Multilevel models

In this post, I want to give a brief introduction to multilevel models, which help when you’re trying to figure out questions like what the average income in each ZIP code is; that is, very specific questions that might be difficult to answer well with even large amounts of data, because they pertain to very narrow slices of that data. A multilevel model can make better use of the data you already have by adding the structure of your data, e.g. ZIP codes in counties, into your model.

The way a multilevel model encodes structure is by taking the parameters that you’re estimating, and adding another model which ‘explains’ those parameters. If you try to estimate average income per ZIP code, you need to estimate however many parameters as there are ZIP codes; in a multilevel model, there will be another model on top of that which says these ZIP codes are in the same county and are the product of the same underlying pattern. By encoding this structure into the model, it allows the model to share information among related groups, and groups that don’t have much data can borrow strength from those that do.

Forecasting elections

A neat example of the use of multilevel models appears in the paper Forecasting elections with non-representative polls, whose authors forecast the 2012 US presidential election with Xbox online poll data. Generally when pollsters forecast elections, they try to put together a representative sample of voters and ask them who they will vote for. The authors on the other hand took an extremely non-representative sample of people who answered polls on the Xbox online service, then used a multilevel model to estimate how different demographic groups would vote.

The Xbox poll data had a huge number of respondents, however they were very different in composition from voters: 93% were men, and 65% were between 18 and 29. This means the simple approach, taking an average and making that your forecast, would be a bad idea. Taking averages by demographic group would be better, but in this data, many demographic groups don’t appear very often, and your resulting group-level averages might not be that good.

The authors resolve this using a multilevel model which estimates shares of the vote for different demographic groups, but where these estimates rely on information shared among related groups — e.g. the estimate for 65+ women relies on data from women and from people 65 or older. And they’re quite successful; compared against exit polls, their estimate for 65+ women was off by only 1% point. And after translating these estimates into election forecasts, they meet or exceed the forecast performance of poll aggregator

A made-up example

In the following made-up example, I want to give you an idea of what is going on when you use a multilevel model. Suppose we’re interested in some topic in Washington DC, Maryland, and Virginia, we want to know what the typical or mean values are, and we’ve collected some data, shown below. We have a lot of data from Maryland and Virginia but sadly only one data point from DC. Also shown are the ‘true’ means, which I know in this case because I generated the data.

Given only the data, we want to figure out the state-level means. To begin with, let’s just directly calculate the sample means state-by-state, even though this might not ultimately be a very good idea. To make it look like science, this is basically the same as estimating a linear model of our data, which we’ll call \( y \), as a function of state, \( x \), to get estimates of our means, parameters \( \beta \):

$$ y = \beta_{state} x_{state} + \epsilon $$

When we actually estimate this, we use 0/1 dummy variables for each state, and correspondingly have three \( \beta_{state} \):

$$ y = \beta_{dc} x_{dc} + \beta_{md} x_{md} + \beta_{va} x_{va} + \epsilon $$

We fit this model in R with lm(y ~ state - 1, data = df)1:

Ok, so that worked for Maryland and Virginia — their estimates are close to the true values. But it did not work so well for DC, unsurprisingly. With only one data point, the estimate for DC couldn’t have been anything else.

We’re pretty sure DC is similar to Maryland and Virginia in many respects. It seems like we should be averaging our DC data with the data from neighboring states somehow, at least as long as we don’t have that much data for DC.

Let’s incorporate that intuition into our model. We can do this by treating our state-level means as data, in a sense, and giving them their own model, in which they share a common region-level mean. But unlike data, which is fixed, we can adjust our state-level means to be close to our region-level mean.

This is our multilevel model: one model for data \( y \), which depends on state-level means \( \beta_{state} \), and one model for \( \beta_{state} \), which depends on region-level mean \( \alpha \).

$$ \begin{align} y =& \ \beta_{state} x_{state} + \epsilon \\ \beta_{state} =& \ \alpha + \eta \end{align} $$

When we fit this multilevel model, we try to make our estimates of \( \beta_{state} \) close to the \( y \) for their state, and at the same time we try to make all the \( \beta_{state} \) close to region-level mean \( \alpha \).

We fit this new model in R using lmer(y ~ (1 | state), data = df)2:

Much better. The multilevel model has pulled the DC estimate towards the region-level mean, and it is now actually quite close to the true value. What’s happening is all of the data contributes to the model’s estimate of the region-level mean, which in turn influences the DC estimate.

What next

Data Analysis Using Regression and Multilevel/Hierarchical Models by Gelman and Hill is the best introduction to multilevel models that I have seen. Chapters 11 and 12 in particular have a very good explanation of what they are, how they work, why and when you’d use them, and have some nice motivating examples.

lme4 is the standard package for estimating multilevel models in R. The Gelman/Hill book and the election forecasting paper above both use it. Most other statistical software packages also support multilevel models, generally under the name mixed-effects models.

  1. We remove the intercept by adding - 1 to the model formula so that our parameter estimates are for each state in themselves, and not compared to some reference state.  

  2. lmer is the linear mixed-effects model function from package lme4. The model formula element (1 | state) tells lmer to add a model for state-level intercepts, which we’re calling \( \beta_{state} \).