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

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.

6. Adapted from Gotelli and Ellison 2004

7. Adapted from Gotelli and Ellison 2004 Source d.f. Sum of squares Mean square F-ratio p-value Among groups a-1 Determined from F-distribution with (a-1),a(n-1) d.f. Within groups a(n-1) Total an-1

8. Adapted from Gotelli and Ellison 2004 Our statistical model

10. Rev. Thomas Bayes 1702-1761 Prior Likelihood

11. Adapted from Clark 2007 Common Risk Independent Risk Hierarchical

12. Adapted from Clark 2007

14. or

15. From Qian and Shen 2007

17. Source d.f. SS MS F-ratio p-value Treatment 3 3.10 1.03 6.73 0.0068 Location 3 1.01 0.34 2.19 0.101 Treatment* Location 9 1.24 .14 .88 0.5543 Residuals 49 7.52 0.16

18. Source d.f. SS MS F-ratio p-value Treatment 3 3.10 1.03 6.73 0.0068 Location 3 1.01 0.34 2.19 0.101 Treatment* Location 9 1.24 .14 .88 0.5543 Residuals 49 7.52 0.16

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

20. Comparison Control v. Foam Control v. Haliclona Control v. Tedania Foam v. Haliclona Foam v. Tedania Orthogonal contrasts p-value 0.0397 0.002 0.0015 0.258 0.0521 Tukey’s HSD p-value 0.16 0.01 0.00001 0.66 0.21 Bonferroni adjusted pairwise t-test p-value 0.238 0.012 0.0009 1.00 0.313 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)

24. Source d.f. SS MS F-ratio p-value Plot 2 209 154 8.9 0.0002 Genotype 6 63 10 0.6 0.72 Plot* Genotype 12 227 19 1.1 0.36 Year 1 113 113 6.5 0.012 Residuals 106 1790 17

25. Source d.f. SS MS F-ratio p-value Plot 2 209 154 8.9 0.0002 Genotype 6 63 10 0.6 0.72 Plot* Genotype 12 227 19 1.1 0.36 Year 1 113 113 6.5 0.012 Residuals 106 1790 17

26. Source d.f. SS MS F-ratio p-value Plot 2 209 154 8.9 0.0002 Genotype 6 63 10 0.6 0.72 Plot* Genotype 12 227 19 1.1 0.36 Year 1 113 113 6.5 0.012 Residuals 106 1790 17

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