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§❻ Hierarchical (Multi-Stage) Generalized Linear Models

§❻ Hierarchical (Multi-Stage) Generalized Linear Models. Robert J. Tempelman. Introduction. Some inferential problems require non-classical approaches; e.g. Heterogeneous variances and covariances across environments.

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§❻ Hierarchical (Multi-Stage) Generalized Linear Models

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  1. §❻ Hierarchical (Multi-Stage) Generalized Linear Models Robert J. Tempelman

  2. Introduction • Some inferential problems require non-classical approaches; e.g. • Heterogeneous variances and covariances across environments. • Different distributional forms (e.g. heavy-tailed or mixtures for residual/random effects). • High dimensional variable selection models • Hierarchical Bayesian modeling provides some flexibility for such problems.

  3. Heterogeneous variance models(Kizilkaya and Tempelman, 2005) • Consider a study involving different subclasses (e.g. herds). • Mean responses are different. • But suppose residual variances are different too. • Let’s discuss in context of LMM (linear mixed model)

  4. Recall linear mixed model • Given: has a certain “heteroskedastic” specification. • determines the nature of heterogeneous residual variances

  5. Modeling Heterogeneous Variances • Suppose • with as a “fixed” intercept residual variance • gk > 0 kthfixed scaling effect. • vl > 0 lthrandom scaling effect.

  6. Subjective and Subjective Priors • “intercept” variance : subjective flat or conjugate vague inverted-gamma (IG) prior • Invoke typical constraints for “fixed effects” • Corner parameterization: gs= 1. • Flat or vague IG prior p(gk); k=1,2,..,s • Structural prior for “random effects” • i.e., vl ~ IG(a, a-1). • E(vl)=1; afunctions like a “variance component” for residual variances.-> hyperparameter

  7. Remaining priors • “Classical” random effects • “Classical” fixed effects • “Classical” random effects VC • Hyperparameter (Albert, 1988) SAS PROC MCMC doesn’t seem to handle this…prior can’t be written as function of corresponding parameter

  8. What was the last prior again??? Uniform(0,1) on Uniform(0,1) on Rosa et al. (2004) Different diffuse priors can have different impacts on posterior inferences!...if data info is poor

  9. Joint Posterior Density • LMM:

  10. Details on FCD • All provided by Kizilkaya and Tempelman (2005) • All are recognizeable except for av: • Use Metropolis-Hastings random walk on using normal as proposal density. • For MH, generally a good idea to transform parameters so that parameter space is entire real line…but don’t forget to include Jacobian of transform.

  11. Small simulation study • Two different levels of heterogeneity: • ae= 5, ae= 15 • = 1 • Two different average random subclass sizes: • ne = 10 vs. ne = 30 • 20 subclasses (habitats) in total • Also modeled fixed effects: • Sex (2 levels) for location and dispersion (g1=2, g2=1). • Additional set of random effects: • 30 levels (e.g. sires) cross-classified with habitats.

  12. PROC MIXED code • “Fixed” effects models for residual variances • REML estimates of “herd” variances expressed relative to average. procmixed data=phenotype; class sireidhabitatidsexid; model y = sexid; random intercept /subject = habitatid ; random intercept /subject = sireid; repeated / local = exp(sexidhabitatid); ods output covparms=covparms; run; Models but treats vlas a fixed effect.

  13. MCMC analyses (code available online)Posterior summaries on ae.

  14. MCMC (₀) and REML (•) estimates of subclass residual variances vs. truth (vl) ae=5;ne=10 ae=15;ne=10 High shrinkage situation ae=5;ne=30 ae=15;ne=30 Low shrinkage situation

  15. Heterogeneous variances for ordinal categorical data • Suppose we had a situation where residual variances were heterogeneous on the underlying latent scale • i.e., greater frequency of extreme vs. intermediate categories in some subclasses

  16. Heterogeneous variances for ordinal categorical data? • On liability scale: has a certain “heteroskedastic” specification. • determines the nature of heterogeneous variances

  17. Cumulative probit mixed model (CPMM) • For CPMM, l maps to Y:

  18. Modeling Heterogeneous Variances in CPMM • Suppose • With as a “fixed” reference residual variance • gk > 0 kthfixed scaling effect. • vl > 0 lthrandom scaling effect. • All other priors same as with LMM

  19. Joint Posterior Density in CPMM • CPMM:

  20. Another small simulation study • Two different levels of heterogeneity: • ae = 5, ae= 15 • Average random subclass size: ne= 30 • 20 subclasses (habitats) in total • Also modeled fixed effects: • Sex (2 levels) for location and dispersion. • Additional set of random effects: • 30 levels (e.g. sires) cross-classified with habitats. • Thresholds: t1 = -1, t1 = 1.5

  21. ae = 5; ne = 30 • ae = 15; ne = 30 ESS = 391 ESS = 1422

  22. Posterior means of subclass residual variances vs. truth (vl) ae=5;ne=30 ae=15;ne=30 No PROC GLIMMIX counterpart Another alternative: Heterogeneous thresholds!!! (Varona and Hernandez, 2006)

  23. Additional extensions • PhD work by Fernando Cardoso • Heterogeneous residual variances as functions of multiple fixed effects and multiple random effects. • Heterogeneous t-error (Cardoso et al., 2007). • Helps separates outliers from high variance subclasses from effects of outliers. • Other candidates for distribution of wj lead to alternative heavy-tailed specifications (Rosa et al., 2004) t-error is outlier robust

  24. Posterior densities of breed-group heritabilities in multibreed Brazilian cattle (Fernando Cardoso) Based on homogeneous residual variance (Cardoso and Tempelman, 2004) Based on heterogeneous residual variances (Fixed: breed additive&dominance,sex; Random: CG (Cardoso et al., 2005) Some of most variable herds were exclusively Herefords • Estimated CV of CG-specific s2e→ 0.72±0.06 • F1s2e = 0.70±0.16 purebred s2e

  25. Heterogeneous G-side scale parameters • Could be accommodated in a similar manner. • In fact, the borrowing of information across subclasses in estimating subclass-specific random effects variances is even more critical. • Low information per subclass? REML estimates will converge to zero.

