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ANOVA Case Study
Automobile Brake Tests

Data Format: Replicated Two-Factor ANOVA
Table 1 shows the format of a data set with two factors and a balanced (equal) number of observations per treatment (each row-column intersection is a treatment). To avoid needless subscripts, the observations in each treatment are represented simply as xxx. Except for the replication within cells, the format is the same as the unreplicated two-factor ANOVA.

Table 1. Data Format of Replicated Two-Factor ANOVA

Levels of Factor B
B1 B2 ... Bc Row Sum Row Mean
Levels of Factor A A1 xxx
xxx
...
xxx xxx
xxx
...
xxx ...
...
...
... xxx
xxx
...
xxx R1

A2 xxx
xxx
...
xxx xxx
xxx
...
xxx ...
...
...
... xxx
xxx
...
xxx R2

... ... ... ... ... ... ...
Ar xxx
xxx
...
xxx xxx
xxx
...
xxx ...
...
...
... xxx
xxx
...
xxx Rr

Col Sum C1 C2 ... Cc T
Col Mean

...


There are three F tests which could be performed: one for each main effect (factors A and B) and a third F test for interaction. The sums of squares may be visualized in the form of a table, shown in Table 2 for an experiment with r rows and c columns, and with m replications per treatment. The total sum of squares is partitioned into four components:

SST = SSA + SSB + SSI + SSE

where

SST = total sum of squared deviations about the mean
SSA = between rows sum of squares (effect of factor A)
SSB = between columns sum of squares (effect of factor B)
SSI = interaction sum of squares (effect of AB)
SSE = error sum of squares (residual variation)


Table 2. Two-Factor ANOVA With Replication
Source of
Variation Sum of Squares Degrees of
Freedom Mean
Square F Ratio
Factor A
(rows) SSA =
r†1


Factor B
(columns) SSB =
r†1


Interaction
(Aï‚´B) SSI =
(r†1)(c†1)


Error SSE =
rc(m-1)

Total SST =
rcm†1

Data: Braking Distance
Table 3 shows a random sample of braking results from...