
Let	O
be	O
the	O
ordered	O
unadjusted	O
p-values	B-General_Concept
.	O
</s>
<s>
The	O
adjusted	O
p-values	B-General_Concept
for	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
are	O
:	O
</s>
<s>
In	O
the	O
earlier	O
example	O
,	O
the	O
adjusted	O
p-values	B-General_Concept
are	O
,	O
,	O
and	O
.	O
</s>
<s>
Similar	O
adjusted	O
p-values	B-General_Concept
for	O
Holm-Šidák	O
method	O
can	O
be	O
defined	O
recursively	O
as	O
,	O
where	O
.	O
</s>
<s>
Due	O
to	O
the	O
inequality	O
for	O
,	O
the	O
Holm-Šidák	O
method	O
will	O
be	O
more	O
powerful	O
than	O
Holm-Bonferroni	B-General_Concept
method	I-General_Concept
.	O
</s>
<s>
The	O
weighted	O
adjusted	O
p-values	B-General_Concept
are	O
:	O
</s>
<s>
A	O
hypothesis	O
is	O
rejected	O
at	O
level	O
α	O
if	O
and	O
only	O
if	O
its	O
adjusted	O
p-value	B-General_Concept
is	O
less	O
than	O
α	O
.	O
</s>
<s>
In	O
the	O
earlier	O
example	O
using	O
equal	O
weights	O
,	O
the	O
adjusted	O
p-values	B-General_Concept
are	O
0.03	O
,	O
0.06	O
,	O
0.06	O
,	O
and	O
0.02	O
.	O
</s>
<s>
The	O
Holm	B-General_Concept
–	I-General_Concept
Bonferroni	I-General_Concept
method	I-General_Concept
is	O
"	O
uniformly	O
"	O
more	O
powerful	O
than	O
the	O
classic	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
meaning	O
that	O
it	O
is	O
always	O
at	O
least	O
as	O
powerful	O
.	O
</s>
<s>
Thus	O
,	O
The	O
Hochberg	O
procedure	O
is	O
uniformly	B-General_Concept
more	I-General_Concept
powerful	I-General_Concept
than	O
the	O
Holm	B-General_Concept
procedure	I-General_Concept
.	O
</s>
<s>
A	O
similar	O
step-up	O
procedure	O
is	O
the	O
Hommel	O
procedure	O
,	O
which	O
is	O
uniformly	B-General_Concept
more	I-General_Concept
powerful	I-General_Concept
than	O
the	O
Hochberg	O
procedure	O
.	O
</s>
<s>
Holm	O
originally	O
called	O
the	O
method	O
the	O
"	O
sequentially	O
rejective	O
Bonferroni	B-General_Concept
test	I-General_Concept
"	O
,	O
and	O
it	O
became	O
known	O
as	O
Holm	O
–	O
Bonferroni	O
only	O
after	O
some	O
time	O
.	O
</s>
