Allometric Crown Width Equations for Northwest Trees

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

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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, ?i Xi ß.
• g() is the link function, it transforms the mean
  onto a scale where the covariates are additive.

Source Schabenberger and Pierce (2002, p. 313)
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Statistical model (continued)

• In my case, g is log and Xi ß 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
ith tree on jth 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.

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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.

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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.