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

2. 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

3. 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)

4. 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

5. 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

6. 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

7. Adjusted Y Adjusted Y values: Adjusted Y means:

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

9. 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

10. 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

11. 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

12. Sex and fruitfly longevity

13. 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

14. 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

15. 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

16. 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