Introduction
Logistic regression is one of the most widely used statistical methods for modeling binary outcomes. Whether you are predicting whether a patient develops a disease, whether a customer churns, or whether a student passes an exam, logistic regression provides a principled framework for relating predictor variables to a yes/no outcome.
Unlike ordinary least squares (OLS) regression, which predicts a continuous value, logistic regression models the log-odds of an event occurring. The output is a probability between 0 and 1, making it ideal for classification problems. This guide walks you through every stage of a logistic regression analysis: from stating your research question, through checking assumptions, fitting the model, and interpreting the results with concrete numbers.
By the end of this article you will be able to set up a logistic regression, understand each coefficient as an odds ratio, evaluate model fit, and report your findings in a publication-ready format. If you want to run logistic regression on your own data right away, try our Logistic Regression Calculator.
When to Use Logistic Regression
Logistic regression is appropriate when:
- Your dependent variable is binary (e.g., 0/1, yes/no, pass/fail).
- You have one or more predictor variables that can be continuous, categorical, or a mix of both.
- You want to estimate the probability of the outcome or understand the effect size of each predictor.
Common research scenarios include medical diagnosis (disease present vs. absent), marketing (purchase vs. no purchase), and education (graduation vs. dropout).
Key Assumptions
Before fitting the model, verify the following assumptions:
1. Binary Dependent Variable
The outcome must have exactly two categories. If you have three or more unordered categories, consider multinomial logistic regression instead.
2. Independence of Observations
Each observation should be independent. Repeated measures on the same subject violate this assumption and require mixed-effects logistic regression.
3. Linearity of the Logit
Each continuous predictor should have a linear relationship with the log-odds of the outcome. You can check this by plotting the predictor against the empirical logits or by adding an interaction term between the predictor and its natural log.
4. No Severe Multicollinearity
Predictors should not be highly correlated with one another. Calculate the Variance Inflation Factor (VIF) for each predictor; values above 10 indicate a problem.
5. Adequate Sample Size
A common rule of thumb is at least 10 events per predictor variable (EPV). With 5 predictors and 50 events in the smaller outcome category you are at the minimum. More conservative guidelines recommend 20 EPV.
6. No Extreme Outliers or Influential Points
Check Cook's distance and leverage values. Observations with Cook's D greater than 1 warrant investigation.
Example Dataset
Suppose we want to predict whether a patient is readmitted to the hospital within 30 days (1 = readmitted, 0 = not readmitted) based on age, length of stay (LOS), and number of comorbidities.
| Patient | Age | LOS (days) | Comorbidities | Readmitted | |---------|-----|------------|---------------|------------| | 1 | 72 | 5 | 3 | 1 | | 2 | 45 | 2 | 0 | 0 | | 3 | 68 | 7 | 4 | 1 | | 4 | 55 | 3 | 1 | 0 | | 5 | 80 | 9 | 5 | 1 | | 6 | 42 | 1 | 0 | 0 | | 7 | 63 | 4 | 2 | 0 | | 8 | 77 | 6 | 3 | 1 | | 9 | 50 | 2 | 1 | 0 | | 10 | 74 | 8 | 4 | 1 | | 11 | 61 | 3 | 2 | 0 | | 12 | 83 | 10 | 6 | 1 | | 13 | 47 | 2 | 0 | 0 | | 14 | 70 | 5 | 3 | 1 | | 15 | 59 | 4 | 1 | 0 | | 16 | 66 | 6 | 2 | 1 | | 17 | 78 | 7 | 4 | 1 | | 18 | 41 | 1 | 0 | 0 | | 19 | 73 | 8 | 5 | 1 | | 20 | 52 | 3 | 1 | 0 |
In this sample of 20 patients, 10 were readmitted and 10 were not.
Step 1: Check Assumptions
Linearity of the Logit
Divide the continuous predictor into quartiles and compute the proportion of events in each quartile. Plot these empirical logits against the quartile midpoints. A roughly linear pattern supports the assumption.
| Age Quartile | Midpoint | Readmission Rate | Empirical Logit | |-------------|----------|-------------------|-----------------| | 41 - 50 | 45.5 | 0.00 | -3.00 (bounded) | | 51 - 61 | 56.0 | 0.00 | -3.00 (bounded) | | 62 - 73 | 67.5 | 0.60 | 0.41 | | 74 - 83 | 78.5 | 1.00 | 3.00 (bounded) |
The pattern is monotonically increasing, supporting linearity.
