Introduction

Medical research applications often involve hierarchical data structures such as patients within hospitals or physicians within hospitals; for example, assessing differences in mortality rates across hospitals relative to a specific condition or procedure. Data are collected on random samples of patients nested within each hospital. In this application, it might be appropriate to adjust for covariates at both the patient-level (such as patient age, patient gender and the severity of the index diagnosis) and at the hospital-level (such as hospital size and hospital teaching status).

Hierarchical linear models, sometimes called multi-level linear models, nested models, mixed linear models or covariance components models, handle these hierarchical data structures. These models have historically been used in educational research where hierarchies occur naturally; students nested within classrooms, classrooms nested within schools and schools nested within

* Correspondence to: Lisa M. Sullivan, Boston University School of Public Health, Department of Epidemiology and Biostatistics, 715 Albany Street, Boston, MA 02115, U.S.A.

Tutorials in Biostatistics Volume 2: Statistical Modelling of Complex Medical Data Edited by R. B. D'Agostino © 2004 John Wiley & Sons, Ltd. ISBN: 0-470-02370-8

districts. Recent advances in statistical computing capabilities have made these models more available to researchers across a variety of disciplines.

In this tutorial we provide an introduction to the technique relative to two-level hierarchical data structures. We provide references for readers interested in three-level structures. In Section 2 we motivate the application with an example and we illustrate the application using two popular statistical computing packages, HLM/2L1 and SAS Proc Mixed.2 In Section 3 we present notation, specify models in detail and discuss assumptions. In Section 4 we describe estimation techniques and hypothesis testing procedures. In Section 5 we provide data handling and programming statements to develop and test hierarchical linear models using HLM/2L and SAS Proc Mixed. In Section 6 we present results of analyses based on data from our example and based on simulated data. In Section 7 we provide a brief summary.

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