We now introduce multilevel linear and generalized linear models, including issues such as varying intercepts and slopes and non-nested models. We view multilevel models either as regressions with potentially large numbers of coefficients that are themselves modeled, or as regressions with coefficients that can vary by group.

This chapter describes a variety of ways in which probabilistic simulation can be used to better understand statistical procedures in general, and the fit of models to data in particular. In Sections 8.1–8.2, we discuss fake-data simulation, that is, controlled experiments in which the parameters of a statistical model are set to fixed “true” values, and then simulations are used to study the properties of statistical methods. Sections 8.3–8.4 consider the related but different method of predictive simulation, where a model is fit to data, then replicated datasets are simulated from this estimated model, and then the replicated data are compared to the actual data.

The difference between these two general approaches is that, in fake-data simulation, estimated parameters are compared to true parameters, to check that a statistical method performs as advertised. In predictive simulation, replicated datasets are compared to an actual dataset, to check the fit of a particular model.

Fake-data simulation

Simulation of fake data can be used to validate statistical algorithms and to check the properties of estimation procedures. We illustrate with a simple regression model, where we simulate fake data from the model, y = α + βx + ∊, refit the model to the simulated data, and check the coverage of the 68% and 95% intervals for the coefficent β.

There are generally many options available when modeling a data structure, and once we have successfully fit a model, it is important to check its fit to data. It is also often necessary to compare the fits of different models.

Our basic approach for checking model fit is—as we have described in Sections 8.3–8.4 for simple regression models—to simulate replicated datasets from the fitted model and compare these to the observed data. We discuss the general approach in Section 24.1 and illustrate in Section 24.2 with an extended example of a set of models fit to an experiment in animal learning. The methods we demonstrate are not specific to multilevel models but become particularly important as models become more complicated.

Although the methods described here are quite simple, we believe that they are not used as often as they could be, possibly because standard statistical techniques were developed before the use of computer simulation. In addition, fitting multilevel models is a challenge, and users are often so relieved to have successfully fit a model with convergence that there is a temptation to stop and rest rather than check the model fit. Section 24.3 discusses some tools for comparing different models fit to the same data.

Posterior predictive checking is a useful direct way of assessing the fit of the model to various aspects of the data. Our goal here is not to compare or choose among models but rather to explore the ways in which any of the models being considered might be lacking.

Simple methods from introductory statistics have three important roles in regression and multilevel modeling. First, simple probability distributions are the building blocks for elaborate models. Second, multilevel models are generalizations of classical complete-pooling and no-pooling estimates, and so it is important to understand where these classical estimates come from. Third, it is often useful in practice to construct quick confidence intervals and hypothesis tests for small parts of a problem—before fitting an elaborate model, or in understanding the output from such a model.

This chapter provides a quick review of some of these methods.

Probability distributions

A probability distribution corresponds to an urn with a potentially infinite number of balls inside. When a ball is drawn at random, the “random variable” is what is written on this ball.

Areas of application of probability distributions include:

• Distributions of data (for example, heights of men, heights of women, heights of adults), for which we use the notation yi, i = 1, …, n.

• Distributions of parameter values, for which we use the notation θj, j = 1, …, J, or other Greek letters such as α, β, γ. We shall see many of these with the multilevel models in Part 2 of the book. For now, consider a regression model (for example, predicting students' grades from pre-test scores) fit separately in each of several schools. The coefficients of the separate regressions can be modeled as following a distribution, which can be estimated from data.

• […]

We start with an overview of classical linear regression and generalized linear models, focusing on practical issues of fitting, understanding, and graphical display. We also use this as an opportunity to introduce the statistical package R.

Chapter 9 discussed situations in which it is dangerous to use a standard linear regression of outcome on predictors and an indicator variable for estimating causal effects: when there is imbalance or lack of complete overlap or when ignorability is in doubt. This chapter discusses these issues in more detail and provides potential solutions for each.

Imbalance and lack of complete overlap

In a study comparing two treatments (which we typically label “treatment” and “control”), causal inferences are cleanest if the units receiving the treatment are comparable to those receiving the control. Until Section 10.5, we shall restrict ourselves to ignorable models, which means that we only need to consider observed pre-treatment predictors when considering comparability.

For ignorable models, we consider two sorts of departures from comparability—imbalance and lack of complete overlap. Imbalance occurs if the distributions of relevant pre-treatment variables differ for the treatment and control groups. Lack of complete overlap occurs if there are regions in the space of relevant pre-treatment variables where there are treated units but no controls, or controls but no treated units.

Imbalance and lack of complete overlap are issues for causal inference largely because they force us to rely more heavily on model specification and less on direct support from the data.

In this chapter we introduce the fitting of multilevel models in Bugs as run from R. Following a brief introduction to Bayesian inference in Section 16.2, we fit a varying-intercept multilevel regression, walking through each step of the model. The computations in this chapter parallel Chapter 12 on basic multilevel models. Chapter 17 presents computations for the more advanced linear and generalized linear models of Chapters 12–15.

Why you should learn Bugs

As illustrated in the preceding chapters, we can quickly and easily fit many multilevel linear and generalized linear models using the lmer() function in R. Functions such as lmer(), which use point estimates of variance parameters, are useful but can run into problems. When the number of groups is small or the multilevel model is complicated (with many varying intercepts, slopes, and non-nested components), there just might not be enough information to estimate variance parameters precisely. At that point, we can get more reasonable inferences using a Bayesian approach that averages over the uncertainty in all the parameters of the model.

