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Analysis of covariance Experimental design and data analysis for biologists (Quinn & Keough, 2002) Environmental sampling and analysis

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Linear models All predictors continuous –regression models –effects measured as regression slopes All predictors categorical –“ANOVA” models –effects measured as differences b/w group means Continuous and categorical predictors –covariance models –effects measured as adjusted differences b/w group means

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Analysis of covariance Covariance: –measure of how much two variables covary, i.e. vary together Analysis of covariance (ANCOVA): –comparing mean values of response variable between groups (single or multifactor design) where response variable covaries with other measured continuous variables (covariates)

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Sex and fruitfly longevity Response variable –longevity of male fruitflies Factor A –“sex” treatment with 5 groups –1 virgin female, 8 virgin females, 1 pregnant female etc. Covariate –thorax length Hypothesis –no effect of treatment on longevity of male fruitflies, adjusting for thorax length

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Shrinking in sea urchins Response variable –suture width in sea urchins Factor A –food treatment with 3 groups –high food, low food, initial sample Covariate –body volume Hypothesis –no effect of food treatment on suture width of sea urchins, adjusting for body volume

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ANCOVA model i is effect of factor A (groups or treatments) is pooled (across groups) regression slope b/w Y and X x ij is value of covariate for jth observation in ith group ij is variation in Y not explained by either factor A or covariate X

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Adjusted Y Adjusted Y values: Adjusted Y means:

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Adjusted means x1x1 x2x2 x Group 1 Group 2 y2y2 y1y1 y 1adj y 2adj

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Assumptions Apply to adjusted response variable Normality and homogeneity of variances –boxplots, residual plots, etc. Linearity of Y and covariate relationship –scatterplot Covariate not different between groups –ANOVA on covariate Homogeneity of within-group regression slopes –test factor by covariate interaction term

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Homogeneity of slopes Fit model: –y = + + x + x Test by x interaction term If not significant –fit usual ANCOVA model If significant –use Wilcox modification of Johnson-Neyman procedure –tedious but informative

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Sex and fruitfly longevity H 0 : 1 = 2 = 3 = i (equal within-group regression slopes) Fit model: (log longevity) ij = mean + (treatment) i + (thorax length) i j + (treatment x thorax length) ij + ij SourcedfMSFP Treatment x thorax length40.0111.560.189 Residual1150.007 No evidence to reject H 0 of equal within-group slopes Refit model with pooled regression slope

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Sex and fruitfly longevity

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H 0 : 1(adj) = 2(adj) = … = i(adj) SourcedfMSFP Treatment40.19627.97<0.001 Thorax length11.017145.44<0.001 Residual1190.007 Reject H 0 of equal adjusted mean log longevity between groups Also reject H 0 of zero pooled regression slope (log longevity against thorax length) ANOVA MS Residual = 0.015 (120df); cf 0.007 above

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Sex and fruitfly longevity TreatmentAdjusted meanUnadjusted mean 1 (8 preg females)1.8081.789 2 (no partners)1.7711.789 3 (1 preg female)1.7941.799 4 (1 virg female)1.7171.737 5 (8 virg females)1.5891.564

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Further tests Planned contrasts and trends on adjusted means –partition SS on adjusted means Unplanned multiple comparisons on adjusted means –use conditional (on covariate) Tukey test

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Complexities More covariates –adjust Y for both covariates –homogeneity of slopes for each covariate –covariates shouldn’t be correlated (collinearity) More factors –nested or factorial or both –testing homogeneity of slopes is tricky interactions b/w covariate and each factor and b/w covariate and factor interactions X by A, X by B, X by A by B

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