Biology Reference
In-Depth Information
6.5. Z-Transform for Combining Test Statistics
Because we cannot in general expect data from different studies to be
commensurable, we combine information farther down the spectrum
from the raw values. In addition, we do not require a common (com-
mensurable) parameter across studies, so we also skip past combining
parameter estimates. But in order to preserve as much information and
flexibility as possible, we do not want to reduce to ranks or decisions.
Since we are able to estimate all single-gene models in each study, we are
in a position to combine the single-gene statistics across studies.
A useful statistic for the purpose of combining is the
z
-score. Within
the GLM framework, this statistic is often straightforward to obtain. In
the simplest case, combining tests of single coefficients, one option is to
combine the
K
study standardized coefficients
= β
·
i
/SE(
β
·
i
),
Z
i
where
i
indexes genes. For sufficiently large sample sizes,
Z
i
are each
approximately distributed as standard normal under the null hypothesis
H
:
0. An alternative which is approximately equivalent is to use the
signed square root of the deviance of the likelihood ratio test for one
additional parameter
28
:
β =
β
·
)
Z
D
=
sgn(
√
D.
Where
z
-statistics are not readily available, either the individual model
statistics may be transformed to yield an approximate standard normal or
the corresponding
p-
values may be probit-transformed to yield a
z
-score.
The single-gene individual study
z
-scores
Z
ij
are then combined
metaanalytically over the
K
studies using equal weighting by the inverse
normal method
7
:
K
i
Â
,
Z
=
Z
/
K
(1)
i
ik
i
k
=
1
where
i
indicates genes,
k
indicates study, and
K
i
is the number of datasets
in which gene
i
is present (that is, any platform missing the gene is
