This paper traces the course of the consequences of viewing test responses as simply providing dichotomous data concerning ordinal relations. It begins by proposing that the score matrix is best considered to be items-plus-persons by items-plus-persons, and recording the wrongs as well as the rights. This shows how an underlying order is defined, and was used to provide the basis for a tailored testing procedure. It also was used to define a number of measures of test consistency. Test items provide person dominance relations, and the relations provided by one item can be in one of three relations with a second one: redundant, contradictory, or unique. Summary statistics concerning the number of relations of each kind are easy to get and provide useful information about the test, information which is related to but different from the usual statistics. These concepts can be extended to form the basis of a test theory which is based on ordinal statistics and frequency counts and which invokes the concept of true scores only in a limited sense.

Applications of item response theory, which depend upon its parameter invariance property, require that parameter estimates be unbiased. A new method, weighted likelihood estimation (WLE), is derived, and proved to be less biased than maximum likelihood estimation (MLE) with the same asymptotic variance and normal distribution. WLE removes the first order bias term from MLE. Two Monte Carlo studies compare WLE with MLE and Bayesian modal estimation (BME) of ability in conventional tests and tailored tests, assuming the item parameters are known constants. The Monte Carlo studies favor WLE over MLE and BME on several criteria over a wide range of the ability scale.