Peter H. Schönemann
Professor Emeritus • Department of Psychological Sciences • Purdue University

[hierarchy]

# Hierarchy

Hierarchy is the term Spearman used to characterize patterns of a correlation matrix that are necessary for the existence of  a general ability "g": A pxp correlation matrix is called hierarchical if, after suitable joint permutations of its rows and columns, its entries diminish as one moves away from the diagonal.

This is the graphic expression of the (necessary) algebraic condition that  all tetrad differences (2x2 determinants not involving diagonal elements) vanish.

Both conditions follow at once from Spearman's Two Factor Theory.  Spearman postulated that any "intelligence test"worthy of the name can be expressed as a sum of a general intelligence factor (g) and a specific factor (that includes measurement error). The theory further postulates that all p specific factor are uncorrelated with each other and also with g. Hence, if one partials out this general factor, one is left with a diagonal partial covariance matrix of the p uncorrelated specifics. In other words,  the numerator of the familiar partial correlation formula must be zero for all pairs of tests, i, j:

rij.g = [rij - rigrgj]/denominator = 0 so that rij = rigrgj for all i,j (i different from j) .

One usually writes ai := rig. The px1 vector of the ai (>0) is called the factor pattern, its p elements ai  loadings.  On arranging them in descending order of magnitude, once finds  that such a correlation matrix satisfies hierarchy. However, in actual fact most empirically observed pxp correlation matrices do not satisfy this strong condition, which means, sadly, that "g does not exist" in the real world.

Some experts  have tried to deflect from this inconvenient fact of nature by perverting Spearman's notion of "g" to suit their own  commercial and ideological agendas. A popular ploy is to substitute the first principal component for g, which does not satisfy Spearman's conditions for the existence of g. Most of these experts  have probably never  read Spearman, who introduced these concepts in:

General intelligence, objectively determined and measured. American Journal of Psychology, 1904, 15, 201-293.

Over the years, this has  led to a great deal of  confusion which probably could have been avoided if the peer review system had functioned the way it is supposed to.