Standardization, Residualization, and Adjusted Statistical Thresholds
for Forest Health Indicators
William A. Bechtold 1
Stanley J. Zarnoch 1
Stand-level CCV Indicators
Michael E. Schomaker
2
1
USDA Forest Service, Southern Research Station, Asheville, NC
Colorado State Forest Service, Ft. Collins, Co
Corresponding author:
W.A. Bechtold, USDA Forest Service, Southern Research Station,
P.O. Box 2680, Asheville, NC 28802
Phone: 828-257-4357 e-mail: wabechtold@fs.fed.us
2
Exam ple
Standardized and Residualized Indicator Values
Standardized Indicators
An indicator can be adjusted for different statistical distributions among species by
standardizing to a mean of zero and a standard deviation of one. Values are thus
expressed in ter ms of standard deviation units from the mean for a given species. This
adjustment results in more meaningful interpretations w hen comparisons are made
among species, and enables summations across species to yield stand-level
indicators. The standardization of an indicator is defined as
I ij′ =
I ij − I j
sj
(1)
where
I′ij
= the standardized indicator for tree i w ithin species j,
Iij
= the non-standardized indicator for tree i w ithin species j,
Ij
sj
= the average for the non-standardized indicator for species j, and
Composite Crow n Volume ( CCV), a composite indicator derived from multiple tree crow n
parameters, w as computed for 6,179 individual trees located on 250 FHM plots distributed
across Virginia, Georgia, and Alabama:
CC V = 0.5 π R 2 C L C D
(4)
w here
π
= 3.14159,
R
= crow n diameter / 2,
H
= total tree length,
CL
= H (crow n ratio) / 100, and
CD
= (crow n density / 100)(1 – dieback / 100)(1 – transparency / 100)
Tree-level CCV Indicators
Raw and Standardized Values
Trees w ere then grouped by species. Mean CCV differed substantially by species,
ranging from 697 cu ft (slash pine) to 6,422 cu ft (American beech), w ith coefficients of
variation often exceeding 100%, thus demonstrating the need to standardize across species.
Equation (1) w as used to obtain adjusted tree-level CCV values that w ere standardized, by
species, to a mean of zero and a standard deviation of one. Indicators standardized in this
manner reflect the deviation of all trees from their species mean (in ter ms of standard
deviation units) and are invariant to the species distribution on the plot. At this point, it is valid
to combine or compare data across species, since all tree-level values are now on the same
basis w ith respect to species.
Residualized and Standardized-Residualized Values
The distributional properties of CCV in its raw (equation 4, figure 1a) and standardized
(equation 1, figure 1b) for ms are show n in figure 1. Although standardization results in a
mean of zero and standard deviation of one, note that standardization alone has little effect
on skew ness. Skew ness is often improved by residualization w ith models, so the follow ing
model designed to adjust CCV for tree size and stand density w as employed:
CCV =
b 0 + b 1 (dbh) + b 2 (BA)
12 0
50
mean = 1905
40
kurtosis = 0.913
20
10
dbh =
tree diameter (inches) at breast height,
BA
=
stand-level basal area per acre, and
bi
=
regression parameters estimated from the data.
Note that the residuals (equation 2, figure 1c) and standardized residuals
(equation 3, figure 1d) associated w ith the model (equation 5) have substantially smaller
skew ness coefficients and are more nor mally distributed than standardization alone
(figure 1b). Reduction in skew ness is advantageous w hen performing statistical tests based
on assumptions of nor mality. It also diminishes reliance on transformations and
nonparametric statistical methods.
0
0
1000
2000
3000
4000
5000
me an = 16 11
sd = 2469
3 500
ske wne ss = 4 .6 66
3 000
ku rto sis = 36.28 8
15 00
10 00
Number of Trees
20 00
Number of Trees
4 000
mean = 0 .0 000
sd = 0.997 6
skewne ss = 4 .1 67
kurtosis = 28.67 1
1
2
800 0
5 percent
r = ile0.46
600 0
400 0
200 0
-1
0
1
2
Fi gu re 3 . Cl as sifi cati on of p lo ts b a sed o n Me a n C om po si te C row n V ol um e a n d its sta n da rd ize d re si du a l (2 50 pl ots).
