bayesian anova

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

History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal 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 , 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.

History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal World 13: Genotype and Frequency

Dependence in an invasive grass.

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d.f. Sum of squares

Mean square

F-ratio p-value

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a-1 Determined from F-distribution with (a-1),a(n-1) d.f.

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Our statistical model

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History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal World 13: Genotype and Frequency

Dependence in an invasive grass.

Rev. Thomas Bayes 1702-1761

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Common Risk Independent Risk Hierarchical

Adapted from Clark 2007

History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal World 13: Genotype and Frequency

Dependence in an invasive grass.

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From Qian and Shen 2007

History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal World 13: Genotype and Frequency

Dependence in an invasive grass.

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

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

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 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)

History of ANOVAThe Math of ANOVABayes TheoremAnatomy of Baysian ANOVACompare and Contrast!Rumble in the Jungle: Advantages of

BayesReal 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!?

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

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

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

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