1

–

2

Apple Blueberry Cherry Dewberry Elderberry

Judge 1 5 2 3 1 4

Judge 2 4 1 2 3 5

438 rank correlation

Dewberry Blueberry Cherry Elderberry Apple

Judge 1 1 2 3 4 5

Judge 2 3 1 2 5 4

as the ranking of pies in a taste-testing competition.

S

PEARMAN

’

S METHOD

and K

ENDALL

’

S METHOD

can be

used to test the degree of association between two dif-

ferent rankings of the same set of objects.

In

MATRIX

theory, the row rank of a matrix is the

maximum number of linearly independent rows the

matrix possesses, and its column rank is the maximum

number of linearly independent columns it has. The pro-

cess of G

AUSSIAN ELIMINATION

shows that these two val-

ues always agree, and the common value of these two

ranks is called the rank of the matrix. An m×nmatrix

is said to be of full rank if its rank equals the smaller of

mand n. A square matrix of full rank is invertible.

See also

INVERSE MATRIX

;

RANK CORRELATION

.

rank correlation Two methods are commonly used

to determine whether or not two ranking schemes are

well-matched. For example, in a pie-baking contest,

two judges might rank five pies as follows:

If the judges followed purely objective criteria, and

were free of personal preference in their choices, then

one would expect two identical ranking choices. If,

on the other hand, the judges followed entirely inde-

pendent criteria, or no criteria at all (randomly

assigning ranks), then one would expect very little

correlation between the two lists. The results of this

competition seem to lie somewhere between these

two extremes.

Kendall’s Coefficient of Rank Correlation

In 1938 M. G. Kendall developed one measure of rank

association given by a single numerical value τ, adopt-

ing values between –1 and 1. A value of 1 indicates a

perfect matching in rank values; a value of 0 indicates

no consistency in the rank assignments; and a value of

–1 perfect disagreement (that is, one judge’s top choice

is the other judge’s least favored choice, and so on.)

The numerical value τis computed by completing the

following steps:

1. Rearrange the order of the entrants so that the ranks

given by the first judge are in order.

2. Looking now only at the second row, compute the

score of each entrant. This is the number of entrants

to its right higher in rank minus the number of

entrants to its right of lower rank. (The rightmost

entrant is assigned a score of zero.)

3. Sum all the scores. Call this sum S.

4. If the number of entrants is n, then the maximum

possible sum is (n– 1) + (n– 2) +…+ 2 + 1 + 0 =

n(n– 1). Kendall’s coefficient of rank correlation

is the ratio of Sto this maximal sum:

In our example, step 1 of the procedure yields the

reordered table:

There are two entrants to the right of dewberry with

rank higher than 3 (elderberry and apple), and two

lower than 3 (blueberry and cherry), thus dewberry has

score 0. Similarly, the scores of the remaining pies are:

blueberry = 3 – 0 = 3, cherry = 2 – 0 = 2, elderberry =

0 – 1 = –1, and apple = 0. The total sum of scores is S=

0 + 3 + 2 + (–1) + 0 = 4. The maximum possible sum is

4 + 3 + 2 + 1 + 0 = 10. Thus Kendall’s rank correlation

coefficient here is:

This value indicates that the rank assignments are pos-

sibly inconsistent.

Spearman’s Coefficient of Rank Correlation

An alternative rank coefficient was developed by

Charles Spearman in 1904. It is denoted ρ, and is com-

puted by summing the differences squared in the rank