This is the home page of Pop 510: Multilevel Models, as offered in the Spring of 2018 (Session II). The course registrar's page is here. For Pop 509: Survival Analysis, click here. For my own research on multilevel models click here.
This half-course, offered in the second session of the spring term, provides an introduction to statistical methods for the analysis of multilevel data, such as data on children, families, and neighborhoods.
We review fixed- and random-effects models for the analysis of clustered and longitudinal data before moving on to multilevel random-intercept and random-slopes models.
We discuss model fitting and interpretation, including centering and estimation of cross-level interactions.
We cover models for continuous as well as binary and count data, reviewing the different approaches to estimation in common use, including Bayesian inference.
A list of my papers on multilevel models including abstracts and links to JSTOR where available, as well as the Rodríguez-Goldman data, may be found in the multilevel section.
We start with simple variance-component models using data on language scores from Snijders and Boskers. Part 2 has random intercept and random slope models, and Part 3 has a model with a level-2 predictor, where the random intercept and slopes depend on school SES.
We illustrate growth curve models by replicating an analysis by Goldstein of the heights of school boys measured on nine occasions between ages 11 and 13. In addition to random intercepts, slopes and curvatures, we consider serial correlation among residuals.
The next example deals with 3-level linear models using growth curve data on 1721 students in 60 schools. We start with simple variance-component models and then move to growth curves with random intercepts and then random slopes. Our analysis concludes with a comparison of growth curves in schools that differ in observed and unobserved characteristics.
We next move to multilevel models for binary data, starting with a dataset on hospital deliveries of births clustered by mother, which we analyze using several methods, starting with maximum likelihood via numeric quadrature here. The Bayesian Models section has various analyses of this dataset using Markov chain Monte Carlo (MCMC) methods.
We continue with an application to contraceptive use in Bangladesh, where we consider random-intercept and random-slope models. We also illustrate the estimation of random effects using maximum likelihood and posterior Bayes estimates.
For a three-level logit model consider the analysis of immunization in Guatemala. The data are available on the multilevel section of the website and the book by Rabe-Hesketh and Skrondal has a substantial analysis.
We illustrate a random-intercept Poisson model using data on lip cancer in Scotland from Rabe-Hesketch and Skrondal (2012).
An important application of Poisson models is to survival data. I describe the calculation of predicted probabilities after fitting a piecewise exponential model using data on infant and child mortality in Kenya, as explained in my chapter i the handbook of Multilevel Analysis.
The notes on how to run multilevel logit models using winBUGS are here, with a link to a compound document that can be run from WinBUGS. See also part 2, showing how to run WinBUGS in batch mode, and how to import CODA output into Stata for further analysis.
JAGS is "Just Another Gibbs Sampler", it uses essentially the same language as winBUGS and works well with R via `rjags`. I try it on the hospital data here
I also recommend you have a look at the MCMC feature in MLwiN, as demostrated in class. This is probably the easiest way to estimate multilevel models using MCMC procedures.
We now have a sample run of Stan, the latest on MCMC estimation using Hamiltonian Monte Carlo and the No U-Turn Sampler (NUTS), applied to the hospital delivery data right here.
Stata can fit some multilevel models using Metropolis-Hastings combined with Gibbs sampling. We illustrate the procedure using the same hospital delivery data used with WinBUGS and Stan and compare resuts of all methods here.