Analysis of Variance (ANOVA) and Multivariate Analysis of Variance (MANOVA) Session 6.
Multivariate Analysis of Variance for Stream Classification in Texas
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Transcript of Multivariate Analysis of Variance for Stream Classification in Texas
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Multivariate Analysis of Variancefor
Stream Classification in Texas
Eric S. HershCE397 – Statistics in Water Resources
Term Project
Cinco de Mayo, 2009
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Can we quantitatively regionalize the streams of Texas?
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East Texas
North-Central Texas
WestTexas
South-CentralTexas
Lower Rio Grande Basin
Hersh, E.S., Maidment, D.R., and W.S. Gordon. “An Integrated Stream Classification System to Support Environmental Flow Analyses in Texas.” J. Am. Water Res. Assoc. Submitted November 2008.
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Revisited - the question posed
Can we improve the way in which we perform the regionalization and thus (potentially)
increase its classification strength?
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Analysis of VarianceANOVAPurpose: test whether group means are different
MANOVAMultivariate Analysis of Variance
Purpose: ANOVA with several
dependent variables
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• Multiple metric dependent variables (n=18)
• Based on categorical (non-metric) independent variables (n=5 regions)
• Manipulate independent variables to determine effect on dependent variables using SAS PROC GLM (general linear model)
Region = DO ± Temp ± TSS ± pH ± Cond ± AirTemp ± Precip ± PET ± MAQ ± MAV ± BFI ± ZeroQ ± IQR ± Slope ± Substrate ± Sand ± Silt ± Clay
The Model
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ANOVA MANOVA
= = … =
where:
p = parameter (dependent variable)
k = factor (independent variable)
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Data Gaps
• Total number of subbasins in Texas = 205• Number with complete data = 103
Uh oh! This test is going to lose a lot of value. Unless…
• Can we fill in the gaps somehow?
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Data Gaps
• Some of the subbasins in Texas have no rivers.
• Many have no gages.
• Many have no WQ sampling stations.– Synthetic data would be difficult and poor.
• But, the MANOVA test requires complete matrices.– Solution: fill in gaps with parameter means
– Dilutes strength of classification (regions tend toward others)
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Hypothesis Test• Null Hypothesis: (vectors of) the group means
are equalOf course not! That’s preposterous! There would be no
regionalization!
But… we don’t care.
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(PRISM, 1971-2000)
West
East
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Evaluating the Model
• Pillai’s trace considered most robust– S.S. Pillai, 1901-1950, Indian mathematician
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Revision Methodology1. Identify bordering subbasins
(n=50, but 10 border multiple, so 60 trials total)2. Switch one subbasin, check for increase in test stat,
record and reset (21 deemed beneficial)3. Rank by improvement4. Implement changes in order, discard if decline (18 kept)5. View in geographic context, apply decision rules (no
islands or peninsulas, 15 kept)
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OLD NEW
SWITCHED
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Possible Future Work
• Write final report