allometric crown width equations for northwest trees nicholas l. crookston rmrs – moscow june 2004
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Crown width of Crown width of Northwest treesNorthwest trees
Allometric Crown Width Equations for Northwest Trees
Nicholas L. Crookston
RMRS – Moscow
June 2004
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Crown width of Crown width of Northwest treesNorthwest trees
Introduction
• Goals
• Data Source
• Model Form
• Statistical Model
• Analysis
• Results and discussion
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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.
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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.
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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.
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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.
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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
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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
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Crown width of Crown width of Northwest treesNorthwest treesComplex model form:
)(0 FBBCW
ElPBACL
HtDBH
BBB
BB
EElPBACL
HtDBH
)1(
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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
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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.
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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.
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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)
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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
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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
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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
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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”
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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.
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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.
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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.
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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.
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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).
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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).
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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.
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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
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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.
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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
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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.