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

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

5. 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. Adapted from Gotelli and Ellison 2004

7. Sourced.f.Sum of squaresMean squareF-ratiop-value Among groups a-1Determined from F- distribution with (a-1),a(n-1) d.f. Within groups a(n-1) Totalan-1 Adapted from Gotelli and Ellison 2004

8. Our statistical model

9. 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. Rev. Thomas Bayes 1702-1761 Prior Likelihood

11. Adapted from Clark 2007 Common RiskIndependent RiskHierarchical

12. Adapted from Clark 2007

13. 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. or

15. From Qian and Shen 2007

16. 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. Sourced.f.SSMSF- ratio p- value Treatment33.101.036.730.0068 Location31.010.342.190.101 Treatment* Location 91.24.14.880.5543 Residuals497.520.16

18. Sourced.f.SSMSF- ratio p- value Treatment33.101.036.730.0068 Location31.010.342.190.101 Treatment* Location 91.24.14.880.5543 Residuals497.520.16

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

20. ComparisonControl v. Foam Control v. Haliclona Control v. Tedania Foam v. Haliclona Foam v. Tedania Orthogonal contrasts p- value 0.03970.0020.00150.2580.0521 Tukey’s HSD p-value0.160.010.000010.660.21 Bonferroni adjusted pairwise t-test p-value 0.2380.0120.00091.000.313 Bayesian credible interval around the difference between 2 means (-0.68, 0.03)(-0.84, -0.12)(-0.91, -0.18)(-0.51, 0.21)(-0.58, 0.14)

21. 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. 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) What’s up now Fisher, Neyman- Pearson null hypothesis testing!?

24. Sourced.f.SSMSF- ratio p- value Plot22091548.90.0002 Genotype663100.60.72 Plot* Genotype 12227191.10.36 Year1113 6.50.012 Residuals106179017

25. Sourced.f.SSMSF- ratio p- value Plot22091548.90.0002 Genotype663100.60.72 Plot* Genotype 12227191.10.36 Year1113 6.50.012 Residuals106179017

26. Sourced.f.SSMSF- ratio p- value Plot22091548.90.0002 Genotype663100.60.72 Plot* Genotype 12227191.10.36 Year1113 6.50.012 Residuals106179017

27. 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) } } Nick Gotelli Robin Collins