ANOVA (Analysis of variance)
Used for comparing sample means of more than two samples.
The series of multiple comparisons you make is named a family, and the Type-1 error associated with it is called
Family-wise Error Rateor FWER.
the
FWER is calculated as:
Alfa = significance level (0.05)
c = number of multiple comparisons
Anova allows us to compare multiple samples in one single procedure rather than in tests that are run separately.
This way the Type-1 error probability remains 5%.
ANOVA tests whether the variance between groups is larger than that within groups. For this we use squared
residuals.
Residuals = closely related to errors. Error of an observed value is the deviation of the observed value
from the (unobservable) true or expected value ( the mean of the entire population). Residual of an
observed value is the difference between the observed value and the (observable) estimated value (the
sample mean).
The Sum of Squares Within are the squared differences (residuals) between each individual measurement and its
group mean. (reflects the variation within a group)
The Sum of Squares between are the squared differences between group means and the overall mean. (reflects
the variation between groups or treatments)
Ultimately a Mean Sum of Squares Within (MSSwithin) and a Mean Sum of Squares Between (MSSBetween)
are calculated. The ratio ↓ is a measure for how much variation in the experiment is explained by the
factor ‘treatment’
F-value > 1: more variation is explained than unexplained.
F-value < 1: More variation unexplained. Bad news when looking for a significant result.
The F-ratio has a F-distribution. This distribution has two degrees of freedom and allows us to assess whether a
calculated ratio of Mean Sums of Squares has an extreme value or not.
Explained variation: variation between groups
Unexplained variation: variation within groups.
th
The i measurement in group j is denoted xi,j . For instance: the second observation in the third group is written as
x
2,3 .
The F-ratio now follows an F-distribution with df 1 = k − 1 and df 2 = N − k degrees of freedom. df 1 is sometimes
also called the degrees of freedom of the numerator, and df 2 that of the denominator of the F-ratio.
Variation between groups (MSSbetween)
Residuals: deviation of each group mean from the overall grand mean.
The squared residuals are multiplied by the number of observations
The variation within groups (MSSwithin)
Used for comparing sample means of more than two samples.
The series of multiple comparisons you make is named a family, and the Type-1 error associated with it is called
Family-wise Error Rateor FWER.
the
FWER is calculated as:
Alfa = significance level (0.05)
c = number of multiple comparisons
Anova allows us to compare multiple samples in one single procedure rather than in tests that are run separately.
This way the Type-1 error probability remains 5%.
ANOVA tests whether the variance between groups is larger than that within groups. For this we use squared
residuals.
Residuals = closely related to errors. Error of an observed value is the deviation of the observed value
from the (unobservable) true or expected value ( the mean of the entire population). Residual of an
observed value is the difference between the observed value and the (observable) estimated value (the
sample mean).
The Sum of Squares Within are the squared differences (residuals) between each individual measurement and its
group mean. (reflects the variation within a group)
The Sum of Squares between are the squared differences between group means and the overall mean. (reflects
the variation between groups or treatments)
Ultimately a Mean Sum of Squares Within (MSSwithin) and a Mean Sum of Squares Between (MSSBetween)
are calculated. The ratio ↓ is a measure for how much variation in the experiment is explained by the
factor ‘treatment’
F-value > 1: more variation is explained than unexplained.
F-value < 1: More variation unexplained. Bad news when looking for a significant result.
The F-ratio has a F-distribution. This distribution has two degrees of freedom and allows us to assess whether a
calculated ratio of Mean Sums of Squares has an extreme value or not.
Explained variation: variation between groups
Unexplained variation: variation within groups.
th
The i measurement in group j is denoted xi,j . For instance: the second observation in the third group is written as
x
2,3 .
The F-ratio now follows an F-distribution with df 1 = k − 1 and df 2 = N − k degrees of freedom. df 1 is sometimes
also called the degrees of freedom of the numerator, and df 2 that of the denominator of the F-ratio.
Variation between groups (MSSbetween)
Residuals: deviation of each group mean from the overall grand mean.
The squared residuals are multiplied by the number of observations
The variation within groups (MSSwithin)
