LATENT ROOTS AND VECTORS 489

squares of the factor-loadings then gives the factor-variance for that
factor. (See Table II(#), where the factor-loadings and variances
for the first factor are calculated explicitly.) Here, as the reader
can easily test for himself, simple summation happens to yield the
same result as weighted summation.

The factor-measurements for each person, Pt9 can be computed
by simply adding his test-measurements, after first weighting them
by the factor-loading for the test concerned (F'tM) : the totals
are then usually normalized, i.e. reduced to unitary standard
measure, by dividing by the respective factor-variances (V) (the
detailed working is shown in Table I). It will be observed that the
calculation is very much simpler than the laborious procedure
required in calculating regression coefficients to obtain estimated
factor-measurements in the ordinary way (described by Thurstone
[84], pp. 226 L, and Thomson [132], pp. 93 f.).

The latent vectors, Lh are obtained by simply normalizing the
columns of the factor-loadings, F/: i.e. dividing each column by
the root of its squares.

When c correlating tests,' therefore, we first obtain the
covariance matrix RL ; on factorizing this by the * least-squares'
method we obtain the factorial matrix, Ft; and from Ft we obtain
(i) the normalized factor-loadings for tests, Lh (ii) the factor-
variances, VD and (iii) the factor-measurements for persons, Pt.

Table II(i) shows how the initial set of measurements can be
reconstructed by multiplying these three component matrices.

Similarly, when i correlating persons' we first obtain the
covariance matrix, Rp; on factorizing this we obtain the factorial
matrix, Fp ; and from Fp we obtain (i) the normalized factor-load-
ings for persons, L^ (ii) the factor-variances, /^, and (iii) the factor-
measurements for tests, Pp.

Table lll(b) shows how the initial set of measurements
can be reconstructed by multiplying these three component
matrices.

It will be observed (i) that the factor-variances are the same in
either case (Fp = 7/), and (ii) that the factor-measurements for *
persons obtained by 'covariating tests' are identical with the normal-
ized factor-loadings for persons obtained by c covariating persons,5
and (iii) that the factor-measurements for tests obtained by ' co-
variating persons ? are identical with the normalized factor-loadings
for tests obtained by * covariating tests?: in short, with a measure-
ment matrix thus standardized and thus factorized, the results
are the same whether we correlate (or rather covariate) persons
or tests,