X.—The Factorial Analysis of Multiple Item Tests

Summary 6. The expected score of an individual on a test consisting of a large number of items is assumed to be given by a formula involving the ability of the individual and also two quantities constant for the test. An expression is then derived for the covariance between two tests measuring different abilities. It appears that if a factorial analysis is performed on a set of tests of unequal difficulty, using the matrix of variances and covariances, a spurious factor will tend to be introduced depending mainly on the differences in difficulty. The effect of this is removed by transforming the variances and covariances to a new set of coefficients. A numerical example of the process is given. In conclusion I should like to thank the Carnegie Trust for the Universities of Scotland for a grant to cover the cost of the setting and printing of mathematical formulæ in a paper previously published in the Society's Proceedings (LXI, A, 1943, 273–287).