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Spatially structured random effects

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1 Spatially structured random effects
by Daniel A. Griffith Ashbel Smith Professor of Geospatial Information Sciences

2 ABSTRACT Researchers increasingly are employing random effects modeling to analyze data. When data are georeferenced, a random effect term needs to be spatially structured in order to account for spatial autocorrelation. Spatial structuring can be achieved in various ways, including the use of semivariogram, spatial autoregressive, and spatial filter models. SAS implements the semivariogram option for linear mixed models. GeoBUGS implements the spatial autoregressive option for either linear or generalized linear mixed modeling. Recently developed spatial filtering methodology can be used in either case, as well as with the SAS generalized linear mix model procedure, and furnishes one means of estimating space-time mixed models. This presentation summarizes comparisons of these three forms of spatial structuring, illustrating implementations with selected ecological data for the municipalities of Puerto Rico.

3 From Legendre Spatial structures in communities indicate that some process hasbeen at work to create them. Two families of mechanisms cangenerate spatial structures in communities: • Autocorrelation model: the spatial structures are generated by thespecies assemblage themselves (response variables). • Induced spatial dependence model: forcing (explanatory) variablesare responsible for the spatial structures found in the speciesassemblage. They represent environmental or biotic control of thespecies assemblages, or historical dynamics To understand the mechanisms that generate these structures, we needto explicitly incorporate the spatial community structures, at all scales, into the statistical model. Spatial autocorrelation (SA) is technically defined as the dependence, due to geographic proximity, present in the residuals of a [regression-type] model of a response variable y whicht akes into account all deterministic effects due to forcing variables.

4 Spatial autocorrelation can be interpreted in different ways
As a spatial process mechanism – the cartoon As a diagnostic tool – the Cliff-Ord Eire example (the model specification should be nonlinear) As a nuisance parameter – eliminating spatial dependency to avoid statistical complications As a spatial spillover effect – georeferencing of pediatric lead poisoning cases in Syracuse, NY As an outcome of areal unit demarcation – the modifiable areal unit problem (MAUP) As redundant information – spatial sampling; map interpolation As map pattern – spatial filtering (to be discussed in this course) As a missing variables indicator/surrogate – a possible implication of spatial filtering As self-correlation – what is discussed next

5 The magic box is a physical model of spatial autocorrelation

6 The permutation perspective

7 The SASIM game

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11 Measures of spatial autocorrelation
MC: Moran Coefficient; GR: Geary Ratio; semivariogram Spherical Exponential Bessel function (1st order, 2nd kind)

12 Georeferenced data scatterplots
The horizontal axis is the measurement scale for some attribute variable The vertical axis is the measurement scale for neighboring values (topological distance-based) of the same attribute variable OR The horizontal axis is (usually) Euclidean distance between geocoded locations The vertical axis is the measurement scale for geographic variability

13 Describing a scatterplot trend
positive relationship: High Y with High X & Medium Y with Medium X & Low Y with Low X negative relationship: High Y with Low X & Medium Y with Medium X & Low Y with High X

14 Description of the Moran scatterplot
Positive spatial autocorrelation - high values tend to be surrounded by nearby high values - intermediate values tend to be surrounded by nearby intermediate values - low values tend to be surrounded by nearby low values 2002 population density MC = 0.49 GR = 0.58

15 Description of the Moran scatterplot
Negative spatial autocorrelation - high values tend to be surrounded by nearby low values - intermediate values tend to be surrounded by nearby intermediate values - low values tend to be surrounded by nearby high values competition for space MC = -0.16 sMC = 0.075 GR = 1.04

16 Graphical portrayals of spatial autocorrelation latent in transformed As data

17 Constructing eigenfunctions for filtering spatial autocorrelation out of georeferenced variables:
Moran Coefficient = (n/1T C1)x YT(I – 11T/n)C (I – 11T/n)Y/ YT(I – 11T/n)Y the eigenfunctions come from (I – 11T/n)C (I – 11T/n)

18 Random effects model is a random observation effect (differences among individual observational units) is a time-varying residual error (links to change over time) The composite error term is the sum of the two.

