Biomedical Engineering Reference
In-Depth Information
We permute the z
i
values each of n! ways to obtain different values of U,
say U
1
;U
2
;:::;U
n!
. Then if U
0
is the observed value of U, a one-sided exact
p-value is
P
n!
i=1
IfU
i
U
0
g
n!
p =
;
where I(A) = 1 if A is true and 0 otherwise. The other one-sided p-value
is found by reversing the inequality in the expression for p. A two-sided p-
value is the minimum of 1 and twice the smaller of the one-sided p-values.
For moderate sample sizes, these calculations may be intractable by com-
plete enumeration, and smarter algorithms are needed (see references in Hirji,
2006). But even using more sophisticated algorithms, the exact p-value may
be intractable. In these cases, we can estimate the exact p-value by Monte
Carlo permutation. We take b Monte Carlo permutation samples, create the
score statistic from each permutation, U
1
;:::;U
b
, and estimate the one-sided
p-value with
1 +
P
i=1
IfU
i
U
0
g
b + 1
p
MC
=
:
We add 1 to the numerator and denominator to ensure validity, as otherwise
p-values of 0 are possible (see, for example, Fay et al., 2007). We can make
p
MC
as accurate as needed by increasing b, and represent the accuracy us-
ing confidence intervals for binomial random variables because the indicators
IfU
i
U
0
g are Bernoulli with parameter equal to the exact p-value, p.
Using a permutational central limit theorem (see, for example, Sen, 1985),
one can show that for large sample sizes under the null, U
0
is approximately
normal with mean 0 and variance V
p
, where V
p
is defined in Equation (13.8).
For tests with g > 2 treatment groups, we let zi
i
be a g1 treatment indicator
vector, with zi
i
= 0 denoting the reference group. Then we reject when Q
0
=
U
0
V
1
U
0
is large, where
p
)
8
<
9
=
(
X
X
1
n 1
(c
i
c)
2
(z
i
z)(z
i
z)
0
V
p
=
;
(13.8)
:
;
i=1
j=1
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