The Concept of Test Equivalence

The finding of a single factor in a correlation matrix can nevertheless be very meaningfull, provided that it is not immediately reified into a specific unitary faculty. It means that, whatever factors play a role in the emergence of the test scores, they play a role in the same way for all tests. In that respect it is much more appropriate to use the concept of test equivalence instead of the concept of unidimensionality. The concept of unidimensionality may easily suggest that a single, unitary faculty is measured by the various tests. The concept of test equivalence does not have this connation. A formal definition of test equivalence has been given by by Lord & Novick (1968, p. 49-50). They distinguish the concepts of -equivalence and essentially -equivalence. The definition of -equivalence is given on page 49 and reads as follows:

Distinct measures and of person i on measurement j and k are -equivalent if, for all i, (Lord & Novick (1968, p. 49, definition 2.13.5).

is the observed score of person i on measurement j and is the true score of person i on measurement j.

The true score of person i on measurement j is defined as the expected value of of the observed score, that is (Lord & Novick (1968, p. 30, definition 2.3.1).

The definition of essentially -equivalence is given on page 50 and reads as follows:

Distinct measures and of person i on measurement j and k are essentially -equivalent if, for all i, , where is a constant (Lord & Novick (1968, p. 50, definition 2.13.8).

Note, that neither in the case of -equivalence, nor in the case of essentially -equivalence , that is, the error variances are not necessarily equal. Both -equivalence and essentially -equivalence are special cases of linearly -equivalence, where linearly -equivalence is defined as follows:

Distinct measures and of person i on measurement j and k are linearly -equivalent if, for all i, , where and both are constants.

For and , one obtains -equivalence and for , one obtains essentially -equivalence. If the additional assumption is made that , then linearly -equivalence is equivalent with the unidimenional factor analytical model. Therefore it is legitimate to use factor analysis to the test the null-hypothesis of linearly -equivalence. Note, that the factor loadings reflect the error variances .