Multicollinearity
| Predictor | VIF | |---------------|------| | Age | 1.85 | | LOS | 2.41 | | Comorbidities | 2.67 |
All VIF values are well below 10, so multicollinearity is not a concern.
Step 2: Fit the Model
The logistic regression equation is:
ln(p / (1-p)) = β₀ + β₁ · Age + β₂ · LOS + β₃ · Comorbidities
After maximum likelihood estimation, suppose we obtain the following coefficients:
| Parameter | Coefficient (B) | Std. Error | Wald Chi-Square | p-value | Odds Ratio (e^B) | 95% CI for OR | |---------------|-----------------|------------|-----------------|---------|-------------------|-----------------| | Intercept | -8.524 | 3.210 | 7.05 | 0.008 | -- | -- | | Age | 0.065 | 0.031 | 4.40 | 0.036 | 1.067 | 1.004 - 1.134 | | LOS | 0.312 | 0.148 | 4.45 | 0.035 | 1.366 | 1.022 - 1.826 | | Comorbidities | 0.487 | 0.223 | 4.77 | 0.029 | 1.627 | 1.051 - 2.519 |
Step 3: Interpret the Coefficients
Odds Ratios
The most intuitive way to interpret logistic regression results is through odds ratios (OR):
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Age (OR = 1.067): For each one-year increase in age, the odds of 30-day readmission increase by 6.7%, holding LOS and comorbidities constant. Over a 10-year age difference, the odds increase by a factor of 1.067^10 = 1.91, nearly doubling.
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LOS (OR = 1.366): Each additional day in the hospital increases the odds of readmission by 36.6%. A patient with a 7-day stay has 1.366^5 = 4.76 times the odds of readmission compared to a patient with a 2-day stay, all else being equal.
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Comorbidities (OR = 1.627): Each additional comorbidity increases the odds of readmission by 62.7%. A patient with 4 comorbidities has 1.627^4 = 7.01 times the odds compared to a patient with 0 comorbidities.
Predicted Probabilities
For a 75-year-old patient with a 6-day stay and 3 comorbidities:
logit(p) = -8.524 + 0.065(75) + 0.312(6) + 0.487(3) = -8.524 + 4.875 + 1.872 + 1.461 = -0.316
p = 1 / (1 + e⁰·³¹⁶) = 1 / (1 + 1.372) = 0.422
This patient has approximately a 42.2% predicted probability of 30-day readmission.
Step 4: Evaluate Model Fit
Overall Model Significance
| Test | Chi-Square | df | p-value | |-----------------------------|------------|----|---------| | Omnibus / Model Chi-Square | 22.14 | 3 | < 0.001 | | Hosmer-Lemeshow | 5.32 | 8 | 0.723 |
- The Omnibus test is significant (p < 0.001), meaning the model with predictors fits significantly better than the null (intercept-only) model.
- The Hosmer-Lemeshow test is non-significant (p = 0.723), indicating good calibration (predicted probabilities align with observed rates).
Pseudo R-Squared
| Measure | Value | |--------------------|-------| | Cox & Snell R-sq | 0.421 | | Nagelkerke R-sq | 0.562 | | McFadden R-sq | 0.387 |
These pseudo R-squared values suggest the model explains a moderate to substantial proportion of the variation in readmission.
Classification Table
Using a 0.50 probability cutoff:
| Observed \ Predicted | Not Readmitted | Readmitted | % Correct | |---------------------|----------------|------------|-----------| | Not Readmitted | 8 | 2 | 80.0% | | Readmitted | 1 | 9 | 90.0% | | Overall | | | 85.0% |
The model correctly classifies 85% of patients. Sensitivity (true positive rate) is 90% and specificity (true negative rate) is 80%.
Step 5: Check for Influential Observations
Examine Cook's distance for each observation. Values exceeding 1.0 are potentially influential.
| Patient | Cook's Distance | Leverage | |---------|-----------------|----------| | 7 | 0.87 | 0.35 | | 16 | 0.62 | 0.28 | | Others | < 0.40 | < 0.25 |
Patient 7 (age 63, LOS 4, 2 comorbidities, not readmitted) has the highest Cook's distance at 0.87 but remains below the threshold of 1.0. No observations need to be removed.
