4a_ mixed mod_pt1 - Motivating texts Advanced Topics in...

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Advanced Topics in Forest Biometrics – FOR6934 Mixed Models – Part I Fal 2007, C. Staudhammer Slide 2 Motivating texts Schabenberger, O. and F.J. Pierce (2002) Contemporary Statistical Models for the Plant and Soil Sciences . CRC Press, NY, NY. Littell, R.C., G.A. Milliken, W.W. Stroup, and R.D. Wolfinger (1996) SAS System for Mixed Models . SAS Institute, Cary,NC. Fal 2007, C. Staudhammer Slide 3 Outline 1. What makes a model mixed? 2. Examples of mixed models Fal 2007, C. Staudhammer Slide 4 Recall: Statistical models ± Statistical models are stochastic models that contain unknown constants that we estimate from data ² These unknown constants are parameters (e.g., τ 1 , 2 , ... ) ± Statistical models describe distributional properties of response variables ² Variability can be decomposed into known and unknown sources Fal 2007, C. Staudhammer Slide 5 Recall: Statistical model terminology ± Response: the outcome of interest ( Y ) ± Parameter: any unknown constant (e.g., i , σ 2 ) ± Prediction: fitted value of the model ± Model errors: difference between the observed response and the fitted value The observed number of fruits from the i th treatment, j th tree: ) ˆ ( Y ij e y ˆ ˆ + = ) ˆ ( e Fal 2007, C. Staudhammer Slide 6 Recall: Example model ± Example: the weight of a brazil nut fruit, Y, under silvicultural treatment i is: Y = µ + i + e where: e is a random variable with mean zero and variance 2 ± The expected value of Y under treatment i is: E[ Y i ] = + i
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Fal 2007, C. Staudhammer Slide 7 Fixed effects models ± An effect is fixed if all possible levels about which inferences will be made are represented ² A level of a fixed effect is an unknown constant, which does not vary ± Fixed effect models contain only fixed effects (apart from a single error term) ² Most regression models are fixed effects models ± If treatments are fixed, then for treatments A and B: n y y y y n y n y y E y E B A B A B A B B A A / 2 ) var( ) var( ) var( : Thus / ) var( / ) var( ) ( and ) ( 2 2 2 σ τ µ = + = = = + = + = ( ) () 2 1 2 1 2 1 1 1 1 1 2 2 2 2 ... var var ) ( var var ) var( n n An A A n Aj n Aj A n Aj n A n e e e e e y y = = + + + = = + + = = Fal 2007, C. Staudhammer Slide 8 Random effects models ± Effects are random if the levels represent only a random sample of possible levels ± Random effect models contain only random effects (apart from intercept) ² Sub-sampling, clustering, and random selection of treatments result in random effects in models ± If treatments are random, then: where: 2 is the variance of the treatment effects (which is non- zero! ) ) / ( 2 ) var( / ) var( ) ( 2 2 2 2 n y y n y y E B A A A A + = + = + = In a random effects model, is not normally of any interest since the only parameter being estimated is . If you are interested in treatment means, then you should have a fixed effects model. A y ( ) ( ) 2 1 2 2 1 2 2 1 1 1 1 1 1 1 2 2 ... var var var var ) ( var var ) var( n n An A A n A n Aj n A n Aj A n Aj n A n e e e n e e y y + = + = + + + + = + = + + = = Fal 2007, C. Staudhammer Slide 9 Mixed effects models ± Mixed effect models contain some random and some fixed effects (apart from intercept and error) ²
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4a_ mixed mod_pt1 - Motivating texts Advanced Topics in...

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