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Mixed and Hierarchical Linear Models

Mixed and Hierarchical Linear Models

This course will teach you the basic theory of linear and non-linear mixed effects models, hierarchical linear models, algorithms used for estimation, primarily for models involving normally distributed errors, and examples of data analysis.


This course explains the basic theory of linear and non-linear mixed-effects models, including hierarchical linear models (HLM). A key feature of mixed models is that, by introducing random effects in addition to fixed effects, they allow you to address multiple sources of variation when analyzing correlated data. The course provides a basic understanding and knowledge of mixed-effect models that will enable you to put what you learn into practice. You will use several software programs to fit mixed-effects models to real data sets; outcomes will be presented and discussed.

  • Intermediate, Advanced
  • 4 Weeks
  • Expert Instructor
  • Tuiton-Back Guarantee
  • 100% Online
  • TA Support

Learning Outcomes

Students who complete this course will gain a basic understanding of mixed-effect models. They will use procedures in several software programs to fit mixed-effects models to real data sets. They will learn basic specifications of linear mixed effects models, techniques for estimation and hypothesis testing, and basic concepts of nonlinear mixed effects models. Participants are strongly encouraged to contribute to discussions on the online course discussion board, where exchanges of examples, software code, and ideas about modeling approaches enhance the conceptual and theoretical material with practical solutions.

  • Specify a linear mixed model
  • Identify the associated marginal model
  • Estimate the covariance and fixed effect parameters
  • Conduct hypothesis tests on models
  • Estimate nonlinear mixed effects models

Who Should Take This Course

Researchers analyzing longitudinal or clustered data sets arising from experiments, clinical trials, or surveys, where the data are not amenable to simple statistical analysis and correlated observations need to be accounted for.

Our Instructors

Course Syllabus

Week 1

Overview of Linear Mixed Effects Models (LMM)

  • Specification of LMM
  • The Marginal Model Implied by a LMM

Week 2

Estimation and Hypothesis Tests in LMMs

  • Estimation of Covariance Parameters (ML vs REML estimation)
  • Estimation of Fixed Effect Parameters
  • Wald and likelihood ratio tests

Week 3

LMM Examples

  • Estimation
  • Hypothesis tests
  • Checking model assumptions

Week 4

Nonlinear Mixed Effects Models (NLMM): Basic Concepts

  • Estimation methods for NLMM
  • Applications of NLMM
  • Examples
  • Guided analysis of participant data sets

Class Dates


09/23/2022 to 10/21/2022


09/22/2023 to 10/20/2023


09/20/2024 to 10/18/2024


Private: Statistics 1 – Probability and Study Design

This course, the first of a three-course sequence, provides an introduction to statistics for those with little or no prior exposure to basic probability and statistics.
  • Skill: Intermediate, Advanced
  • Credit Options: CEU

Private: Statistics 2 – Inference and Association

This course, the second of a three-course sequence, will teach you the use of inference and association through a series of practical applications, based on the resampling/simulation approach, and how to test hypotheses, compute confidence intervals regarding proportions or means, computer correlations, and use of simple linear regressions.
  • Skill: Intermediate, Advanced
  • Credit Options: CEU

Matrix Algebra

This course will teach you the basics of vector and matrix algebra and operations necessary to understand multivariate statistical methods, including the notions of the matrix inverse, generalized inverse and eigenvalues and eigenvectors.
  • Skill: Intermediate, Advanced
  • Credit Options: CEU
Karolis Urbonas
Susan Kamp
Stephen McAllister
Amir Aminimanizani
Elena Rose
Leonardo Nagata
Richard Jackson

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Mixed and Hierarchical Linear Models

Additional Information


Homework in this course consists of short answer questions to theoretical problems (some of which involve matrix algebra), guided data analysis problems using existing software, guided data modeling problems using existing software, and an end-of-course modeling project.

In addition to assigned readings, this course also has an end of course data modeling project, example software files, practice exercises, and supplemental readings available online.

Course Text

The course text is Linear Mixed Models: A Practical Guide using Statistical Software – third edition – by B. T. West, K. B. Welch, and A. T. Galecki, with contributions from B. W. Gillespie, which you can order online.


Concepts of mixed models will be illustrated with examples analyzed using PROC/MIXED in SAS and functions in R. Most of the illustrations for the linear case will have parallel examples in Stata and/or SPSS (not all features are available in those packages).  Exercises should be doable with SAS, R, Stata, or SPSS.

Note:  If you are planning to use R in this course and are not already familiar with it, please consider taking one of our courses where R is introduced from the ground up:  “Introduction to R: Data Handling,”  “Introduction to R: Statistical Analysis,” or “Introduction to Modeling.”  R has a learning curve that is steeper than that of most commercial statistical software.

Supplemental Information

Literacy, Accessibility, and Dyslexia

At, we aim to provide a learning environment suitable for everyone. To help you get the most out of your learning experience, we have researched and tested several assistance tools. For students with dyslexia, colorblindness, or reading difficulties, we recommend the following web browser add-ons and extensions:







  • Navidys (for colorblindness, dyslexia, and reading difficulties)
  • HelperBird for Safari (for colorblindness, dyslexia, and reading difficulties)

Register For This Course

Mixed and Hierarchical Linear Models