Reliability and validity of tests are varied by selection of subjects in some of the tests. The correction formulas have been algebraically derived in Chapters 10, 11, 12, and 13 of Gulliksen's text, “Theory of Mental Tests (1950, John Wiley)” (Chapter 13 is an exception). The same formulas can be derived by the concept of test vectors. It is assumed here as in a conventional model in test theory that a test vector has two orthogonal components, T and E. The length of the test vector is adjusted to be equal to the standard deviation of the test scores in a given group, and the reliability coefficient is defined by the square of the cosine of the angle between the test vector X and its true component T. The correlation coefficient between two distinct test vectors, X₁ and X₂, is defined by the cosine of the angle between X₁ and X₂.
Change of reliability coefficients. Let R and r be reliability coefficients of a test in an original group and in a selected group, respectively, and σ and s be standard deviations of the test in these two groups. A formula (1-R)σ²=(1-r)s² can be derived from the assumptions that only the true component of the test vector T is varied by selection and the error component E is invariant under selection.
Validity under univariate selection. When the number of explicit selection variables is one and that of incidental selection variables is n, the relationship between the variance-covariance matrix of tests for the original group and that for the selected group is given by C=P^TDP, where C and D are the (n+1)-variance-covariance matrices of tests for the original and the selected group, respectively, including an explicit selection variable X in the first entry, and where P=[(σ_X/s_x)(σ_X/s_x-1)(s_y1/s_x)r_xy1(σ_X/s_x-1)(s_y2/s_x)r_xy2…(σ_X/s_x-1)(s_yn/s_x)r_xyn 0 1 0 0 0 0 1 0 0 0 0 1] in which σ_X is the standard deviation of the explicit selection test X in the original group, s_x, s_y1, …s_yn are standard deviations of explicit selection test x and incidental selection tests, y₁, …, y_n in the selected group, and r_xy1, etc., are correlation coefficients between tests in the selected groups. The underlying assumptions are that the component of an incidental selection vector which is orthogonal to the explicit selection vector X is invariant under the selection for X and that the component which is parallel to X is changed to the same direction with the same proportion as the change of X.
Validity under multivariate selection. The general formulas correcting correlation coefficients under multivariate selection have been given by Aitken (1934, Proc. Edinburgh Math. Soc.) in a matrix form and they are also treated in Chapter 13 of Gulliksen's text, The same formulas are derived from the assumptions below as an extension of the vector concepts referred in the preceding paragraphs. The incidental selection vectors are divided into two components. The first one is normal to the hyperplane spanned by explicit selection vectors and is assumed to be