Analysis of
Variance
(ANOVA)
When ANOVA is used..
• All the explanatory variables are
categorical (factors)
• Each factor has two or more levels
• Example: You have 60 DNA samples from
3 plants: A, B, and C. And you measured
DNA concentration once for each sample.
Explanatory variable will be plants.
One-way ANOVA
• A single factor with three or more levels.
• Example: see previous example
Multi-way ANOVA
• Two or more factors.
• Example: You measured concentration of
DNA samples from 3 plants (A, B, and C)
that were grown in four different soils (K,
M, N, P) - two-way ANOVA
Multi-way ANOVA
• Null hypotheses: The results of a twoway anova include tests of two null
hypotheses: that the means of
observations grouped by one factor are
the same; that the means of observations
grouped by the other factor are the same;
• Model=concentration~plant+soils
Factorial ANOVA
• When there is replication at each
combination of levels in a multi-way
ANOVA.
• Example: You measured 3 times
concentration of DNA samples from 3
plants (A, B, and C) that were grown in
four different soils (K, M, N, P) - two-way
ANOVA with replication
Factorial ANOVA (cont.)
• Null hypotheses: The results of a two-way anova
with replication include tests of three null
hypotheses: that the means of observations
grouped by one factor are the same; that the
means of observations grouped by the other
factor are the same; and that there is no
interaction between the two factors. The
interaction test tells you whether the effects of
one factor depend on the other factor.
• Model=concentration~plant+soils+plant:soils
ANOVA
ANOVA compares the mean values by
comparing variances.
It calculates the total variation (SSY)
and partitioning it into two
components: explained variation
(SSA) and unexplained variation
(SSE)
SSA
SSY
SSE
Total variation
• SSY- the total sum of squares is the sum of
squares of the differences between the data
points and the overall mean, n is number of
samples per treatment, k is the number of
treatments
k
SSY 
n

i 1
j 1
( y ij  y )
2
Unexplained variation
• SSE- error sum of squares is the sum of the
squares of the differences between the data
points and their individual treatments mean
k
SSE 
n

i 1
j 1
( y ij  y j )
2
Explained variation
• SSA- treatment sum of squares is the sum of
the squares of the differences between the
individual treatment means and the overall
mean
k
SSA  n  ( y i  y )
2
i 1
• The amount of the variation explained by
differences between the treatment means
Explained variation (cont.)
• SSA=SSY-SSE
• The larger difference between total
variation and unexplained variation (SSYSSE) the larger explained variation (SSA)
• the greater the deference between
treatment means
Before starting ANOVA
1. Check for constancy
of variance
Is the variances differ by more
than factor of 2?
2. Test homogeneity of variance
Is Fligner-Killeen test showing significant pvalue?
Analysis of sample Assumptions
•
•
•
•
Independence of samples elements
Normality
Homogeneity
Sufficient sample sizes, equal sample
sizes is the best