Two-factor ANOVA with equal replications Experimental design: 2
2 (or 2 2 ) factorial with n = 5 replicate Total number of
observations: N = 2 2 5 = 20 Equal replications also termed
orthogonality 2
Slide 3
The hypothesis H 0 : There is on effect of hormone treatment on
the mean plasma concentration H 0 : There is on difference in mean
plasma concentration between sexes H 0 : There is on interaction of
sex and hormone treatment on the mean plasma concentration Why not
just use one-way ANOVA with for levels? 3
Slide 4
How to do a 2-way ANOVA with equal replications Calculating
means Calculate cell means: Calculate the total mean (grand mean)
Calculating treatment means 4
Slide 5
How to do a 2-way ANOVA with equal replications Calculating
general Sum of Squares Calculate total SS: Calculate the cell SS
Calculating treatment error SS 5
Slide 6
How to do a 2-way ANOVA with equal replications Calculating
factor Sum of Squares Calculating factor A SS: Calculating factor B
SS Calculating A B interaction SS A B interaction SS = cell SS
factor A SS factor B SS = 4,9005 A B DF = cell DF factor A DF
factor B DF = 1 6
Slide 7
How to do a 2-way ANOVA with equal replications Summary of
calculations 7
Slide 8
How to do a 2-way ANOVA with equal replications Hypothesis test
H 0 : There is on effect of hormone treatment on the mean plasma
concentration F = hormone MS/within-cell MS = 1386,1125/18,8370 =
73,6 F 0,05(1),1,16 = 4,49 H 0 : There is on difference in mean
plasma concentration between sexes F = sex MS/within-cell MS = 3,73
F 0,05(1),1,16 = 4,49 H 0 : There is on interaction of sex and
hormone treatment on the mean plasma concentration F = A B
MS/within-cell MS = 0,260 F 0,05(1),1,16 = 4,49 8
Slide 9
Visualizing 2-way ANOVA Table 12.2 and Figure 12.1 9
Slide 10
2-way ANOVA in SPSS 10
Slide 11
2-way ANOVA in SPSS 11 Click Add
Slide 12
Visualizing 2-way ANOVA without interaction 12
Slide 13
Visualizing 2-way ANOVA with interaction 13
Slide 14
2-way ANOVA Random or fixed factor Random factor: Levels are
selected at random Fixed factor: The value of each levels are of
interest and selected on purpose. 14
Slide 15
2-way ANOVA Assumptions Independent levels of the each factor
Normal distributed numbers in each cell Equal variance in each cell
Bartletts homogenicity test (Section 10.7) s 2 ~ within cell MS; ~
within cell DF The ANOVA test is robust to small violations of the
assumptions Data transformation is always an option (see chpter 13)
There are no non-parametric alternative to the 2-way ANOVA 15
Slide 16
2-way ANOVA Multiple Comparisons Multiple comparesons tests ~
post hoc tests can be used as in one-way ANOVA Should only be
performed if there is a main effect of the factor and no
interaction 16
Slide 17
2-way ANOVA Confidence limits for means 95 % confidence limits
for calcium concentrations on in birds without hormone treatment
17
Slide 18
2-way ANOVA With proportional but unequal replications
Proportional replications: 18
Slide 19
2-way ANOVA With disproportional replications Statistical
packges as SPSS has porcedures for estimating missing values and
correcting unballanced designs, eg using harmonic means Values
should not be estimated by simple cell means Single values can be
estimated, but remember to decrease the DF 19
Slide 20
2-way ANOVA With one replication Get more data! 20
Slide 21
2-way ANOVA Randomized block design 21
Slide 22
3-way ANOVA 22
Slide 23
3-way ANOVA H 0 : The mean respiratory rate is the same for all
species H 0 : The mean respiratory rate is the same for all
temperatures H 0 : The mean respiratory rate is the same for both
sexes H 0 : The mean respiratory rate is the same for all species H
0 : There is no interaction between species and temperature across
both sexes H 0 : There is no interaction between species and sexes
across temperature H 0 : There is no interaction between sexes and
temperature across both spices H 0 : There is no interaction
between species, temperature, and sexes 23
