Douglas-fir mortality estimation with generalized linear mixed models
Jeremy Groom, David Hann, Temesgen Hailemariam
2012 Western Mensurationists’ MeetingNewport, OR
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
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
DATALevels: Installations – plots – trees - revisits
Yr 1 Yr 5 Yr 10…
Measuring & modeling
• Single-tree regular mortality models– FVS, ORGANON
• Logistic models– Revisits = equally spaced
• Problems– Lack of independence!• Datum = revisit?• Nested design (levels)
Our goals
• Account for overdispersion– Level: tree
• Revisit data: mixed generalized linear vs. non-linear– Random effect level = installation
• Predictive abilities for novel data
Setting
• SW BC, Western Washington & Oregon• Revisits: 1-18• 3-7 yrs between revisits• Plots = 0.041 – 0.486 ha (x = 0.069)• Excluded installations with < 2 plots
Installations Plots DF Trees Revisits201 753 58,099 157,473
Coping with data
• Hann et al. 2003, 2006Nonlinear model:
PM = 1.0 – [1.0 + e-(Xβ)]-PLEN +εPM
PM = 5 yr mortality rate
PLEN = growth period in 5-yr increments
εPM = random error on PM
Weighted observations by plot area
Predictors = linearGeneralized Linear Model OK
Parameterization
PM = 1.0 – [1.0 + e-(Xβ)]-PLEN +εPM
Originally: Xβ = β0 + β1DBH + β2CR + β3BAL + β4DFSI
Ours: Xβ = β0 + β1DBH + β2DBH2 + β3BAL + β4DFSI
With random intercept, data from Installation i, Observations j :
Xβ + Zγ = β0 + bi+ β1DBHij + β2DBH2ij + β3BALij + β4DFSIij
Four Models
• NLS: PM = 1.0 – [1.0 + e-(Xβ)]-PLEN +εPM
(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 +εPM
• GXFE (Prediction): PM = 1.0 – [1.0 + e-(Xβ + Zγ)]-PLEN +εPMX
Tests
• Parameter estimation – Parameter & error
• Predictive ability– Leave-one-(plot)-out– Needed at least 2 plots/installation – Examined bias, AUC
Linear: y = Xβ + Zγ
Non-linear: y = 1.0 – [1.0 + e-(Xβ + Zγ)]-1
Xβ + Zγ = β0 + bi+ Xijβ1
3 2 1 0 -1 -2 -30
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Linear Non-linear
b1
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β0 b1 Xijβ1
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Mean = 0
How did the models do?Parameter Estimation
NLS GXR GXME Estimate StdError Estimate StdError Estimate StdError
Fixed Effects
Intercept -4.5118 0.02807 -4.5118 0.09267 -5.0958 0.2891
DBH -0.2105 0.00251 -0.2105 0.00829 -0.2719 0.00677
DBH2 0.00168 7.8E-05 0.00168 0.00026 0.00279 0.00017
BAL 0.00421 1.8E-05 0.00421 6.1E-05 0.00495 8.3E-05
DFSI 0.04897 0.00068 0.04897 0.00224 0.05996 0.00804
Random Effects
Residual (Subject = Tree) 10.884 0.03879 10.275 0.03665
Intercept (Subject = Installation) 0.6353 0.07953
How did the models do?Prediction
Models Bias (P5-year mort) AUC H-L Test
NLS 0.002643908 0.845 366.8
GXME -0.000604775 0.864 388.8
GXFE 0.0110345 0.844 1505.6
Bias by BAL
PM5 by BAL
Prediction vs. observation for DBH
Findings
• R-sided random effects & overdispersion
• Prediction– Informed random effects– Conditional model RE = 0
• ‘NLS’ is the winner• FEM 2012
GLIMMIX = bad?
• Subject-specific vs. population-average model
• When would prediction work?– BLUP
• Why didn’t I do that??
Acknowledgements
• Stand Management Cooperative
• Dr. Vicente Monleon
Bias by DBH
Bias by DFSI
PM5 by Diameter Class
PM5 by DFSI
• 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 (?)
Mixed models to the rescue (?)
Linear mixed-effectsY = 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 +εPM