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 , 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 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.
a
n
  (Y
i 1
j 1
SStotal
Adapted from Gotelli and Ellison 2004
__
a
ij  Y )  
2
i 1
n
__
a
n
i 1
j 1
 (Y  Y )    (Y
j 1
i
SS among groups
2
__
2

Y
)
ij
i
SS within groups
a
n
__
  (Y
i 1
j 1
a
ij  Y )  
2
i 1
SStotal
Source
d.f.
Among
groups
a-1
Within
groups
Sum of squares
a
n
__
 (Y  Y )
i 1
j 1
a
n

an-1
a
n
i 1
j 1
 (Y  Y )    (Y
j 1
i
2
2
i
a

__
(Yij  Yi )2
j 1
n
__
 (Yij  Y )2
j 1
__
2

Y
)
ij
i
SS within groups
Mean square
F-ratio
p-value
SS among groups
MS among groups
(a-1 )
MS within groups
Determined from Fdistribution with
(a-1),a(n-1) d.f.
a(n-1)
i 1
Adapted from Gotelli and Ellison 2004
__
SS among groups
i 1
Total
n
SS within groups
a(n-1 )
SS total
an-1
a
n
  (Y
i 1
j 1
SStotal
__
a
ij  Y )  
2
i 1
n
__
n
i 1
j 1
 (Y  Y )    (Y
j 1
i
SS among groups
Our statistical model
yij    1i   ij
Adapted from Gotelli and Ellison 2004
a
2
__
2

Y
)
ij
i
SS within groups
 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.
p( , y ) p( ) p( y |  )
p( | y ) 

p( y )
p( y )
Rev. Thomas Bayes 1702-1761
p( | y )  p( ) p( y |  )
Prior
Likelihood
Common Risk
y1 , y2 , y3 .... y10

Independent Risk
Hierarchical
y1 , y2 , y3 .... y10
y1 , y2 , y3 .... y10
1 , 2 ,3 ....10
1 , 2 ,3 ....10
, 
Adapted from Clark 2007
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.
yij    1i   ij
1i ~ N (0,  2 )
or
yij ~ N (  1i , y2 )
sm 
 ~ N (0,  2 )
1 J m ( m)
( m) 2
(



)

j
j
J m1 j 1
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.
Source
d.f.
SS
MS
Fratio
pvalue
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
Fratio
pvalue
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 pvalue
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 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.
What’s up now
Fisher, NeymanPearson null
hypothesis
testing!?
• 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)
Source
d.f.
SS
MS
Fratio
pvalue
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
Fratio
pvalue
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
Fratio
pvalue
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 {
Robin Collins
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[,])
Nick Gotelli
#compute
for(i in
e.y[i]
s.y <-
finite population standard deviation
1:n){
<- y[i] - y.mu[i]}
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)
}
}