FACTOR ANALYSIS AND NEGATIVE FIT RESIDUAL CORRELATION VALUES 

            Question:

What role can Factor Analysis play in a Rasch analysis?

            Explanation:

There is an important difference between carrying out a factor analysis of the responses of a person-by-item matrix compared to carrying out a factor, or principal component, analysis of the residuals of the responses arising from Rasch model predictions.

A traditional factor analysis of the responses runs the risk of finding factors that are a function of item difficulty, and to some degree the distribution of the persons; items which are difficult and those which are easy have higher correlations amongst themselves.

In a factor, or principal component, analysis of residuals from the Rasch model predictions, account is taken of both the item difficulty and the person locations. Conducting a factor analysis on these residuals will then reveal if there is any systematic relationship between subsets of items after minimizing the occurrence of difficulty factors.

In RUMM2030, such an analysis can be undertaken by selecting the Residual Principal Component feature from the Display Control form. This is often referred to as a Principal Component Analysis, or PCA.  Loadings on the first and second factors can indicate subsets of items that might be more similar in their responses than accounted for by the model.

            Action:

• Refer to Chapter 3 of the "Displaying the RUMM2030 Analysis" manual for details.

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            Question:

Why are correlations between fit residuals generally negative?

            Explanation:

An examination of the correlation matrix of residuals can also be informative.  However, it is important to be aware that the expected correlation among the residuals is negative; in the case of just two items, the residual correlation would be -1.0.  With a typical number of dichotomous items, say 30 or so, the expected correlations may be close to zero and, as a consequence, comparison of observed correlations with zero may not be very misleading.  If the number of persons is very large, however, all observed correlations will be statistically, significantly different from 0, even when items fit the Rasch model perfectly.  Clearly, in this case, this significance is simply a function of sample size.

It is better to pay attention to positive correlations that are unusually high. This is evidence, generally, of some kind of local dependence between items.  An examination of the content of the items generally indicates why responses to them might be more similar than expected under the model with independence.

            Action:

• RUMM2030 provides a new facility for assessing the magnitude of the effect of local dependence in terms of a change in difficulty of the dependent item.
• For details on the conduct of this routine, available in the Licence Edition only, refer to Chapter 14 of the "Extending the RUMM2030 Analysis" manual. 

NOTE: the independent item within the pair of dependent items to be investigated, may be referred to as the "Base" item in an earlier edition of this manual.

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