Logistic regression is a statistical modeling tool used to predict the probability of a binary outcome, meaning the result can only fall into one of two categories, such as “yes” or “no.” Unlike standard linear regression, which predicts a continuous number, logistic regression uses a specialized function to ensure its output is always a probability between zero and one. Interpreting the output translates complex mathematical estimates into clear insights about which factors increase or decrease the likelihood of the specific outcome occurring. Although the output, often filled with log-odds and significance tests, can seem intimidating, a systematic approach makes the information understandable.
Assessing Overall Model Performance
Before examining individual predictor variables, analysts must determine if the model as a whole provides a good fit for the data. Logistic regression uses Maximum Likelihood Estimation (MLE) to generate metrics evaluating the model’s overall explanatory power. These metrics compare a “Null Model,” which contains no predictors, against the “Residual Model,” which includes all tested variables.
The Deviance statistic, often labeled as the Log-Likelihood multiplied by negative two, measures how poorly the model fits the data; a lower value indicates a better fit. Comparing the Null Deviance to the Residual Deviance allows for a model Chi-Square test. A statistically significant result suggests the set of predictors significantly improves the model’s ability to explain the outcome compared to a model with no predictors.
Since the standard R-squared metric is not applicable, several “Pseudo R-squared” measures exist, such as McFadden’s R-squared. These metrics quantify the proportion of the outcome’s variation explained by the predictors, conceptually similar to linear R-squared. They range from zero to one, with higher values suggesting a stronger model. Information Criteria, such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), also assess fit by penalizing models with too many variables, aiding in the selection of the most efficient model version.
Interpreting the Coefficient Table
The coefficient table contains the raw statistical estimates for each independent variable used to predict the binary outcome. The central value is the regression coefficient (\(\beta\)), which represents the influence of that predictor on the outcome variable. This coefficient describes the change in the log-odds of the outcome for every one-unit increase in the predictor, assuming all other variables are held constant. A positive \(\beta\) indicates higher log-odds, while a negative \(\beta\) suggests the opposite relationship.
The raw log-odds coefficient is difficult to interpret practically because it operates on a non-linear log scale. To determine the reliability of this relationship, the output includes the Standard Error, which quantifies the variability of the coefficient estimate. The Z-score or Wald statistic is calculated by dividing the coefficient by its Standard Error, providing a test statistic used to assess significance.
The P-value indicates the probability of observing a relationship this strong if no true relationship existed in the population. If the P-value falls below a predetermined threshold, typically 0.05, the predictor is considered statistically significant, suggesting the variable has a non-zero effect on the log-odds of the outcome. The sign of the \(\beta\) coefficient indicates the direction of the relationship, and the P-value confirms its statistical meaning.
Translating Coefficients into Odds Ratios
The most common and accessible way to interpret logistic regression results is by converting the raw coefficient (\(\beta\)) into an Odds Ratio (OR). This transformation is accomplished by exponentiating the coefficient (\(e^\beta\)), which moves the result from the logarithmic scale to a multiplicative, linear scale. The Odds Ratio directly measures how much the odds of the outcome change for a one-unit increase in the predictor variable.
An Odds Ratio of exactly 1.0 signifies that the predictor has no effect on the odds of the outcome. If the OR is greater than 1.0, the predictor increases the odds of the outcome. For example, an OR of 1.5 means the odds are 1.5 times higher (a 50% increase) for every one-unit increase in the predictor. Conversely, an OR less than 1.0, such as 0.75, indicates the predictor decreases the odds by a factor of 0.75 (a 25% reduction).
Analysts must consider the 95% Confidence Interval (CI) associated with the Odds Ratio, which provides a range where the true population Odds Ratio likely falls. This interval measures precision. If the range includes the value 1.0, the result is not considered statistically significant. A narrow confidence interval suggests a precise estimate of the effect, while a wide interval indicates greater uncertainty about the relationship’s true magnitude.
Evaluating Predictive Accuracy
Beyond assessing statistical significance, analysts must evaluate how well the model performs at classifying new observations. This assessment begins with the Confusion Matrix, a table comparing the model’s predictions to the actual outcomes. The matrix separates results into four categories: True Positives (correctly predicting the event), True Negatives (correctly predicting the non-event), False Positives (incorrectly predicting the event), and False Negatives (incorrectly predicting the non-event).
Several metrics are derived from the Confusion Matrix, including Sensitivity and Specificity. Sensitivity (True Positive Rate) measures the proportion of actual positive cases the model correctly identified. Specificity (True Negative Rate) measures the proportion of actual negative cases correctly classified as negative. These metrics are influenced by the probability cutoff chosen by the analyst, which determines the threshold for classifying a predicted probability as a “positive” outcome.
The Receiver Operating Characteristic (ROC) curve and its Area Under the Curve (AUC) offer a threshold-independent assessment of the model’s discriminative power. The ROC curve plots Sensitivity against the False Positive Rate (1-Specificity) across all possible cutoff values, illustrating the trade-off between correctly identifying positive cases and incorrectly flagging negative cases. The AUC summarizes this performance into a single value, ranging from 0.5 (random guessing) to 1.0 (perfect discrimination). An AUC value greater than 0.8 is often considered an indication of a good classifier.