Mixed Effects Implement

Mixed Effects Implement - Linear mixed model implementation...

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Linear mixed model implementation in lme4 Douglas Bates Department of Statistics University of Wisconsin – Madison May 5, 2009 Abstract We describe the form of the linear mixed-effects and generalized linear mixed-effects models fit by lmer and give details of the repre- sentation and the computational techniques used to fit such models. These techniques are illustrated on several examples. 1 A simple example The Rail data set from the MEMSS package is described in Pinheiro and Bates (2000) as consisting of three measurements of the travel time of a type of sound wave on each of six sample railroad rails. We can examine the structure of these data with the str function > str(Rail) 'data.frame': 18 obs. of 2 variables: $ Rail : Factor w/ 6 levels "A","B","C","D",. .: 1 1 1 2 2 2 3 3 3 4 . .. $ travel: num 55 53 54 26 37 32 78 91 85 92 . .. Because there are only three observations on each of the rails a dotplot (Figure 1) shows the structure of the data well. > print(dotplot(reorder(Rail, travel) ~ travel, Rail, xlab = "Travel time (ms)", + ylab = "Rail")) In building a model for these data > Rail 1
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Travel time (ms) Rail B E A F C D 40 60 80 100 ●● Figure 1: Travel time of sound waves in a sample of six railroad rails. There were three measurements of the travel time on each rail. The order of the rails is by increasing mean travel time. Rail travel 1 A 55 2 A 53 3 A 54 4 B 26 5 B 37 6 B 32 7 C 78 8 C 91 9 C 85 10 D 92 11 D 100 12 D 96 13 E 49 14 E 51 15 E 50 16 F 80 17 F 85 18 F 83 we wish to characterize a typical travel time, say μ , for the population of such railroad rails and the deviations, say b i , i = 1 , . . . , 6 of the individual rails from this population mean. Because these specific rails are not of interest by themselves as much as the variation in the population we model the b i , which are called the “random effects” for the rails, as having a normal (also called “Gaussian”) distribution of the form N (0 , σ 2 b ). The j th measurement on the i th rail is expressed as y ij = μ + b i + ± ij b i ∼ N (0 , σ 2 b ) , ± ij ∼ N (0 , σ 2 ) i = 1 , . . . , 6 j = 1 , . . . , 3 (1) The parameters of this model are μ , σ 2 b and σ 2 . Technically the b i , i = 1 , . . . , 6 are not parameters but instead are considered to be unobserved ran- dom variables for which we form “predictions” instead of “estimates”. 2
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To express generalizations of models like (1) more conveniently we switch to a matrix/vector representation in which the 18 observations of the travel time form the response vector y , the fixed-effect parameter μ forms a 1- dimensional column vector β and the six random effects b i , i = 1 , . . . , 6 form the random effects vector b . The structure of the data and the values of any covariates (none are used in this model) are used to create model matrices X and Z . Using these vectors and matrices and the 18-dimensional vector ± that represents the per-observation noise terms the model becomes y = + Zb + ± , ± ∼ N , ( 0 , σ 2 I ) , b ∼ N ( 0 , σ 2 Σ ) and b ± (2) In the general form we write p for the dimension of β , the fixed-effects parameter vector, and q
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This note was uploaded on 12/26/2011 for the course ECON 245a taught by Professor Staff during the Fall '08 term at UCSB.

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Mixed Effects Implement - Linear mixed model implementation...

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