A modification of the PC algorithm yielding order-independent skeletons

We consider the PC algorithm for estimating a Markov equivalence class of directed acyclic graphs. This algorithm is known to be order-dependent, in the sense that the output depends on the order in which the variables are given. We show that this order-dependence is not just an aesthetical problem, and can lead to highly variable results in high-dimensional settings. We propose a simple modification, called PC-stable, that yields order-independent adjacencies in its output graph and is consistent in high-dimensional settings under the same conditions as the original PC algorithm. We compare the PC and PC-stable algorithms in simulation studies and on a yeast gene expression data set, and show that PC-stable yields significantly improved performance in high-dimensional settings. The modification used in PC-stable can be easily combined with other adaptations of the PC algorithm, including hybrid algorithms and the FCI algorithm. All software is implemented in the R-package pcalg.