What is a Multi-Scale Analysis? Implications for Modeling Presence/Absence of Bird Species
Kathryn M.
1
Georgitis ,
Alix I.
1
Gitelman ,
1Department
Don L.
1
Stevens ,
Nick
2
Danz ,
and JoAnn
of Statistics Oregon State University Corvallis, Oregon
2 Natural Resources Research Institute University of Minnesota-Duluth Duluth, Minnesota
2
Hanowski
R82-9096-01
We used logistic regression to explore the association of Pine
Warbler with landscape composition variables at each spatial extent
separately.
500 (m)
Lowland conifer, Pine and Oak-Pine, Spruce-Fir,
Spruce-Fir*Pine and Oak-Pine
1000 (m)
Pine and Oak-Pine, Spruce-Fir, Spruce-Fir*Pine and Oak-Pine
5000 (m)
Pine and Oak-Pine, foresta, foresta*Spruce-Fir, Spruce-Fir
* Response Variable: presence/absence of Pine Warbler (PIWA) in the Western Great Lakes
-Methods described in Howe et al 1998
* Explanatory Variables: percentage of each land cover type within 4 different buffer widths i.e. spatial extents (Figure 1)
Figure 1: Diagram of G.I.S. buffering
scheme used to create explanatory
variables from TM Imagery.
Table 1: Correlations at each spatial extent for subset of
explanatory variables used in model selection procedure.
Spatial Extent
100 m
500 m
1000 m
5000 m
1000m
500m
a:The forest variable is an indicator for stands located in the Chequamegon national forest.
100 m
500 m
1000 m
5000 m
100m
Drawbacks with Option 1:
- 210 possible models for each spatial extent
- Multi-collinearity of explanatory variables
- Does not account for possible relationships between
spatial extents
Pine and Oak-Pine
Lowland Conifer
-0.233
-0.280
-0.269
-0.161
Pine and Oak-Pine
Aspen-Birch
-0.464
-0.585
-0.623
-0.531
Table 3: Explanatory variables included in logistic regression model for
estimated log odds of Pine Warbler using variables from the 100m, 500m, and
1000m spatial extent in model selection procedure.
Spatial Extent
100m
500m
1000m
100m*1000m
Explanatory Variables Selected Using BIC criteria
Aspen-Birch, Northern Hardwoods and Oaks, Pine and Oak-Pine,
Spruce-Fir
None
Spruce-Fir
Pine and Oak-Pine*Spruce-Fir
Drawbacks with Option 2:
- minimum of 230 possible models
- Multi-collinearity of explanatory variables within and between
spatial extents
As spatial extent increases the correlation between
land cover types changes (Table 1). This will affect
the inclusion and exclusion of variables in the final
model (see option 1).
We created new explanatory variables to investigate if the presence of
PIWA is associated with a change in land cover variables from the 100m to
the 500m spatial extent.
Table 4: Results for Proportional and Difference logistic regression models.
X500m is the % of land cover type X at the 500m spatial extent,
and X100m is the % at the 100m spatial extent.
Model
Proportional
Difference
New Variable
X 500m/X100m
X 500m- X100m
Explanatory Variables
Aspen-Birch, Pine and Oak-Pine
Pine and Oak-Pine
Drawbacks with Option 3:
- Model selection problem
- Correlation structure?
- Is this a biologically meaningful function of the variables?
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Another Alternative: Graphical Models
Pine and Oak-Pine500
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Parameter
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
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Superior Forest
Stand 3
Stand 1
Stand 2
Stand 1
Stand 6
Stand 2
Chequamegon Forest
Stand 1
Stand 3
Chippewa Forest
Stand 5
Stand 5
Stand 3
Stand 4
Mean
-1.164
-0.0813
0.0026
0.00364
We are investigating the use of Graphical Models to
address multi-scale questions in ecological studies. The
conditional modeling approach allows for specification of
correlation between variables at multiple scales. Also, this
approach can be used to explore how the presence/absence
of a species is related to vegetation characteristics at
multiple scales, similar to Bayesian Hierarchical models.
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95%
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Stand 4
Lowland Conifer100
Diagram 3: Schematic of Graphical Model for
presence/absence of PIWA. For example, Pine and OakPine500 is the % of Pine and Oak-Pine at the 500m spatial
extent. Lines without arrows represent correlation between
two variables. Lines with arrows represent a directional
association.
95% Confidence Interval
-1.719, -0.633
-0.5654, 0.1523
-0.009755, 0.02342
-0.01925, 0.04036
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95%
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Confidence
Interval
for

Pine and Oak-Pine100
Presence/Absence of PIWA
Results Model 1
Bayesian Hierarchical Models are a useful method for multi-scale analysis because they can be used to
explore relationships between processes operating at different scales in ecological systems (e.g. Monleon et al
2002). A typical hierarchical structure is displayed in diagram 2.
