1. Allometric Crown Width Equations for Northwest Trees Nicholas L. Crookston RMRS – Moscow June 2004

2. Introduction Goals Data Source Model Form Statistical Model Analysis Results and discussion

3. Goals To construct biologically and statistically sound models for inventoried tree species. To provide models of varying complexity to support varying uses. In FVS, predicted CW is used to estimate canopy cover.

4. Data Source First installment of the Oregon and Washington CVS plots. A grid system of 11,000 plots on public land. 19 National Forests. 250,000 observations of CW spread over 34 species.

5. Plot design A cluster of 5 subplots centered on a grid point; further subdivided into plots of varying sizes where large trees were tallied on larger plots and small trees on smaller plots. CW was measured on GSTs: –live trees, age 5, DBH 1 inch for softwood species and 3 inches for hardwood species.

6. Crown width measurement: Measure a horizontal distance across the widest part of the crown, perpendicular to a line extending from the stake position [at plot center] to the tree bole. Recorded to the last whole foot.

7. Model Formulation CW increases with DBH

8. Simple model form: Based on the allometric relationship between CW and DBH. Basic model fits observed trends.

9. Complex model form:

10. Statistical model Observations are not independent, GSTs from the same plot are more alike than trees are in general. CW measurements are right-skewed; never less than zero but can be quite a bit larger than the mean

11. Standard deviation of CW is proportional to mean DBH.

12. Statistical model (continued) A generalized linear mixed effects model (GLMM) can be used to address the statistical properties. CW is modeled as Gamma distributed with a log link function.

13. Statistical model (continued) Two components of a GLMM are specified. –The systematic component is a linear combination of covariates, η i =X i β. –g() is the link function, it transforms the mean onto a scale where the covariates are additive. Source: Schabenberger and Pierce (2002, p. 313)

14. Statistical model (continued) In my case, g is log and X i β is the log transform of the allometric equation. This is different than linear regression.

15. Statistical model (continued) Applying the inverse link, exp(), we get the following: where is the predicted mean CW for tree i.

16. Statistical model (continued) Include plot-level random effects. where i th tree on j th plot

17. Statistical model (continued) Fitting was done with glmmPQL from R (Venables and Ripley 2002, p. 298). McCulloch and Searle (2001, p. 283) have said that the development of PQL methods –have had an air of ad hocery –modern methods may be better performing –have not been fully tested

18. Statistical model (continued) … McCulloch and Searle (2001, p. 283)… –get better as the conditional distribution of the response variable given the random effects gets closer to normal. –binary data are the worse case The conditional distribution of the CW data does approach the normal. The method seems to have worked well.

19. Statistical model (continued) Alternatives to GLMM: –Directly fit the nonlinear model using nonlinear mixed effects. –Ignore the plot effects. –Fit the log transformed linear model. GLMM addressed all the problems in a single step.

20. Statistical tests The simple model was always acceptable (based on t-tests and theory). The complex model was compared to the simple using a likelihood ratio test. This test requires nested models. Individual terms in the complex model were tested using partial t-tests.

21. Statistical tests AIC was also used. For nested models AIC and the likelihood ratio test will lead to the same conclusions, but they are based on different ideas. An improvement in AIC of about 2 corresponds to a likelihood ratio test at the 0.05 level of significance.

22. Results Species specific equations using of DBH are presented for 34 species. Complex equations are presented for 29 species. Predictor variables include –crown length (CL), –tree height (Ht), –plot basal area, –elevation, and –geographic location (National Forest).

23. Results DBH is the most important predictor of CW Implications of the complex equation: –CWs increase with DBH and CL but decrease with Ht when DBH and CL are also in the equation. –CWs are smaller at higher elevations (the one exception is western larch).

24. Results Implications (continued) –CWs, generally, increase with density for shade tolerant species and decrease with density for some shade intolerant species. –The effect of density on CW was weak perhaps because density also influences other covariates.

26. Discussion The allometric equation is better than recently published linear and polynomial equations. The bias at the extremes of the distribution can be large. When the equation is used to predict canopy cover, the bias in CW can imply a 10-20 percent bias in canopy cover.

28. Closing comments Remember the basics. Im not sure the glammPQL was worth the effort, but I really like R. The manuscript is in review at the online journal Forest Biometry, Modelling and Information Sciences.