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 ratePLEN = growth period in 5-yr incrementsεPM = random error on PMWeighted 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 + β2DBH2

ij + 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 + ZγNon-linear: y = 1.0 – [1.0 + e-(Xβ + Zγ)]-1

Xβ + Zγ = β0 + bi+ Xijβ1

3 2 1 0 -1 -2 -30

1

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0

0.1

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Linear Non-linear

b1

Line

ar Y

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ue

Non

linea

r Y

Val

ue

β0 b1 Xijβ1

2 3 1

2 2 1

2 1 1

2 0 1

2 -1 1

2 -2 1

2 -3 1

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) = XβNot true for non-linear models!PM = 1.0 – [1.0 + e-(Xβ + Zγ)]-PLEN +εPM