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N-way ANOVA

N-way ANOVA. 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 o rthogonality. The hypothesis.

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N-way ANOVA

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  1. N-way ANOVA

  2. 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

  3. 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?

  4. How to do a 2-way ANOVA with equal replicationsCalculating means Calculatecellmeans: Calculate the total mean (grand mean) Calculatingtreatmentmeans

  5. How to do a 2-way ANOVA with equal replicationsCalculating general Sum of Squares Calculate total SS: Calculate the cell SS Calculatingtreatmenterror SS

  6. 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

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

  8. 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

  9. Visualizing 2-way ANOVA Table 12.2 and Figure 12.1

  10. 2-way ANOVA in SPSS

  11. 2-way ANOVA in SPSS ClickAdd

  12. Visualizing 2-way ANOVA without interaction

  13. Visualizing 2-way ANOVA with interaction

  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.

  15. 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

  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

  17. 2-way ANOVA Confidence limits for means 95 % confidence limits for calcium concentrations on in birds without hormone treatment

  18. 2-way ANOVA With proportional but unequal replications Proportional replications:

  19. 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

  20. 2-way ANOVA With one replication Get more data!

  21. 2-way ANOVA Randomized block design

  22. 3-way ANOVA

  23. 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

  24. 3-way ANOVA Latin Square

  25. Exercises 12.1, 12.2, 14.1, 14.2

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