1. Using Structural Equation Modeling to Analyze Monitoring Data Jim Grace NWRC

2. What is structural equation modeling? A framework for using statistical methods to ask complex questions of data. Macrohabitat η c1 Microhabitat η c2 Diversity η e3 Litter η e2 Herbaceous η e1 lake, x 1 impound, x 2 swale, x 3 vhit1, y 1 vhit2, y 2 herbl, y 3 herbc, y 4 wlitr, y 5 litrd, y 6 litrc, y 7 rich, y 8 0 0.26.75.98.92.64.90.81 -.47.95 1.13.44 ns 1.0.80 1.00.59.77

3. The Origin of Structural Equation Modeling Sewell Wright 1897-1988 1st paper in: 1920

4. The Wright Idea Y 1 = α 1 + β 1 X + ε 1i Y 2 = α 2 + β 2 X + β 3 Y 1 + ε 2i XY1Y1 ε 1i Y2Y2 ε 2i

5. The LISREL Synthesis Karl Jöreskog 1934 - present Key Synthesis paper- 1973

6. LISREL: A flexible, multiequational framework y 1 = α 1 + β 1 x + ε 1i y 2 = α 2 + β 2 x + β 3 y 1 + ε 2i y 3 = α 3 + β 4 y 1 + β 5 y 2 + ε 3i y 4 = α 4 + β 6 y 1 + β 7 y 3 + ε 4i Can include observed, latent, and composite variables. x = Λ x ξ + δ y = Λ y η + ε η = α + Β η + Γξ + ζ The LISREL Equations Jöreskög 1973

7. Σ = { σ 11 σ 12 σ 22 σ 13 σ 23 σ 33 } Implied Covariance Matrix compare Absolute Model Fit x1 y1 y2 Hypothesized Model Observed Covariance Matrix { 1.3.24.41.01 9.7 12.3 } S = + Estimation and Evaluation Parameter Estimates estimation LS, ML, and BA

8. 1. It is a “model-oriented” method, not a null-hypothesis-oriented method. Some Properties of SEM 2. Highly flexible modeling toolbox. 3. Can be applied in either confirmatory (testing) or exploratory (model building) mode. 4. Variety of estimation approaches can be used, including likelihood and Bayesian.

9. 1. Seeks to model uncertainty rather than probabilities. A Bit about the Bayesian Approach 2. Philosophically well suited for supporting decision making. 3. Popularity partly based on new algorithms that create great flexibility in modeling. 4. It's indeterminant solution procedure, contributes to some uncertainty about results for more complex models(?)

10. Why do we need multivariate modeling?

11. 179 variables 73 variables 129 variables

12. Do the conventional methods meet your needs? All your great scientific ideas! ANOVA result you hope to get!

13. modified from Starfield and Bleloch (1991) How do data relate to learning? Understanding of Processes univariate descriptive statistics exploration, methodology and theory development realistic predictive models abstract models multivariate descriptive statistics more detailed theoretical models univariate data modeling multivariate data modeling Data SEM

14. Example #1: Theodore Roosevelt Natl. Park Weed Problem

15. Larson & Grace (2004) Biol. Ctl. 29:207-214; Larson et al. (2007) Biol. Ctl. 40:1-8. spurge flea beetles Aphthona nigriscutus Aphthona lacertosa - beetles released since 1989 The Use of Biocontrol Insects On Leafy Spurge - data collected since 1999

16. Based on the Available Data, What Have the Beetles Been Doing? Spurge is in decline

17. r = -0.21, p = 0.01 Change in spurge density as a function of A. nigriscutis density r = -0.40, p < 0.001 Change in spurge density as a function of A. lacertosa density How Does Spurge Decline Relate to Beetle Density?

18. Multivariate View: Hypothesized Model A. lacertosa 2001 Number of Stems 2000 A. nigriscutis 2000 A. lacertosa 2000 A. nigriscutis 2001 Change in Stems 2000-2001

19. Change in Stems 2000-2001 A. nigriscutis 2001 Number of Stems 2000 A. nigriscutis 2000 A. lacertosa 2001 A. lacertosa 2000 R 2 = 0.54 R 2 = 0.61 -.20.66.57.31 -.14 -.55.23.51.17 -.24 R 2 = 0.42.08 Results for 2000 - 2001 note: raw correlation was r = -.21 note: raw correlation was r = -.40 ns

20. Example #2: Coastal Prairie Vegetation and Soil Properties Summary of Community Characteristics using Ordination Axis 1 Axis 2

21. Mg Mn N Ca Zn K P pH C Axis1 elev Axis2.77 -.23 -.35 Results from Stepwise Regression Analysis

22. SEM model results Mg Mn N Ca Zn K P pH C elev ELEV MINRL HYDR -.67.71 -.33 AXIS2 AXIS1 a2 a1.57 -.30 -.28 -.23 R 2 =.63 R 2 =.09

23. The Problem: A variety of theories about diversity lead to a similar set of bivariate expectations Example #3: Evaluating Theories of Diversity

24. Suspected Underlying Processes competitive exclusion competitive inhibition sampling effect Abiotic Conditions stress filtering Hetero- geneity facilitation of coexistence expansion of niche space Disturbance damage mortality recruitmentextinction Species Richness Species Lost Local Species Pool Biomass production biomass loss Biomass Removed Net Photosyn. niche complementarity

25. National Center for Ecological Analysis and Synthesis Project Finnish meadows Kansas prairie Louisiana prairie Minn. prairie Texas grasslands Louisiana marsh2 Indian tropical savanna Louisiana marsh1 Wisconsin prairie Miss. prairie Utah grassland Africa grassland

26. Abiotic Conditions stress filtering Disturbance damage mortality recruitment extinction Species Richness Species Lost Local Species Pool Biomass production biomass loss Biomass Removed Net Photosyn. Interpretations niche complementarity competitive exclusion

27. Collaborative Applications of Multivariate Modeling USGS - Numerous units and individuals Univ. California - Davis Univ. Northern Arizona Univ. North Carolina Univ. Alabama Univ. Minnesota Nat. Ctr. Ecol. Analysis Univ. New Mexico Purdue Univ. Univ. Texas - Arlington Michigan State Univ. Univ. Groenegen (The Netherlands) Syracuse Univ. Rice Univ. Univ. Houston LSU US Forest Service Colorado State Univ. Univ. California - Irving Oregon State Univ. Yale Univ. Univ. Wisc. - Eau Claire Univ. Connecticut Univ. Newcastle - UK Univ. Montpellier - France