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

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.

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.

Excerpt from Studies in Crop Variation: An examination of the yield of dressed grain from Broadbalk Journal of Agriculture Science, , 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.

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.

Adapted from Gotelli and Ellison 2004

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

Our statistical model

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.

Rev. Thomas Bayes Prior Likelihood

Adapted from Clark 2007 Common RiskIndependent RiskHierarchical

Adapted from Clark 2007

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.

or

From Qian and Shen 2007

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.

Sourced.f.SSMSF- ratio p- value Treatment Location Treatment* Location Residuals

Sourced.f.SSMSF- ratio p- value Treatment Location Treatment* Location Residuals

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

ComparisonControl v. Foam Control v. Haliclona Control v. Tedania Foam v. Haliclona Foam v. Tedania Orthogonal contrasts p- value Tukey’s HSD p-value Bonferroni adjusted pairwise t-test p-value 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)

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.

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!?

Sourced.f.SSMSF- ratio p- value Plot Genotype Plot* Genotype Year Residuals

Sourced.f.SSMSF- ratio p- value Plot Genotype Plot* Genotype Year Residuals

Sourced.f.SSMSF- ratio p- value Plot Genotype Plot* Genotype Year Residuals

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