New Advances in Statistical Modeling and Applications

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AStA - Advances in Statistical Analysis, a journal of the German Statistical Society, is published quarterly and presents original contributions on statistical methods and applications and review articles. Authors submitting a manuscript should indicate for which section their work is intended.

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Description: AStA - Advances in Statistical Analysis, a journal of the German Statistical Society, is published quarterly and presents original contributions on statistical methods and applications and review articles. Statistical Applications: The Statistical Application section provides a forum for innovative use of statistical modeling and analysis techniques in a wide range of application areas.

Additional examples of BEAR maps are provided later in this chapter. The representation in Figure 4—2 corresponds to a class of measurement models called item response models, which are discussed below.

First, how-. Boxes indicate observable variables; oval indicates a latent variable. Early studies of student testing and retesting led to the conclusion that although no tests were perfectly consistent, some gave more consistent results than others. Classical test theory CTT was developed initially by Spearman as a way to explain certain of these variations in consistency expressed most often in terms of the well-known reliability index.

In CTT, the construct is represented as a single continuous variable, but certain simplifications were necessary to allow use of the statistical methods available at that time. The observation model is simplified to focus only on the sum of the responses with the individual item responses being omitted see Figure 4—3.

For example, if a CTT measurement model were used in the BEAR example, it would take the sum of the student scores on a set of assessment tasks as the observed score. The reliability is then the ratio of the variance of the true score to the variance of the observed score. This type of model may be sufficient when one is interested only in a single aspect of student achievement the total score and when tests are considered only as a whole.

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Scores obtained using CTT modeling are usually translated into percentiles for norm-referenced interpretation and for comparison with other tests. The simple assumptions of CTT have been used to develop a very large superstructure of concepts and measurement tools, including reliability indices, standard error estimation formulae, and test equating practices used to link scores on one test with those on another.

CTT modeling does not allow. Formally, CTT does not include components that allow interpretation of scores based on subsets of items in the test. Historically, CTT has been the principal tool of formal assessments, and in part because of its great simplicity, it has been applied to assessments of virtually every type. Because of serious practical limitations, however, other theories—such as generalizability theory, item response modeling, and factor analysis—were developed to enable study of aspects of items.

The purpose of generalizability theory often referred to as G-theory is to make it possible to examine how different aspects of observations—such as using different raters, using different types of items, or testing on different occasions—can affect the dependability of scores Brennan, ; Cronbach, Gleser, Nanda, and Rajaratnam, In G-theory, the construct is again characterized as a single continuous variable. However, the observation can include design choices, such as the number of types of tasks, the number of raters, and the uses of scores from different raters see Figure 4—4.

These are commonly called facets 4 of measurement. Facets can be treated as fixed or random. When they are treated as random, the observed elements in the facet are considered to be a random sample from the universe of all possible elements in the facet.

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For instance, if the set of tasks included on a test were. When facets are treated as fixed, the results are considered to generalize only to the elements of the facet in the study. In practice, researchers carry out a g-study to ascertain how different facets affect the reliability generalizability of scores. This information can then guide decisions about how to design sound situations for making observations—for example, whether to average across raters, add more tasks, or test on more than one occasion. To illustrate, in the BEAR assessment, a g-study could be carried out to see which type of assessment—embedded tasks or link items—contributed more to reliability.

Such a study could also be used to examine whether teachers were as consistent as external raters. Generalizability models offer two powerful practical advantages. First, they allow one to characterize how the conditions under which the observations were made affect the reliability of the evidence. Second, this information is expressed in terms that allow one to project from the current assessment design to other potential designs. Perhaps the most important shortcoming of both CTT and G-theory is that examinee characteristics and test characteristics cannot be separated; each can be interpreted only in the specific context of the other.

Whether an item is difficult or easy depends on the ability of the examinees being measured, and the ability of the examinees depends on whether the test items are difficult or easy. Item response modeling IRM was developed to enable comparisons among examinees who take different tests and among items whose parameters are estimated using different groups of examinees Lord and Novick, ; Lord, Furthermore, with IRM it is possible to predict the properties of a test from the properties of the items of which it is composed.

