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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.
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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
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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
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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
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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!
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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
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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
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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
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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
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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
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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
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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
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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.
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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!
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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
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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.
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effect df SS F p treatment weight <0.0001 age <0.0001 box age blueberry error
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effect df SS F p treatment weight <0.0001 age <0.0001 blueberry error
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effect df SS F p treatment ,7 weight <0.0001 age <0.0001 error
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