1. Two-way analysis on variance
observations classified by two criteria simultaneously
… or we study the effect of two factors simultaneously.
We use a two-way ANOVA to handle the situation.

2. Results are presented as an ANOVA table
Dependent Variable: kaal
Sum of
Source DF Squares Mean Square F Value Pr > F
Model <.0001
Error
Corrected Total
R-Square Coeff Var Root MSE kaal Mean
Source DF Type III SS Mean Square F Value Pr > F
taim <.0001
sugu <.0001
Results are presented as an ANOVA table
_________________________________
effect df SS F p
plant <0.0001
sex <0.0001
error

3. _________________________________
effect df SS F p
plant <0.0001
sex <0.0001
error
because MS=SS/df, no need to report
R2 = SSeffect/SStotal iga efekti kohta eraldi
Or as a sentence:
“the effect of host plant on pupal weight was statistically confirmed (F1,17=32.3, p<0.0001; two-way ANOVA with sex as an additional factor).“
... are not the same as for one-way ANOVA,
no dependent variable, no direction
separately for each factor

4. interaction – the effect of one factor depends on the level of another factor,
There is an interaction in these data:
parallel lines,
with a change in the
direction or not,
symmetric
Source DF SS MS F P
plant
sex <.0001
plant*sex <.0001

5. There is no interaction when the effects are additive.
_________________________________________
effect df SS F p
plant , , ,8
sex ,2 131,2 <0.0001
plant*sex ,0 33,1 <0.0001
error ,7
There is no interaction when the effects are additive.
Logarithmic transformation turns the model to multiplicative!
In the case of interaction,
there may be main effects but need not be. Be careful!

6. Fill in the blank so that there is no interaction
male on birch 50 mg
female on birch 80 mg
male on alder 70 mg
female on alder mg

7. Fill in the blank so that there were an interaction
„with a change in the direction“
crocodile in the sea 8 m
crocodile in a lake 6 m
snake in the sea 9 m
snake in a lake m

8. Fill in the blank so that there is no interaction
male on birch 50 mg
female on birch 80 mg
male on alder 70 mg
female on alder 100 mg
male on willow 100 mg
female on willow mg

9. Fill in the blank so that there is no interaction
male on birch 50 mg
female on birch 80 mg
male on alder 70 mg
female on alder 80 mg
male on willow 100 mg
female on willow mg

10. Fill in the blank so that there is no interaction
after logarithmic transformation:
male on birch 50 mg
female on birch 100 mg
male on alder 80 mg
female on alder mg

11. Fill in the blank so that there is no main effect of
tree species:
male on birch 50 mg
female on birch 80 mg
male on alder 70 mg
female on alder mg

12. Fill in the blank so that there is no main effect of
either of the factors:
male on birch 50 mg
female on birch 80 mg
male on alder mg
female on alder mg

13. It is well possible to have a more than two-way ANOVA,
Source DF Type III SS Mean Square F Value Pr > F
taim
sugu <.0001
taim*sugu <.0001
varv
taim*varv
sugu*varv
taim*sugu*varv
auk
taim*auk
sugu*auk
taim*sugu*auk
varv*auk
taim*varv*auk
sugu*varv*auk
taim*sugu*varv*auk
It is well possible to have a more than two-way ANOVA,
it will be complicated with interactions :
An interaction between three factors – the natuure of a the
two-way interaction depends on the level of the third factor.

14. Regression anlysis is based on the same principles -
can be combined:
covariate, analysis of covariance (ANCOVA),
does not really matter whether the independent variables is continuous or categorical!
We will present in the same way, df of the covariate always 1.
Often just to eliminate a confounding effect!

15. With the covariate included: p = 0.0032
control
egg size
with snails
weight of the mother
With the covariate included:
p =
the covariate itself: p <
egg size
p = 0.10

16. Why does it help to find the difference?
F=MSmodel/MSerror – covariate reduced error variance!
You can add any number of covariates but:
- too many reduce the power of the test
- will not include if non-significant – unless we know in advance
- ethical problem – playing with the model we can get what we want by chance;
- if depends on non-significant covariates – too little data!
- backward elimination
model simplification procedure
The effect of covariate eliminated – LSMEANS.

17. _________________________________________
effect df SS F p
treatment
weight <0.0001
age <0.0001
box age
blueberry
error

18. _________________________________________
effect df SS F p
treatment
weight <0.0001
age <0.0001
blueberry
error

19. _________________________________________
effect df SS F p
treatment ,7
weight <0.0001
age <0.0001
error