In IRM, the construct model is still represented as a single continuous variable, but the observation model is expressed in terms of the items as in Figure 4— 5.

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The model is usually written as an equation relating the probability of a. The student parameter is usually translated into a scaled score 5 for interpretation.

The item parameters express, in a mathematical equation, characteristics of the item that are believed to be important in determining the probabilities of observing different response categories. Most applications of IRM use unidimensional models, which assume that there is only one construct that determines student responses.

Indeed, if one is interested primarily in measuring a single main characteristic of a student, this is a good place to start.


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However, IRMs can also be formulated for multidimensional contexts discussed below. Formal components of IRM have been developed to help diagnose test and item quality. For instance, person and item fit indices help identify items that do not appear to work well and persons for whom the items do not appear to work well. As an example of how IRM can be used, consider the map of student progress for the variable Designing and Conducting Investigations displayed in Figure 4—6.

Although hidden in this image of the results, the underlying foundation of this map i. One can now, additionally, talk about what tends to happen with specific items, as expressed by their item parameters. This formulation allows for observation situations in which different students can respond to different items, as in computerized adaptive testing and matrix-sampling designs of the type used in the National Assessment of Educational Progress NAEP.

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Figure 4—5 can also be used to portray a one-dimensional factor analysis, although, in its traditional formulation, factor analysis differs somewhat from IRM. Like IRM, unidimensional factor analysis models the relationship between a latent construct or factor e. In factor analysis, the observable variables are strictly continuous rather than ordered categories as in IRM. More recent formulations relax these limitations. Similar to ways in which G-theory has extended CTT, elements of the observations, such as raters and item features, can be added to the basic item response framework see Figure 4—7 in what might be called faceted IRMs.

Examples of facets are 1 different raters, 2 different testing conditions, and 3 different ways to communicate the items. One foundational difference is that in IRMs the items are generally considered fixed, whereas in G-theory they are most often considered random.


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  • That is, in G-theory the items are considered random samples from the universe of all possible similarly generated items measuring the particular construct. In practice very few tests are constructed in a way that would allow the items to be truly considered a random sampling from an item population. In the measurement approaches described thus far, the latent construct has been assumed to be a continuous variable. In contrast, some of the research on learning described in Chapter 3 suggests that achievement in certain domains of the curriculum might better be characterized in the form of discrete classes or types of understanding.

    That is, rather than assuming. Models based on this approach are called latent class models. The classes themselves can be considered ordered or unordered. When the classes are ordered, there is an analogy with the continuum models: each latent class can be viewed as a point on the continuum see Figure 4—8. An argument could be made for using latent classes in the BEAR example discussed earlier. For interpretation purposes, this map would probably be just about as useful as the current one.

    One might ask, which assumption is right—continuous or discrete? The important question is, given the decisions one has to make and the nature of cognition and learning in the domain, which approach provides the most interpretable information? Investigating this question may indeed reveal that one approach is better than the other for that particular purpose, but this finding does not answer the more general question. Each of the four general classes of models described above—classical, generalizability, item response, and latent class—can be extended to incorporate more than one attribute of the student.

    Doing so allows for connections to a richer substantive theory and educationally more complex interpretations. In multidimensional IRM, observations are hypothesized to correspond to multiple constructs Reckase, ; Sympson, For instance, performance on mathematics word problems might be attributable to proficiency in both mathematics and reading. In the IEY example above, the progress of students on four progress variables in the domain of science was mapped and monitored see Box 4—2 , above. Note that in this example, one might have analyzed the results separately for each of the progress variables and obtained four independent IRM estimations of the student and item parameters, sometimes referred to as a consecutive approach Adams, Wilson, and Wang, There are both measurement and educational reasons for using a multidimensional model.

    In measurement terms, if one is interested, for example, in finding the correlation among the latent constructs, a multidimensional model allows one to make an unbiased estimate of this correlation, whereas the consecutive approach produces smaller correlations than it should. Educationally dense longitudinal data such as those needed for the IEY maps can be difficult to obtain and manage: individual students may miss out on specific tasks, and teachers may not use tasks or entire activities in their instruction.