17DesignLME - Experimental Designs and LME's LME models...

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Experimental Designs and LME’s I LME models provide one way to model correlations among observations I Very useful for experimental designs where there is more than one size of experimental unit I Or designs where the observation unit is not the same as the experimental unit. I My philosophy (widespread at ISU and elsewhere) is that the way an experiment is conducted specifies the random effect structure for the analysis. I Observational studies are very different No randomization scheme, so no clearcut random effect structure Often use model selection methods to help chose the random effect structure I NOT SO for a randomized experiment. c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 17 1 / 25
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I One example: I study designed as an RCBD. I treatments are randomly assigned within a block I analyze data, find out block effects are small I should you delete block from the model and reanalyze? I above philosophy says “tough”. Block effects stay in the analysis I for the next study, seriously consider: blocking in a different way or using a CRD I Following pages work through details of LME’s for some common study designs c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 17 2 / 25
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Mouse muscle study I Mice grouped into blocks by litter. 4 mice per block, 4 blocks. I Four treatments randomly assigned within each block I Collect two replicate muscle samples from each mouse I Measure response on each muscle sample I 16 eu., 32 ou. I Questions concern differences in mean response among the four treatments I The model usually used for a study like this is: I y ijk = μ + τ i + l j + m ij + e ijk , where y ijk is the k th measurement of the response for the mouse from litter j that received treatment i , ( i = 1 , 2 , 3 , 4 ; j = 1 , 2 , 3 , 4 ; k = 1 , 2 ) . c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 17 3 / 25
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I The fixed effects: β = μ τ 1 τ 2 τ 3 τ 4 IR 5 is an unknown vector of fixed parameters I u = [ l 1 , l 2 , l 3 , l 4 , m 11 , m 21 , m 31 , m 41 , m 12 , ... , m 34 , m 44 ] 0 is a vector of random effects describing litters and mice. I ± = [ e 111 , e 112 , e 211 , e 212 , ... , e 441 , e 442 , ... , e 441 , e 442 ] 0 is a vector of random errors. I with y = [ y 111 , y 112 , y 211 , y 212 , ... , y 411 , y 412 , ... , y 441 , y 442 ] 0 , the model can be written as a linear mixed effects model y = X β + Z u + ± , where c ± 2011 Dept. Statistics (Iowa State University) Stat 511 section 17 4 / 25
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X = 1 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 0 0 0 1 1 0 0 0 1 Z = 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 1 0 . . . 0 1 0 0 0 0 0 1 0
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17DesignLME - Experimental Designs and LME's LME models...

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