## The Annals of Applied Statistics

### Bayesian shrinkage methods for partially observed data with many predictors

#### Abstract

Motivated by the increasing use of and rapid changes in array technologies, we consider the prediction problem of fitting a linear regression relating a continuous outcome $Y$ to a large number of covariates $\mathbf{X}$, for example, measurements from current, state-of-the-art technology. For most of the samples, only the outcome $Y$ and surrogate covariates, $\mathbf{W}$, are available. These surrogates may be data from prior studies using older technologies. Owing to the dimension of the problem and the large fraction of missing information, a critical issue is appropriate shrinkage of model parameters for an optimal bias-variance trade-off. We discuss a variety of fully Bayesian and Empirical Bayes algorithms which account for uncertainty in the missing data and adaptively shrink parameter estimates for superior prediction. These methods are evaluated via a comprehensive simulation study. In addition, we apply our methods to a lung cancer data set, predicting survival time ($Y$) using qRT-PCR ($\mathbf{X}$) and microarray ($\mathbf{W}$) measurements.

#### Article information

Source
Ann. Appl. Stat., Volume 7, Number 4 (2013), 2272-2292.

Dates
First available in Project Euclid: 23 December 2013

https://projecteuclid.org/euclid.aoas/1387823319

Digital Object Identifier
doi:10.1214/13-AOAS668

Mathematical Reviews number (MathSciNet)
MR3161722

Zentralblatt MATH identifier
1283.62049

#### Citation

Boonstra, Philip S.; Mukherjee, Bhramar; Taylor, Jeremy M. G. Bayesian shrinkage methods for partially observed data with many predictors. Ann. Appl. Stat. 7 (2013), no. 4, 2272--2292. doi:10.1214/13-AOAS668. https://projecteuclid.org/euclid.aoas/1387823319

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#### Supplemental materials

• Supplementary material: Supplemental article. Here we give the full derivation of the Gibbs steps, computational details and the results from the simulation study. The data from Section 5 and the code for its analysis are available at http://www-personal.umich.edu/~philb.