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ANCOVA Regression with more than one line

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

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ANCOVA Regression with more than one line

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

  12. An alternative dataset • 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

  13. The Data Covariate Response Fixed 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 totest statistically to address our hypothesis?

  15. Testing parallel lines • Pick one line to be the reference (e.g. Grazed) • What is the equation for the Grazed line? • Seed = b0 + b1Root • What is the equation for Ungrazed line? • Seed = b0 + bug + b1Root

  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. Testing non-parallel lines • Pick one line to be the reference (e.g. Grazed) • What is the equation for the Grazed line? • Seed = b0 + b1Root • What is the equation for Ungrazed line? • Seed = b0 + bug + (b1+b2)Root • Seed = b0 + bug + b1Root + b2Root_UG

  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

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