1. Statistical Modeling and Analysis of MOFEP Chong He ( with John Kabrick, Xiaoqian Sun, Mike Wallendorf) Department of Statistics University of Missouri-Columbia

2. Outline Review current statistical analysis Review current statistical analysis Spatial structure Spatial structure Bayesian multivariate spatial modeling Bayesian multivariate spatial modeling Our progress and challenge Our progress and challenge New research: sampling design? New research: sampling design?

3. Current statistical analysis models for MOFEP studies Complete random block (Sheriff & He): Complete random block (Sheriff & He): -- using compartment as unit, -- using compartment as unit, -- 9 data points each year, -- 9 data points each year, -- 5 unknown parameters: 2 for blocks, 2 for -- 5 unknown parameters: 2 for blocks, 2 for treatments, and 1 for variance; treatments, and 1 for variance; Split-plot (Sheriff & He): Split-plot (Sheriff & He): -- using ELT as unit, -- using ELT as unit, -- to test treatment effect: 9 data points and 5 unknown -- to test treatment effect: 9 data points and 5 unknown -- to test ELT related effects: 18 data points & 10 unknown -- to test ELT related effects: 18 data points & 10 unknown (assume 2 ELT per compartment). (assume 2 ELT per compartment). Split-plot with repeated measurements (Sheriff & He): Split-plot with repeated measurements (Sheriff & He): -- using ELT as unit & repeat over year. -- using ELT as unit & repeat over year.

4. Current statistical analysis models for MOFEP studies (cont.) Meta-analysis (Gram et al): Meta-analysis (Gram et al): -- using compartment as unit, -- using compartment as unit, -- based on effective size d j =(M T - M C )/SD TC -- based on effective size d j =(M T - M C )/SD TC cumulative effective size d + cumulative effective size d + Others, such as regression & ANOVA: Others, such as regression & ANOVA: -- using sample plot as unit, -- using sample plot as unit, -- lots of data ( assume data points are independent), -- lots of data ( assume data points are independent), -- resulting large type I error (indicate a significant -- resulting large type I error (indicate a significant treatment effect when there is not), the error rate could treatment effect when there is not), the error rate could be as high as 40%. α =.05 is based on independency be as high as 40%. α =.05 is based on independency assumption. assumption.

5. Spatial structure Physical and biological variables observed in nature display spatial patterns (gradients and patches); Physical and biological variables observed in nature display spatial patterns (gradients and patches); Patterns may result either from deterministic processes or from processes causing spatial autocorrelation, or both; Patterns may result either from deterministic processes or from processes causing spatial autocorrelation, or both; Model 1 (spatial dependence): Model 1 (spatial dependence): y j = µ j + f (explanatory variables j ) + ε j y j = µ j + f (explanatory variables j ) + ε j Model 2 (spatial autocorrelation): Model 2 (spatial autocorrelation): y j = µ j + Σ i f (y i - µ y ) + ε j y j = µ j + Σ i f (y i - µ y ) + ε j Model 3 (combination of model 1&2): Model 3 (combination of model 1&2): y j = µ j + f 1 (explanatory variables j ) + Σ i f 2 (y i - µ y ) +ε j y j = µ j + f 1 (explanatory variables j ) + Σ i f 2 (y i - µ y ) +ε j Model 4 : explanatory variables j may themselves be modeled Model 4 : explanatory variables j may themselves be modeled by model 3. by model 3.

6. Bayesian multivariate spatial model Bayesian method Bayesian method likelihood f(y| θ) + prior (θ)  posterior (θ|y) likelihood f(y| θ) + prior (θ)  posterior (θ|y) -- all the inference are based on the posterior -- all the inference are based on the posterior -- informative & non-informative priors -- informative & non-informative priors Bayesian multivariate spatial model Bayesian multivariate spatial model y j = µ j + f 1 (explanatory variables j ) y j = µ j + f 1 (explanatory variables j ) + Σ i f 2 (y i - µ y ) +ε j, y j, µ j are vectors + Σ i f 2 (y i - µ y ) +ε j, y j, µ j are vectors priors on unknown parameters priors on unknown parameters -- latent variables: response variable and explanatory variables -- latent variables: response variable and explanatory variables may be measured at difference location or scale. may be measured at difference location or scale. -- Please discuss your research questions with us and we can -- Please discuss your research questions with us and we can help you! help you!

7. Our progress and challenge One Ph.D. student started to work on the modeling this semester. One Ph.D. student started to work on the modeling this semester. Transfer geo-data from GIS system to Splus Transfer geo-data from GIS system to Splus system. system. Start developing Bayesian spatial model on Start developing Bayesian spatial model on vegetation data. vegetation data. Challenge: too many variables to work with.

8. New research: sampling design? We may use the developed model to address the sampling problem such as: We may use the developed model to address the sampling problem such as: -- do we need more or less sample points? -- do we need more or less sample points? -- where to add more sample points? -- where to add more sample points? -- how often? -- how often?