<s>
The	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
by	O
ranks	O
,	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
H	O
test	O
(	O
named	O
after	O
William	O
Kruskal	O
and	O
W	O
.	O
Allen	O
Wallis	O
)	O
,	O
or	O
one-way	B-General_Concept
ANOVA	I-General_Concept
on	O
ranks	O
is	O
a	O
non-parametric	B-General_Concept
method	I-General_Concept
for	O
testing	O
whether	O
samples	O
originate	O
from	O
the	O
same	O
distribution	O
.	O
</s>
<s>
It	O
extends	O
the	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
U	I-General_Concept
test	I-General_Concept
,	O
which	O
is	O
used	O
for	O
comparing	O
only	O
two	O
groups	O
.	O
</s>
<s>
The	O
parametric	O
equivalent	O
of	O
the	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
is	O
the	O
one-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
(	O
ANOVA	O
)	O
.	O
</s>
<s>
A	O
significant	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
indicates	O
that	O
at	O
least	O
one	O
sample	O
stochastically	O
dominates	O
one	O
other	O
sample	O
.	O
</s>
<s>
For	O
analyzing	O
the	O
specific	O
sample	O
pairs	O
for	O
stochastic	O
dominance	O
,	O
Dunn	O
's	O
test	O
,	O
pairwise	O
Mann	B-General_Concept
–	I-General_Concept
Whitney	I-General_Concept
tests	I-General_Concept
with	O
Bonferroni	B-General_Concept
correction	I-General_Concept
,	O
or	O
the	O
more	O
powerful	O
but	O
less	O
well	O
known	O
Conover	O
–	O
Iman	O
test	O
are	O
sometimes	O
used	O
.	O
</s>
<s>
Since	O
it	O
is	O
a	O
nonparametric	B-General_Concept
method	O
,	O
the	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
does	O
not	O
assume	O
a	O
normal	O
distribution	O
of	O
the	O
residuals	O
,	O
unlike	O
the	O
analogous	O
one-way	B-General_Concept
analysis	I-General_Concept
of	I-General_Concept
variance	I-General_Concept
.	O
</s>
<s>
This	O
is	O
the	O
same	O
issue	O
that	O
happens	O
also	O
with	O
the	O
Mann-Whitney	B-General_Concept
test	I-General_Concept
.	O
</s>
<s>
Finally	O
,	O
the	O
decision	O
to	O
reject	O
or	O
not	O
the	O
null	O
hypothesis	O
is	O
made	O
by	O
comparing	O
to	O
a	O
critical	O
value	O
obtained	O
from	O
a	O
table	O
or	O
a	O
software	O
for	O
a	O
given	O
significance	B-General_Concept
or	O
alpha	O
level	O
.	O
</s>
<s>
If	O
a	O
table	O
of	O
the	O
chi-squared	O
probability	O
distribution	O
is	O
available	O
,	O
the	O
critical	O
value	O
of	O
chi-squared	O
,	O
,	O
can	O
be	O
found	O
by	O
entering	O
the	O
table	O
at	O
g−1	O
degrees	O
of	O
freedom	O
and	O
looking	O
under	O
the	O
desired	O
significance	B-General_Concept
or	O
alpha	O
level	O
.	O
</s>
<s>
Therefore	O
,	O
a	O
researcher	O
might	O
use	O
sample	O
contrasts	O
between	O
individual	O
sample	O
pairs	O
,	O
or	O
post	O
hoc	O
tests	O
using	O
Dunn	O
's	O
test	O
,	O
which	O
(	O
1	O
)	O
properly	O
employs	O
the	O
same	O
rankings	O
as	O
the	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
,	O
and	O
(	O
2	O
)	O
properly	O
employs	O
the	O
pooled	O
variance	O
implied	O
by	O
the	O
null	O
hypothesis	O
of	O
the	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
in	O
order	O
to	O
determine	O
which	O
of	O
the	O
sample	O
pairs	O
are	O
significantly	O
different	O
.	O
</s>
<s>
When	O
performing	O
multiple	O
sample	O
contrasts	O
or	O
tests	O
,	O
the	O
Type	O
I	O
error	O
rate	O
tends	O
to	O
become	O
inflated	O
,	O
raising	O
concerns	O
about	O
multiple	B-General_Concept
comparisons	I-General_Concept
.	O
</s>
<s>
A	O
large	O
amount	O
of	O
computing	O
resources	O
is	O
required	O
to	O
compute	O
exact	O
probabilities	O
for	O
the	O
Kruskal	B-General_Concept
–	I-General_Concept
Wallis	I-General_Concept
test	I-General_Concept
.	O
</s>
<s>
The	O
Kruskal-Wallis	B-General_Concept
test	I-General_Concept
finds	O
a	O
significant	B-General_Concept
difference	I-General_Concept
(	O
p	O
=	O
6.901e-06	O
)	O
indicating	O
that	O
ozone	O
differs	O
among	O
the	O
5	O
months	O
.	O
</s>
<s>
The	O
post-hoc	O
tests	O
indicate	O
that	O
,	O
after	O
Bonferroni	B-General_Concept
correction	I-General_Concept
for	O
multiple	B-General_Concept
testing	I-General_Concept
,	O
the	O
following	O
differences	O
are	O
significant	O
(	O
adjusted	O
p	O
<	O
0.05	O
)	O
.	O
</s>
