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More complicated ANOVA models: two-way and repeated measures Chapter 12 Zar Chapter 11 Sokal & Rohlf First, remember your ANOVA basics………. -Total SS in 1-way ANOVA -Deviations around total mean 8 Yield (tonnes) 7 6 5 Overall mean 4 3 2 Fert 1 1 Fert 3 Fert 2 0 0 10 20 Plot number 30 Within group SS= deviations around group means Group means 8 Yield (tonnes) 7 6 5 4 3 2 Fert 1 1 Fert 3 Fert 2 0 0 10 20 Plot number 30 Among groups SS=deviations of group means from overall mean Group means 8 Yield (tonnes) 7 6 5 Overall mean 4 3 2 Fert 1 1 Fert 3 Fert 2 0 0 10 20 Plot number 30 Mean squares Combine information on SS and df Total mean squares = total SS/ total df total variance of data set Within group mean squares = within SS/ within df variance (per df) among units given Error MS same treatment Unfortunate word usage Among groups mean squares = among SS / among df variance (per df) among units given different treatments The question: Does fitting the treatment mean explain a significant amount of variance? F= Among groups mean squares Within group mean squares Compare calculated F to critical value from table (B4) If calculated F as big or bigger than critical value, then reject H0 But remember……. H0: m1 = m2 = m3 Need separate test (multiple comparison test) to tell which means differ from which Factorial ANOVA= simultaneous analysis of the effect of more than one factor on population means -- Effect of light (or music) and water on plant growth -- Effect of drug treatment and gender on patient survival --Effect of turbidity and prey type on prey consumption by yellow perch --Effect of gender and income bracket on # pairs of shoes owned Two-way ANOVA vs a nested (hierarchical) ANOVA see chapter 10 S& R Example: the effect of drug on quantity of skin pigment in rats. 5 rats per drug 3 skin samples per rat Each sample divided in to 2 lots, each hydrolyzed 2 optical density readings per hydrolyzed sample Random effects 5 drugs + 1 control= 6 groups (fixed effect) Drug is the main factor of interest All other levels are subordinate Rat1 in drug treatment 1 is not the same as Rat1 in drug treatment 2 Above design is nested. Rats are nested within drug treatment, skin sample is nested within rat etc……. Can be mixed model (as in example) where primary effect is fixed (drug) but subordinate levels are random Or can be completely random model if the levels (eg drugs) were truly a random sample of all possible drugs Two-way ANOVA, Two-factor ANOVA There must be correspondence across classes --Effect of turbidity level and prey type on prey consumption by yellow perch High and low turbidity must be the same across all prey types Turbidity could be random or fixed Prey type probably always fixed? -- Effect of drug treatment and gender on patient survival Drug treatments must be same for both genders Drug could be random or fixed Gender always fixed? Terminology --Two factors A and B -- a = number of levels of A; starting with i -- b = number of levels of B; starting with j -- n = number replicates; starting with l -- Each combination of a level of A with a level of B is called a cell -- Cell analogous to groups in 1-way ANOVA --If there are 2 levels of 2 factors analysis called 2 x 2 factorial Low A High A Low B Low A Low B High A Low B High B Low A High B High A High B cell a b n Total SS = (Xijl –X)2 i=1 j=1 l=1 = (all deviations from grand mean)2 Total DF = N-1 Among Cell SS = variability between cell means and grand mean --among cell DF= ab-1 --Analogous to among groups SS in 1-way ANOVA Within Cell SS = deviations from each cell mean --within cell DF = ab (n-1) --analogous to within groups SS in 1-way ANOVA But……. Goal of 2-way ANOVA is to assess the affects of each of the 2 factors independently of each other --Consider A to be the only factor in a 1-way ANOVA (ignore B) a Factor A SS = bn (Xi –X)2 i=1 Then --Consider B to be the only factor in a 1-way ANOVA b Factor B SS = an (Xj –X)2 j=1 Now the tricky part…………… -- Among cell variability usually variability among levels of A + variability among levels of B -- The unaccounted for variability is due to the effect of interaction -- Interaction means that the effect of A is not independent of the presence of a particular level of B --Interaction effect is in addition to the sum of the effects of each factor considered separately With zm Without zm Low light With zm Low light Without zm Low light High light With zm High light Without zm High light Grow algae two levels of light and with and without