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

2. 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

3. 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

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

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

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

7. 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 InstallationsPlotsDF TreesRevisits 20175358,099157,473

8. 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

9. 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

10. 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

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

12. 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 1 231 221 211 201 21 2-21 2-31 Mean = 0

13. How did the models do? Parameter Estimation NLSGXRGXME EstimateStdErrorEstimateStdErrorEstimateStdError Fixed Effects Intercept -4.51180.02807-4.51180.09267-5.09580.2891 DBH-0.21050.00251-0.21050.00829-0.27190.00677 DBH 2 0.001687.8E-050.001680.000260.002790.00017 BAL 0.004211.8E-050.004216.1E-050.004958.3E-05 DFSI 0.048970.000680.048970.002240.059960.00804 Random Effects Residual (Subject = Tree) 10.8840.0387910.2750.03665 Intercept (Subject = Installation) 0.63530.07953

14. How did the models do? Prediction ModelsBias (P 5-year mort ) AUCH-L Test NLS0.002643908 0.845366.8 GXME-0.000604775 0.864388.8 GXFE0.0110345 0.8441505.6

15. Bias by BAL

16. PM5 by BAL

17. Prediction vs. observation for DBH

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

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

20. Acknowledgements Stand Management Cooperative Dr. Vicente Monleon

22. Bias by DBH

23. Bias by DFSI

24. PM5 by Diameter Class

25. PM5 by DFSI

26. 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 (?)

27. 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