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3_GLMs - Motivating texts Advanced Topics in Forest...

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Advanced Topics in Forest Biometrics – FOR6934 Review of General Linear Models 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. What is a model? A scientific model is an abstraction of reality Models can be further classified: Mathematical models Stochastic models Statistical models Mathematical models Mathematical models are mechanistic (deterministic) devices A given set of inputs gives an output that is predicted with certainty Example: the weight of a brazil nut fruit, Y, under silvicultural treatment i is: Y = µ + τ i where: τ i is the i th treatment effect Deterministic models have no random components Stochastic models Mathematical models ignore uncertainty, whereas stochastic models include random effects Example: the weight of a brazil nut fruit, Y, under silvicultural treatment i is: Y = µ + τ i + e where: e is a random variable with mean zero and variance σ 2 The expected value of Y under treatment i is: E[ Y i ] = µ + τ i Why do we need stochastic elements? (Adapted from Schabenberger and Pierce 2002) The model is not correct for a particular observation, but correct on average Assumptions are necessary to abstract phenomenon It is often impossible to measure/observe all variables Stochastic models are often more parsimonious When you have real data, you have sampling error What is sampling error?
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Statistical models Statistical models are stochastic models that contain
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