  26. Heterogeneous bivariate G-side and R-side inferences! Bello et al. (2010, 2012) Investigated herd-level and cow-level relationship between 305-day milk production and calving interval (CI) as a function of various factors CG (herd-year) effects Residual (cow) effects

  27. Herd-Specific and Cow- Specific (Co)variances Herd k Let Cow j and

  28. Rewrite this Herd k Cow j Model each of these different colored terms as functions of fixed and random effects (in addition to the classical b and u)!

  29. bST effect on Herd-Level Association betweenMilk Yield and Calving Interval a,b P < 0.0001 -1.37b 0.01a 0.07a days per 100 kg milk yield bST: Bovine somatotropin 0% <50% ≥50% % herd on bST supplementation

  30. Number of times milking/day on Cow-level Association betweenMilk Yield and Calving Interval Overall Antagonism 0.51±0.01 day longer CI per 100 kg increase in cumulative 305-d milk yield 0.57a 0.45b days per 100 kg milk yield 2X 3+X Daily Milking Frequency a,b P < 0.0001

  31. 0.0 0.2 0.4 0.6 0.8 1.0 Increase in # of days of CI / 100 kg herd milk yield Variability between Herds for(Random effects) • DICM0 – DICM1= 243 • Expected range between extreme herd-years ± 2 = 0.7 d / 100 kg 0.7 d/100kg 0.16 0.86 Ott and Longnecker, 2001

  32. Whole Genome Selection (WGS) m >>>>>n • Model: i= 1,2,…,n. Residual effects (e.g. age, parity) Phenotypes SNP allelic substitution effects Polygenic Effects Genotypes Genotypes Phenotype Animal LD (linkage disequilibrium)

  33. Typical WGS specifications • Random effects spec. on g (Meuwissen et al. 2001) • BLUP: • BayesA/B: BayesA = BayesB with p = 0. • “Random effects/Bayes” modeling allows m >> n • Borrowing of information across genes.

  34. First-order antedependence-specifications (Yang and Tempelman, 2012) • Instead of independence, specify first order antedependence: Correlation SNP 1 SNP 2 SNP 3 SNP 4 Random effects modeling: facilitates borrowing of information across SNP intervals Ante-BayesB Ante-BayesA = Ante-BayesB with p = 0

  35. Results from a simulation study Accuracy of Genomic EBV vs. LD level • Advantage of Ante-BayesA/B over conventional BayesA/B increases with increasing marker density (LD = linkage disequilbrium) P<.001 Bayes A/B vs. AnteBayesA/B (r2)

  36. Other examples of multi-stage hierarchical modeling? • Spatial variability in agronomy using t-error (Besag and Higdon, 1999) • Ecology (Cressie et al., 2009). • Conceptually, one could model heterogeneous and spatially correlated overdispersion parameters in Poisson/binomial GLMM as well!

  37. What I haven’t covered in this workshop • Model choice criteria • Bayes factors (generally, too challenging to compute) • DIC (Deviance information criteria) • Bayesian model averaging • Advantage over conditioning on one model (e.g. for multiple regression involving many covariates) • Posterior predictive checks. • Great for diagnostics • Residual diagnostics based on latent residuals for GLMM (Johnson and Albert, 1999).

  38. Some closing comments/opinions • Merit of Bayesian inference • Marginalfor LMM with classical assumptions. • GLS with REML seems to work fine. • Of greater benefit for GLMM • Especially binary data with complex error structures • Greatestbenefit for multi-stage hierarchical models. • Larger datasets nevertheless requiredthan with more classical (homogeneous assumptions).

  39. Implications • Increased programming capabilities/skills are needed. • Cloud/cluster computing wouldn’t hurt. • Don’t go in blind with canned Bayesian software. • Watch the diagnostics (e.g. trace plots) like a hawk! • Don’t go on autopilot. • WinBugs/PROC MCMC works nicely for the simpler stuff. • Highly hierarchical models require statistical/algorithmic insights…do recognize limitations in parameter identifiability (Cressie et al., 2009)

  40. National Needs PhD FellowshipsMichigan State University • Focus: • Integrated training in quantitative, statistical and molecular genetics, and breeding of food animals • Features: • Research in animal genetics/genomics with collaborative faculty team • Industry internship experience • Public policy internship in Washington, DC • Statistical consulting center experience • Teaching or Extension/outreach learning opportunities • Optional affiliation with inter-departmental programs in Quantitative Biology, Genetics, others • Faculty Team: • C. Ernst, J. Steibel, R. Tempelman, R. Bates, H. Cheng, T. Brown, B. Alston-Mills • Eligibility is open to citizens and nationals of the US. Women and underrepresented groups are encouraged to apply.

  41. Thank You!!! • Any more questions??? http://actuary-info.blogspot.com/2011/05/homo-actuarius-bayesianes.html

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