Analysis of Variance
2-Way ANOVA
MARE 250
Dr. Jason Turner
Two-Way – ANOVA
Two-way ANOVA - procedure to test the equality of
population means when there are two factors
2-Sample T-Test (1R, 1F, 2 Levels)
One-Way ANOVA (1R, 1F, >2 Levels)
Two-Way ANOVA (1R, 2F, >1 Level)
Two-Way – ANOVA
For Example…
One-Way ANOVA – means of urchin #’s from
each distance (shallow, middle, deep) are equal
Response – urchin #, Factor – distance
Two-Way ANOVA – means of urchin’s from each
distance collected with each quadrat (¼m, ½m)
are equal
Response – urchin #, Factors – distance,
quadrat
Two-Way – ANOVA
Factor 1
Location
(S, M, D)
Deep
Intermed.
Shallow
SeaWall
Factor 2
Quad Size
(¼m, ½m)
Two-Way – ANOVA
Factor 1
Location
(S, M, D)
Deep
Intermed.
Shallow
SeaWall
Factor 2
Quad Size
(¼m, ½m)
Two-Way – ANOVA
Factor 1
Location
(S, M, D)
Deep
Intermed.
Shallow
INTERACTION
SeaWall
Factor 1 X Factor 2
Location X Quad Size
Factor 2
Quad Size
(¼m, ½m)
Two-Way – ANOVA
Results
If the effect of a fixed factor is significant, then
the level means for that factor are significantly
different from each other (just like a one-way
ANOVA)
If the effect of an interaction term is significant,
then the effects of each factor are different at
different levels of the other factor(s)
Two-Way – ANOVA
Results
Two-Way – ANOVA
Results
Urchins
Location
Quad Size
Two-Way – ANOVA
Results
Two-Way ANOVA : Analysis of Variance Table
Source
DF
SS
MS
F
P
Location
1 228.17 228.167 8.99 0.008
Quadsize 2 308.33 154.167 6.07 0.010
Interaction 2
76.33 38.167 1.50 0.249
Error
18 457.00 25.389
Total
23 1069.83
For the urchin analysis, the results indicate the following:
The effect of Location (p = 0.008) is significant
This indicates that urchin populations numbers were
significantly different a different distances from shore
The effect of Quad Size (p = 0.010) is significant
This indicates quadrat type had a significant effect upon the
number of urchins collected
The interaction between Distance and Quadrat (p
= 0.249) is not significant
This means that the distance and quadrat size results were
not influencing the other
Thus, it is okay to interpret the individual
effects of either factor alone
Two-Way – ANOVA
Results
Two-Way ANOVA : Analysis of Variance Table
Source
DF
SS
MS
F
P
Location
1 228.17 228.167 8.99 0.008
Quadsize 2 308.33 154.167 6.07 0.010
Interaction 2
76.33 38.167 1.50 0.009
Error
18 457.00 25.389
Total
23 1069.83
For the urchin analysis, the results indicate the following:
The effect of Location (p = 0.008) is significant
This indicates that urchin populations numbers were
significantly different a different distances from shore
The effect of Quad Size (p = 0.010) is significant
This indicates quadrat type had a significant effect upon the
number of urchins collected
The interaction between Distance and Quadrat (p
= 0.009) is not significant
This means that the distance and quadrat size results WERE
INFLUENCING the other
Thus, the individual
Factors must be analyzed alone
Interactions
Use interactions plots to assess the two-factor
interactions in a design
Evaluate the lines to determine if there is an
interaction:
If the lines are parallel, there is no interaction
If the lines cross, there IS Interaction
The greater the lines depart from being parallel,
the greater the degree of interaction
Interactions Plots
Interactions Plots
Interactions Plots
Why is there interaction?
Because we get a different answer regarding
#Urchins by Location (S,M,D) when using different
Quadrats (¼m, ½m)
Interactions Plots
Why is there interaction?
Because we get a different answer regarding
#Urchins by Quad Size (¼m, ½m) at different
Locations (S,M,D)
Two-Way – ANOVA
The two-way ANOVA procedure does not support
multiple comparisons
To compare means using multiple comparisons, or if
your data are unbalanced – use a General Linear Model
General Linear Model - means of urchin #’s and
species #’s from each distance (shallow, middle, deep)
are equal
Responses – urchin #, Factor – distance, quadrat
Unbalanced…No Problem!
Or multiple factors…
General Linear Model - means of urchin #’s and
species #’s from each distance (shallow, middle, deep)
are equal
Two-Way – ANOVA
Two-Way ANOVA is a statistical test – there is a
parametric (Two-Way ANOVA) and
nonparametric version (Friedman’s)
There are 3 ways to run a Two-Way ANOVA in
minitab:
1) Two-Way ANOVA – for parametric (normal)
balanced (equal n among levels) data
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
3) Friedman – nonparametric (not normal) data
Two-Way – ANOVA
1) Two-Way ANOVA – for parametric (normal)
balanced (equal n among levels) data
- See examples of Two-Way ANOVA above
* Note – Two-Way ANOVA program cannot run
Multiple Comparisons Tests (Tukey)
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Location
Quad Size
Two-Way – ANOVA
2) General Linear Model (GLM) – for all
parametric (normal) data – balanced or not
Location*Quad Size
Two-Way – ANOVA
3) Friedman – nonparametric (not normal) data
Two-Way – ANOVA
3) Friedman – nonparametric (not normal) data