Linear Mixed Models for Longitudinal and Clustered Data Lecture Overview
Explore the lecture on Linear Mixed Models for Longitudinal and Clustered Data by Dr. Daniel Zhao from the Department of Biostatistics and Epidemiology at the University of Oklahoma. Learn about data model setup, estimation, prediction, and interpretation using the lme4 R package with in-class exercises. Discover the types of data suitable for Linear Mixed Models, including clustered data (such as a cardiovascular study on families) and longitudinal data (like a clinical trial comparing hypertension medications). Dive into an example trial on treating lead-exposed children with Succimer, analyzing blood lead levels over time. View tables and plots illustrating blood lead levels for insightful visualization.
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Presentation Transcript
Linear Mixed Models for Longitudinal and Clustered Data 4/20/2021 Daniel Zhao, PhD Department of Biostatistics and Epidemiology Hudson College of Public Health University of Oklahoma Health Sciences Center
Outline Lecture (45 min) Data Model set up Estimation Prediction Interpretation R demo (30 min) lme4 R package In-class exercise (45 min) 2
Typical data suitable for LMM 1. Clustered Data (can be cross-sectional) 1. A cardiovascular study conducted on a group of families (clusters). 2. Outcomes (blood pressures) of patients from different families are independent, but from the same families are correlated. 2. Longitudinal Data (sequential in time) 1. A clinical trial to compare two medications on treating hypertension 2. Outcomes (blood pressures) from different patients are independent, but from the same patients are correlated. Note: Classical linear regression models ignoring the correlation structure will produce biased parameter estimates. 3
Example: Treatment of Lead-Exposed Children Trial Exposure to lead during infancy is associated with substantial deficits in tests of cognitive ability Chelation treatment of children with high lead levels usually requires injections and hospitalization A new agent, Succimer, can be given orally Randomized trial examining changes in blood lead level during course of treatment 100 children randomized to placebo or Succimer Measures of blood lead level at baseline, 1, 4 and 6 weeks 4
Table 1: Blood lead levels (g/dL) at baseline, week 1, week 4, and week 6 for 8 randomly selected children. 5
Table 2: Mean blood lead levels (and standard deviation) at baseline, week 1, week 4, and week 6. 6
Figure 1: Spaghetti plot of blood lead levels at baseline, week 1, week 4, and week 6 7
Figure 2: Mean profile plot of mean blood lead levels at baseline, week 1, week 4, and week 6 8
Linear Mixed Effects Model (LMM) ? + ??? ??+ ??? ???= ??? ? = 1, ,?; ? = 1, ,?? ??? : the observed continuous response variable for the ?th occasion of the ?th subject (or the ?th individual in the ?th cluster) ??? : the observed (time-varying) covariates for the fixed effects ? : the unknown population fixed effects ?? : the unknown subject-specific random effects, ??~?(0,?) ??? : the observed (time-varying) covariates for the random effects ?? : the within-subject error vector, ??~?(0,??). 9
Mean and Variance In matrix form, the LMM is ??= ??? + ????+ ?? ? ?? = ??? ??? ?? = ????? An LMM is completely specified by the mean and variance. Many choices of ? and ?? are available, including unstructured, compound symmetry, AR(1), etc Models without specifying the random effects ?? are called Mixed Models for Repeated Measures (MMRM) Note that in SAS, ?? can be very flexible In R package lme4, ??= ?2???. And random effects are required. + ?? 10
Estimation: Maximum Likelihood ML estimator of ? is the generalized least squares (GLS) estimator and depends on marginal covariance among the repeated measures. In general, there is no simple expression for ML estimator of the covariance components - ? and ?2 (or ?) - requires iterative techniques. Because ML estimation of covariance is known to be biased in small samples, use restricted ML (REML) estimation instead. 11
Prediction For the general case, the prediction of ?? is given by: ??= ? ????; ?, ?, ?2= ??? where ?= ??? ???? = ????? When the unknown covariance parameters have been replaced by their ML or REML estimates, the resulting predictor is often referred to as the Empirical BLUP or the Empirical Bayes (EB) estimator. Finally, the ?th subject s predicted response profile is, ??= ?? ? + ?? ??= ?? ? + ?? ??? =( R? ? 1(?? ?? ?), + ??= ????? ? + ?2? 1?? ?? ? 1 )?? ? 1)?? ? + (? R? ? 12
Model Selection for LMM Both the mean model and covariance structure need to be selected A practical approach is to fit a saturated mean model and use AIC or BIC to pick the best covariance structure. Using the chosen covariance structure, find the final mean model through backward elimination 13
Handling of time in longitudinal studies If the time points for everyone are the same (say, weeks 1, 3, 4, 5, etc) Can treat time as a continuous variable if response is linear in time or functional form can be modeled Otherwise, treat time as a categorical variable and specify a reference time If the time points for everyone are not the same Time can only be treated as continuous variable 14
R packages for LMM foreign package, gls function: less often used. nlme package, nlme function: can be used for nonlinear mixed effects models lme4 package, lmer function: most commonly used 15
Syntax of lmer function in lme4 package In lmer the model is specified by the formula argument. As in most R model-fitting functions, this is the first argument. The model formula consists of two expressions separated by the ~ symbol. The expression on the left of ~ is the response variable. The RHS consists of one or more terms separated by `+' symbols. A random-effects term consists of two expressions separated by the vertical bar |. The expression on the right of the `|' is evaluated as a factor, which we call the grouping factor for that term. 16
