HIERARCHICAL CRITERIA 349

sums of the squares of the new residuals will be equal to (u — i) f2.

TT i, <.- r i ZEr,?4.i tt— I W — J — * — I

Hence the ratio of the two sums

u n — s — t

and the ratio of the standard deviations of the two successive sets

of residuals will be A/-----------------. Accordingly, when the ratio

T TI ~~~ S ~~~ t

reaches this value, it might be supposed that the residual correlations
whose squares have been summed for the denominator are deter-
mined entirely by chance.

Thurstone plots a curve for the standard deviations of the first
13 sets of residuals obtained in his research ([17], p. 64) ; and the
foregoing result fits the prolongation of his curve sufficiently well.
He offers, however, no criterion based on these standard deviations ;
but proposes instead an empirical rule based on a similar comparison
of sums or means, i.e. upon the mean deviations. When s significant
factors have been extracted, " so that only chance variation remains

in the residuals,"1 then, he suggests, A/ r^ + * =-------» where

T 2^ 2*iT$ ft

ZZrs and ZErs + 1 denote the numerical sum of the values of the
residuals, calculated regardless of sign. This rule, however, does
not make allowance for the fact that we are dealing with * reduced '
matrices. To indicate the approach to equality, the foregoing
argument suggests we should rather take

t u — I n — s — t — I

_ _

22rs ~~ u ~~ n — s — - t

Let us test this formula by applying it to Thurs tone's own ^ data.
His correlations are based on 57 tests. Consequently the minimum
rank of the initial correlation matrix will be 47. After extracting
13 factors, the rank of the residual matrix should be 34. Hence the
foregoing formula suggests that the ratio of the sums of the two
successive residuals will be approximately f£ = *97JjProvided tl^e
remaining factor-variances are equal. Actually, the observed ratio
(based on factorization by simple, not weighted, summation) is given
by Thurstone as -960, and is apparently still rising : the preceding
figures are -921, -945, so that it would seem that equality has not
yet been reached.

For Thurstone's two 8-variable tables with no common factors
we should have 4 as the minimum rank ; the ratio would therefore
be | = -750 : Thurstone's calculations give -774 and -742^ respec-
tively, averaging -757. For his three 2O-variable tables with I, 2,
and 4 significant factors to be extracted, his own criterion gives the

1 Loc* tit., p. 66.