• Corpus ID: 170079209

Essential regression

@inproceedings{Bing2019EssentialR,
  title={Essential regression},
  author={Xin Bing and Florentina Bunea and Marten H. Wegkamp and Seth Strimas-Mackey},
  year={2019}
}

• Xin Bing, F. Bunea, +1 author Seth Strimas-Mackey
• Published 29 May 2019
• Computer Science

Essential Regression is a new type of latent factor regression model, where unobserved factors Z ∈ R influence linearly both the response Y ∈ R and the covariates X ∈ R with K p. Its novelty consists in the conditions that give Z interpretable meaning and render the regression coefficients β ∈ R relating Y to Z – along with other important parameters of the model – identifiable. It provides tools for high dimensional regression modelling that are especially powerful when the relationship between a response and essential representatives Z of the Xvariables is of interest. Since in classical factor regression models Z is often not identifiable, nor practically interpretable, inference for β is not of direct interest and has received little attention. We bridge this gap in E-Regressions models: we develop a computationally efficient estimator of β, show that it is minimax-rate optimal (in Euclidean norm) and component-wise asymptotically normal, with small asymptotic variance. Inference in Essential Regression is performed after consistently estimating the unknown dimension K, and all the K subsets of the X-variables that explain, respectively, the individual components of Z. It is valid uniformly in β ∈ R , in contrast with existing results on inference in sparse regression after consistent support recovery, which are not valid for regression coefficients of Y on X near zero. Prediction of Y from X under Essential Regression complements, in a low signal-to-noise ratio regime, the battery of methods developed for prediction under different factor regression model specifications. Similarly to other methods, it is particularly powerful when p is large, with further refinements made possible by our model specifications. Moreover, Essential Regression provides a statistical platform for analysis in regression with clustered covariates, with or without overlap. This allows us to address possible inferential questions in post clustering-inference, and subsequently provide guidelines regarding the use and misuse of cluster averages as popular dimension reduction devices in high-dimensional regression. We benchmark the Essential Regression methodology on a recently collected data set for the study of the effectiveness of a new SIV vaccine. Our analysis enables the determination of the most important antibody-centric mechanisms associated with the vaccine response.