Questions: Logistic Regression for Binary Outcomes

Question 1

A researcher reports that a logistic regression coefficient for education (in years) on voting behavior is β = 0.15. A colleague concludes that each additional year of education raises the probability of voting by 15 percentage points. What is wrong with this interpretation?

• A. Nothing is wrong — logistic regression coefficients directly represent probability changes
• B. The coefficient 0.15 represents a change in log-odds, not probability; the actual probability change is non-constant and depends on the baseline probability
• C. The coefficient must first be squared before interpreting it as a probability change
• D. The interpretation is wrong because logistic regression reports marginal effects at the mean automatically

Answer: (B)

Logistic regression coefficients are changes in log-odds per unit increase in the predictor — not probability changes. Because the logistic function is S-shaped, the same log-odds change corresponds to very different probability changes depending on where you are on the curve. Near 50% probability, a 0.15 log-odds change might shift probability ~3–4 points; near 5% or 95% it shifts much less. To communicate results honestly, researchers should report predicted probabilities at substantively meaningful predictor values.

Question 2

Why does logistic regression model the log-odds of an outcome rather than the probability directly?

• A. Log-odds are easier to compute than probabilities on modern hardware
• B. It is a historical convention with no mathematical justification
• C. Predicted probabilities from a linear model can fall outside [0,1], and the relationship between predictors and probability is rarely linear across the full range
• D. The logistic function eliminates the need for maximum likelihood estimation, simplifying inference

Answer: (C)

A linear model applied directly to a binary outcome has two fatal flaws: it can predict probabilities below 0 or above 1, and it assumes the effect of a predictor on probability is constant from 0% to 100%, which is unrealistic. Probabilities compress near their bounds. The logistic function maps any real-valued linear predictor to (0,1) and produces an S-shaped curve that naturally captures this compression. The log-odds transformation is the mathematical device that converts the bounded probability scale to the unbounded real line where linear modeling is valid.

Question 3

An odds ratio greater than 1 for a predictor in a logistic regression model means that subjects with higher values of that predictor are more likely than not (probability > 50%) to experience the outcome.

• T. True
• F. False

Answer: False

An odds ratio greater than 1 means the odds of the outcome *increase* with the predictor — relative to a baseline. Whether the outcome probability exceeds 50% depends on the baseline probability, which is not captured by the odds ratio alone. For example, if the baseline probability is 5%, an OR of 3.0 raises the odds from 0.053 to 0.158, corresponding to a probability of ~14% — still well below 50%. Odds ratios are relative measures; absolute probability requires knowing where you start.

Question 4

A logistic regression coefficient can be converted to an odds ratio by exponentiating it (e^β).

• T. True
• F. False

Answer: True

This is exactly correct. The logistic regression model is log(odds) = β₀ + β₁X, so β₁ = log(odds₁) − log(odds₀) = log(odds₁/odds₀). Exponentiating both sides gives e^β₁ = odds₁/odds₀ — the odds ratio. An OR of 1.5 means the odds are 50% higher for a one-unit increase in X. This transformation is standard in reporting logistic regression results, though predicted probabilities are often more interpretable.

Question 5

Why are predicted probabilities often more informative than odds ratios when communicating logistic regression results to a non-technical audience?

Model answer: Odds ratios are relative measures that depend on a baseline and are often misread as relative risks (which they are not, especially when baseline probabilities are high). Predicted probabilities, computed by plugging specific predictor values into the logistic function, give concrete, bounded quantities — 'a person with 16 years of education has a 73% predicted probability of voting' — that are intuitive and honest about the non-linearity of the relationship. They make the S-shaped nature of the model visible and avoid the common misinterpretation of OR as a percentage change in probability.

The non-linearity of logistic regression means that a predictor's effect on probability is large in the middle of the distribution and small at the extremes. Odds ratios hide this variation. Predicted probabilities at representative or meaningful values of the predictors (e.g., average-income vs. high-income voter) show the practical magnitude of the effect in ways that are directly interpretable.