19 Random effects model: normally distributed intercept term
~ N(0, ) and uncorrelated with covariates supports inference beyond the nonrandom sample analyzed simplest is where intercept is allowed to vary across areal units (repeated observations are individual time series) The random effect variable is integrated out (with numerical methods) of the likelihood fcn accounts for missing variables & within unit correlation (commonality across time periods)

20 Spatial structuring of random effects
CAR: conditional autoregressive model ICAR: improper conditional autoregressive model (spatial autocorrelation set to 1,and a spatially structured and a spatially unstructured variance component is estimated)─should be specified as a convolution prior (spatially structured & unstructured random effects) SF: spatial filter identified with a frequentist GLM

21 Frequentist Bayesian Definition of probability Long-run expected frequency in repeated (actual or hypothetical) experiments (Law of LN) Relative degree of belief in the state of the world Point estimate Maximum likelihood estimate Mean, mode or median of the posterior probability distribution Uncertainty intervals for parameters “confidence intervals” based on the Likelihood Ratio Test (LRT)  i.e., the expected probability distribution of the maximum likelihood estimate over many experiments “credible intervals” based on the posterior probability distribution

22 Uncertainty intervals of non-parameters
Based on likelihood profile/LRT, or by resampling from the sampling distribution of the parameter Calculated directly from the distribution of parameters Model selection Discard terms that are not significantly different from a nested (null) model at a previously set confidence level Retain terms in models, on the argument that processes are not absent simply because they are not statistically significant Difficulties Confidence intervals are confusing (range that will contain the true value in a proportion α of repeated experiments); rejection of model terms for “non-significance” Subjectivity; need to specify priors

23 Impact of sample size prior distribution likelihood distribution As the sample size increases, a prior distri-bution has less and less impact on results; BUT effective sample size for spatially autocorrelated data

24 Bayesian inference Using Gibbs Sampling
What is BUGS? Bayesian inference Using Gibbs Sampling is a piece of computer software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods. It grew from a statistical research project at the MRC BIOSTATISTICAL UNIT in Cambridge, but now is developed jointly with the Imperial College School of Medicine at St Mary’s, London.

25 BUGS Classic BUGS WinBUGS (Windows Version) GeoBUGS (spatial models) PKBUGS (pharmokinetic modeling) The Classic BUGS program uses text-based model description and a command-line interface, and versions are available for major computer platforms (e.g., Sparc, Dos). However, it is not being further developed.

26 What is WinBUGS? WinBUGS, a windows program with an option of a graphical user interface, the standard ‘point-and-click’ windows interface, and on-line monitoring and convergence diagnostics. It also supports Batch-mode running (version 1.4). GeoBUGS, an add-on to WinBUGS that fits spatial models and produces a range of maps as output. PKBUGS, an efficient and user-friendly interface for specifying complex population pharmacokinetic and pharmacodynamic (PK/PD) models within the WinBUGS software.

27 What is GeoBUGS? Available via bugs/winbugs/geobugs.shtml Bayesian inference is used to spatially smooth the standardized incidence ratios using Markov chain Monte Carlo (MCMC) methods. GeoBUGS implements models for data that are collected within discrete regions (not at the individual level), and smoothing is done based on Markov random field models for the neighborhood structure of the regions relative to each other.

28 where qi is known but C is unknown and too horrible to calculate.
What is MCMC? MCMC is used to simulate from some distribution p known only up to a constant factor, C: pi = Cqi where qi is known but C is unknown and too horrible to calculate. MCMC begins with conditional (marginal) distributions, and MCMC sampling outputs a sample of parameters drawn from their posterior (joint) distribution.

29 The geographic distribution of elevation across the island of Puerto Rico
From a USGS DEM containing 87,358,136 points. Darkness of gray scale is directly proportional to elevation.

30 SAS PROC MIXED summary results for a quadratic gradient LMM: LN( + 17
Semivario-gram model none spherical expo-nential Gaussian power Bessel Variance (nugget) --- 0.0331 0.2151 0.2210 0.2514 0.2450 Spatial correlation < 0.7643 0.5730 0.2702 0.6089 Residual 0.217 0.169 0.0253 0.0057 b0 6.080*** 6.157*** 6.201*** 6.202*** bu2 -0.349*** -0.463*** -0.486*** buv -0.263*** -0.254** -0.255** -0.257** bv -0.270*** -0.114 -0.168 bv2 -0.527*** -0.529*** -0.561*** -0.569***

31 The average random effects term example MCMC chain from a WinBUGS run
ICAR spatial filter (SF)

32 WinBUGS: geographic distributions of unstructured (left) and spatially structured (right) random effects spatial filter (SF) WinBUGS: ICAR

33 Comparative parameter estimates for a LMM quadratic gradient description of LN( + 17.5)
SAS semivariogram (Bessel) model SAS SF GeoBUGS-ICAR WinBUGS-SF (100 weeded replications) estimate se b0 6.1906 0.287 6.1101 0.055 6.5175 0.168 0.061 bu2 0.122 0.031 0.153 0.035 buv 0.123 0.030 0.101 bv2 0.125 0.032 0.037 var 0.0055 0.019 --- 0.0049 0.007 0.0305 0.024 varure 0.2856 0.091 0.0001 0.0047 0.0318 0.025 varssre 0.7205 0.221 0.0282 0.4854 0.093 0.0301 varure denotes the variance of the unstructured random effects varssre denotes the variance of the spatially structured random effects

