Uncommon Ground

Monthly Archive: September 2019

A Bayesian approach to variable selection using horseshoe priors

Variable selection in multiple regression

The Lasso has been very widely used, particularly in high-dimensional problems where the number of observations is less than the number of covariates.1 In fact, when I checked Google Scholar on Saturday, it had been cited nearly 30,000 times.2 Bayesians didn’t want to be left out, so Trevor Park and George Casella developed the Bayesian Lasso.3 The Bayesian Lasso overcomes what to my mind is one of the great disadvantages of the original Lasso, the difficulty of providing an assessment of how reliable the regression coefficients are. Like any other Bayesian method that uses MCMC methods, it’s just as easy to get credible intervals on parameters as it is to get posterior means. The Bayesian Lasso also estimates \(\lambda\) as part of the procedure rather than relying on cross-validation. The R package monovm provides an implementation of the Bayesian Lasso in addition to other shrinkage regression methods.

I haven’t explored monovm, but if you’re interested in the Bayesian Lasso, you might want to check it out. Instead of exploring the Bayesian Lasso, the R notebook I’ve put together here explores the use of “horseshoe priors” in rstanarm. The basic idea is the same. We’d like to “shrink” some parameter estimates towards zero, and we’d like to have the data tell us which estimates to shrink. The nice thing about “horseshoe priors” in rstanarm is that if you know how to set up a regression in stan_glm() or stan_glmer() you can use a horseshoe prior very easily in your analysis simply by changing the prior parameter in your call to one of those functions.

  1. This is often referred to as an \(n \ll p\) problem. I’m not going to address that problem here, but if you deal with genomic data, you’ll want to familiarize yourself with the problem and the approaches typically used for addressing it.
  2. 29,558 times to be exact.
  3. Park, T. and G. Casella. 2008. The Bayesian Lasso. Jornal of the American Statistical Association. 103:681-686. doi: 10.1198/016214508000000337

Using the Lasso for variable selection

Variable selection in multiple regression

If you’ve been following along, you’ve now seen some fairly simple approaches for reducing the number of covariates in a linear regression. It shouldn’t come as a shock that statisticians have been worried about the problem for a long time or that they’ve come up with some pretty sophisticated approaches to the problem.1 The first one we’ll explore is the Lasso (least asolute shrinkage and selection operator), which Rob Tibshirani introduced the Lasso to statistics and machine learning more than 20 years ago.2 You’ll find more details in the R notebook illustrating using the Lasso to select covariates, but here are the basic ideas.

The “shrinkage” part of the name refers to the idea that we don’t expect all of the covariates we’re including in the model to be important. And if a covariate isn’t important, we want the magnitude of the regression coefficient associated with that component to be zero (or nearly zero). In other words, we want the estimate to be “shrunk” towards zero rather than taking the value it would if we included it in the full multiple regression.

The “selection” part of the name refers to the idea that we don’t know ahead of time which of the covariates are important (and shouldn’t be shrunk towards 0) and which are important (and should be shrunk towards 0). We want the data to tell us which covariates are important and which aren’t, i.e., we want the data to “select” important covariates.

The Lasso accomplishes this by adding a penalty to the typical least squares estimates. Instead of simply minimizing the sum of squared deviations from the regression line, we do so subject to a constraint that the total magnitude of all regression coefficients is less than some value. We’ll use glmnet() to fit the Lasso. If you explore the accompanying documentation, you’ll see that the Lasso is just one method along a continuum of constrained optimization approaches. I’ll let you explore those on your own if you’re interested.

  1. I’m not going to discuss forward, backward, or all subsets approaches to selecting variables. They don’t seem to be used much anymore (for good reason). If you’re interested in them, take a look at the Wikipedia page on stepwise regression.
  2. Wikipedia points out that it was originally introduced 10 years earlier in geophysics, but Tibshirani discovered it independently, and it was his discovery that led to its wide use in statistics and machine learning.

A simple introduction to Bayesian linear regression

I referred in passing to rstanarm and Bayesian linear regression in the R notebook on reducing the number of covariates. We’ll encounter Bayesian approaches again soon, and I just happened to run across a nice, simple introduction to Bayesian linear regression. It uses Python, with which I am only glancingly familiar, but you don’t need to run Python to read the discussion and understand what’s going on. If you’re unfamiliar with Bayesian inference or if you’d just like to check your understanding, take a look at this Introduction to Bayesian Linear Regression.

An update on principal components regression

Variable selection in multiple regression

In the R notebook on principal component regression I noted that interpreting principal components can be a challenge. When I wrote that, I hadn’t seen a nice paper by Chong et al.1 The method they describe is presented specifically in the context of interpreting selection gradients after a principal components regression, but the idea is general. Once you’ve done the regression on principal components, transform the regression coefficients back to the original scale. Doing this does require, however, fitting all of the principal components, not just the first few.

Here’s the citation and abstract:

Chong, V. K., H. F. Fung, and J. R. Stinchcombe. 2018. A note on measuring natural selection on principal component scores. Evolution Letters. 2-4: 272–280 dos: 10.1002/evl3.63

Measuring natural selection through the use of multiple regression has transformed our understanding of selection, although the methods used remain sensitive to the effects of multicollinearity due to highly correlated traits. While measuring selection on principal component (PC) scores is an apparent solution to this challenge, this approach has been heavily criticized due to difficulties in interpretation and relating PC axes back to the original traits. We describe and illustrate how to transform selection gradients for PC scores back into selection gradients for the original traits, addressing issues of multicollinearity and biological interpretation. In addition to reducing multicollinearity, we suggest that this method may have promise for measuring selection on high‐dimensional data such as volatiles or gene expression traits. We demonstrate this approach with empirical data and examples from the literature, highlighting how selection estimates for PC scores can be interpreted while reducing the consequences of multicollinearity.

  1. John Stinchcombe pointed me to the paper. Thanks John.

Principal components regression

Variable selection in multiple regression

In the last installment of this series we explored a couple of simple strategies to reduce the number of covariates in a multiple regression.1, namely retaining only covariates that have a “real” relationship with the response variable2 and selecting one covariate from each cluster of (relatively) uncorrelated covariates.3 Unfortunately, we found that neither approach worked very well in our toy example.4.

One of the reasons that the second approach (picking “weakly” correlated covariates) may not have worked very well is that in our toy example we know that both x1 and x3 contribute positively to y, but our analysis included only x1. Another approach that is sometimes used when there’s a lot of association among covariates is to first perform a principal components analysis and then to regress the response variable on the scores from the first few principal components. The newest R notebook in this series explores principal component regression.

Spoiler alert: It doesn’t help the point estimates much either, but the uncertainty around those point estimates is so large that we can’t legitimately say they’re different from one another.

  1. If you’ve forgotten why we might want to reduce the number of covariates, look back at this post.
  2. The paradox lurking here is that if we knew which covariates these were, we probably wouldn’t have measured the others (or at least we wouldn’t have included them in the regression analysis).
  3. There isn’t a good criterion to determine how weak the correlation needs to be to regard clusters as “relatively” uncorrelated.
  4. If you’re reading footnotes, you’ll realize that the situation isn’t quite as dire as it appears from looking only at point estimates. Using rstanarm() for a Bayesian analysis shows that the credible intervals are very broad and overlapping. We don’t have good evidence that the point estimates are different from one another.