p-Values for High-Dimensional Regression

By Nicolai Meinshausen, Lukas Meier, Peter Buhlmann. 2008

469 citations recorded by the source corpus.

Assigning significance in high-dimensional regression is challenging. Most computationally efficient selection algorithms cannot guard against inclusion of noise variables. Asymptotically valid p-values are not available. An exception is a recent proposal by Wasserman and Roeder that splits the data into two parts. The number of variables is then reduced to a manageable size using the first split, while classical variable selection techniques can be applied to the remaining variables, using the data from the second split. This yields asymptotic error control under minimal conditions. This involves a one-time random split of the data, however. Results are sensitive to this arbitrary choice, which amounts to a “p-value lottery” and makes it difficult to reproduce results. Here we show that inference across multiple random splits can be aggregated while maintaining asymptotic control over the inclusion of noise variables. We show that the resulting p-values can be used for control of both family-wise error and false discovery rate. In addition, the proposed aggregation is shown to improve power while reducing the number of falsely selected variables substantially.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/c3620866f75cf4d2cd498abc91b598ae1ddd1300

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Statistics and Probability

Citations in the corpus

Confidence intervals and hypothesis testing for high-dimensional regression (2013, methodology): “Table 2 summarizes the false positive rates and the statistical powers achieved by our proposed method, the multisample-splitting method (Meinshausen et al., 2009), and the ridge-type projection estimator (Bühlmann, 2013) for several configurations.”
Two-Stage Procedures for High-Dimensional Data (2011, methodology): “Meinshausen et al. (2009) and Wasserman and Roeder (2009) considered variable selection for the regression model by using hypothesis testing to eliminate some variables such as H0j : βj = 0 vs.”
A Unified Theory of Confidence Regions and Testing for High-Dimensional Estimating Equations (2015, unknown): “P-values and confidence intervals based on sample splitting and subsampling are suggested by Meinshausen et al. (2009); Meinshausen and Bühlmann (2010); Shah and Samworth (2013); Wasserman and Roeder (2009).”
A General Theory of Hypothesis Tests and Confidence Regions for Sparse High Dimensional Models (2014, background): “Recently, in the high dimensional setting, several authors including [25, 26, 39] considered p-values based on the sample splitting technique or subsampling.”
Statistical inference for model parameters in stochastic gradient descent (2016, methodology): “, [22, 38, 6, 4]), the Lasso estimator can be used as a screening method to reduce the set of the variables to Ŝ, a subset which contains the true support S with probability tending to 1.”
Statistical significance in high-dimensional linear models (2012, methodology): “Using their idea and extending it to multiple sample splitting, Meinshausen, Meier and B¨uhlmann (2009) develop a much more stable method for construction of p -values for hypotheses H 0 ,j : β 0 j = 0 ( j = 1 ,... ,p ) and for adjusting them in a non-naive way for multiple testing over p…”
On asymptotically optimal confidence regions and tests for high-dimensional models (2013, methodology): “For multiple sample splitting [32], we do variable screening with the Lasso whose regularization parameter is chosen by 10-fold crossvalidation.”
Identifying a Minimal Class of Models for High-dimensional Data (2015, methodology): “The multiple sample splitting method to get p-values, Meinshausen et al. (2009), yields only one signiﬁcant predictor.”
High-dimensional econometrics and regularized GMM (2018, background): “tions following the de-biasing approach are [124], [109], and [79]. Rather than focus on a pre-speciﬁed low-dimensional parameter, one may also do inference for high-dimensional parameters. [117] and [92] use sample splitting to provide proceduresformultipleinferenceinhigh-dimensionalsettingsthatcan”
Two-Step Lasso Estimation of the Spatial Weights Matrix (2015, background): “[53] which allow for controlling the false discovery rate.”
High-dimensional inference in misspecified linear models (2015, background): “the preferred multiple sample instead of single sample splitting [24], can be used for the random design misspecified case.”
A General Framework for Robust Testing and Confidence Regions in High-Dimensional Quantile Regression (2014, background): “Wasserman and Roeder (2009) and Meinshausen et al. (2009) proposed the sample-splitting approaches to obtaining valid p-values in high-dimensional problems.”
Estimation for High‐Dimensional Linear Mixed‐Effects Models Using ℓ1‐Penalization (2010, methodology): “Having some variable screening property as in (4), we can reduce the false positive selections by various methods, besides the adaptive Lasso mentioned above, including also stability selection (Meinshausen and Bühlmann, 2010) based on subsampling or via assigning p-values (Wasserman and Roeder, 200”