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Two-factor ANOVA with equal replications

Experimental design: 2 2 (or 22) factorial with n = 5 replicate

Total number of observations:

N = 2 2 5 = 20

Equal replications also termed orthogonality

The hypothesis

H0: There is on effect of hormone treatment on the mean plasma concentration

H0: There is on difference in mean plasma concentration between sexes

H0: There is on interaction of sex and hormone treatment on the mean plasma concentration

Why not just use one-way ANOVA with for levels?

How to do a 2-way ANOVA with equal replicationsCalculating means

Calculatecellmeans:

Calculate the total mean (grand mean)

Calculatingtreatmentmeans

How to do a 2-way ANOVA with equal replicationsCalculating general Sum of Squares

Calculate total SS:

Calculate the cell SS

Calculatingtreatmenterror SS

How to do a 2-way ANOVA with equal replicationsCalculating 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

How to do a 2-way ANOVA with equal replicationsSummary of calculations

How to do a 2-way ANOVA with equal replicationsHypothesis test

H0: There is on effect of hormone treatment on the mean plasma concentration

F = hormoneMS/within-cell MS = 1386,1125/18,8370 = 73,6

F0,05(1),1,16 = 4,49

H0: There is on difference in mean plasma concentration between sexes

F = sex MS/within-cell MS = 3,73

F0,05(1),1,16 = 4,49

H0: There is on interaction of sex and hormone treatment on the mean plasma concentration

F = A B MS/within-cell MS = 0,260

F0,05(1),1,16 = 4,49

Visualizing 2-way ANOVA

Table 12.2 and Figure 12.1

2-way ANOVA in SPSS

2-way ANOVA in SPSS

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Visualizing 2-way ANOVA without interaction

Visualizing 2-way ANOVA with interaction

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.

2-way ANOVA Assumptions

- Independent levels of the each factor
- Normal distributed numbers in each cell
- Equal variance in each cell
- Bartlettshomogenicity test (Section 10.7)
- s2 ~ 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)
- Therearenonon-parametric alternative to the 2-way ANOVA

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

2-way ANOVA Confidence limits for means

95 % confidence limits for calcium concentrations on in birds without hormone treatment

2-way ANOVA With proportional but unequal replications

Proportional replications:

2-way ANOVA With disproportional replications

Statisticalpackges as SPSS has porcedures for estimating missing values and correctingunballanced designs, eg usingharmonicmeans

Valuesshould not beestimated by simple cellmeans

Single valuescanbeestimated, but remember to decrease the DF

2-way ANOVA With one replication

Get more data!

2-way ANOVA Randomized block design

3-way ANOVA

H0: The mean respiratory rate is the same for all species

H0: The mean respiratory rate is the same for all temperatures

H0: The mean respiratory rate is the same for both sexes

H0: The mean respiratory rate is the same for all species

H0: There is no interaction between species and temperature across both sexes

H0: There is no interaction between species and sexes across temperature

H0: There is no interaction between sexes and temperature across both spices

H0: There is no interaction between species, temperature, and sexes

3-way ANOVA Latin Square

Exercises

12.1, 12.2, 14.1, 14.2

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