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N-way ANOVA 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 2 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? 3 How to do a 2-way ANOVA with equal replications Calculating means Calculate cell means: X ab n l 1 X abl eg n 5 l 1 X 11 l 16 ,3 20 , 4 12 , 4 15 ,8 9 ,5 5 14 ,88 n Calculate the total mean (grand mean) X a b n i 1 j 1 l 1 X ijl 21 ,825 N Calculating treatment means X i b n j 1 l 1 bn X ijl eg X 1 13 ,5 4 How to do a 2-way ANOVA with equal replications Calculating general Sum of Squares Calculate total SS: total SS X a b n i 1 j 1 l 1 ijl X 2 1762 , 7175 total DF N 1 19 Calculate the cell SS cells SS n a i 1 X b j 1 ij 1461 ,3255 i 1 X X 2 cells DF ab 1 3 Calculating treatment error SS within - cells (error) SS n within - cells (error) DF ab n 1 16 a b n j 1 l 1 X ij 301 , 3920 2 ijl 5 How to do a 2-way ANOVA with equal replications Calculating factor Sum of Squares Calculating factor A SS: factor A SS bn a i 1 X i X X 2 1386 ,1125 factor A DF a 1 1 Calculating factor B SS factor B SS an b j 1 X j 2 70 ,3125 factor B DF b 1 1 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 How to do a 2-way ANOVA with equal replications Summary of calculations 7 How to do a 2-way ANOVA with equal replications Hypothesis test H0: There is on effect of hormone treatment on the mean plasma concentration F = hormone MS/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 8 Visualizing 2-way ANOVA Table 12.2 and Figure 12.1 9 2-way ANOVA in SPSS 10 2-way ANOVA in SPSS Click Add 11 Visualizing 2-way ANOVA without interaction 12 Visualizing 2-way ANOVA with interaction 13 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 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) • 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) There are no non-parametric alternative to the 2-way ANOVA 15 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 2-way ANOVA Confidence limits for means 95 % confidence limits for calcium concentrations on in birds without hormone treatment s 95 % CI X 1 t 0 , 05 ( 2 ), 2 bn within cell DF; s within cell MS 2 17 2-way ANOVA With proportional but unequal replications Proportional replications: n ij # row i # col j N 18 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 Xˆ ijl aA i bB j a b n ij i 1 j 1 l 1 X ijl N 1 a b 19 2-way ANOVA With one replication Get more data! 20 2-way ANOVA Randomized block design 21 3-way ANOVA 22 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 23 3-way ANOVA Latin Square 24 Exercises 12.1, 12.2, 14.1, 14.2 25