What is Spearman correlation?

Charles Spearman, answered from the texts and cited to the page.

Charles Spearman answers

The question conflates two things that share a name but are not the same instrument. The Spearman rank-order correlation — the familiar rho, computed on ranked data — is a statistical convenience that predates my theoretical work and bears my name only by the accident of priority. What I built, and what matters to the theory of mind, is something altogether more specific: the use of correlation coefficients as the raw material from which the structure of ability is to be inferred.

The start of the whole inquiry was a curious observation made in the correlations calculated between measurements of different abilities — scores for tests, marks for school subjects, estimates made on general impression.¹ These correlations were noticed to tend toward a peculiar arrangement expressible in a definite mathematical formula, what I called the tetrad equation.

The left-hand side of that equation — the tetrad difference — is formed by taking four abilities, a, b, p, q, and computing r_ap × r_bq minus r_aq × r_bp.² When that quantity vanishes throughout a table of correlations, something mathematically remarkable follows: every individual measurement of every ability in that table can be divided into exactly two independent parts.³

One part, g, remains the same for any one individual across all the correlated abilities; the other, s, varies from ability to ability even within the same person.⁴ That is the engine. The correlations are not merely described; they are interrogated. When the tetrad equation is satisfied, a single common factor is demonstrated, not guessed. When it fails — when significant tetrad differences persist — a group factor is established and can be measured.⁵

The method thus cuts both ways: it proves the presence of g where the equation holds, and proves the presence of something beyond g where it does not. So the honest answer is this: Spearman correlation, in the sense that matters to the science of ability, is the entire apparatus of tetrad differences brought to bear on a matrix of intercorrelations among cognitive performances. The rank-order rho is a tool; the tetrad criterion is a discovery.

Sources

1. These correlations were noticed to tend towards a peculiar arrangement, which could be expressed in a definite mathematical formula.

   The Abilities of Man

2. r_bq − r_aq × r_bp = 0. This formula has been termed the tetrad equation and the value constituting the left side of it is the tetrad difference.

   The Abilities of Man

3. whenever the tetrad equation holds throughout any table of correlations, and only when it does so, then every individual measurement of every ability... can be divided into two independent parts

   The Abilities of Man

4. The one part has been called the general factor and denoted by the letter g; it is so named because, although varying freely from individual to individual, it remains the same for any one individual in respect of all the correlated abilities.

   The Abilities of Man

5. nothing but the presence of significant tetrad differences can prove that any specific correlation is present. Hence... we now want to discover all cases where it fails to be so. Whenever this happens, there must necessarily exist some 'specific correlation'; and often this can actually be measured. Therewith is also established and measured the group factor.

   The Abilities of Man