St.Croix Forest
Lowland Conifer500
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Figure 2 and Figure 3 display the % of Pine and Oak-Pine and % of Spruce-fir for nine stands versus spatial
extent. Notice the stands located in Chippewa forest have very different %’s of Pine and Oak-Pine (and
Spruce-Fir) at the 100m scale but at the 5000m scale they have similar values. These figures raise the
question perhaps it is the change in the % of Pine and Oak-Pine from one spatial extent to another that is
what is important when trying to model the presence of Pine Warbler (see option 3).
Option 4: Analysis Using Bayesian Hierarchical Models
- Multi-collinearity of explanatory variables within and
between spatial extents
* affects inclusion/exclusion of variables
* inflates variance of coefficients
* unclear interpretation of coefficients
- What form of the explanatory variables are biologically
meaningful?
Possible statistical methods that can be used to address the question of interest:
Option 1: Perform an analysis for each spatial extent separately
Option 2: Use the variables from all spatial extents in one analysis
Option 3: Use summary variables that combine information from multiple spatial extents
Option 4: Use Bayesian Hierarchical Models
Stand 4
Option 3: Analysis Using “New” Explanatory Variables
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Pine and Oak-Pine
Spruce-Fir
-0.311
0.026
0.110
0.209
Lowland Conifer
Spruce-Fir
-0.068
-0.137
-0.263
-0.633
Option 2: Analysis Using all the Explanatory Variables
We used all the variables from the 100m, 500m, and 1000m spatial
extent in the logistic regression model selection procedure.
- Variable selection procedures do not account for number
of possible explanatory variables (Fred Ramsey,
personal communication).
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Table 2: Significant explanatory variables included in logistic regression model for
estimated log odds of Pine Warbler at each spatial extent.
Spatial Extent
Explanatory Variables Selected Using BIC Criteria
100 (m)
Lowland conifer, Pine and Oak-Pine
Discussion:
Problems with Options 1-4:
The Data:
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Option 1: Analysis at Each Spatial Extent Separately
What statistical methods can be used to address multi-scale questions such as:
How is the presence of Pine Warbler related to vegetation characteristics at different spatial scales?
Stand 2
Stand 2
Stand 1
Stand 3
Diagram 2: Diagram of hierarchical structure present in a forest ecosystem. Stands are nested within the forests.
Likelihood model for the observations-
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Yi.j~ Bernoulli(pi,j)
Logit(pi,j) = ,j +,j X1,i,j + 2,j X2,i,j
where Yi,j =1 if PIWA was present in the ith stand located in the jth forest and 0 otherwise,
and where X1,i,j is the percentage of Lowland Conifer at the 100m spatial extent for the ith stand in the jth forest
X2,i,j is the percentage of Pine and Oak-Pine at the 100m spatial extent for the ith stand in the jth forest
For the next level of the hierarchy we used:
Model 1- ~N(mj,t)
Model 2- ,~N(mj,t)
mj = 0 +Z1,j +2Z2,j
where Z1,j is the mean percentage of Pine and Oak-Pine for the jth forest
Z2,j is the mean percentage of Lowland conifer for the jth forest
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Results Model 2
95% Wald C.I. for 
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Mean
-0.06529
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-0.4674
-0.02388
-0.05129
95% Confidence
Interval
-0.1361, -0.01759
0.008479, 0.03178
-5.35, 3.006
-0.1865, 0.2031
-0.4004, 0.3651
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Acknowledgements:
I would like to thank Jerry Niemi for the opportunity to work on an interesting
ecological problem. Also, Fred Ramsey who elucidated the problem with model selection
criteria.
3
Drawbacks with Option 4:
References:
Howe, R. W., G. J. Niemi, S. J. Lewis, and D. A. Welsh. 1998. A standard method for
monitoring songbird populations in the Great Lakes region. Loon 70:188-197.
Monleon, V. A.I. Gitelman, and A.N. Gray. 2002. Multi-scale relationships between coarse
woody debris and presence/absence of Western Hemlock in the Oregon Coast Range.
In Case Studies in Bayesian Statistics Vol. VI Gatsonis et al (Eds.). Springer-Verlag, New
York, Inc.
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- Forest level covariates not independent measurements
- Concentric circle sampling design not a “true” nested
hierarchy
- Multi-collinearity of explanatory variables within same
spatial extent
1
Funding/Disclaimer:
The research described in this presentation has been funded by the U.S. Environmental
Protection Agency through the STAR Cooperative Agreement CR82-9096-01 Program on
Designs and Models for Aquatic Resource Surveys at Oregon State University. It has not
been subjected to the Agency's review and therefore does not necessarily reflect the views
of the Agency, and no official endorsement should be inferred.