Analysis of covariance
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1 Analysis of covariance Statistical modelling: theory and practice Gilles Guillot October 2, 2013 Gilles Guillot ANCOVA October 2, / 12
2 1 Introductory example 2 3 Reading Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
3 Introductory example The travel stylized data (from Sheather) A small travel agency has retained your services to help them better understand two important customer segments. The first segment, which we will denote by A, consists of those customers who have purchased an adventure tour in the last twelve months. The second segment, which we will denote by C, consists of those customers who have purchased a cultural tour in the last twelve months. Data are available on 925 customers (i.e. on 466 customers from segment A and 459 customers from segment C). Note that the two segments are completely separate in the sense that there are no customers who are in both segments. Interest centres on identifying any differences between the two segments in terms of the amount of money spent in the last twelve months. In addition, data are also available on the age of each customer, since age is thought to have an effect on the amount spent. Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
4 Introductory example EDA travel=read.table( " header=true) plot(travel$age,travel$amount, col=travel$c+1,xlab="age",ylab="amount") legend(col=c(1,2),pch=1,legend=c("adventure","culture"), x="top") Gilles Guillot ANCOVA October 2, / 12
5 A model We need a linear regression with segment-specific coefficients or one-way ANOVA with continuous effect. Notation: y ij spending of j-th customer in segment i (i = 1, 2) x ij age of j-th customer in segment i with x ij = 0 Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
6 ANCOVA model without interaction: y ij = µ + α i + βx ij + ε ij µ α i segment-specific departure from intercept β effect of age ε ij iid N (0, σ 2 ) Identifying constraint: α 1 = 0. For a I-level factor, we have 1 + (I 1) = I + 2 unknown independent parameters. Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
7 Parameter inference Business as usual: LSE. As seen with other models, the interpretation and expression of entries in θ depend on identifying constraints. Gilles Guillot ANCOVA October 2, / 12
8 Parameter inference Inference with R. Example with travel spending data: ## ANCOVA without interaction: lm.noint <- lm(amount ~ 1 + Segment + Age,data=travel) ## same as call above: lm(amount ~.,data=travel) summary(lm.noint) plot(travel$age,travel$amount, col=travel$c+1,xlab="age",ylab="amount") legend(col=c(1,2),pch=1,legend=c("adventure","culture"), x="top") ## fitted lines abline(sum(lm.noint$coefficients[1:2]), lm.noint$coefficients[3],col=2) abline(lm.noint$coefficients[1], lm.noint$coefficients[3],col=1) In the above, Segment has to be a factor. Intrepretation of results... Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
9 ANCOVA model with interaction: y ij = µ + α i + βx ij + γ i x ij + ε ij µ α i segment-specific departure from intercept β common effect of age γ i segment-specific effect of age ε ij iid N (0, σ 2 ) Identifying constraint: α 1 = 0 and γ 1 = 0. For a I-level factor, we have 1 + (I 1) (I 1) + 1 = 2I + 1 unknown independent parameters Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
10 Parameter inference with interaction Example with travel spending data: lm.int <- lm(amount ~ 1 + Segment + Age + ## same as call above ## lm(amount ~ Segment*Age,data=travel) summary(lm.int) Segment*Age,data=travel) anova(lm.int) ## fitted lines abline(sum(lm.int$coefficients[1:2]), sum(lm.int$coefficients[3:4]),col=2,lty=2) abline(lm.int$coefficients[1], lm.int$coefficients[3],col=1,lty=2) Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
11 ANCOVA model in matrix form ANCOVA model without interaction: With θ = (µ, α 1,..., α I, β) t, the model y ij = µ + α i + βx ij + ε ij can be written Y = Xθ + E where X is an n (I + 2) matrix a bit cumbersome to write in L A TEX. Gilles Guillot (gigu@dtu.dk) ANCOVA October 2, / 12
12 Reading Suggested reading Sheather, section 5.3, pdf [here] Gilles Guillot ANCOVA October 2, / 12
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