We recommend the following strategy for multilevel modeling:

1. Start by fitting classical regressions using the lm() and glm() functions in R. Display and understand these fits as discussed in Part 1 of this book.

2. Set up multilevel models—that is, allow intercepts and slopes to vary, using non-nested groupings if appropriate—and fit using lmer(), displaying as discussed in most of the examples of Part 2A.

3. […]

Most of this book concerns the interpretation of regression models, with the understanding that they can be fit to data fairly automatically using R and Bugs. However, it can be useful to understand some of the theory behind the model fitting, partly to connect to the usual presentation of these models in statistics and econometrics.

This chapter outlines some of the basic ideas of likelihood and Bayesian inference and computation, focusing on their application to multilevel regression. One point of this material is to connect multilevel modeling to classical regression; another is to give enough insight into the computation to allow you to understand some of the practical computational tips presented in the next chapter.

Least squares and maximum likelihood estimation

We first present the algebra for classical regression inference, which is then generalized when moving to multilevel modeling. We present the formulas here without derivation; see the references listed at the end of the chapter for more.

Least squares

The classical linear regression model is yi = Xiβ + ∊i, where y and ∊ are (column) vectors of length n, X is a n × k matrix, and β is a vector of length k. The vector β of coefficients is estimated so as to minimize the errors ∊i.

Multilevel modeling is applied to logistic regression and other generalized linear models in the same way as with linear regression: the coefficients are grouped into batches and a probability distribution is assigned to each batch. Or, equivalently (as discussed in Section 12.5), error terms are added to the model corresponding to different sources of variation in the data. We shall discuss logistic regression in this chapter and other generalized linear models in the next.

State-level opinions from national polls

Dozens of national opinion polls are conducted by media organizations before every election, and it is desirable to estimate opinions at the levels of individual states as well as for the entire country. These polls are generally based on national randomdigit dialing with corrections for nonresponse based on demographic factors such as sex, ethnicity, age, and education.

Here we describe a model developed for estimating state-level opinions from national polls, while simultaneously correcting for nonresponse, for any survey response of interest. The procedure has two steps: first fitting the model and then applying the model to estimate opinions by state:

1. We fit a regression model for the individual response y given demographics and state. This model thus estimates an average response θl for each cross-classification l of demographics and state. In our example, we have sex (male or female), ethnicity (African American or other), age (4 categories), education (4 categories), and 51 states (including the District of Columbia); thus l = 1, …, L = 3264 categories.

2. […]

Choices in the design of data collection

Multilevel modeling is typically motivated by features in existing data or the object of study—for example, voters classified by demography and geography, students in schools, multiple measurements on individuals, and so on. Consider all the examples in Part 2 of this book. In some settings, however, multilevel data structures arise by choice from the data collection process. We briefly discuss some of these options here.

Unit sampling or cluster sampling

In a sample survey, data are collected on a set of units in order to learn about a larger population. In unit sampling, the units are selected directly from the population. In cluster sampling, the population is divided into clusters: first a sample of clusters is selected, then data are collected from each of the sampled clusters.

In one-stage cluster sampling, complete information is collected within each sampled cluster. For example, a set of classrooms is selected at random from a larger population, and then all the students within each sampled classroom are interviewed. In two-stage cluster sampling, a sample is performed within each sampled cluster. For example, a set of classrooms is selected, and then a random sample of ten students within each classroom is selected and interviewed. More complicated sampling designs are possible along these lines, including adaptive designs, stratified cluster sampling, sampling with probability proportional to size, and various combinations and elaborations of these.

Fit many models

Think of a series of models, starting with the too-simple and continuing through to the hopelessly messy. Generally it's a good idea to start simple. Or start complex if you'd like, but prepare to quickly drop things out and move to the simpler model to help understand what's going on. Working with simple models is not a research goal—in the problems we work on, we usually find complicated models more believable—but rather a technique to help understand the fitting process.

A corollary of this principle is the need to be able to fit models relatively quickly. Realistically, you don't know what model you want to be fitting, so it's rarely a good idea to run the computer overnight fitting a single model. At least, wait until you've developed some understanding by fitting many models.

Do a little work to make your computations faster and more reliable

This sounds like computational advice but is really about statistics: if you can fit models faster, you can fit more models and better understand both data and model. But getting the model to run faster often has some startup cost, either in data preparation or in model complexity.

Data subsetting

Related to the “multiple model” approach are simple approximations that speed the computations. Computers are getting faster and faster—but models are getting more and more complicated! And so these general tricks might remain important.

We now discuss how to go beyond simply looking at regression coefficients, first by using simulation to summarize and propagate inferential uncertainty, and then by considering how regression can be used for causal inference.

Logistic regression is the standard way to model binary outcomes (that is, data yi that take on the values 0 or 1). Section 5.1 introduces logistic regression in a simple example with one predictor, then for most of the rest of the chapter we work through an extended example with multiple predictors and interactions.

Logistic regression with a single predictor

Example: modeling political preference given income

Conservative parties generally receive more support among voters with higher incomes. We illustrate classical logistic regression with a simple analysis of this pattern from the National Election Study in 1992. For each respondent i in this poll, we label yi = 1 if he or she preferred George Bush (the Republican candidate for president) or 0 if he or she preferred Bill Clinton (the Democratic candidate), for now excluding respondents who preferred Ross Perot or other candidates, or had no opinion. We predict preferences given the respondent's income level, which is characterized on a five-point scale.

The data are shown as (jittered) dots in Figure 5.1, along with the fitted logistic regression line, a curve that is constrained to lie between 0 and 1. We interpret the line as the probability that y = 1 given x—in mathematical notation, Pr(y = 1|x).