**Bayesian modeling for ordinal substrate size using EPA** stream data A spatial model for ordered categorical data Megan Dailey Higgs Jennifer Hoeting Brian Bledsoe* Department of Statistics, Colorado State University *Department of Civil Engineering, Colorado State University

**Substrate size in streams** • Influences in-stream physical habitat • Often indicative of stream health • EPA collected data at 485 sites in Washington and Oregon between 1994 and 2004

**Data Collection Protocol** • At a site: • 11 transects x 5 points along each transect • Choose particle under the sharp end of a stick • Visually estimate and classify size

**Creating the response** • For a site: • Transform the original size classes to log10(Geometric Mean) for all sample points • Find the median for the site • Geometric mean

**The response** • Yi = median[log10(geometric mean)] for site i • Transformation provides a more symmetric, continuous-like variable • Typically modeled as a continuous variable • Predictive models have performed poorly • Response is an ordered categorical variable • 12 categories (6 with very few observations)

**Ordered categorical data** • Yi is a categorical response variable with K ordered values: {1,…,K} • Modeling objectives: • Explain the variation in the ordered response from covariate(s) • Incorporate the spatial dependence • Estimate, predict, and create maps of Pr(Yi≤ k) and Pr(Yi = k)

**Non-spatial model for ordered categorical data** Spatial model for binary and count data • Diggle, Tawn, & Moyeed (1998) • Gelfand & Ravishanker (1998) Albert & Chib (1993, 1997) • Generalized geostatistical models with a latent • Gaussian process • Metropolis Hastings within Gibbs sampling • approach Formulating the spatial model Spatial model for ordered categorical data + =

**Latent variable formulation ** • Define latent variable, Zi, such that Zi = Xi’β + εi • εi ~ N(0,1) for the probit model • εi ~ Standard Logistic for logit model • Define the categorical response, Yi = {1,…,K}, using Zi and ordered cut-points, θ= (θ1 , … ,θK-1), where 0 = θ1 < θ2 < … < θK-1 < θK = ∞ Yi = 1 if Zi < θ1 Yi = k if θk-1 ≤ Zi < θk Yi = K if Zi ≥ θK-1

**Latent variable formulation** • Thus, Pr(Yi ≤ k | θ, β) = Pr(Zi< θk) Pr(Yi = k | θ, β) = Pr(θk-1 ≤ Zi < θk) • If Z ~ N(Xi’β, 1), then Pr(Yi ≤ k | θ, β) = Φ(θk – Xi’β) Pr(Yi = k | θ, β) = Φ(θk – Xi’β) - Φ(θk-1 – Xi’β) where Φ is the N(0,1) cdf

**Spatial cumulative model** • Zi = Xi’β + Wi + εi is the latent variable • where εi ~ N(0,1) W~ N(0, s2H(d)) (H(d))ij = r(si-sj;d) Zi | β, Wi ~ N(Xi’β + Wi , 1) • Pr(Yi≤ k | β, θ, Wi) = Pr(Zi < θk) = Φ(θk – Xi’β - Wi) Where θ= (θ1 , … ,θK) is a vector of cut-points such that 0 = θ1 < θ2 < … < θK-1 < θK = ∞

**Fitting the spatial model** • The likelihood • Estimating b= (b0, b1), g= (s2, d) , θ= (θ2, … ,θK-1) • Transform θ to a real-valued, unrestricted cut-points: a = (a2 , ... , aK-1) where a2 = log(θ2) ak= log(θk – θk-1) • MCMC sampling • Metropolis-Hastings within Gibbs sampling • Prior: • b – flat and conjugate Normal • s2 and d – Independent uniform priors • a - multivariate normal

**Simulated data** • Simulated data at a subset of the original locations (n = 82) • Cluster infill around the 82 sites (n=120) • Spatial process: • W is a stationary Gaussian process with E[W(s)]=0 and Cov[W(si),W(sj)] = s2r(si-sj;d) • Exponential correlation function: r(d) = exp(-dd) • Covariate: • Distance weighted stream power

**Preliminary Results** • Posterior quantities • Based on 1000 iterations (burn-in = 1000)

**Posterior mean of the spatial process**

**Posterior SD of the spatial process**

**Posterior mean and SD for Pr(Yi = 2)**

**Posterior mean and SD for Pr(Yi = 5)**

**Posterior mean and SD for Pr(Yi≤ 5)**

**Future Work** • Convergence and mixing for the spatial model • Models and methods for large data sets • Spectral parameterization of the spatial process • Wikle (2002), Paciorek & Ryan (2005), Royle & Wikle (2005) • Importance sampling • Gelfand & Ravishanker (1998), Gelfand, Ravishanker, & Ecker (2000) • Sub-sampling • Investigate different spatial correlation functions and distance metrics • Traditional • Stream based • Model selection for the spatial model

**CR-829095** Funding and Affiliations FUNDING/DISCLAIMER The work reported here was developed under the STAR Research Assistance Agreement CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA. The views expressed here are solely those of the authors and STARMAP, the Program they represent. EPA does not endorse any products or commercial services mentioned in this presentation. Megan’s research is also partially supported by the PRIMES National Science Foundation Grant DGE-0221595003.

**Thank you**

**Subset of data (nsmall = 82)**

**Sample path plot - Example**

**Surface for estimating g=(s2,d)**

**Sample path plot – Avoiding plateau**