110 A Practical Guide to Cancer Systems Biology
fit with data fromLsamples, the AIC statistics can be written as follows6,8:
AIC(Ni)=log(
1
L
(Xi−Xˆi)T(Xi−Xˆi))
+2 Ni
L
(9.13)whereXˆidenotes the estimated expression profile of thei-th target gene,
i.e.Xˆi=Φi·θˆi,andˆσ^2 i =L^1 (Xi−Xˆi)T(Xi−Xˆi) is the estimated residual
error. As the residual error ˆσ^2 i decreases, the AIC decreases. In contrast,
while the number of regulatory genes (or parameters)Niincreases, the AIC
increases. Therefore, there is a tradeoff between residual error and model
order. As the expected residual error decreases with increasing number of
regulatory genes in models of inadequate complexity, there should be a
minimum around the optimal number of regulatory genes. The minimization
achieved in Eq. (9.13) will indicate the ideal model order (i.e. the optimal
number of gene that regulate the target gene) of the discrete dynamic model.
With the statistical selection ofNiregulatory genes by minimization of the
AIC, the question of whether a regulatory gene is a significant one or just
a false positive for thei-th target gene can be determined. Hence, AIC
can be adopted to select model order, filtering out insignificant regulations
in the gene regulatory network based on the estimated regulatory abilities
(aij’s). Due to computational efficiency, it is impractical to compute the AIC
statistics for all possible regression models. Stepwise methods such as forward
selection method and backward elimination method are developed to avoid
the complexity of exhaustive search.9,10However, in the case of backward
selection method, a variable once eliminated can never be reintroduced
into the model, and in the case of forward selection, once included can
never be removed.^10 Thus, the stepwise regression method which combines
forward selection method and backward elimination method is suggested to
be used to compute the AIC statistics. In addition to AIC model selection
criteria, the Student’st-test^3 is further employed to calculate thep-values
for the regulatory abilities under the null hypothesisH 0 : aij =0to
determine the significant regulatory relationships. The regulations withp-
value 0 .05 are determined as significant regulations and preserved in the
gene regulatory network. According to these modeling procedures, the gene
regulatory network for the specific experimental condition is constructed
from the transcriptomics data.
- Summary and perspective
In this chapter, we demonstrate the use of dynamic modeling techniques to
analyze high-throughput data for understanding the interactions/regulations