Worksheet - Satistical Comparison of Proximity Matrices

Example:

Proximity Matrix 1	Proximity Matrix 2
A1 B1 C1 D1 E1 a1 21 40 32 28 12 b1 10 32 45 24 21 c1 0 13 16 11 5 d1 8 -4 12 17 25 e1 -4 -14 6 19 17	A2 B2 C2 D2 E2 a2 1 1 1 1 1 b2 0 1 1 1 1 c2 0 0 1 1 1 d2 0 0 0 1 1 e2 0 0 0 0 1

Calculate a Z-statistic as the sum of products of corresponding elements in Proximity Matrix 1 and 2

	A	B	C	D	E
a	21	40	32	28	12
b	0	32	45	24	21
c	0	0	16	11	5
d	0	0	0	17	25
e	0	0	0	0	17

Z = 346

Now calculate Kendall's tau for matrices 1 and 2. Kc = S |(X_ij-X_ik)(Y_ij-Y_ik)|

First for rows (Kr)

	AB	AC	AD	AE	BC	BD	BE	CD	CE	DE	S
a	0	0	0	0	0	0	0	0	0	0	0
b	1	1	1	1	0	0	0	0	0	0	4
c
d
e

First for columns (Kc)

	A	B	C	D	E
ab
ac
ad
ae
bc
bd
be
cd
ce
de
S

tau = Kc+Kr

Now permute rows and columns of Table 2 in all possible ways, calculate a Z for each and plot the distribution of Z-Values. How many generated ones are higher than the actual Z?

Worksheet:

Proximity Matrix 1	Proximity Matrix 3
A1 B1 C1 D1 E1 a1 21 40 32 28 12 b1 10 32 45 24 21 c1 0 13 16 11 5 d1 8 -4 12 17 25 e1 -4 -14 6 19 17	A2 B2 C2 D2 E2 a2 0 0 1 1 1 b2 0 0 0 1 1 c2 -1 0 0 0 1 d2 -1 -1 0 0 0 e2 -1 -1 -1 0 0

Proximity Matrix 3

Calculate a Z-statistic as the sum of products of corresponding elements in Proximity Matrix 1 and 2

	A	B	C	D	E
a
b
c
d
e

Z =

Now consider the following permutation of rows and columns for Table 3

Proximity Matrix 1	Permutation Proximity Matrix 3
A1 B1 C1 D1 E1 a1 21 40 32 28 12 b1 10 32 45 24 21 c1 0 13 16 11 5 d1 8 -4 12 17 25 e1 -4 -14 6 19 17	C2 E2 A2 D2 B2 b2 0 1 0 1 0 e2 -1 0 -1 0 -1 c2 0 1 -1 0 0 d2 0 0 -1 0 -1 a2 1 1 0 1 0

	A	B	C	D	E
a
b
c
d
e

Zp =

Now calculate Kendall's tau (Kc) for matrices 1 and 3

	A	B	C	D	E
a
b
c
d
e

Kc =