Mixed Models

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Mixed and Hierarchical Linear Models

taught by James Hardin

Aim of Course:

This course will explain the basic theory of linear and non-linear mixed-effects models, including hierarchical linear models (HLM). It will outline the algorithms used for estimation, primarily for models involving normally distributed errors, and will provide examples of data analysis. The course aims at providing a basic understanding and knowledge of mixed-effect models that will allow you to use them in practice. To achieve this aim, the use of procedures in SAS, SPSS, R, Stata, and HLM to fit mixed-effects models to real data sets will be presented and discussed. The instructors will also demonstrate model fitting using other procedures as requested.

Mixed-effects models are a powerful class of models used for the analysis of correlated data. Examples of correlated data include (but are not limited to) clustered data, repeated observations, longitudinal data, multiple dependent variables, spatial data or data from population pharmacokinetic/pharmacodynamic studies. 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. For example, in a longitudinal study, they allow you to analyze both within- and between-subject variation.

The course will detail a basic theory of mixed models and will provide an overview of estimation methods, with emphasis on the linear case. Concepts of mixed models will be illustrated using the aforementioned software packages. 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.

This course may be taken individually (one-off) or as part of a certificate program.

Course Program:

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


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.

Mixed and Hierarchical Linear 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.



Organization of the Course:

This course takes place online at the Institute for 4 weeks. During each course week, you participate at times of your own choosing - there are no set times when you must be online. Course participants will be given access to a private discussion board. In class discussions led by the instructor, you can post questions, seek clarification, and interact with your fellow students and the instructor.

At the beginning of each week, you receive the relevant material, in addition to answers to exercises from the previous session. During the week, you are expected to go over the course materials, work through exercises, and submit answers. Discussion among participants is encouraged. The instructor will provide answers and comments, and at the end of the week, you will receive individual feedback on your homework answers.

Time Requirement:
About 15 hours per week, at times of  your choosing.

Students come to the Institute for a variety of reasons. As you begin the course, you will be asked to specify your category:

  1. You may be interested only in learning the material presented, and not be concerned with grades or a record of completion.
  2. You may be enrolled in PASS (Programs in Analytics and Statistical Studies) that requires demonstration of proficiency in the subject, in which case your work will be assessed for a grade.
  3. You may require a "Record of Course Completion," along with professional development credit in the form of Continuing Education Units (CEU's).  For those successfully completing the course,  CEU's and a record of course completion will be issued by The Institute, upon request.

Course Text:

The course text is Linear Mixed Models: A Practical Guide using Statistical Software - second edition - by B. T. West, K. B. Welch, and A. T. Galecki, with contributions from B. W. Gillespie, which you can ordered online, or by using this form. CRC Press typically gives students a generous discount when students order the text using the above form (not by ordering the text 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. For information on obtaining software, click here.

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.

Mixed and Hierarchical Linear Models

April 28, 2017 to May 26, 2017April 27, 2018 to May 25, 2018

Course Fee: $589