17.3 Model evaluation and interpretation

Logistic regression models are powerful tools for predicting binary outcomes. Evaluating these models is crucial to understand their performance and make informed decisions. This topic covers key metrics and techniques used to assess and interpret logistic regression models.

From confusion matrices to ROC curves, we'll explore various ways to measure model accuracy and effectiveness. We'll also dive into coefficient interpretation, statistical tests, and validation strategies to ensure our models are robust and reliable.

Performance Metrics

Confusion Matrix and Accuracy

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• Confusion matrix displays predicted vs actual class outcomes in a tabular format
• Contains four key components: True Positives (TP), True Negatives (TN), False Positives (FP), False Negatives (FN)
• Accuracy measures overall correctness of predictions calculated as $(TP + TN) / (TP + TN + FP + FN)$
• Provides a general overview of model performance but can be misleading for imbalanced datasets
• Accuracy alone may not suffice for evaluating models with uneven class distributions

Precision and Recall

• Precision quantifies the proportion of correct positive predictions out of all positive predictions
• Calculated as $TP / (TP + FP)$
• Useful when minimizing false positives is crucial (spam detection)
• Recall measures the proportion of actual positive cases correctly identified
• Computed as $TP / (TP + FN)$
• Important when identifying all positive instances is critical (disease diagnosis)
• Precision and recall often involve a trade-off depending on the specific problem

F1 Score

• F1 score combines precision and recall into a single metric
• Calculated as the harmonic mean of precision and recall: $2 * (Precision * Recall) / (Precision + Recall)$
• Provides a balanced measure of model performance especially for imbalanced datasets
• Ranges from 0 to 1 with higher values indicating better performance
• Particularly useful when seeking a balance between precision and recall

Model Evaluation Curves

ROC Curve and AUC

• Receiver Operating Characteristic (ROC) curve plots True Positive Rate against False Positive Rate
• Illustrates model performance across various classification thresholds
• True Positive Rate (TPR) equals recall calculated as $TP / (TP + FN)$
• False Positive Rate (FPR) computed as $FP / (FP + TN)$
• Area Under the Curve (AUC) summarizes ROC curve performance in a single value
• AUC ranges from 0 to 1 with higher values indicating better discrimination
• AUC of 0.5 suggests no better than random guessing while 1.0 indicates perfect classification

Log-likelihood and Deviance

• Log-likelihood measures how well the model fits the observed data
• Calculated as the sum of log probabilities for each observation given the model
• Higher log-likelihood values indicate better model fit
• Deviance quantifies the difference between the current model and a perfect model
• Computed as -2 times the log-likelihood
• Lower deviance values suggest better model performance
• Used in model comparison and selection processes

Regression Interpretation

Odds Ratios and Coefficient Interpretation

• Odds ratio represents the change in odds for a one-unit increase in the predictor variable
• Calculated as the exponential of the logistic regression coefficient $exp(β)$
• Odds ratio > 1 indicates increased odds while < 1 suggests decreased odds
• Coefficient interpretation involves understanding the direction and magnitude of predictor effects
• Positive coefficients increase the log-odds of the outcome while negative coefficients decrease them
• Magnitude of coefficients indicates the strength of the relationship with the outcome variable

Statistical Tests for Coefficients

• Wald test assesses the significance of individual predictor variables
• Calculated by dividing the coefficient estimate by its standard error
• Tests the null hypothesis that the coefficient equals zero
• Likelihood ratio test compares nested models to evaluate overall model fit
• Computed as the difference in deviance between two models
• Used to assess the significance of adding or removing predictor variables
• McFadden's R-squared measures the improvement of the full model over the null model
• Calculated as $1 - (Log-likelihood(full model) / Log-likelihood(null model))$
• Ranges from 0 to 1 with higher values indicating better model fit

Validation Techniques

Cross-validation Strategies

• Cross-validation assesses model performance on unseen data
• K-fold cross-validation divides data into K subsets or folds
• Trains model on K-1 folds and tests on the remaining fold repeating K times
• Common choices for K include 5 or 10
• Leave-one-out cross-validation uses N-1 observations for training and 1 for testing repeating N times
• Stratified cross-validation maintains class proportions in each fold
• Useful for imbalanced datasets to ensure representative sampling
• Provides a more robust estimate of model performance compared to a single train-test split