1 Aligned-Rank Transform Methods
We have seen that rank-based methods such as the Kruskal-Wallis test for a
completely randomized design are useful nonparametric alternatives to the
analysis of variance (ANOVA). These methods were further advanced by
Iman and Conover (1981), who discussed the concept of using standard para-
metric tests applied to ranked data. However, the rank-transform method
can fail when applied to more complicated models, such as a factorial ANOVA
model with interaction.
An alternative to the rank-transform method for multifactor experiments
is the aligned-rank transform method. We will describe the aligned-rank
transform method for a completely randomized factorial design with two
factors and balanced data (replications within treatment combinations are
all equal). The standard model for a completely randomized factorial design
with two factors, including interaction, is:
Yijk =µ+αi+βj+γij +εijk ,
where Yijk is the kth replicate observation from level iof factor Aand level
jof factor B. With (typical) sum-to-zero restrictions on the parameters,
the least-squares estimates are:
bαi=bµi. −bµ, b
βj=bµ.j −bµ, bγij =bµij −bµi. −bµ.j +bµ,
where
bµi. =1
nb X
j
X
k
Yijk ,bµ.j =1
na X
i
X
k
Yijk ,bµ=1
nab X
i
X
j
X
k
Yijk
The idea behind the aligned-rank transform method is to create a sep-
arate term for each effect in the model (main effects and interactions) by
subtracting out all other effects. Then these separate terms (the aligned
effects) are ranked and analyzed. For the interaction term and the main
effects of factors Aand B, the aligned effects are:
ABijk =Yijk −bµ−bαi−b
βj, Aijk =Yijk −bµ−b
βj−bγij ,and Bij k =Yijk −bµ−bαi−bγij.
Note that these aligned effects are always calculated using the estimates
of µ, α, β, and γfrom the full model above which includes all terms. Each of
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