What is GFLassoInference?
GFLassoInference is an R package for testing for a difference in means between a pair of connected components resulting from the graph fused lasso.
How do I install the package?
To download the GFLassoInference package, use the code below.
require("devtools")
devtools::install_github("yiqunchen/GFLassoInference")
library(GFLassoInference)Why do we need GFLassoInference?
Testing for equality in means between two groups is one of the most fundamental task in statistics with numerous applications. If the groups are defined a priori, i.e., without making use of the observed data, then classical hypothesis tests will control the (selective) Type I error; that is, the probability of a false rejection, given that the null hypothesis is tested.
However, in practice, data analysts often find themselves testing the null hypothesis of equality in means for two groups that are a function of the same data used for testing. Consider the graph fused lasso, a widely popular optimization problem used to reconstruct underlying signals that are piecewise constant on a graph. The solution can be segmented into connected components — that is, elements of the solution that share a common value, and are connected in the original graph. Suppose we want to test the equality of the signal across two connected components. Because now the groups are defined through the data, naive procedures such as a two-sample z-test will lead to an inflated selective Type I error.
To tackle this problem, Hyun et al. (2018) proposed a test that controls the selective Type I error based on a p-value (henceforth referred to as pHyun that quantifies the probability of observing such a large difference in the sample means, conditional on all outputs of the algorithm used to obtain the graph fused lasso solution. However, pHyun conditions more information than is needed to determine the null hypothesis under consideration, which leads to extremely low power in practice.
In this work, we propose an alternative p-value pĈ1, Ĉ2 which conditions only on the pair of connected components being tested. The test based on the resulting p-value has higher power than the test based on pHyun, while still guaranteeing the selective Type I error control.
As an example, consider the graph fused lasso on a grid graph, constructed by connecting each node to its four closest neighbors (up, down, left, right). This leads to the two-dimensional fused lasso problem, also known as total-variation denoising when applied to an image (Rudin et al. 1992, Tibshirani and Taylor 2011). In the leftmost panel, we display the piecewise mean structure of the signal; in the middle panel, we see that both tests based pHyun or pĈ1, Ĉ2 (i.e., rejecting H0 when the p-value is less than α) control the selective Type I error, but the test based z-test pNaive = ℙ(|ν⊤Y|≥|ν⊤y|) leads to inflated selective Type I error. In the rightmost panel, we see that the test based on pĈ1, Ĉ2 has higher power than that based on pHyun. More detailed simulation results can be found in Section 5.2 of our manuscript.
Link to additional resources
- You can learn more about the technical details in our manuscript and in the technical details section.
- You can learn more about how to use our software in the tutorials section.
- Finally, code and steps to reproduce the figures in our manuscript can be found in the GitHub repo https://github.com/yiqunchen/GFLassoInference-experiments.
Citation
If you use GFLassoInference for your analysis, please cite our manuscript:
Chen YT, Jewell SW, Witten DM. (2021+) More powerful selective inference for the graph fused lasso. arXiv preprint. https://arxiv.org/abs/2109.10451.
Bug Reports / Change Requests
If you encounter a bug or would like to make a change request, please file it as an issue here.
References
Chen YT, Jewell SW, Witten DM. (2021+) More powerful selective inference for the graph fused lasso. arXiv preprint. https://arxiv.org/abs/2109.10451.
Fithian W, Sun D, Taylor J. (2014) Optimal Inference After Model Selection. arXiv:1410.2597 [mathST].
Hyun S, G’Sell M, Tibshirani RJ. (2018) Exact post-selection inference for the generalized lasso path. Electron J Stat.
Lee J, Sun D, Sun Y, Taylor J. Exact post-selection inference, with application to the lasso. Ann Stat. 2016;44(3):907-927. doi:10.1214/15-AOS1371
Rudin L, Osher S, Fatemi E. Nonlinear total variation based noise removal algorithms. Physica D. 1992;60(1):259-268. doi:10.1016/0167-2789(92)90242-F
Tibshirani RJ, Taylor J. The solution path of the generalized lasso. Ann Stat. 2011;39(3):1335-1371. doi:10.1214/11-AOS878