1. Using mixed effects models to quantify dependency among repeated measures Dr. Christopher Franck LISA Short Course August 5, 2015

2. Laboratory for Interdisciplinary Statistical Analysis Collaboration: Visit our website to request personalized statistical advice and assistance with: Experimental Design Data Analysis Interpreting Results Grant Proposals Software (R, SAS, JMP, SPSS...) LISA statistical collaborators aim to explain concepts in ways useful for your research. Great advice right now: Meet with LISA before collecting your data. All services are FREE for VT researchers. We assist with research—not class projects or homework. LISA helps VT researchers benefit from the use of Statistics www.lisa.stat.vt.edu LISA also offers: Educational Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: M-F 1-3 PM Old Security Building room 103 for questions requiring <30 mins Also 10-noon GLC Meeting Room A and 10-noon AM 403-J Hutcheson Hall. 2

3. LISA Summer 2015 Short Courses Materials available online! DateCourse TitleInstructor Wed, 06/24/2015Designing ExperimentsThomas Metzger Wed, 07/01/2015Basics of RAna Maria Ortega Villa Wed, 07/08/2015Generalized Linear Models (GLMs) and Categorical Data Analysis (CDA) Lin Zhang Wed, 07/15/2015Graphics in RWill DeShong Wed, 07/22/2015Multivariate Clustering Analysis in R Yuhyun Song Wed, 07/29/2015Power Analyses and Sample Size Calculations for Research Celia Rose Eddy Wed, 08/05/2015Using mixed effects models to quantify dependency among repeated measures Dr. Chris Franck

4. A few notes on the course We will learn/review some introductory and more advanced concepts. We will cover material in some technical detail. I will strive to summarize key technical points graphically and in writing. I’ll put these summaries in blue. We will also see data examples on the computer using the MIXED procedure in SAS, and graphics using R. My questions for you are in red. Let’s interact! Questions welcome at any time.

5. Data example 1: Effect of aluminum on growth of sugar maple [1],[2] Outcome Y is height measurements (mm) of 68 sugar maple seedlings. Treatment is four solutions of aluminum concentrations, 0, 100, 300, and 600 μM. Each tree is measured four times, once each from weeks 4-7 of life. 17 trees * 4 treatments * 4 time points = 272 total observations. Do you think measurements from different trees are independent? How about measurements from the same tree over time?

6. Rule 1: plot your data What observations do you have about these data? Effect of time? Effect of treatment?

7. Linear model review

8. ANOVA-style model

9. The class of linear models can be expressed very generally using matrices

10. The ordinary regression model assumes independence among error terms This is the zoomed in view of the top 4 rows and columns. The ijth entry of a covariance matrix includes the dependence (aka covariance) between the ith row and jth column. Here, i and j are observation numbers between 1 and n. What is iith entry? Obs1Obs2Obs3Obs4 Obs1σ2σ2 000 Obs20σ2σ2 00 Obs300σ2σ2 0 Obs4000σ2σ2

11. A brief review of Maximum likelihood Estimation -

12. What does maximum likelihood ‘look’ like? The likelihood function has the parameters as inputs. The values of the parameters that maximize the likelihood function are MLEs The picture on right is for 1,000 normal observations with mean=3, sd=2. Higher dimensions, not possible to visualize, but identical concept. See MLE visual demo.R

13. Mixed effects models contain both fixed and random effects. A mixed effects model includes both fixed and random effects – fixed effects correspond to factors with levels that we plan to conduct inference on. We might want to compare different treatments which are randomly assign to units. Which effect(s) in the maple example is(are) fixed? – Random effects correspond to factors whose levels are thought of as a sample from a conceptually infinite population. If I have repeated measures on individuals who are each subjected to multiple diet conditions, I might use a random effect to model person- to-person variability. Which effect(s) in the maple example is(are) random?

14. Mixed effects models notation

15. Restricted maximum likelihood overview Restricted maximum likelihood (or REML) is an optimization technique that is frequently used in the analysis of mixed models. Many software packages automatically default to REML when you add random effects into your model (see output, documentation for your favorite software). REML optimizes a likelihood where fixed effects have been removed (integrated out) in order to estimate random effects. Used to reduce bias in estimators of variance components

16. Quantities we estimate

17. Assuming the random effect on subject results in a block diagonal dependency among measurements from the same unit. σ 2 +σ 2 B σ2Bσ2B σ2Bσ2B σ2Bσ2B 0000… σ2Bσ2B σ2Bσ2B σ2Bσ2B 0000… σ2Bσ2B σ2Bσ2B σ2Bσ2B 0000… σ2Bσ2B σ2Bσ2B σ2Bσ2B 0000… 0000 σ2Bσ2B σ2Bσ2B σ2Bσ2B … 0000σ2Bσ2B σ2Bσ2B σ2Bσ2B … 0000σ2Bσ2B σ2Bσ2B σ2Bσ2B … 0000σ2Bσ2B σ2Bσ2B σ2Bσ2B … ⁞⁞⁞⁞⁞⁞⁞⁞ Dependency among 68 observations Dependency among first 8 observations

18. Fitting a linear mixed effects model to the Maple data in SAS

19. What patterns from our model results are reflected in this plot? Plot produced using R. Least square means from model output using SAS code above.

20. Intraclass (ICC) correlation is a measure of similarity among repeated measurements

22. Homework Perform a similar analysis on the CD4 data set.

23. Mixed model matrix notation

24. Final thoughts Mixed models add flexibility by allowing for random effects, which model unknown parameters with a probability distribution (e.g. normal) Can be used for repeated measures analysis. Mixed effects models are a member of the class of hierarchical models. We examined one dependency structure called compound symmetry. Many others exist e.g. autoregression in time series, exponential strucure, Matern, and many others.

25. References [1]Meredith, M. P., & Stehman, S. V. (1991). Repeated measures experiments in forestry: focus on analysis of response curves. Canadian Journal of Forest Research, 21(7), 957-965. [2]Thornton, F. C., Schaedle, M., & Raynal, D. J. (1986). Effect of aluminum on the growth of sugar maple in solution culture. Canadian Journal of Forest Research, 16(5), 892-896. R Core Team (2015). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/. Bates D, Maechler M, Bolker B and Walker S (2015). _lme4: Linear mixed- effects models using Eigen and S4_. R package version 1.1-8,. Bates D, Maechler M, Bolker BM and Walker S (2015). “Fitting Linear Mixed-Effects Models using lme4.” ArXiv e-print; in press, _Journal of Statistical Software_,.