Consideraciones a la hora de diseñar e implementar Wi-Fi sobre LTE para crear

Indeed, the measurement of stage-governed development is of such importance that it has been the focus of Mark Wilson (University of California, Berkeley), who has developed the Saltus model (a latent group model) from the basic

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FIG. 9.2.

Map of subjects and items on the ligit scale for the three Noeling tasks. The 3A items of caskets task with a concrete operational difficults level are in bold. Items marked by * do not fir (t > 2.0).

Rasch model to detect discontinuity in development (Wilson, 1985). Two key papers by Wilson (1989) and Mislevy and Wilson (1996) cover the key theoretical and technical issues in the application of the Saltus model. The term "Saltus," from the Latin meaning "leap," is designed to represent Wilson's conception of the relatively abrupt changes in the relative difficulty of items for aperson who moves from one stage to the next. Although the tradition in developmental and other research had been to use Guttman scalogram analyses, Wilson (1989) summarized:

This technique, derived from attitude questionnaires, assumes that there is just one item per level and assigns subjects to levels according to the following crite--

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ria. (a) A subject is in level n if he or she has responded correctly up to and including the item representing level n and has failed the remainder of the items: Such aresponse is said to be scalable. (b) A subject with any other response is of a

nonscale type. (p. 277)

A far more realistic method would involve the development of a testing situation having many items or tasks at each of the developmental levels or stages, with the expectation that as a person moves into a new stage, discontinuity in development would be marked by a rather rapid transition from passing no, or few, tasks to passing many or most tasks indicative of the new level. Indeed, the Guttman requirements for ordinal scaling are rarely met even with continual refinement of testing procedures. There will always be some error in our recorded observations of human behavior. Some errors will be the result of deficiencies in our observation methods, whereas others will result from the nature of human behavior itself: the role of individual differences and the sheer unpredictability of humans. Under these pervasive conditions, Guttman scaling, while fundamentally correct on the crucial issue of order but ignoring the equally important aspect of stochasticity, has much more stringent requirements about human performance than the developmental theories it is being used to assess (Kofsky, 1966; Wilson, 1989).

The results from Noelting and Rousseau (in press) reported earlier in this chapter reveal what Wilson terms

segmentation, prima facie evidence for first-order developmental discontinuity: When tests with many items at each

level are submitted to Rasch analysis, items representing different stages should be contained in different segments of the scale (Fig. 9.2), with a nonzero distance between segments, and, of course, the items should be mapped in the order predicted by the theory. When item estimates are being tested for segmentation, the idea of a segmentation index must reflect the distance between the levels in terms of the errors of measurement of the item difficulty estimates. Given that the t test used to calculate the between-stages differences in the Significant Gap column in Fig. 9.2 is the ratio of the "difference between adjacent item estimates" and the "sum of the measurement errors for those two items," then Noelting's results provide evidence of stage like developmental discontinuity (Noelting, Coudé, Rousseau, & Bond, 2000; Wilson, 1989).

The more subtle type of leap that the Saltus model is uniquely designed to detect and model (i.e., second-order

developmental discontinuity) can be appreciated readily from consideration of the following Saltus principles. Although the Saltus method can be applied to a number of hierarchical developmental levels in one analysis, the key points are encapsulated more easily in a situation of less complexity: one in which person abilities and item difficulties can be seen as representing either of two developmental levels, as shown in Table 9.2.

For each group of subjects in turn, we shall presume to view the two types of items from their perspectives. In each situation, we summarize the ability levels of the persons at a certain level by the mean of the ability estimates and summa--

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TABLE 9.2

Subjects and Items Grouped for Saltus Example

rize the difficulty levels of the items at a certain level by the mean of the difficulty estimates. The vertical arrow indicates increasing person ability or item difficulty (see Fig. 9.3).

First, we present the perspective of the lower level group:

The reasoning is that from the perspective of the mean ability of the lower level subjects (β Group I), the mean difficulty of the lower level items (θA)appears to be relatively easy, whereas from that same perspective (βI), the mean difficulty of the higher level items (θA) appears to be hard (i.e., [θB ] - [βI ] >> [θA ] - [βI ]). Wilson referred to the difference between these two values as DI (the group I gap): DI = ([θB ] - [βI ]) -([θA ] - [βI ]). This represents the relative difference in difficulty between higher (θB) and lower (θA) items for the lower level group of persons (βI).

Now, we present the perspective of the higher level group (see Fig. 9.4). From the perspective of the mean ability of the higher level subjects (βII ) the mean difficulty of the lower level items (θA ) appears to be relatively easy, and

FIG. 9.3.

Group I subjects regard items grouped for Saltus example.

FIG. 9.4.

Group II subjects regard items grouped for Saltus example.

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the mean difficulty of the higher level items (θB ) appears to be relatively easy as well (i.e., [θB ] - [βII ] > [θA ] - [βII ]). Wilson referred to the difference between these two values as DII (the group II gap): DII = ([θB ] - [βII ]) - ([θA ] - [βII ]). This represents the relative difference in difficulty between higher (θB ) and lower (θA ) items for the higher level group of persons (βII ). The asymmetry index, D = DI - DII, is used then in the Saltus model to detect a less immediately obvious form of developmental discontinuity.

Therefore, a positive asymmetry index (i.e., DI > DII) indicates that the item types are closer together in item difficulty terms for group II than for group I, and a large DI - DII difference is the indicator of second-order developmental discontinuity: Type B items are much harder than type A items for subjects in group I (i.e., their probability of success on type B items is low). Once the leap is made across the developmental levels (i.e., for subjects in group II), the difference between the two item types becomes much less.

When the asymmetry index is 0, second-order developmental discontinuity has not been detected, and the principles of the basic Rasch model can be applied to the detection of first-order developmental discontinuity (as in the Noelting example). Wilson (1989) claimed that the further the asymmetry index is from 0, the greater the need for the Saltus model to interpret what the Rasch model could detect only as misfit. Importantly, Saltus results include both a probability of a person being in each stage and a location in each.