34 binomial GLMM random effects
SAS SF WinBUGS SF WinBUGS ICAR

35 WinBUGS (100 weeded replications)
SF GLMM SAS NLMIXED (SF) WinBUGS (100 weeded replications) SF ICAR statistic estimate se b0 0.2624 0.2852 0.2419 0.0013 0.0014 3.0646 1.1559 3.0600 1.3632 *** 3.2182 1.3433 3.0116 1.4256 0.0015 0.0054 0.0066 0.7045 0.7144 0.3727 0.9787 0.9783 0.9583 P(S-W) <0.0001 < MCss 0.967 0.975 0.787 GRss 0.158 0.154 0.177 0.119 0.132 0.036 1.045 1.000 1.129 0.356 0.357 0.388 0.739 0.696 0.001 0.011 -0.001 -0.009 0.022

36 Graphical diagnostics of residuals for the GLMM estimated with SAS

37 Scatterplot of the SAS and mean WinBUGS estimated spatially structured random effects terms

38 Individual GLMM estimation results for each Puerto Rican sugar cane crop year
Individ-ual SF eigen-vector #s Raw per-centages # 0s Point-in-time estimation MC GR -a - P(S-W) 1965/66 1,4,6,24 0.484 0.458 3 1.1959 0.0064 0.0020 1.0135 0.0055 1966/67 0.490 0.454 4 1.3786 0.0060 0.0021 1.1138 0.0027 1967/68 0.474 0.434 6 1.7354 0.0050 0.0018 1.0856 0.0017

39 Space-time data: preliminaries
random effect (re) re + ss: 1965/66 re + ss: 1966/67 re + ss: 1967/68 Random effects term is constant across time; spatial structuring changes over time

40 Space-time GLMM: Puerto Rican sugar cane crop years 1965/ /68 when all fixed effects are year-specific statistic crop year 1965/66 crop year 1966/7 crop year 1967/68 estimate se b0 0.2336 0.0009 4.5040 1.1684 4.5785 4.9226 4.9713 1.2432 5.3372 5.8053 1.1620 1.1621 1.0994 1.0995 pseudo-R2 0.9950 0.9976 0.9929 0.449 0.467 0.493 0.580 0.562 0.537 MCresiduals 0.019 0.042 0.009 GRresiduals 0.916 0.839 0.808

41 Discussion & Implications
All three common specifications of spatial structuring—semivariogram, spatial autoregressive and SF models—for a random effect term in mixed statistical models perform in an equivalent fashion. Matching Bayesian model priors with their implicit frequentist counterparts yields estimation results from both approaches that are essentially the same. making use of spatially structured random effects tends to furnish an alternative to quasi-likelihood estimation techniques for GLMMs

42 Semivariogram models offer a geostatistical theoretical basis and have been implemented in SAS for LMMs. A spatial statistics practitioner with the necessary computer programming skills can employ WinBUGS in order to utilize them with GLMMs. Spatial autoregressive modeling offers a theoretical basis for spatial structuring, and is available in GeoBUGS. This would be very difficult to trick SAS into doing. Spatial filtering, which can be derived from spatial autoregressive model specifications, tends to be more exploratory in nature (being akin to principal components analysis) can be implemented in either SAS or WinBUGS for either LMMs or GLMMs, and can be easily extended to space-time datasets with either of these software packages.

43 Illustrative Puerto Rico sugar cane examples tend to have a random effect term that virtually equates to the corresponding LMM/GLMM residual variate. This is not always the case, as is highlighted by the extension of a GLMM specification to a space-time sugar cane dataset. All of the estimated random effects terms for the various Puerto Rico examples tend to be non-normal.

44 once a random effect term has been estimated with a frequentist approach, using it when calculating a deviance statistic allows its number of degrees of freedom to be approximated for GLMMs. Although n values are estimated, because they are correlated, the resulting number of degrees of freedom is less than n. This particular finding should help spatial statistics practitioners better understand the cost of employing a statistical mixed model.

45 A df aside: future research
Spiegelhalter et al. (2002) address the df problem for complex hierarchical models in which the number of parameters is not clearly defined because, for instance, of the presence of random effects. An information-theoretic argument is used to approximate the effective number of parameters in a model, equivalent to the trace of the product of the Fisher information and the posterior covariance matrices. this particular approximation is equivalent to the trace of the ‘hat’ matrix for linear models with a normally distributed error term.

46 k dfs for random effects


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