1. ANCOVA Regression with more than one line Andrew Jackson a.jackson@tcd.ie

2. Focus on gene / environment effects Prof. Donal Manahan’s seminar “Evolution and development: an ecological perspective” 3/11/2011 Growth rates are affected by extrinsic environmental conditions Growth rates are affected by intrinsic physiological factors which may be governed by genetic factors

3. Experiments Rear larval oysters at different temperatures Record simple growth rate as mm/week Repeat the experiment with different genotypes

4. The effect of temperature Intercept = 5, slope = 1.3

5. The effect of genotype

6. The effect of both together

7. How do these lines differ? Temperature affects both genotypes equally There is a fixed effect of genotype – Constant for all temperatures The red genotype grows faster than the black one coefficients – Slopes = 1.3 – red intercept = 8 – black intercept=5

8. A different genotype

9. How do these lines differ? There is still an effect of temperature But, now it is different for each genotype The effect of genotype is no longer fixed for all temperatures There is an interaction between temperature (environment) and genotype Coefficients – Green slope = 1.8, intercept = 8 – Black slope = 1.3, intercept = 5

10. A slightly different question And why its important to consider the linear covariate when comparing between groups

11. Known as: Analysis of Covariance: ANCOVA Also a GLM with fixed factors and linear covariates How do we compare two lines statistically?

12. Experiment to study effect of herbivores on primary productivity in ecosystems Series of in situ exclusion experiments Measured: – Seed mass (g) – Grazed / Ungrazed – Root diameter at start of experiment An alternative dataset

13. The Data RootFruitGrazing 6.22559.77Ungrazed 6.48760.98Ungrazed 4.91914.73Ungrazed 5.1319.28Ungrazed 5.41734.25Ungrazed 5.35935.53Ungrazed 8.64378.28Grazed 7.91641.48Grazed 9.35198.47Grazed 7.06640.15Grazed 8.15852.26Grazed 7.38246.64Grazed 8.51571.01Grazed 8.5383.03Grazed CovariateResponseFixed Factor

14. Questions to ask How does grazing affect seed production? Why was root diameter recorded? – How might this have changed the picture if it were omitted? What do we need to test statistically to address our hypothesis?

15. Pick one line to be the reference (e.g. Grazed) What is the equation for the Grazed line? – Seed = b 0 + b 1 Root What is the equation for Ungrazed line? – Seed = b 0 + b ug + b 1 Root Testing parallel lines

16. Testing parallel Lines in R Call: glm(formula = Fruit ~ Root + Grazing, data = mydata) Deviance Residuals: Min 1Q Median 3Q Max -17.1920 -2.8224 0.3223 3.9144 17.3290 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -127.829 9.664 -13.23 1.35e-15 *** Root 23.560 1.149 20.51 < 2e-16 *** GrazingUngrazed 36.103 3.357 10.75 6.11e-13 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 AIC: 271.13 Number of Fisher Scoring iterations: 2

17. Pick one line to be the reference (e.g. Grazed) What is the equation for the Grazed line? – Seed = b 0 + b 1 Root What is the equation for Ungrazed line? – Seed = b 0 + b ug + (b 1 +b 2 )Root – Seed = b 0 + b ug + b 1 Root + b 2 Root_UG Testing non-parallel lines

18. Testing parallel Lines in R Call: glm(formula = Fruit ~ Root * Grazing, data = mydata) Deviance Residuals: Min 1Q Median 3Q Max -17.3177 -2.8320 0.1247 3.8511 17.1313 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -125.173 12.811 -9.771 1.15e-11 *** Root 23.240 1.531 15.182 < 2e-16 *** GrazingUngrazed 30.806 16.842 1.829 0.0757. Root:GrazingUngrazed 0.756 2.354 0.321 0.7500 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 AIC: 273.01 Number of Fisher Scoring iterations: 2