2010 Volume 22 Issue 2 Pages 91-108
The present paper discussed two related problems, i.e., 1) Do the so-called horseshoe effects exist in Hayashi's second method of quantification or Quantification II? If the answer is yes, how do they appear? 2) How can we avoid the influence of the horseshoe effects in Quantification II? As it was found that horseshoe effects existed, we proposed two-step methods of linear discriminant analysis after unidimensional scaling. For unidimensional scaling we proposed two different types of partial canonical correlation analysis or partial correspondence analysis suitable for the cases in which predictor variables were mutually highly correlated and were only slightly correlated. Then we applied Quantification II and the two-step methods to artificial data sets in which predictor variables had unidimensional structure and compared the performances with correct classification rates. The results for the training data sets showed that Quantification II was superior to the two-step methods when the sample size N was small but the result of comparison was opposite when N was large and that the change point of the sample size increased as K increased, where K indicates the number of categories of categorized predictor variables. The results for the test data sets showed that the two-step method was superior for all N. Considering the situations where test data were not available, we studied whether 0.632 bootstrap estimation could be used for estimating the comparative performances for the test data sets and found that the bootstrap estimation was useful for choosing the method with higher performance.
