A theory and procedure of scale analysis with applications by R. J. Mokken

A theory and procedure of scale analysis with applications by R. J. Mokken

By R. J. Mokken

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The early standard works on the subject do not give an unambiguou� definition of error. This has led to a rather unfortunate formulation of cocnicienh of scalabiltiy. 5. One of the simplest definition'> of error gives equal weight to every imperfect pallem: 1 hey all count as o11c error. Thi::. simple method of defining error has hardly been actually put into practice. >ed. however, some imperfect paucrns arc judged "more wrong that others", as we shall sec. Fir'>t there i� the definition by Guuman himself.

K. (I. I) where X; is the score for item i. There are theoretically Zk different response pallerns. We shall further suppose that each subject has a gi v en hut unknown . g. attitude). Let that unknown value be(}. Our measurement problem amounts to getting information about IJ from the observable response vector x. In our discussion of the trace lines of item� we noted that they represented the probability of a scale answ er (score I) a� a function of this unknown value(}. We might formulate this proba b ility as* P{x; = lje} *Stochastic variables will be underlined in this study.

It is the population equivalent of the sample difficulty of an item and should again be distinguished from the item difficulty 8. We shall denote* the population difficulty of item i by Tr;. This population difficulty Tr; may be interpreted as that fraction of the population of subjects that will respond positively (will give the scale response) to item i. 3. 5 illustrate the effect of different population distributions in the case of (perfect) G-items. 4, for instance, Tr;. ;s 8; along the 8-axis, is represented by the shaded area under the di�trihution curve to the right of the point 8;.

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