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Discriminant Analysis (LDA & QDA)

Created:December 24, 2025
Last Updated:December 24, 2025

This calculator performs comprehensive Discriminant Analysis, including both Linear Discriminant Analysis (LDA) and Quadratic Discriminant Analysis (QDA). These are powerful classification techniques used to predict group membership based on predictor variables and to understand which variables best discriminate between groups.

What You'll Get:

  • Model Comparison: Compare LDA and QDA performance side-by-side
  • Classification Accuracy: Overall accuracy and per-group performance metrics
  • Confusion Matrix: Detailed breakdown of correct and incorrect classifications
  • Discriminant Functions: Coefficients showing how each variable contributes to classification
  • Decision Boundaries: Visual representation of how groups are separated
  • Cross-Validation: Unbiased accuracy estimates using k-fold cross-validation
  • Feature Importance: Ranking of variables by their discriminating power
  • APA-Formatted Report: Professional statistical reporting ready for publication

💡 Pro Tip: LDA assumes equal covariance matrices across groups (homogeneity of variance-covariance), while QDA allows different covariances. Use LDA when groups have similar spread, and QDA when they differ. LDA is more stable with small samples, while QDA is more flexible but requires more data. For regression-like problems, consider Logistic Regression.

Ready to classify your data? (Iris flower species classification) to see discriminant analysis in action, or upload your own data to predict group membership and understand what distinguishes your groups.

Calculator

1. Load Your Data

2. Select Variables

The variable that defines the groups you want to classify (e.g., species, diagnosis, category)

Selected: 0 of 0 features

3. Analysis Options

Proportion of data to use for testing (0.1 - 0.5)

How to weight group sizes in classification

For reproducible results (default: 42)

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Definition

Discriminant Analysis is a multivariate statistical technique used to classify observations into predefined groups based on predictor variables. It finds linear (LDA) or quadratic (QDA) combinations of features that best separate the groups, making it ideal for prediction and understanding which variables distinguish between groups.

When to Use Discriminant Analysis

Use Discriminant Analysis when you want to:

  • Classify observations: Predict which group a new observation belongs to (e.g., disease diagnosis, species identification)
  • Understand group differences: Identify which variables best discriminate between groups
  • Reduce dimensionality: Create discriminant functions that capture group differences in fewer dimensions
  • Alternative to logistic regression: When you have multiple groups (3+) or want to visualize group separation
  • Complement MANOVA: After finding significant group differences, determine which variables drive those differences

LDA vs QDA: Choosing the Right Method

Linear Discriminant Analysis (LDA):

  • Assumes equal covariance matrices across all groups
  • Creates linear decision boundaries
  • More stable with smaller sample sizes
  • Better when groups have similar variance structures
  • Fewer parameters to estimate (more parsimonious)

Quadratic Discriminant Analysis (QDA):

  • Allows different covariance matrices for each group
  • Creates quadratic (curved) decision boundaries
  • More flexible but requires larger sample sizes
  • Better when groups have different variance structures
  • Can overfit with small samples

💡 Recommendation: Run both and compare! This calculator provides both analyses so you can see which performs better for your data.

Assumptions

  • Multivariate normality: Predictor variables should be approximately normally distributed within each group
  • Independence: Observations should be independent of each other
  • No multicollinearity: Predictor variables should not be highly correlated with each other
  • Homogeneity of covariance (LDA only): Groups should have similar variance-covariance matrices (not required for QDA)
  • Adequate sample size: At least 20 observations per group, preferably more for QDA

How to Perform Discriminant Analysis with R

R
library(MASS)
library(tidyverse)

# Iris dataset (species classification)
data <- iris

# Linear Discriminant Analysis
lda_model <- lda(Species ~ Sepal.Length + Sepal.Width + 
                 Petal.Length + Petal.Width, data = data)

# View results
print(lda_model)

# Predictions
predictions <- predict(lda_model, data)
table(Predicted = predictions$class, Actual = data$Species)

# Quadratic Discriminant Analysis
qda_model <- qda(Species ~ Sepal.Length + Sepal.Width + 
                 Petal.Length + Petal.Width, data = data)

# View results
print(qda_model)

# Predictions
qda_predictions <- predict(qda_model, data)
table(Predicted = qda_predictions$class, Actual = data$Species)

# Plot LDA
plot(lda_model)

How to Perform Discriminant Analysis with Python

Python
import pandas as pd
import numpy as np
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis, QuadraticDiscriminantAnalysis
from sklearn.model_selection import train_test_split, cross_val_score, KFold
from sklearn.metrics import classification_report, confusion_matrix
import matplotlib.pyplot as plt
import seaborn as sns

# Load iris dataset
from sklearn.datasets import load_iris
iris = load_iris()
X = iris.data
y = iris.target

# Split data (random_state for reproducibility)
X_train, X_test, y_train, y_test = train_test_split(
    X, y, test_size=0.3, random_state=42, stratify=y
)

# Linear Discriminant Analysis
lda = LinearDiscriminantAnalysis()
lda.fit(X_train, y_train)

# Predictions
y_pred_lda = lda.predict(X_test)

# Accuracy
print(f"LDA Accuracy: {lda.score(X_test, y_test):.3f}")

# Cross-validation (with reproducible seed)
cv_scores = cross_val_score(
    lda, X_train, y_train,
    cv=KFold(n_splits=5, shuffle=True, random_state=42)
)
print(f"LDA CV Accuracy: {cv_scores.mean():.3f} (+/- {cv_scores.std():.3f})")

# Confusion Matrix
print("\nLDA Confusion Matrix:")
print(confusion_matrix(y_test, y_pred_lda))

# Classification Report
print("\nLDA Classification Report:")
print(classification_report(y_test, y_pred_lda, 
                          target_names=iris.target_names))

# Quadratic Discriminant Analysis
qda = QuadraticDiscriminantAnalysis()
qda.fit(X_train, y_train)

# Predictions
y_pred_qda = qda.predict(X_test)

# Accuracy
print(f"\nQDA Accuracy: {qda.score(X_test, y_test):.3f}")

# Confusion Matrix
print("\nQDA Confusion Matrix:")
print(confusion_matrix(y_test, y_pred_qda))

# Plot decision boundaries (for 2D projection)
X_lda = lda.transform(X)
plt.figure(figsize=(10, 6))
for i, target_name in enumerate(iris.target_names):
    plt.scatter(X_lda[y == i, 0], X_lda[y == i, 1], 
                label=target_name, alpha=0.8)
plt.xlabel('LD1')
plt.ylabel('LD2')
plt.title('LDA Projection')
plt.legend()
plt.show()

Interpretation Guidelines

  • Accuracy: Overall classification accuracy above 70% is generally considered good, but this depends on your field and number of groups
  • Confusion Matrix: Look for high values on the diagonal (correct classifications) and low values off-diagonal (misclassifications)
  • Discriminant Coefficients: Larger absolute values indicate variables that contribute more to group separation
  • LDA vs QDA: If QDA performs much better, groups have different covariance structures; if similar, LDA is preferred for parsimony

Verification