Ch. 14: Cluster Analysis (CA)
I. Situation
A. Given proximity or distance data obtained from all possible
pairwise combinations of stimuli or regular subject-by-
variable data from a usual data collection, CA makes
clusters
based on the distance between stimuli (variables).
B. Many similar techniques have been developed in different
areas (biology, sociology, psychology). SAS lists 11 methods.
C. CA methods can be classified into hierarchical or non-
hierarchical cluster analyses.
1. Hierarchical Cluster Analysis
a) Starts with n-clusters (each stimulus is a cluster).
b) The two closest clusters are merged to form a new
cluster that replaces the two old clusters.
c) Merging two closest clusters is repeated until only
one cluster is left.
d) Different methods have different ways to compute the
distance between two clusters.
2. Non-hierarchical Cluster Analysis: n-clusters will be
separated into g-clusters without using a hierarchical
method.
II. Similarity(Proximity) and Dissimilarity(Distance) data
A. Euclidean distance
d(x, y) =
)(
)'()()()'(
1
1
2
yx
Syxyxyxyx
p
iii
−−=−=−−
−
=
∑
.
where
x = (x1, x2, . . xp)’,
y = (y1, y2, . . yp)’,
p = the number of stimuli, and
S = sample covariance matrix.
B. Minkowski metric (general formula)
d(x, y) =
r
p
i
r
ii
yx
/1
1
||
−
∑
=
.
where p = the number of stimuli, and r = the order of power.
If r=2, then it is the Euclidean distance. If p=2, r=1, then
it is the city block (Manhattan distance)
C. Correlation(closeness of the shapes)
The squared Euclidean distance d2(x, y) =
∑
=
−
p
iii yx
1
2
)(
can be
expressed as,
d2(x, y) =
)1(2
)()(
22 xyyxyx
rvvyxp
vv −+−+−
,
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