The Relationship Between Delta and r-Biserial

Abstract

The negative relationship we have observed in earlier chapters between discrimination and difficulty is at the heart of the problems reported by item writers, namely, that hard items that are also discriminating “are difficult to write.” Naturally, it is important to have items of a wide range of difficulty. If the entire range of difficulty is not represented in the test, there will be problems in having the test scale properly because of ceiling effects. For example, there may not be sufficient data at the upper and lower extremes of the scale to determine the appropriate equating transformation that ensures that test scores are comparable among forms. (For a review of equating methodology see, e.g., Holland & Rubin, 1982; for an introduction see Hambleton & Swaminathan, 1985.) Similarly, to ensure that the same composite of abilities is being reflected in the total score, the contribution from each of the subtests to the overall test specifications should remain constant among forms. In short, having an understanding of the causes behind the apparent difficulties in writing items at a particular level of difficulty is essential to maintaining, and possibly improving, the psychometric characteristics of a test. In this chapter we will explore several explanations for the negative relationship between delta and r-biserial.