Binary logistic regression predicts a yes/no outcome: did the patient develop diabetes, did the customer churn, did the student pass? It is the go-to method when your dependent variable has two categories, and its results are usually reported as odds ratios, which many readers find confusing.
This guide explains odds and odds ratios in plain language, walks through a real-style output, and shows how to report it.
In this guide
Why not ordinary regression for a yes/no outcome?
If you code the outcome 0/1 and run multiple regression, predicted values can fall below 0 or above 1, residuals cannot be normal, and their variance depends on the predicted value. Logistic regression solves this by modelling the log-odds of the outcome, which can take any value, and converting it back to a probability between 0 and 1.
Probability, odds and log-odds
| Probability (p) | Odds = p / (1 − p) | Log-odds = ln(odds) |
|---|---|---|
| .20 | 0.25 (1 to 4) | −1.39 |
| .50 | 1.00 (1 to 1) | 0.00 |
| .80 | 4.00 (4 to 1) | 1.39 |
The model is linear on the log-odds scale:
ln(p / (1 − p)) = b₀ + b₁X₁ + b₂X₂ + …
Each coefficient B is the change in log-odds for a one-unit increase in the predictor. Exponentiating it, Exp(B), gives the odds ratio (OR).
How to interpret an odds ratio
- OR = 1: the predictor makes no difference to the odds.
- OR > 1: higher values of the predictor increase the odds. OR = 1.12 means the odds are 12% higher for each one-unit increase.
- OR < 1: higher values decrease the odds. OR = 0.55 means the odds are 45% lower (1 − 0.55 = 0.45).
An odds ratio of 2 does not mean "twice as likely". It means twice the odds. When the outcome is rare (below about 10%), odds ratios and risk ratios are similar; when it is common, the odds ratio exaggerates the risk ratio. Say "odds", not "likelihood" or "risk", in your write-up.
Worked example
A public health researcher studies 500 adults and predicts type 2 diabetes (1 = yes, 0 = no) from age (years), BMI (kg/m²) and physical activity (1 = meets guidelines, 0 = does not).
| Predictor | B | SE | Wald | p | OR | 95% CI for OR |
|---|---|---|---|---|---|---|
| Age | 0.039 | 0.010 | 15.21 | < .001 | 1.04 | [1.02, 1.06] |
| BMI | 0.113 | 0.025 | 20.43 | < .001 | 1.12 | [1.07, 1.18] |
| Physically active | −0.598 | 0.220 | 7.39 | .007 | 0.55 | [0.36, 0.85] |
Interpretation, holding the other predictors constant:
- Age: each additional year increases the odds of diabetes by about 4%. Over 10 years, that compounds to 1.04¹⁰ ≈ 1.48, so 48% higher odds.
- BMI: each additional BMI point increases the odds by 12%.
- Physical activity: people who meet activity guidelines have 45% lower odds of diabetes than those who do not (OR = 0.55).
The 95% confidence intervals for the ORs do not include 1, so all three effects are significant.
Model fit statistics
| Statistic | Value | What it tells you |
|---|---|---|
| Omnibus (likelihood ratio) test | χ²(3) = 68.4, p < .001 | The model predicts better than an intercept-only model |
| Nagelkerke R² | .21 | A pseudo-R²; a rough guide to explanatory power, not "% of variance explained" |
| Hosmer-Lemeshow test | χ²(8) = 6.42, p = .60 | A non-significant result suggests predicted probabilities match observed rates |
| Classification accuracy | 78% (at a 0.5 cut-off) | Share of cases predicted correctly |
Two warnings. First, pseudo-R² values are typically much lower than linear-regression R² values, so do not compare them directly. Second, classification accuracy depends on the cut-off and on how common the outcome is; if 80% of people are "no", predicting "no" for everyone already gives 80%. For a fuller picture of discrimination, look at the ROC curve and AUC.
Assumptions
- Binary outcome with independent observations.
- Linearity of the logit: continuous predictors relate linearly to the log-odds. Check with the Box-Tidwell test or by adding a squared term.
- No severe multicollinearity among predictors.
- Enough events: a common rule is at least 10 events (cases in the smaller outcome category) per predictor. With 3 predictors you want at least 30 people with the outcome, and preferably more.
- No complete separation: if a predictor perfectly predicts the outcome, coefficients and SEs explode.
How to report logistic regression in APA 7
A binary logistic regression was performed to examine the effects of age, BMI and physical activity on the likelihood of type 2 diabetes. The model was statistically significant, χ²(3) = 68.4, p < .001, Nagelkerke R² = .21, and the Hosmer-Lemeshow test indicated adequate fit, χ²(8) = 6.42, p = .60. Older age (OR = 1.04, 95% CI [1.02, 1.06]) and higher BMI (OR = 1.12, 95% CI [1.07, 1.18]) were associated with higher odds of diabetes, whereas meeting physical activity guidelines was associated with lower odds (OR = 0.55, 95% CI [0.36, 0.85]).
Present the coefficients table (B, SE, Wald, p, OR and 95% CI) as an APA table.
Related methods
- Outcome with three or more unordered categories: multinomial logistic regression.
- Ordered outcome (low/medium/high): ordinal logistic regression.
- You need marginal effects (changes in probability rather than odds): see probit and logit with average marginal effects.
- Comparing two groups on a yes/no outcome without other predictors: a chi-square test is enough.
Frequently asked questions
How do you interpret an odds ratio less than 1?
An odds ratio below 1 means the odds of the outcome decrease as the predictor increases. Subtract it from 1 to get the percentage decrease: an OR of 0.55 means 45% lower odds.
What is the difference between odds ratio and relative risk?
The odds ratio compares odds (p / (1 − p)); the relative risk compares probabilities. They are close when the outcome is rare, but the odds ratio is further from 1 than the relative risk when the outcome is common.
What is a good Nagelkerke R squared?
There is no universal cut-off. Pseudo-R² values in logistic regression are usually lower than R² in linear regression; values of .20 to .40 often indicate a reasonably useful model.
What does a significant Hosmer-Lemeshow test mean?
It suggests that the model's predicted probabilities do not match the observed outcome rates well. Note that with very large samples the test can flag trivial differences.