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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.

Trying out a couple of simple strategies for reducing the number of covariates

Variable selection in multiple regression

If you’ve been following this series, you now know that multiple regression can be very useful but that its usefulness depends on overcoming several challenges. One of those challenges is that if we use all of the covariates available to us and some of them are highly correlated with one another, our assessment of which covariates have an association with the response variable may be misleading and any prediction we make about new observations may be very unreliable. That leads us to the problem of variable selection. Rather than using all of the covariates we have available, maybe we’d be better off if we used only a few.

In this R notebook, I explore a couple of approaches to variable selection:

  1. Restricting the covariates to those we know have an association with the response variable.1
  2. Identifying clusters of covariates that are highly associated with one another, (relatively) unassociated with those in other clusters, and picking one covariate from each cluster for the analysis.2

As you’ll see for the sample data set we’ve been exploring in which there are two clusters of covariates having strong associations within clusters and weak to non-existent associations between clusters, neither of these approaches serves us particularly well. The next installment will explore another commonly used approach – principal components regression.

  1. There’s at least one obvious problem with this approach that I don’t discuss in the notebook. In the work I’ve been involved with, we rarely know ahead of time which covariates, if any, have “real” relationships with the response variable. Most often we’ve measured covariates because we anticipate that they have some relationship to what we’re interested in and we’re trying to figure out which one(s) are most important.
  2. This approach has some practical problems that I don’t discuss in the notebook. How strong do associations have to be to be “highly associated”? How weak do they have to be to be “(relatively) unassociated”? What do we do if there isn’t a clear cutoff between “highly associated” and “(relatively) associated”?

The Marist Mindset List for the Class of 2023

Yes, you read that right. It’s the Marist Mindset List, not the Beloit Mindset List. It’s the same Mindset list as before, but it now has a new home. If you’ve never heard of the Mindset List before, here’s the full press release. The short version is

The Marist Mindset List is created by Ron Nief, Director Emeritus of Public Affairs at Beloit College, along with educators McBride and Westerberg, Shaffer, and Zurhellen. Additional items on the list, as well as commentaries and guides, can be found at www.marist.edu/mindset-list and www.themindsetlist.com.

As always, I enjoy looking over the list, even though it makes me feel really old. Here are a few of the items I found particularly striking this year.

  • Like Pearl Harbor for their grandparents, and the Kennedy assassination for their parents, 9/11 is an historical event.
  • The primary use of a phone has always been to take pictures.
  • The nation’s mantra has always been: “If you see something, say something.”
  • They are as non-judgmental about sexual orientation as their parents were about smoking pot.
  • Apple iPods have always been nostalgic.

You can find the full list at www.marist.edu/mindset-list. Enjoy!

Challenges of multiple regression (or why we might want to select variables)

Variable selection in multiple regression

We saw in the first installment in this series that multiple regression may allow us to distinguish “real” from “spurious” associations among variables. Since it worked so effectively in the example we studied, you might wonder why you would ever want to reduce the number of covariates in a multiple regression.

Why not simply throw in everything you’ve measured and let the multiple regression sort things out for you? There are at least a couple of reasons:

  1. When you have covariates that are highly correlated, the associations that are strongly supported may not be the ones that are “real”. In other words, if you’re using multiple regression in an attempt to identify the “important” covariates, you may identify the wrong ones.
  2. When you have covariates that are highly correlated, any attempt to extrapolate predictions beyond the range of covariates that you’ve measured may be misleading. This is especially true if you fit a linear regression and the true relationship is curvilinear.1

This R notebook explores both of these points using the same set of deterministic relationships we’ve used before to generate the data, but increasing the residual variance.2

  1. The R notebook linked here doesn’t explore the problem of extrapolation when the true relationship is curvilinear, but if you’ve been following along and you have a reasonable amount of facility with R, you shouldn’t find it hard to explore that on your own.
  2. The R-squared in our initial example was greater than 0.99. That’s why multiple regression worked so well. The example you’ll see here has an R-squared of “only” 0.42 (adjusted 0.36). The “only” is in quotes because in many analyses in ecology an evolution, an R-squared that large would seem pretty good.

What is multiple regression doing?

Not long after making my initial post in this series on variable selection in multiple regression, I received the following question on Twitter:

The short answer is that lm() isn’t doing anything special with the covariates. It’s simply minimizing the squared deviation between predictions and observations. The longer version is that it’s able to “recognize” the “real” relationships in the example because it’s doing something analogous to a controlled experiment. It is (statistically) holding other covariates constant and asking what the effect of varying just one of them is. The trick is that it’s doing this for all of the covariates simultaneously.

I illustrate this in a new R notebook by imagining a regression analysis in which we look for an association between, say, x9 and the residuals left after regressing y on x1.

What is multiple regression doing?

Collecting my thoughts about variable selection in multiple regression

I was talking with one of my graduate students a few days ago about variable selection in multiple regression. She was looking for a published “cheat sheet.” I told her I didn’t know of any. “Why don’t you write one?” “The world’s too complicated for that. There will always be judgment involved. There will never be a simple recipe to follow.” That was the end of it, for then.

From the title you can tell that I decided I needed to get my own thoughts in order about variable selection. If you know me, you also know that I find one of the best ways to get my thoughts straight is to write them down. So that’s what I’m starting now.

Expect to see a new entry every week or so. I’ll be posting the details in R notebooks so that you can download the code, run it yourself, and play around with it if you’re so inclined.1 As I develop notebooks, I’ll develop a static page with links to them. Unlike the page on causal inference in ecology, which links to blog posts, these will link directly to HTML versions of R notebooks that will show discuss the aspect of the issue I’m working through that week along with the R code that facilitated my thinking. All of the source code will be available in a Github repository, but you’ll also be able to download the .Rmd file when you have the HTML version open simply by clicking on the “Code” button at the top right of the page and selecting “Download Rmd” from the dropdown.

If you’re still interested after all of that. Here’s a link to the first installment:

Why multiple regression is needed

  1. You’ll get the most out of R notebooks if you work with them through RStudio. Fortunately, the open source version is likely to serve your needs, so all it will cost you is a little bit of disk space.