Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.5. Two-Way ANOVA 537

Coarse Aggregate Content

Binder-Grade 38% 41% 44%

PG58 0.835 0.822 0.785

0.845 0.826 0.795

PG64 0.855 0.832 0.790

0.865 0.836 0.800

PG70 0.815 0.800 0.770

0.825 0.820 0.790

The data are also in the fileconductivity.rda. Assuming this file has been loaded

into the R work area, the mean profile plot is computed by

interaction.plot(Binder,Aggregate,Conductivity,legend=T)

and it is displayed in Figure 9.5.1. Note that the mean profiles are almost parallel,

a graphical indication of little interaction between the factors. The ANOVA for

the study is computed by the following two commands. It yields the tabled results

(which we have abbreviated). The next to last column shows theF-test statistics

discussed in this section.

fit=lm(Conductivity ~ factor(Binder) + factor(Aggregate) +

factor(Binder)*factor(Aggregate))

anova(fit)

Analysis of Variance Table

Df Sum Sq F value Pr(>F)

factor(Binder) 2 0.0020893 14.1171 0.001678

factor(Aggregate) 2 0.0082973 56.0631 8.308e-06

factor(Binder):factor(Aggregate) 4 0.0003253 1.0991 0.413558

As the interaction plot suggests, interaction is not significant (p=0.4135). In prac-

tice, we would accept the additive (no interaction) model. The main effects are

both highly significant. So both factors have an effect on conductivity. See Devore

(2012) for more discussion.

EXERCISES

9.5.1.For the two-way interaction model, (9.5.15), show that the following decom-

position of sums of squares is true:

∑a

i=1

∑b

j=1

∑c

k=1

(Xijk−X...)^2 =bc

∑a

i=1

(Xi..−X...)^2 +ac

∑b

j=1

(X.j.−X...)^2

+c

∑a

i=1

∑b

j=1

(Xij.−Xi..−X.j.+X...)^2

+

∑a

i=1

∑b

j=1

∑c

k=1

(Xijk−Xij.)^2 ;

that is, the total sum of squares is decomposed into that due torow differences,
that due tocolumndifferences, that due tointeraction,andthatwithin cells.