Testing for Differences in Gaussian Graphical Models: Applications to Brain Connectivity
Functional brain networks are well described and estimated from data with Gaussian Graphical Models (GGMs), e.g. using sparse inverse covariance estimators. Comparing functional connectivity of subjects in two populations calls for comparing these estimated GGMs. Our goal is to identify differences in GGMs known to have similar structure. We characterize the uncertainty of differences with confidence intervals obtained using a parametric distribution on parameters of a sparse estimator. Sparse penalties enable statistical guarantees and interpretable models even in high-dimensional and low-sample settings. Characterizing the distributions of sparse models is inherently challenging as the penalties produce a biased estimator. Recent work invokes the sparsity assumptions to effectively remove the bias from a sparse estimator such as the lasso. These distributions can be used to give confidence intervals on edges in GGMs, and by extension their differences. However, in the case of comparing GGMs, these estimators do not make use of any assumed joint structure among the GGMs. Inspired by priors from brain functional connectivity we derive the distribution of parameter differences under a joint penalty when parameters are known to be sparse in the difference. This leads us to introduce the debiased multi-task fused lasso, whose distribution can be characterized in an efficient manner. We then show how the debiased lasso and multi-task fused lasso can be used to obtain confidence intervals on edge differences in GGMs. We validate the techniques proposed on a set of synthetic examples as well as neuro-imaging dataset created for the study of autism.
After the rebuttal: I thank the authors for answering my many questions. There are important technical/experimental details left to the rebuttal that should have been in the submitted manuscript (and will hopefully be included in the final appendix). Also, I am not as optimistic as the authors about the ease of analysis if one replaces the individual sparsity assumption with another norm, but I don't think it is a major issue. I suggest that the authors modify the final text appropriately to clarify this detail. Overall, I still think this is a technically solid paper with some potential to impact the field. I have updated my scores slightly to reflect this. ===================================== The authors propose to improve a parametric (asymptotic) test for differences between precision matrices by replacing the de-biased lasso with a "multitask fused lasso". The proposed regularizer is constructed specifically for the case where the differences between the two precision matrices are sparse, in addition to separate sparsity of each precision matrices. The primary innovation is the (asymptotic) de-biasing of the estimator, resulting in a parametric (asymptotically normal) test. The technique is motivated by applications to testing differences in brain connectivity between two populations. Results on simulated data suggest that the proposed de-biasing improves statistical power to detect differences. Result on real data are focused on comparison between the proposed parametric test and the permutation based standard, suggesting similar performance without the expense of permutation.
The goal of improving testing for differences between graphs is clearly relevant to neuroimaging and other application domains. While the specifics are somewhat incremental, I think this is a great idea, and reasonably well executed.  Major issues: * Please explain specifically which gradients are used to get from (4) to (5). This derivation seems incorrect if one takes separate derivatives with respect to \beta_1 and \beta_2. How do you end up with a sum of terms (and not two separate terms)? Or is this the sum of the two gradient terms? If accepted, I strongly suggest more a detailed derivation in the appendix.  * While the authors spend significant effort motivating the need to test sparse differences between "dense" graphs (e.g. repeatedly referring to [19] to suggest that brain networks are not sparse), the resulting techniques are in-fact designed to test sparse differences between "sparse" graphs. I think this is unfortunate, because the idea of improving tests for edge differences is sufficiently interesting and potentially useful without the mismatch between motivation and algorithm. I strongly suggest that the authors modify the motivation to better reflect the proposed test.  * One of the issues with the regression approach to precision matrix estimation is that one gives up finite sample guarantees of both symmetry and positive semi-definiteness. Please add some details on how you handle this e.g. do you symmetrize before testing for differences? How do you do this? Do you implement cross-validation using the log likelihood? If so, how do you implement the determinant? I think these details are important for reproducibility and practical utility.  * I am not convinced of the motivation to (significantly) over-sample one of the groups in the two group test e.g. for simulations n1=800, n2=20 - 150. The authors suggest that this reflects of practice, but I am not aware of many situations where this occurs. Instead, most datasets collected for two sample testing tend to be balanced. Could the authors add more detail or a reference?  * On a related note, Could the authors comment on how the results change if the two groups are balanced? Is there still a gain from using the fused Lasso? How much?  * The algorithms are designed for individual sample level data, while the presented experiments are at the group level. As the fMRI data is collected for each individual, how do the authors construct samples used in the presented experiments? Are the data concatenated for all subjects in each group? Please clarify.  Minor issues: * Please note the number of brain regions used for the fMRI experiments. This is an important detail that is easy to add.   Suggestions: * It would be quite informative to compare to "multitask fused lasso" results with no debasing at all, to clarify what is gained.