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

Crown width of Crown width of Northwest treesNorthwest trees

Allometric Crown Width Equations for Northwest Trees

Nicholas L. Crookston

RMRS – Moscow

June 2004

Crown width of Crown width of Northwest treesNorthwest trees

Introduction

• Goals

• Data Source

• Model Form

• Statistical Model

• Analysis

• Results and discussion

Crown width of Crown width of Northwest treesNorthwest treesGoals

• 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.

Crown width of Crown width of Northwest treesNorthwest treesData 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.

Crown width of Crown width of Northwest treesNorthwest treesPlot 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.

Crown width of Crown width of Northwest treesNorthwest treesCrown 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.

Crown width of Crown width of Northwest treesNorthwest treesModel Formulation• CW increases with DBH

0

5

10

15

0 30 60 90 120 150

DBH Class (cm)

Mea

n C

W w

ithin

DB

H

Cla

ss

Crown width of Crown width of Northwest treesNorthwest trees

Simple model form:

• Based on the allometric relationship between CW and DBH.

• Basic model fits observed trends.

DBHbDBHbCW 0

Crown width of Crown width of Northwest treesNorthwest treesComplex model form:

)(0 FBBCW

ElPBACL

HtDBH

BBB

BB

EElPBACL

HtDBH

)1(

Crown width of Crown width of Northwest treesNorthwest trees

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

Crown width of Crown width of Northwest treesNorthwest trees

0

2

4

6

8

10

12

0 10 20 30 40 50

DBH Class

Sta

ndar

d D

evia

tion

of

CW

Standard deviation of CW is proportional to mean DBH.

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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.

Crown width of Crown width of Northwest treesNorthwest treesStatistical model (continued)• Two components of a GLMM are specified.

– The systematic component is a linear combination of covariates, ηi =Xi β.

– g() is the link function, it transforms the mean onto a scale where the covariates are additive.

iiiYg X ])[( E

Source: Schabenberger and Pierce (2002, p. 313)

Crown width of Crown width of Northwest treesNorthwest treesStatistical model (continued)

• In my case, g is log and Xi β is the log transform of the allometric equation.

)log()()log(])[log( 0 iDBHi DBHBBCW

• This is different than linear regression.

iiDBHi DBHBBCW )log()()log()log( 0

Crown width of Crown width of Northwest treesNorthwest treesStatistical model (continued)

• Applying the inverse link, exp(), we get the following:

DBHBii

iDBHi

DBHB

DBHBBˆ

0

0

ˆˆ

))log()ˆ()ˆexp(log(ˆ

where is the predicted mean CW for tree i. i

Crown width of Crown width of Northwest treesNorthwest treesStatistical model (continued)

• Include plot-level random effects.

)ˆˆexp((ˆ ,00, jji bB

where

),0(N~

),0(N~

2,

20,0

DBHjDBH

j

Db

Dbith tree on

jth plot

))log()ˆˆ( ,, jijDBHDBH DBHbB

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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”

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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.

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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.

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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.

Crown width of Crown width of Northwest treesNorthwest treesStatistical 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.

Crown width of Crown width of Northwest treesNorthwest treesResults• 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).

Crown width of Crown width of Northwest treesNorthwest treesResults• 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).

Crown width of Crown width of Northwest treesNorthwest treesResults• 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.

Crown width of Crown width of Northwest treesNorthwest trees

Douglas fir

0

4

8

12

16

0 50 100 150 200

DBH (cm)

Cro

wn

wid

th (m

)

Observed

Complex

Simple

Crown width of Crown width of Northwest treesNorthwest treesDiscussion• 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.

Crown width of Crown width of Northwest treesNorthwest trees

Douglas fir

0

8

16

24

0 100 200 300

DBH (cm)

Cro

wn

wid

th (m

)

Polynomial

Allomet ric

Crown width of Crown width of Northwest treesNorthwest treesClosing comments

• Remember the basics.• I’m 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.