### Exact Power Under Independence for the False Discovery Rate in Gene Expression Array Experiments

Lawrence Hunter^{1}, Deborah H. Glueck^{2}, Anis Karimpour-Fard and Keith E. Muller

^{1}Larry.Hunter@uchsc.edu, U. Colorado School of Medicine; ^{2}, U. Colorado School of Medicine

The false discovery rate is widely used for multiple comparison
problems, including gene expression array studies. In that case,
choosing the number of chips is an important but poorly treated
problem. We have been unable to find published reports of exact,
analytic expressions for power and sample size for the Benjamini and
Hochberg (1995) False Discovery Rate procedure (FDR). For absolutely
continuous, independent, but not necessarily identically distributed
test statistics, we derive the distribution of the total number of
rejections, the number of false rejections conditioning on the total
number of rejections, and the joint probability distribution of the
number of total and false rejections. Thus we provide methods for
exact small sample power and sample size for gene expression array
experiments. The results also apply to any other use of FDR which
meets the same assumptions. Simulation studies are used to confirm the
analytic results. An example is given for research on the genetic
basis of breast cancer.
*Keywords*: False Discovery Rate, Power, Independent But Not
Identically Distributed

*Support* D.H. Glueck's work was supported by NCI K07CA88811.
L. Hunter was supported by NIAAA 1U01 AA13524-02 and NCI 5 P30
CA46934-15. K.E. Muller's work is supported in part by NCI P01 CA47
982-04, NCI R01 CA095749-01A1 and NIAID 9P30 AI 50410.