2 000
1 500
Conclusions
0
0
5000
10000
15000
2000 0
2500 0
3000 0
3500 0
400 00
-2
0
2
4
6
8
10
12
14
St andardiz
ed C omposite
C rown
Volume
1b) Compos
ite C rown
Volume (St
andardized)
C omposite
C olume
rowm Volume
1a) C om posit
e C row n V
(R aw)
(2)
35 00
(3)
4 000
me an = 0.00 00
mean = 0 .00 00
30 00
sd = 1577
3 500
sd = 0.997 6
25 00
ske wne ss = 3 .5 18
3 000
skewne ss = 3 .0 67
2 500
kurtosis = 26.91 5
Number of Trees
Number of Trees
Like raw indicator values, residualized indicators can be standardized for
comparisons across species:
ku rto sis = 39.52 3
20 00
15 00
10 00
2 000
1 500
1 000
5 00
500
0
0
- 10000
0
10000
20000
300 00
1c) C ompos ite C rown
Residual
Volume (R es idual)
-5
0
5
10
15
1d) Compos ite SCtandardized
rown Volume
R es
(Stidual
andardized Residual)
Rij = the residualized indicator for tree i w ithin species j,
= the average for the residualized indicator for species j, and
= the standard deviation for the residualized indicator for species j.
0
2 500
500
0
Standardized-Residualized Indicators
Rj
sj
-1
2b) C ompos ite Crow n Volume (Standardized Res idual)
Standardized R es idual
1 000
= the predicted value of the indicator for tree i w ithin spec ies j based
= the standardized-residualized indicator for tree i w ithin species j,
-2
2a) Compos ite Crow n Volume (R aw )
Composite Crown Volume
5 00
on the appropr iate model.
Rij′
7000
Standardized Residual
The associated residualized indicator is then defined as
where
6000
5 percenti le
25 00
= an indicator for tree i w ithin species j, and
sj
40
0
Another method to adjust an indicator is to define the indicator as its residual
from a model based on tree and/or stand conditions. Each tree is thus adjusted for
its specific competitive situation, resulting in an indicator more suitable for detecting
abnor malities because such adjustment identifies individuals that do not confor m to
model predictions. Spec ifically, let
Rij − Rj
kurtosi s = 1.891
60
20
0
-2
Rij′ =
80
Figure 2. Stand-level distributions of Composite Crown Volume and its standardized residual (250 plots).
(5)
= the standard deviation for the non-standardized indicator for species j.
Rij = Yij − Yˆij
sd = 0.4659
skewness = 0.361
skewness = 1.0 06
30
where
Residualized Indicators
Yij
Yˆij
mean = -0.0139
10 0
sd = 1236
Composite Crown Volume
Thresholds are important w hen assessing forest health because they separate
the sampled population into categor ies of good and poor. Ideally, thresholds should
be developed on a biological basis. Biological thresholds, the point at w hich a tree
becomes noticeably stressed and begins to decline, are difficult to pinpoint. This
requires extensive research designed to establish correlations betw een indicator
variables and other signs of tree stress such as damage symptoms, reduced grow th,
and prospective mortality. Threshold establishment is further complicated because
thresholds are often species dependent, and because the effect of normal stand
dynamics must be partitioned from the analysis. The ultimate goal in establishing any
threshold is to identify signals that appear to be beyond the range of w hat is
expected.
Statistical thresholds have advantages and disadvantages w hen compared to
biological thresholds. They are disadvantaged because they are somew hat arbitrary,
and alw ays result in a set of observations designated as poor, even in the absence
of a problem. On the other hand, they are easily established by isolating
observations at the tails of statistical distributions, and they are quite useful for
detecting spatial patterns, establishing empirical correlations, and measuring change
over time.
