ABSTRACT
In chapter 4 we presented a method of dealing with dichotomous items which is simplified by the fact that tied-ranks scores with these items are perfectly linearly correlated with the 1 or 0 scores usually used with these items. The use of measures of consistency, W and α, to determine whether items are ordering people consistently was shown to determine the extent to which these items are ordering persons on the same dimension. It was also noted that these same measures could be used to determine whether persons were reacting to a single dimension of items by consistently ordering them. In practice, the consistency of ordering persons may be high but that of ordering items may be low or vice versa. If both are very high the condition known as conjoint ordering is satisfied and this is a necessary but not sufficient condition to form an interval scale. For sufficiency, a condition known as cancellation is required as shown by Luce and Tukey (1964) and mentioned in chapter 2. This condition is almost never tested in applications of Item Response Theory, hence the development of Ordinal Test Theory is necessary.