zebra mussels, 15 reps in each cell, N=60 Measure net primary production of the algae (NPP) We will now graphically examine a range of outcomes of this 2x2 factorial ANVOA Some of the possible outcomes have below. Be prepared to discuss the meaning –ie, your interpretation of the graph with your name on it. Erin H. NPP (mgO2/m2/2hr) No difference of either factor and no interaction Low light High light 20 10 0 With zm Without zm Dave H. Significant main effect of light NPP (mgO2/m2/2hr) Low light High light 20 10 0 With zm Without zm Jhonathon Significant main effect of ZM NPP (mgO2/m2/2hr) Low light High light 20 10 0 With zm Without zm Both main effects are significant, but no interaction NPP (mgO2/m2/2hr) Josh S. Anthony Low light High light 20 10 0 With zm Without zm Significant interaction, but no significant main effect NPP (mgO2/m2/2hr) Colin Xiao-Jain Low light High light 20 10 0 With zm Without zm Interaction and the main light effect are significant NPP (mgO2/m2/2hr) Rajan Coleen Low light High light 20 10 0 With zm Without zm Interaction and the main zm effet are significant NPP (mgO2/m2/2hr) Chen-Lin Nan Low light High light 20 10 0 With zm Without zm the interaction and both main effects are significant NPP (mgO2/m2/2hr) Reza Malak Low light High light 20 10 0 With zm Without zm the interaction and both main effects are significant NPP (mgO2/m2/2hr) Chenxi Damien Low light High light 20 10 0 With zm Without zm How to in SAS: Data X; set Y; proc glm; class gender salary; model shoepair=gender salary gender*salary; interaction Main effects Analysis of covariance (ANCOVA) -Testing for effects with one categorical and one continuous predictor variable -Testing for differences between two regressions -Some of the features of both regression and analysis of variance. -A continuous variable (the covariate) is introduced into the model of an analysis-of-variance experiment. Initial assumption that there is a linear relationship between the response variable and the covariate If not, ANCOVA no advantage over simple ANOVA Ex. Test of leprosy drug Variables = Drug PreTreatment - two antibiotics (A and D) & control (F) - a pre-treatment score of leprosy bacilli PostTreatment - a post-treatment score of leprosy bacilli -10 patients selected for each drug) -6 sites on each measured for leprosy bacilli. -Covariate = pretreatment score included in model for increased precision in determining the effect of drugs on the posttreatment count of bacilli. data drugtest; input Drug $ PreTreatment PostTreatment @@; datalines; A 11 6 A 8 0 A 5 2 A 14 8 A 19 11 A 6 4 A 10 13 A 6 1 A 11 8 A 3 0 D 6 0 D 6 2 D 7 3 D 8 1 D 18 18 D 8 4 D 19 14 D 8 9 D 5 1 D 15 9 F 16 13 F 13 10 F 11 18 F 9 5 F 21 23 F 16 12 F 12 5 F 12 16 F 7 1 F 12 20 ; Different way to read in data proc glm; class Drug; Define categorical variable model PostTreatment = Drug PreTreatment Drug*PreTreatment / solution; run; Model dependent var=categorical variable covariate and categorical * covariate interaction First, slopes must be equal to proceed with other comparisons. If interaction term significant- end of test ** use Type III SS If interaction term not significant can compare intercepts (means) Source DF Type I SS Mean Square F Value Pr > F Drug 2 293.6000000 146.8000000 9.15 0.0010 PreTreatment 1 577.8974030 577.8974030 36.01 <.0001 Source DF Type III SS Mean Square F Value Pr > F Drug 2 68.5537106 34.2768553 2.14 0.1384 PreTreatment 1 577.8974030 577.8974030 36.01 <.0001 Standard Error t Value Pr > |t| Parameter Estimate Intercept -0.434671164 B 2.47135356 -0.18 0.8617 Drug A -3.446138280 B 1.88678065 -1.83 0.0793 Drug D -3.337166948 B 1.85386642 -1.80 0.0835 Drug F 0.000000000 B . . . PreTreatment 0.987183811 0.16449757 6.00 <.0001 Type I SS for Drug gives the between-drug sums of squares for ANOVA model PostTreatment=Drug. Measures difference between arithmetic means of posttreatment scores for different drugs, disregarding the covariate. The Type III SS for Drug gives the Drug sum of squares adjusted for the covariate. Measures differences between Drug LS-means, controlling for the covariate. The Type I test is highly significant (p=0.001), but the Type III test is not. Therefore, while there is a statistically significant difference between the arithmetic drug means, this difference is not significant when you take the pretreatment scores into account. light shade rock -2 log periphyton chlorophyll-a (mg m ) 3 2 rock: r2 = 0.19, p<0.13 light: r2 = 0.47, p<0.02 1 shade: r2 <0.01, p<0.78 0 0 1 -1 log-TP in water (μg L-1) 2 3