I was finally able to run a principle component analysis of Rasch residuals on a large dataset of about 4000 people and 457 items. However, the output look very strange. It produced 3 factors: factor 1 has explained 68.48 of 457 variance units, factor 2 has explained -56.11, and factor 3 has explained -107.27. I am not sure what is the main cause of the negative (and huge) eigenvalues. The data have the weakness that nobody took every item. I also tried to factor analyze the original data with SPSS, but it stopped because there were not enough subjects to analyze. What do you recommend?
Missing data always pose a problem in factor analysis because the basis of the methodology is the decomposition of correlations or covariances. There are two main approaches to the problem. Listwise deletion removes from the data every case with a missing data value. A drawback, apparently observed in your SPSS run, is that a large proportion of the cases may be omitted, skewing the results or preventing successful completion of the analysis.
Pairwise deletion skips over individual computations involving missing values. So correlations between pairs of items are computed based on all cases for which data is present for both items. Pairwise deletion can lead to contradictory results:
Inconsistent matrices of correlations produce negative eigenvalues.
In the analysis of Rasch residuals, there is a useful solution. For each missing residual, impute its expected value of zero. This will force the correlations to be consistent. The zero residuals will dampen the size of factors in the residuals, but will have little effect on the factor structure.
From a small data set, I randomly eliminated 54% of the responses of each respondent. In the residuals from the original data, factors 1 and 2 had eigenvalues of 3.0 and 2.4. Listwise deletion would have eliminated the entire data set. Using pairwise deletion, the eigenvalues were -6.0 and -11.1 with a meaningless factor structure. After imputing zero residuals, the eigenvalues climbed back to 2.0 and 1.5, with a weaker, but still recognizable, factor structure.
John Michael Linacre
Residual Analysis with Missing Data Linacre, J.M. Rasch Measurement Transactions, 1999, 13:1 p. 679
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