Bayesian ANOVA

1. Bayesian ANOVA Or how to learn what you know all over again but different

2. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

3. History of ANOVA Ronald Fisher, 1956 John Bennet Lawes: Founder Rothamsted Experimental station 1843 Harvesting of Broadbalk field, the source of the data for Fisher’s 1921 paper on variation in crop yields.

4. History of ANOVA Cover page from his 1925 book formalizing ANOVA methods Excerpt from Studies in Crop Variation: An examination of the yield of dressed grain from Broadbalk Journal of Agriculture Science , 11 107-135, 1921 Table from chapter 8 of Statistical Methods for Research Workers, On the analysis of randomize block designs.

5. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

6. The Math of ANOVA Adapted from Gotelli and Ellison 2004

7. The Math of ANOVA Adapted from Gotelli and Ellison 2004

8. The Math of ANOVA Our statistical model Adapted from Gotelli and Ellison 2004

9. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

10. Bayes Theorem Rev. Thomas Bayes 1702-1761 Prior Likelihood

11. Bayes Theorem:Hierarchical Bayes Common Risk Independent Risk Hierarchical Adapted from Clark 2007

12. Bayes Theorem:Hierarchical Bayes Adapted from Clark 2007

13. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

14. Anatomy of Bayesian ANOVA or

15. Anatomy of Bayesian ANOVA From Qian and Shen 2007

16. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

17. Compare and Contrast!

18. Compare and Contrast!

19. Compare and Contrast! Lines represent 95% credible intervals for Bayesian estimates and confidence intervals for frequentist.

20. Compare and Contrast!

21. Where we're going • History of ANOVA • The Math of ANOVA • Bayes Theorem • Anatomy of Baysian ANOVA • Compare and Contrast! • Rumble in the Jungle: Advantages of Bayes • Real World 13: Genotype and Frequency Dependence in an invasive grass.

22. Rumble in the Jungle:the advantages of Bayes What’s up now Fisher, Neyman-Pearson null hypothesis testing!? • Avoids the muddled idea of fixed vs. random effects, treating all effects as random. • Provides estimates of effects as well as variance components with corresponding uncertainty. • Allows more flexibility in model construction (e.g. GLM’s instead of just normal models) • Issues such as normality, unbalanced designs, or missing values are easily handled in this framework. • You just don’t believe in p-values (uniformative, etc, see Anderson et al 2000)

23. Real World 13:genotype frequency dependence

24. Real World 13:genotype frequency dependence

25. Real World 13:genotype frequency dependence

26. Real World 13:genotype frequency dependence

27. Thanks! model { for( i in 1:n){ y[i] ~ dnorm(y.mu[i],tau.y) y.mu[i] <- mu + delta[plottype[i]] + gamma[studyyear[i]] + nu[gens[i]] + interact[plottype[i],gens[i]] } mu ~ dnorm(0,.0001) tau.y <- pow(sigma.y,-2) sigma.y ~ dunif(0,100) mu.adj <- mu + mean(delta[])+mean(gamma[]) +mean(nu[])+mean(interact[,]) #compute finite population standard deviation for(i in 1:n){ e.y[i] <- y[i] - y.mu[i]} s.y <- sd(e.y[]) xi.d ~dnorm(0,tau.d.xi) tau.d.xi <- pow(prior.scale.d,-2) for(k in 1:n.plottype){ delta[k] ~ dnorm(mu.d,tau.delta) d.adj[k] <- delta[k] - mean(delta[]) for(z in 1:n.gens) { interact[k,z]~dnorm(mu.inter,tau.inter) } } Robin Collins Nick Gotelli