TWO-FACTOR ANOVA:
A one-factor ANOVA looks at one categorical variable (factor) and how
it affects a response.
A two-factor ANOVA extends this to two categorical variables (factors),
each with multiple levels:
• If Factor A has a levels & Factor B has b levels, the design is called an
a x b factorial design.
• If we have the same number of observations for each combination of
the levels of the two factors, we call this design the balanced design.
• A factor: a controlled, independent variable, whose levels are set by the
experimenter (Eg. Fertilizer type).
• A level: a category within the factor (Eg. Fertilizer 1 vs Fertilizer 2 vs
Fertilizer 3).
A factor = general type of treatment & levels are the actual treatments
applied (each level is 1 treatment group under the factor).
• Different treatments = different levels of a factor.
Two-factor ANOVA model:
We have:
• 1 numerical response variable X (outcome you measure).
• 2 categorical explanatory variables (Factor A and Factor B).
The model is written as:
, Under this model, an observation is considered as a random deviation
from a population mean. However, depending on the levels of factors A &
B, the mean of X changes.
• In other words, an observation isk is a random error of Cijk away from the
mean of El ist El A i3 it
We want to determine if these factors explain changes in the population
mean of X.
• However, the factors might have a combined effect on the mean of X.
• Thus, we first test if there is an interaction between the factors A & B
in the way that they affect the mean of X.
We test the following hypothesis:
No interaction: the effects of A and B are independent.
At least one interaction is present.
Only if H0 is not rejected (no evidence of interaction), we move on to
test the main effects:
• For Factor A:
No effect of A: all levels of A have the same mea
At least 1 level of A differs from the others.
• For Factor B:
No effect of B.
At least 1 level of B differs.
A one-factor ANOVA looks at one categorical variable (factor) and how
it affects a response.
A two-factor ANOVA extends this to two categorical variables (factors),
each with multiple levels:
• If Factor A has a levels & Factor B has b levels, the design is called an
a x b factorial design.
• If we have the same number of observations for each combination of
the levels of the two factors, we call this design the balanced design.
• A factor: a controlled, independent variable, whose levels are set by the
experimenter (Eg. Fertilizer type).
• A level: a category within the factor (Eg. Fertilizer 1 vs Fertilizer 2 vs
Fertilizer 3).
A factor = general type of treatment & levels are the actual treatments
applied (each level is 1 treatment group under the factor).
• Different treatments = different levels of a factor.
Two-factor ANOVA model:
We have:
• 1 numerical response variable X (outcome you measure).
• 2 categorical explanatory variables (Factor A and Factor B).
The model is written as:
, Under this model, an observation is considered as a random deviation
from a population mean. However, depending on the levels of factors A &
B, the mean of X changes.
• In other words, an observation isk is a random error of Cijk away from the
mean of El ist El A i3 it
We want to determine if these factors explain changes in the population
mean of X.
• However, the factors might have a combined effect on the mean of X.
• Thus, we first test if there is an interaction between the factors A & B
in the way that they affect the mean of X.
We test the following hypothesis:
No interaction: the effects of A and B are independent.
At least one interaction is present.
Only if H0 is not rejected (no evidence of interaction), we move on to
test the main effects:
• For Factor A:
No effect of A: all levels of A have the same mea
At least 1 level of A differs from the others.
• For Factor B:
No effect of B.
At least 1 level of B differs.
