Fitting Linear Mixed-Effects Models using lme4

Fitting Linear Mixed-Effects Models using lme4 Douglas BatesUniversity of Wisconsin-MadisonMartin M chlerETH ZurichBenjamin M. BolkerMcMaster UniversitySteven C. WalkerMcMaster UniversityAbstractMaximum likelihood or restricted maximum likelihood (REML) estimates of the pa-rameters in Linear Mixed-Effects Models can be determined using thelmerfunction in thelme4package forR. As for most model- Fitting functions inR, the model is described inanlmercall by a formula, in this case including both fixed- and random-effects formula and data together determine a numerical representation of the model fromwhich the profiled deviance or the profiled REML criterion can be evaluated as a functionof some of the model parameters. The appropriate criterion is optimized, using one ofthe constrained optimization functions inR, to provide the parameter estimates. We de-scribe the structure of the model, the steps in evaluating the profiled deviance or REML criterion, and the structure of classes or types that represents such a model.

At present, the main alternative to lme4 for mixed modeling in R is the nlme package (Pin-heiro, Bates, DebRoy, Sarkar, and R Core Team 2014). The main features distinguishing lme4 from nlme are (1) more efficient linear algebra tools, giving improved performance on

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