Since biological thresholds have yet to be developed, arbitrary thresholds for
some indicators (especially visual crow n ratings) have been used in a variety of
recent forest health reports. Such analyses have the virtue of simplicity, but contain
no adjustment for species differences, other tree attr ibutes, or stand conditions.
These types of adjustments and related analyses can be accomplished w ith
statistical distributions and thresholds. This poster utilizes data from the Crow n
Indicator to demonstrate how such adjustments are accomplished by standardization
and residualization of indicator values. These techniques w ork for individual indicator
variables, as w ell as composite values that may be derived from multiple indicator
variables.
N umber of Plots
Thresholds and Statistical Distributions
Standardization of indicators makes it feasible to produce plot- level indicators
(by averaging tree- level values across all trees on each plot). Plot-level averages
(across all species) for 250 plots w ere calculated from the raw and standardizedresidualized tree-level CCV values. Distributions of these stand-level means are
show n in figures 2a and 2b. Note that the standardized residuals do not have a
mean of 0 and standard dev iation of 1 (as seen w ith tree level-indicators). This is
because standardization w as initially performed at the individual tree and species
level, and spec ies distributions differ across the plots. As w ith the tree- level
indicators, residualization reduces the s kew ness coefficients of the stand-level
indicator.
Further comparison of average raw stand-level values and their standardizedresidualized counterparts w as performed by classifying 250 plots into good and
poor categor ies. For demonstration, a threshold for the poor class w as set at the
low er 5-percentile of the statistical distribution. Each pair of raw and standardizedresidual values w as then plotted on a scatter diagram, w ith the threshold value
indicated by reference lines (figure 3). Agreement betw een raw values and their
standardized residuals is attained for all plots located in the upper right and low er
left quadrants. The other tw o quadrants represent opposite classifications by the
indicators. The raw indicator and its standardized-residual counterpart classify the
same plots as poor only tw ice, confirming that the raw and adjusted values are
measuring different aspects of crow n condition.
The reason for differences in classification is based on the adjustment
potential of the regression models and its effect on the creation of the
standardized-residual indicators. When using raw CCV values, stands w ith high
percentages of trees w ith small crow ns are classif ied as poor, even if crow ns are
nor mally small for those particular species in those types of stands. How ever,
when using the adjusted values, stands are classified as poor only if they have
high percentages of trees at the low er end of their respective species statistical
distributions after adjusting for tree and stand conditions.
N umber of Plots
Abstract. Recent forest health reports have focused on
arbitrary thresholds for indicator values that are not adjusted for
differences among species. To circumvent this difficulty, we
show how indicators can be standardized, by species, to a
mean of zero and variance of one. This enables direct
comparison among species and permits tree-level indicators to
be summed across species to yield plot-level (i.e., stand-level)
indicators. We also demonstrate how to produce residualized
indicators (which can also be standardized) that adjust for tree
size and local stand conditions. Indicators based on crown
volume are given as examples.
Figure 1. Tree-level distributions of Composite Crown Volume and its adjusted counterparts (6,167 trees).
FHM Posters home page
| FHM 2002 Posters
The utility of raw indicators can be extended by standardization across species.
Analyses can be further enhanced w ith model residuals that adjust for the effect of
tree size, stand density, and other parameters. Model residuals can also be
enhanced by standardization. Combinations of these techniques result in a variety
of analytical tools that can be tailored to address issues concerning forest health.
The appropriateness of standardized, res idualized, or raw values depends on the
circumstances. Standardized values are useful w hen compar ing or summing
across individuals w ith different statistical distributions for the indicator of interest.
Res idualized indicators are useful for detecting deviations from expected values.
Raw values are useful for detecting unadjusted net change.
Like raw indicator values, adjusted values can be correlated w ith other
indicators (e.g., lichen diversity, or soil erosion) to establish biological thresholds.
They can be correlated w ith other plot-based data or spatial over lays such as
elevation, forest type, or physiographic class to deter mine if there are statistically
significant differences betw een categories. They can be used for spatial analyses
designed to detect clusters or gradients of unusually good or poor indicator values.