ANCOVARegression with more than one

line

Andrew Jacksona.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

• Growth rates are affected by intrinsic physiological factors which may be governed by genetic factors

Experiments

• Rear larval oysters at different temperatures• Record simple growth rate as mm/week• Repeat the experiment with different

genotypes

The effect of temperature

Intercept = 5, slope = 1.3

The effect of genotype

The effect of both together

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

A different genotype

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

A slightly different question

And why its important to consider the linear covariate when comparing

between groups

• Known as:• Analysis of Covariance:

ANCOVA• Also a GLM with fixed

factors and linear covariates

How do we compare two lines statistically?

• 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

The Data

Root Fruit Grazing6.225 59.77 Ungrazed6.487 60.98 Ungrazed4.919 14.73 Ungrazed5.13 19.28 Ungrazed5.417 34.25 Ungrazed5.359 35.53 Ungrazed8.643 78.28 Grazed7.916 41.48 Grazed9.351 98.47 Grazed7.066 40.15 Grazed8.158 52.26 Grazed7.382 46.64 Grazed8.515 71.01 Grazed8.53 83.03 Grazed

Covariate Response Fixed Factor

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?

• 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

Testing parallel lines

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

• 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

Testing non-parallel lines

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