合否判定データによる判別分析の問題点

摘要

本稿では，マークセンス試験データを用いて判別分析の問題点を議論する．rn試験は10択100問である．正解と不正解を1／0の備に変換し，自明な線形分離可rn能な合否判定の2群判別の説明変数として用いる．また中間試験は9項目，期末試rn験は6項目の設問群の合計得点を説明変数として判別分析する．rnフィッシャーの線形判別関数（LDF），二次判別関数とロジスティック回帰分析を，rn誤分類数最小化（MNM）基準による最適線形判別関数（改定IP-OLDF）と比較する．rn100個の説明変数の判別では次の問題点が発見された．逐次変数選択法は28変数rnから54変数を選んだが，改定IP-OLDFは6変数から32変数で線形分離可能であrnる．また二次判別関数は一方の群の全員が誤分類されることがある．線形分離可能rnな場合，ロジスティック回帰の係数の推定値は不安定になった．9変数と6変数のrn判別では，改定IP－OLDFでは合否判定できるが，LDFと二次判別関数は合否判定rnできなかった．ロジスティック回帰分析は合否判定できたがロジスティック回帰係rn数は必ず不安定になった．%In this paper, we discuss problems of discriminant analysis by mark sense test data. The test consists of 100 questions with 10 choices. The correct or incorrect answers are converted to 1/0 values. Therefore, this data is the discrimination of two groups (pass and fail) with 100 independent variables x_i. And 100 questions are summarized in six or nine sub-total scores.rnTwo groups are trivial linear separable data. Linear discriminant function such as y = f(x) = Score(Σ_ix_i) - pass/fail score. If y ≥ 0, students pass the examination. Otherwise, students don't pass. Therefore, the number of misclassification by this linear discriminant function is 0.rnFisher's linear discriminant function (LDF), quadratic discriminant function and logistic regression are compared with optimal linear discriminant function (Revised IP-OLDF) based on MNM (Minimum number of misclassifications) criterion by these data.rnIn the cases of 100 independent variables discrimination, the following problems are found. The stepwise variable selection methods chose over 28 independent variables, nevertheless Revised IP-OLDF find that these data is linear separable less than 12 independent variables. In some cases, quadratic discriminant function misclassified all pass/fail students to other group. The standard error of coefficients of logistic regression becomes very big.rnIn the cases of summarized sub-total scores discrimination, the number of misclassifications of LDF, quadratic discriminant function are mostly greater than 0, nevertheless MNM of Revised IP-OLDF are 0.