**4. Concluding remarks**

We reconsider a variable selection problem in PCA for qualitative data based on the idea of Mori et al. [2]. For the problem of how to deal with qualitative data, we apply optimal scaling with the ALS algorithm [4] to the qualitative data. For the variable selection in PCA, we use the criteria in M.PCA of Tanaka and Mori [1] for optimally quantified data. That is, the proposed method is an extension of M.PCA by implementing optimal scaling into M.PCA so as to select a subset of qualitative variables. Using this method, since the quantification is done separately for each variable, we can select a subset of variables from mixed measurement level data.

We apply this method to real data from a customer engagement study [3] to select a subset of qualitative variables by using a criterion that maximizes the prediction efficiency. For a case where there is no preassigned number of variables to be selected, it can be suggested to specify the number in such a way that the maximum loss of the efficiency is not over a certain percentage.

As a result, variables are selected in a well-balanced manner from questions asking similar contents, and the selected subset, therefore, provides as much information as possible. It is expected that the nonlinear M.PCA works well for any mixed measurement level data.

*Advances in Principal Component Analysis*
