Biology Reference
In-Depth Information
two-sample tests. In this case, the standardized mean difference
d
(see
Hedges and Olkin
7
) is often used as the effect size of interest. As we
demonstrate below, combining
z
-scores provides a flexible means of
combining information from more general testing scenarios.
5.5. Combining p-values
Combining results using transformed
p
-values has a long history. It does
not require knowledge of the composition of the individual datasets,
which may be highly heterogeneous and with no common parameter.
This level of combination is closely related to the previous level, as trans-
formations (e.g. the probit) can yield test statistics that are straightfor-
ward to combine.
Perhaps the most widely used method for
p
-value combination is
attributed to Fisher.
19
This method is based on the fact that under the
null, the test statistic
p
-value is distributed as a uniform (0,1) random
variable, so that the log
p
-value is distributed exponentially. The
p
-values
are combined by summing the logs. Twice this sum is distributed as a
χ
2
random variable with degrees of freedom two times the number of study
p
-values. A microarray example is given in Rhodes
et al
.,
20
who use
Fisher's method to combine results for genes measured in all studies.
Although it can perform well in a variety of circumstances, a draw-
back of Fisher
p-
value combination is its treatment of conflicting studies.
Given a sufficiently large
2
value for the first
k
- 1 studies, an additional
study — regardless of its size, quality, or extent of negative findings —
cannot produce an overall nonsignificant result with this method, as the
smallest value that can be added to the overall
χ
2
statistic is 0. In addi-
tion, conclusions may depend more heavily on the within-study
choice of statistic rather than on global data characteristics. Combining
inverse normal (probit)-transformed
p
-values (i.e.
z
-scores)
7,16
χ
avoids
these limitations.
5.6. Combining Statistic Ranks
In many microarray studies, the main result consists of an ordered
gene list. One way to combine such lists across studies is by averaging
