Bayesian anova
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Transcript of 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.
upswithin gropsamong groutotal
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SS SS SS
)Y(Y)Y(Y)Y(Y 2
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Adapted from Gotelli and Ellison 2004
upswithin gropsamong groutotal
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SS SS SS
)Y(Y)Y(Y)Y(Y 2
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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-12
11
)Y(Yn
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SS psamong grou
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SS upswithin gro
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upswithin gro
psamong grou
MS
MS
Adapted from Gotelli and Ellison 2004
Adapted from Gotelli and Ellison 2004
upswithin gropsamong groutotal
n
j
__
iij
a
i
n
j
__
i
a
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n
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a
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SS SS SS
)Y(Y)Y(Y)Y(Y 2
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Our statistical model
ijiijy 1
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
)(
)|()(
)(
),()|(
yp
ypp
yp
ypyp
)|()()|( yppyp
Prior Likelihood
Adapted from Clark 2007
10321 ....,, yyyy 10321 ....,, yyyy
10321 ....,,
,
10321 ....,,
10321 ....,, yyyy
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.
or
ijiijy 1
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),0(~ 21 Ni
2)()(
1
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mj
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jmm
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),0(~ 2 N
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