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Discriminant Analysis – Concept & Interpretation
1. Introduction
📌 Discriminant Analysis (DA) is a statistical technique used for classification and prediction by distinguishing between two or more groups based on predictor variables.
📌 It finds a linear combination of independent variables that best separates predefined categories.
📌 Used in economics, finance, marketing, medicine, and social sciences for group classification.
✔ Example: A bank wants to classify loan applicants as “high risk” or “low risk” based on income, credit score, and debt level. Discriminant Analysis helps identify which factors best separate these groups.
2. Concept of Discriminant Analysis
🔹 (1) What Discriminant Analysis Does?
✔ Identifies a function (Discriminant Function) that maximizes the separation between groups.
✔ Projects data into a new space where group differences are maximized.
✔ Similar to regression analysis, but the dependent variable is categorical rather than continuous.
🔹 (2) Discriminant Function
A linear discriminant function for two groups is given by: Z=b1X1+b2X2+…+bpXp+CZ = b_1 X_1 + b_2 X_2 + … + b_p X_p + C
where:
- ZZ = Discriminant score (used for classification).
- X1,X2,…,XpX_1, X_2, …, X_p = Independent (predictor) variables.
- b1,b2,…,bpb_1, b_2, …, b_p = Coefficients (weights) assigned to each variable.
- CC = Constant term.
✔ Higher absolute values of coefficients → Stronger influence of the variable on classification.
3. Types of Discriminant Analysis
🔹 (1) Linear Discriminant Analysis (LDA)
✔ Used when groups have equal variance.
✔ Finds a linear boundary between categories.
✔ Example: Classifying customers into “loyal” vs. “non-loyal” groups based on purchase behavior.
🔹 (2) Quadratic Discriminant Analysis (QDA)
✔ Used when groups have different variances.
✔ Allows for curved decision boundaries.
✔ Example: Classifying stocks as “high risk” vs. “low risk” based on financial metrics.
🔹 (3) Fisher’s Discriminant Analysis
✔ Maximizes the ratio of between-group variance to within-group variance.
✔ Used when the number of predictors is large.
4. Steps in Discriminant Analysis
✔ Step 1: Define Groups → Specify categories (e.g., “High-income vs. Low-income”).
✔ Step 2: Select Predictor Variables → Choose independent variables (e.g., “Age, Education, Salary”).
✔ Step 3: Compute Discriminant Function → Find coefficients for classification.
✔ Step 4: Assign Cases to Groups → Compute scores and classify individuals.
✔ Step 5: Evaluate Model Performance → Use accuracy metrics and confusion matrices.
5. Interpretation of Discriminant Analysis Results
🔹 (1) Discriminant Function Coefficients
✔ Indicate which variables contribute most to classification.
📌 Example Output (Predicting Loan Default Risk)
| Variable | Coefficient (b) | Interpretation |
|---|---|---|
| Income | 0.85 | Higher income → Less risk of default |
| Debt-to-Income Ratio | -1.20 | Higher ratio → More risk of default |
| Credit Score | 1.10 | Higher credit score → Less risk of default |
✔ Interpretation:
- A higher income and credit score reduce default risk.
- A higher debt-to-income ratio increases default risk.
🔹 (2) Wilks’ Lambda
✔ Measures how well the discriminant function separates groups.
✔ Values range from 0 to 1:
- Closer to 0 → Strong separation between groups.
- Closer to 1 → Weak separation.
🔹 (3) Classification Matrix (Confusion Matrix)
✔ Shows how well the model classifies cases into correct groups.
📌 Example Output (Loan Default Prediction):
| Actual \ Predicted | High Risk | Low Risk |
|---|---|---|
| High Risk | 80 | 20 |
| Low Risk | 15 | 85 |
✔ Accuracy = (80 + 85) / (80 + 20 + 15 + 85) = 82.5%
✔ Model correctly classified 82.5% of cases.
6. Applications of Discriminant Analysis
✔ Economics → Classifying countries based on economic performance.
✔ Finance → Predicting loan defaults and credit scoring.
✔ Marketing → Identifying customer segments based on behavior.
✔ Medical Research → Diagnosing diseases based on patient data.
✔ Human Resources → Predicting employee turnover.
7. Discriminant Analysis vs. Other Classification Methods
| Method | Discriminant Analysis | Logistic Regression | Decision Trees |
|---|---|---|---|
| Dependent Variable | Categorical | Categorical | Categorical |
| Assumptions | Multivariate normality | No normality assumption | No assumption |
| Decision Boundary | Linear (LDA) or Quadratic (QDA) | Non-linear possible | Non-linear |
| Use Case | Economic & financial classification | Probability estimation | Complex patterns |
✔ Discriminant Analysis is best when predictor variables follow a normal distribution.
8. Conclusion
✔ Discriminant Analysis is a powerful classification tool for separating groups.
✔ Helps in economic research, finance, marketing, and social sciences.
✔ Discriminant function coefficients & Wilks’ Lambda are key for interpretation.
✔ Can be combined with machine learning models for better predictions.