Step 6: Report Your Results
A well-formatted report might read:
A binary logistic regression was performed to examine the effects of age, length of stay, and number of comorbidities on the likelihood of 30-day hospital readmission (N = 20). The overall model was statistically significant, chi-square(3) = 22.14, p < .001, Nagelkerke R-squared = .562. The model correctly classified 85.0% of cases. All three predictors were statistically significant: age (OR = 1.067, 95% CI [1.004, 1.134], p = .036), length of stay (OR = 1.366, 95% CI [1.022, 1.826], p = .035), and number of comorbidities (OR = 1.627, 95% CI [1.051, 2.519], p = .029). Increasing age, longer hospital stays, and more comorbidities were each associated with higher odds of readmission.
Common Pitfalls and How to Avoid Them
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Complete or quasi-complete separation: When a predictor perfectly predicts the outcome, maximum likelihood estimates become infinite. Solution: use Firth's penalized likelihood or exact logistic regression.
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Ignoring the linearity-of-logit assumption: A significant non-linear relationship can bias coefficients. Solution: test with polynomial terms or restricted cubic splines.
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Overfitting with too many predictors: With small samples, each additional predictor increases the risk of overfitting. Solution: follow the 10-20 EPV guideline and use information criteria (AIC, BIC) for variable selection.
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Confusing odds ratios with relative risk: An odds ratio of 2.0 does not mean the outcome is twice as likely. When the outcome is rare (< 10%), OR approximates RR, but for common outcomes the OR overestimates the RR.
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Using R-squared for model evaluation: Pseudo R-squared values in logistic regression do not have the same interpretation as in linear regression. Use the C-statistic (AUC) and calibration plots instead.
Advanced Considerations
Interaction Terms
If you suspect that the effect of LOS depends on age, include an interaction term (Age x LOS). A significant interaction means the OR for LOS changes at different ages.
Model Comparison with AIC
| Model | AIC | |--------------------------------|-------| | Age only | 24.8 | | Age + LOS | 20.1 | | Age + LOS + Comorbidities | 17.3 | | Full model + Age x LOS | 18.9 |
The model with all three main effects (AIC = 17.3) has the best fit. The interaction term does not improve the model.
ROC Curve and AUC
The area under the receiver operating characteristic curve (AUC) for our model is 0.92, indicating excellent discriminatory ability. An AUC of 0.5 represents chance, while 1.0 represents perfect discrimination.
Try It Yourself
Ready to run logistic regression on your own data? Use our Logistic Regression Calculator to input your variables and get instant results with odds ratios, model fit statistics, and classification metrics.
For related analyses, you might also explore our Multiple Regression Calculator for continuous outcomes or our Chi-Square Test Calculator for examining associations between categorical variables.
FAQ
What is the minimum sample size for logistic regression?
A widely cited guideline is 10 events per variable (EPV). If your smaller outcome group has 50 observations and you have 5 predictors, you meet the minimum. Some methodologists recommend 20 EPV for stable estimates. With fewer events, consider reducing the number of predictors or using penalized methods like Firth logistic regression.
Can I use logistic regression with more than two outcome categories?
Standard binary logistic regression requires exactly two outcome levels. For three or more unordered categories, use multinomial logistic regression. For ordered categories (e.g., mild/moderate/severe), use ordinal logistic regression.
How do I handle missing data in logistic regression?
Complete-case analysis (listwise deletion) is the default in most software but can introduce bias if data are not missing completely at random (MCAR). Multiple imputation is generally preferred. Impute missing values, fit the model on each imputed dataset, and pool the results using Rubin's rules.
What is the difference between the Wald test and the likelihood ratio test?
The Wald test evaluates each coefficient individually and is reported in standard output. The likelihood ratio test compares nested models and is generally more reliable, especially with small samples. When the two disagree, prefer the likelihood ratio test.
How do I report logistic regression results in APA format?
Report the overall model chi-square test, degrees of freedom, and p-value. Include Nagelkerke (or Cox & Snell) R-squared. For each predictor, report B, SE, Wald chi-square, p-value, OR, and 95% CI for the OR. State the classification accuracy if relevant.
Can logistic regression handle continuous predictors?
Yes. Continuous predictors enter the model directly. The resulting odds ratio represents the change in odds for a one-unit increase in the predictor. If a one-unit change is not meaningful (e.g., income in dollars), consider rescaling the variable (e.g., per $1,000 increase) for more interpretable ORs.
What should I do if the Hosmer-Lemeshow test is significant?
A significant Hosmer-Lemeshow test (p < 0.05) suggests poor model calibration. Consider adding non-linear terms (quadratic, spline), interaction terms, or additional predictors. Also check for outliers and influential observations. Note that with very large samples, even trivial miscalibration can yield significance.