Using mixed effects models to quantify dependency among repeated measures

Dr. Christopher FranckLISA Short Course

August 5, 2015

Laboratory for Interdisciplinary Statistical Analysis

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2

LISA Summer 2015 Short CoursesMaterials available online!Date Course Title Instructor

Wed, 06/24/2015 Designing Experiments

Thomas Metzger

Wed, 07/01/2015 Basics of R Ana Maria Ortega Villa

Wed, 07/08/2015 Generalized Linear Models (GLMs) and Categorical Data Analysis (CDA)

Lin Zhang

Wed, 07/15/2015 Graphics in R Will DeShong

Wed, 07/22/2015 Multivariate Clustering Analysis in R

Yuhyun Song

Wed, 07/29/2015 Power Analyses and Sample Size Calculations for Research

Celia Rose Eddy

Wed, 08/05/2015 Using mixed effects models to quantify dependency among repeated measures

Dr. Chris Franck

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.

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?

Rule 1: plot your data• What observations

do you have about these data? Effect of time? Effect of treatment?

Linear model review

• Linear models are an incredibly flexible class.• Example: Multiple regression:• • What are the usual assumptions?

– Correctly specified model– Representative sample– (although this can be relaxed.)– The above errors are independent. (WE will relax

this assumption about the error structure)

ANOVA-style model

• For categorical predictors such as treatment and time point, an ANOVA-style representation might be helpful:

– the number of levels of the treatment.– the number of levels of another treatment (or time point).– represent overall mean, treatment effects, time effects,

interaction effects.– and are the same as above.

• What interpretation does the interaction term have?• Can anyone tell me the relationship between the ANOVA

model and multiple linear regression model?

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

– is an nx1 vector that holds the outcome data.– is a px1 vector of parameters.– is an nxp matrix known as the Design matrix.– ~Multivariate normal– Expressing models in this way greatly eases mathematical

derivations and unifies many common techniques and models (e.g. t tests (paired and unpaired), regression, ANOVA, ANCOVA). Less relevant for this talk.

– For our purposes, examination of the covariance matrix elucidates dependency (or lack thereof) among repeated measures.

The ordinary regression model assumes independence among error terms

σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 σ2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 σ2 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 σ2

Above is for n=68 observations. The diagonal contains the error variance for each obs, each row and column have 0 covariance, indicated independence under normality.

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?

Obs1 Obs2 Obs3 Obs4

Obs1 σ2 0 0 0Obs2 0 σ2 0 0Obs3 0 0 σ2 0Obs4 0 0 0 σ2

A brief review of Maximum likelihood Estimation -

• Colloquially “Likelihood” is used to indicate uncertainty, probability of an event, odds, etc.

• In statistics, the likelihood function is an incredibly important and very specific concept.

• The likelihood function relates the data to the parameters of interest.

• Often appears mathematically complex. E.g. for the regression example:

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

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?

Mixed effects models notation

– represent fixed effects and interactions.– i=1,…, a; – represents a random effect. Generally – .

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

Quantities we estimate

• Fixed effect coefficients (as before)

• Variance components

– represents variability among units upon which repeated measures are made. What is that in this case?

Assuming the random effect on subject results in a block diagonal dependency among measurements from the same unit.

σ2+σ2B σ2

B σ2B σ2

B 0 0 0 0 …

σ2B σ2+σ2

B σ2B σ2

B 0 0 0 0 …

σ2B σ2

B σ2+σ2B σ2

B 0 0 0 0 …

σ2B σ2

B σ2B σ2+σ2

B 0 0 0 0 …

0 0 0 0 σ2+σ2B σ2

B σ2B σ2

B …

0 0 0 0 σ2B σ2+σ2

B σ2B σ2

B …

0 0 0 0 σ2B σ2

B σ2+σ2B σ2

B …

0 0 0 0 σ2B σ2

B σ2B σ2+σ2

B …⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞ ⁞

Variance of any single observation is

Covariance between two observations from the same unit is .

Covariance between two observations from different units is zero.

In this way we have relaxed assumption of independence among all units!

Dependency among 68 observations

Dependency among first 8 observations

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

• proc mixed data=maple plots=all;• class time TRT id;• model y=time|trt;• random ID;• lsmeans time*trt/slice=trt cl;• lsmeans time*trt/slice=time cl;• ods output LSMeans=lsmeanout;• run;quit;

• Corresponds to the model

– represent treatment, time, interaction effects. Fixed or random?

– represents effect of kth tree. Fixed or random? – .

• See output from live demo ‘maple import and model.sas’

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.

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

• For the simple dependency structure we consider here

• An estimate of ICC is– – (i.e., plug in variance component estimates from

computer output)

How does compare to ?

Homework

• Perform a similar analysis on the CD4 data set.

Mixed model matrix notation

– is an nx1 vector that holds the outcome data.– is a px1 vector of parameters.– is an nxp matrix known as the Design matrix for

fixed effects.– in an nxq Design matrix for random effects– is a qx1 vector effect parameters.– ~Multivariate normal

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.

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, <URL: http://CRAN.R-project.org/package=lme4>.

• 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_, <URL: http://arxiv.org/abs/1406.5823>.