# Structure in Rasch Residuals: Why Principal Components Analysis (PCA)?

"The principal concern of factor analysis is the resolution of a set of variables [items] linearly in terms of (usually) a small number of categories or `factors'. This resolution can be accomplished by the analysis of the correlations among the variables" (Harmon, p.4).

"If one were to make his choice entirely upon statistical considerations, a rather natural approach would be to represent the original set of variables in terms of a number of factors, determined in sequence so that at each successive stage the factor would account for the maximum of variance. This statistically optimal solution - the method of principal axes [components] - was first proposed by Pearson [1901], and in the 1930's, Hotelling provided the full development of the method." (Harmon, p.4).

An ideal of the Rasch model is that all the information in the data be explained by the latent measures. Then the unexplained part of the data, the residuals, is, by intention, random noise. In particular, after standardization of each residual by its model standard deviation, the noise should follow a random normal distribution. Further, the noise associated with one item is modelled to be independent of the noise associated with any other item. Consequently, all off-diagonal elements of the inter-item correlation matrix are expected to be 0. Empirical inter-item correlations, of course, depart from 0.

What is the expected correlation of an item with itself, the correlations which comprise the main diagonal of the correlation matrix? "The values put in the diagonal of the observed correlation matrix determine what portions of the unit variances are factored into common factors" (Harmon, p. 35). If the values are 1's, then "principal components" analysis results. Lesser values lead to "common factor" analysis. Under Rasch model conditions, the residuals associated with any item have no relationship to any other item. Consequently each item's residuals comprise a unique factor meaning that the values in the diagonal of the correlation matrix would be set to zero. Empirically, however, this is asserting the conclusion that the data conform to the Rasch model. Let us test this conclusion by asserting that all residual variance is due to common factors and place 1's in the diagonal. If the resulting common factors explain nothing more than chance variations in noise across items, then the assertion of communality is refuted and the data conform to the Rasch model. The existence of substantive common factors, however, would indicate the data harbor more than one latent variable.

The aim of the factor analysis of Rasch residuals is thus to attempt to extract the common factor that explains the most residual variance under the hypothesis that there is such a factor. If this factor is discovered to merely "explain" random noise, then there is no meaningful structure in the residuals. In this kind of investigation, neither factor rotation nor oblique axes appear to be relevant. This approach accords with philosopher Karl Popper's "falsification" test of scientific hypotheses.

If the biggest factor is perceived to partition the items into a meaningful structure according to their factor loadings, then there is structure within the items not explained by the primary measurement system. This structure merits further investigation, and also prompts inspection of lesser factors.

The impact of a residual factor on the measurement system can be easily determined. Extract two subsets of items representing the opposite poles of the factor. Measure each subject on each subset of items. Cross-plot the subject measures. The shape of the plot indicates the extent to which the structure in the residuals is perturbing the measurement of the subjects. Plot each item's subset difficulty against its original difficulty to see in what manner the item construct is being distorted. Only perturbation that has impact on the empirical meaning or use of the measures is of concern.
John Michael Linacre

Harmon, H. H. (1960) Modern Factor Analysis. Chicago: University of Chicago Press.

Structure in Rasch residuals: Why principal components analysis (PCA)? Linacre J.M. … Rasch Measurement Transactions, 1998, 12:2 p. 636.

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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