unitinfo

The unit used for a summary of information measures used to assess model performance. The log likelihood is provided. The model dimension is C if is negative definite. Otherwise, the model dimension is the rank of . The estimated expected log penalty per presented item

is supplied. The measure is between 0 and the logarithm of the largest number of observed categories associated with an item, so that the measure cannot exceed log(2) if all items are dichotomous. This measure has been applied to item-response theory in the case of equal weights and constant (Sinharay, Haberman, Lee, 2011). Without the normalization by number of items, the measure has been employed for model evaluation outside of item-response theory (Gilula & Haberman, 1994, 1995, 2001). The logarithmic penalty function itself has a much longer history in statistics (Mosteller & Wallace, 1964, Savage, 1971). The estimated asymptotic standard error of PE is the same as in the case of chained log-linear models for multinomial responses (Gilula & Haberman, 1995). In addition, a version AK of PE based on the Akaike approach (Akaike, 1974) is provided in which the model dimension is added to the numerator in the above equation, and a version GH of PE based on the Gilula-Haberman approach (Gilula & Haberman, 1994, 1995) is provided in which the trace of is added to the numerator in the above equation if is nonsingular. If the Louis approximation to the negative Hessian matrix is used in computation of maximum likelihood estimates, the GH and AK are the same. If complex sampling is used, then the Gilula-Haberman version of PE is also calculated with replaced by the estimated covariance matrix of based on complex sampling. Complex sampling options are described in dataspec. In listening.txt, unitinfo is not specified and takes its default value of 10 which corresponds to listening1.csv. For sample output, see rows 23 to 28 of listening.csv. The penalty label corresponds to the estimated expected log penalty per presented item. The label "SE_Penalty" corresponds to the standard error of this estimate. This standard error applies to the Gilula-Haberman and Akaike measures as well. Due to the large sample size, all penalty estimates are quite similar and close to 0.5, a value somewhat lower than the upper bound of log(2)=0.693. The Gilula-Haberman and Akaike measures always exceed the basic penalty estimate. As in the example, the Gilula-Haberman and Akaike measures are usually quite close, especially in large samples. They sometimes differ more noticeably when models fit badly.