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Or how to learn what you know all over again but different
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
Ronald Fisher, 1956 John Bennet Lawes: Founder Rothamsted  Experimental station 1843 Harvesting of Broadbalk field, the so...
Excerpt from  Studies in Crop Variation: An examination of the yield of dressed grain from Broadbalk  Journal of Agricultu...
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
Adapted from Gotelli and Ellison 2004
Adapted from Gotelli and Ellison 2004 Source d.f. Sum of squares Mean square F-ratio p-value Among groups a-1 Determined f...
Adapted from Gotelli and Ellison 2004 Our statistical model
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
Rev. Thomas Bayes 1702-1761 Prior Likelihood
Adapted from Clark 2007 Common Risk Independent Risk Hierarchical
Adapted from Clark 2007
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
or
From Qian and Shen 2007
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
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 ...
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 ...
Lines represent 95% credible intervals for Bayesian estimates and confidence intervals for frequentist.
Comparison Control v.  Foam Control v. Haliclona Control v.  Tedania Foam v. Haliclona Foam v. Tedania Orthogonal contrast...
<ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of ...
<ul><li>Avoids the muddled idea of fixed vs. random  </li></ul><ul><li>effects, treating all effects as random. </li></ul>...
 
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 Ye...
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 Ye...
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 Ye...
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]]...
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Bayesian anova

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Bayesian anova

  1. 1. Or how to learn what you know all over again but different
  2. 2. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  3. 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. 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. 5. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  6. 6. Adapted from Gotelli and Ellison 2004
  7. 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. 8. Adapted from Gotelli and Ellison 2004 Our statistical model
  9. 9. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  10. 10. Rev. Thomas Bayes 1702-1761 Prior Likelihood
  11. 11. Adapted from Clark 2007 Common Risk Independent Risk Hierarchical
  12. 12. Adapted from Clark 2007
  13. 13. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  14. 14. or
  15. 15. From Qian and Shen 2007
  16. 16. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  17. 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. 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. 19. Lines represent 95% credible intervals for Bayesian estimates and confidence intervals for frequentist.
  20. 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)
  21. 21. <ul><li>History of ANOVA </li></ul><ul><li>The Math of ANOVA </li></ul><ul><li>Bayes Theorem </li></ul><ul><li>Anatomy of Baysian ANOVA </li></ul><ul><li>Compare and Contrast! </li></ul><ul><li>Rumble in the Jungle: Advantages of Bayes </li></ul><ul><li>Real World 13: Genotype and Frequency Dependence in an invasive grass. </li></ul>
  22. 22. <ul><li>Avoids the muddled idea of fixed vs. random </li></ul><ul><li>effects, treating all effects as random. </li></ul><ul><li>Provides estimates of effects as well as </li></ul><ul><li>variance components with corresponding </li></ul><ul><li>uncertainty. </li></ul><ul><li>Allows more flexibility in model construction </li></ul><ul><li>(e.g. GLM’s instead of just normal models) </li></ul><ul><li>Issues such as normality, unbalanced </li></ul><ul><li>designs, or missing values are easily handled in this framework. </li></ul><ul><li>You just don’t believe in p-values (uniformative, etc, see Anderson et al 2000) </li></ul>What’s up now Fisher, Neyman-Pearson null hypothesis testing!?
  23. 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
  24. 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
  25. 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
  26. 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

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