The Confidence Interval that Wasn't: Bootstrapped "Confidence Intervals" in L1-Regularized Partial Correlation Networks [post]

Donald Ray Williams
2021 unpublished
I shed much needed light upon the default measure of parameter uncertainty in network psychometrics; that is, "confidence intervals" (CI) computed from bootstrapping $\ell_1$-regularized partial correlations. Due to the nature of the $\ell_1$-penalty, however, bootstrapping does not provide an accurate sampling distribution. Although this has long been known in the statistical literature, I set out to determine whether the intervals can at least be considered \emph{approximate}. In multiple
more » ... ession, I first describe the fundamental tension between model selection and estimation consistency inherent to the $\ell_1$-penalty---in the pursuit of sparsity, the sampling distribution of the non-zero coefficients is necessarily compromised which translates into coverage far below nominal levels.With the foundation laid, I proceed to investigate coverage for non-zero relations in partial correlation networks. At best, average coverage was around 0.65 for 90\% CIs. With increasing sample sizes, average coverage decreased to 0.30, perhaps approaching 0 if larger sample sizes were explored. Further, coverage was heavily influenced by the mere position of an edge in the network, ranging from essentially 0 to 0.90, with an average of around 0.50. Meanwhile, for the same simulation conditions, simply bootstrapping the sample covariance matrix provided coverage at the nominal level. In light of the results, I then demonstrate how to judiciously use the bootstrap in both regularized and non-regularized networks: the former can provide a useful summary of data-mining, whereas the latter allows for making inference on network parameters. To ensure network researchers have the option of computing valid CIs, I implemented a non-regularized bootstrap for various types of partial correlations in the {\tt R} package \textbf{GGMnonreg}.
doi:10.31234/ fatcat:woqpsnpb6ncvvjfk2u5hpzcm2i