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Douglas-fir mortality estimation with generalized linear mixed models Jeremy Groom, David Hann, Temesgen Hailemariam 2012 Western Mensurationists’ Meeting Newport, OR

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How it all came to be… Proc GLIMMIX Stand Management Cooperative Douglas-fir Improve ORGANON mortality equation? What happened: – Got GLIMMIX to work – Suspected bias would be an issue – It was! – Not time to change ORGANON

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Mortality Good to know about! – Stand growth & yield models – Regular & irregular (& harvest) Regular: competition, predictable Irregular: disease, fire, wind, snow. Less predictable Death = inevitable, but hard to study – Happens exactly once per tree – Infrequently happens to large trees

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DATA Levels: Installations – plots – trees - revisits Yr 1 Yr 5 Yr 10…

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Measuring & modeling Single-tree regular mortality models – FVS, ORGANON Logistic models – Revisits = equally spaced Problems – Lack of independence! Datum = revisit? Nested design (levels)

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Our goals Account for overdispersion – Level: tree Revisit data: mixed generalized linear vs. non- linear – Random effect level = installation Predictive abilities for novel data

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Setting SW BC, Western Washington & Oregon Revisits: yrs between revisits Plots = – ha (x = 0.069) Excluded installations with < 2 plots InstallationsPlotsDF TreesRevisits ,099157,473

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Coping with data Hann et al. 2003, 2006 Nonlinear model: PM = 1.0 – [1.0 + e -(Xβ) ] -PLEN +ε PM PM = 5 yr mortality rate PLEN = growth period in 5-yr increments ε PM = random error on PM Weighted observations by plot area Predictors = linear Generalized Linear Model OK

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Parameterization PM = 1.0 – [1.0 + e -(Xβ) ] -PLEN +ε PM Originally: Xβ = β 0 + β 1 DBH + β 2 CR + β 3 BAL + β 4 DFSI Ours : Xβ = β 0 + β 1 DBH + β 2 DBH 2 + β 3 BAL + β 4 DFSI With random intercept, data from Installation i, Observations j : Xβ + Zγ = β 0 + b i + β 1 DBH ij + β 2 DBH 2 ij + β 3 BAL ij + β 4 DFSI ij

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Four Models NLS: PM = 1.0 – [1.0 + e -(Xβ) ] -PLEN +ε PM (Proc GLIMMIX = same result as Proc NLS) GXR: NLS + R-sided random effect (overdispersion; identity matrix) GXME: PM = 1.0 – [1.0 + e -(Xβ + Zγ) ] -PLEN +ε PM GXFE (Prediction): PM = 1.0 – [1.0 + e -(Xβ + Zγ) ] -PLEN +ε PM X

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Tests Parameter estimation – Parameter & error Predictive ability – Leave-one-(plot)-out – Needed at least 2 plots/installation – Examined bias, AUC

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Linear: y = Xβ + Zγ Non-linear: y = 1.0 – [1.0 + e -(Xβ + Zγ) ] -1 Xβ + Zγ = β 0 + b i + X ij β 1 β0β0 b1b1 X ij β Mean = 0

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How did the models do? Parameter Estimation NLSGXRGXME EstimateStdErrorEstimateStdErrorEstimateStdError Fixed Effects Intercept DBH DBH E BAL E E E-05 DFSI Random Effects Residual (Subject = Tree) Intercept (Subject = Installation)

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How did the models do? Prediction ModelsBias (P 5-year mort ) AUCH-L Test NLS GXME GXFE

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Bias by BAL

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PM5 by BAL

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Prediction vs. observation for DBH

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Findings R-sided random effects & overdispersion Prediction – Informed random effects – Conditional model RE = 0 ‘NLS’ is the winner FEM 2012

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GLIMMIX = bad? Subject-specific vs. population-average model When would prediction work? – BLUP Why didn’t I do that??

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Acknowledgements Stand Management Cooperative Dr. Vicente Monleon

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Bias by DBH

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Bias by DFSI

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PM5 by Diameter Class

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PM5 by DFSI

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Generalized/nonliner model: Y=f(X, β, Z, γ) + ε; E(γ) = E(ε) = 0 Conditional on installation: E(y|γ) = f(X, β, Z, γ) Unconditionally: E(y) = E[E(y|γ)] = E[f(X, β, Z, γ] Unconditional model not the same as conditional model with random effects set to 0! Mixed models to the rescue (?)

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Linear mixed-effects Y = Xβ + Zγ + ε where E(γ) = E(ε) = 0 Then, conditional on random effect & because expectation = linear E(y|γ) = Xβ + Zγ Unconditionally, E(y) = Xβ Not true for non-linear models! PM = 1.0 – [1.0 + e -(Xβ + Zγ) ] -PLEN +ε PM

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