1. Spatial Prediction of Coho Salmon Counts on Stream Networks
Dan Dalthorp
Lisa Madsen
Oregon State University
September 8, 2005

2. Sponsors
U.S. EPA STAR grant # CR
U.S. EPA Program for Cooperative Research on Aquatic
Indicators at Oregon State University grant # CR

3. Outline
• Introduction
(i) Coho salmon data
(ii) GEEs for spatial data
• Latent process model for spatially correlated counts
• Estimation and results
• Cross-validation
• Simulation study
• Conclusions and future research

4. Coho Salmon Data
Adult Coho salmon counts at selected points in Oregon coastal
stream networks for 1998 through 2003.
Euclidean distance between sampled points.
Stream distance between sampled points.

5. Coastal Stream Networks and Sampling Locations

6. GEEs for Spatially Correlated Data
Liang and Zeger’s (1986) pioneering paper in Biometrika
introduced GEEs for longitudinal data.
Zeger (1988) developed GEE analysis for a time series of counts
using a latent process model.
McShane, Albert, and Palmatier (1997) adapted Zeger’s model and
analysis to spatially correlated count data.
Gotway and Stroup (1997) used GEEs to model and predict spatially
correlated binary and count data.
Lin and Clayton (2005) develop asymptotic theory for GEE estimators
of parameters in a spatially correlated logistic regression model

7. The Latent Process Model
Suppose:
The latent process allows for overdispersion and
spatial correlation in .

8. The Marginal Model
These assumptions imply:
For now, we assume a simple constant-mean model and
a one-parameter exponential correlation function:

9. Estimating the Model Parameters
To estimate parameters solve estimating equations:
where

10. Iterative Modified Scoring Algorithm
Step 0: Calculate initial estimates
Step 1: Update .

11. Step 2: Update .
Step 3: Update .
Iterate steps 1, 2, and 3 until convergence.

12. Assessing Model Fit–Estimating the Mean
Assessing Model Fit–Estimating the Mean
Year
Sample
Mean
Euclidean
Distance
Stream
1998
6.2451
6.0
6.4941
1999
9.0025
8.7286
9.0765
2000
11.92
10.898
11.481
2001
31.359
31.597
34.541
2002
46.494
46.782
46.725
2003
44.453
41.005
41.829

13. Assessing Model Fit – Estimating the Variance
Assessing Model Fit – Estimating the Variance
Year
Sample
Std. Dev.
Euclidean
Distance
Stream
1998
222.07
221.59
1999
443.65
442.61
442.54
2000
384.59
384.75
383.90
2001
2508.6
2502.3
2512.3
2002
9286.6
9265.4
2003
3650.2
3653.4
3648.4

14. Assessing Model Fit – Estimating the Range (Euclidean Distance)

15. Assessing Model Fit – Estimating the Range (Stream Distance)

16. Cross validation to compare predictions based on three different assumptions about the underlying spatial process:
1. Null model (spatial independence) :
2. Spatial correlation as a function of Euclidean distance (ed):
3. Spatial correlation as a function of stream network distance (id)

17. 1. Bias? Not an issue... 2. Precision? Covariance model _
Euclidean Stream distance
1. Bias? Not an issue...
2. Precision?
Covariance model _
Null Euclidean Stream distance

18. Variances of predicteds Odds(|Eed| < |Eid|)
Null Euclidean Stream
Odds(|Eed| < |Eid|)
Year Odds
:152
:132
:171
:198
:171
:197
Total :1021

21. For each year, 8 scenarios that mimic the sample means,
Simulations
For each year, 8 scenarios that mimic the sample means,
variances, and ranges from the data were simulated.
Mean and variance constant
1. Euclidean spatial correlation
2. Stream network spatial correlation
Mean varies randomly by stream network; variance = 3.66 m 1.741
3. Euclidean spatial correlation; long range
4. Euclidean spatial correlation; medium range
5. Euclidean spatial correlation; short range
6. Stream network spatial correlation; long range
7. Stream network spatial correlation; medium range
8. Stream network spatial correlation; short range

22. Simulation proceedure
1. Simulate vector Z of correlated lognormal-Poissons
to cover all sampling sites (n ≈ 400)
2. Estimate parameters (m, s2, range) via latent process
regression from simulated data for a subset of the sampling
sites (blue)
3. Predict Z at the remaining sites (red, m ≈ 400) using:
(Gotway and Stroup 1997)
4. Repeat 100 times for each scenario (8) and year (6)

23. Use Euclidean distance or stream distance in covariance model?
Evaluation of predictions via two measures:
where:

26. Summary of Findings Cross-validations:
1. MSPEs same for Euclidean distance and stream network distance;
2. Errors usually smaller with Euclidean distance;
3. Population spikes more likely to be detected with Euclidean distance.
Simulations:
1. Euclidean spatial process: Euclidean covariance gives smaller MSPE than does
stream network distance covariance;
2. Stream network process: Euclidean covariance model MSPEs comparable to
those of stream distance model EXCEPT when network means varied and range
of correlation was large.

27. Future work -- Incorporate covariates (with some misaligned data);
-- Incorporate downstream distances/flow ratios;
-- Spatio-temporal modeling;
-- Rank correlations in place of covariances;
-- Model selection;
-- Non-random data;