Graphics Reference
In-Depth Information
Interaction Plots
6.6
Two factors A and B are said to interact if the changes in a response variable Y as
factor A goes from one level to another differ depending on the level of the second
factor B.henotationintheANOVAsettingis:“
.”
he standard display of an interaction uses separate lines for each level of one
factor, the trace factor, and by convention connects the points for each level of the
second factor, the x-factor. Connecting the levels of the trace factor is an interesting
convention because the levels are usually - as in this example - on a nominal scale
and the implied legitimacy of interpolation is not meaningful. Parallel trace lines
indicate lack of interaction. Nonparallel trace lines indicate interaction. he p-value
in the ANOVA table is used to determine how far from parallel the lines must be to
rejectthenull hypothesis ofnointeraction. Ifinteraction isdetermined tobepresent,
then the main effects are usually not interpretable and we must use simple effects
(Sect.
.
.
) instead.
(
μ
ij
−
μ
i
′
j
)
differsfrom
(
μ
ij
′
−
μ
i
′
j
′
)
Two-factor Rhizobium Example
6.6.1
Erdman (
)discusses experiments to determine if antibiosis occursbetween Rhi-
zobium meliloti and Rhizobium trifolii. Rhizobium isa bacteria, growing on the roots
of clover and alfalfa, that fixes nitrogen from the atmosphere into a chemical form
plants can use. he research question for Erdman was whether there was an interac-
tion between the twotypesof bacteria, onespecialized foralfalfa plants andtheother
for clover plants. If there were an interaction, it would indicate that clover bacteria
mixed with alfalfa bacteria changed the nitrogen-fixing response of alfalfa to alfalfa
bacteria or of clover to clover bacteria. he biology of the experiment says that inter-
action indicates antibiosis or antagonism of the two types of rhizobium. hat is, the
goal was to test whether the two types of rhizobium killed each other off. If they did,
then there would be less functioning bacteria in the root nodules and consequently
nitrogen fixation would be slower.
A portion of Erdman's study involves a two-way factorial layout with factors
strain
at sixlevelsofrhizobiumculturesand
comb
,afactorwithtwo distinctbac-
teria as the two levels. he univariate response is a function of the nitrogen content
per milligram of plants grown under these
conditions. Erdman was specifically
interested in whether two factors interact, as this would have implications for best
choice of
strain
. he standard interaction plot for these data is in Fig.
.
.
Extended Two-way Interaction Plot
6.6.2
Figure
.
isatrellisplotthat illustrates allthemaineffectsandtwo-wayinteractions
foramultifactormodel.Boxplotsforthemaineffectsareshownalongthemain(SW-
NE) diagonal of the matrix. Two standard interaction plots, with interchanged roles
for the trace and x-factors, are shown along the off-diagonals. In our experience it is
not redundant to show the interchanged roles because usually one of the interaction
