13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.11 Logistic and multiple regression
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Written by Resident Doctor Committee · Published
New in the 2027 syllabus
The College added this topic for exams from January 2027.
Choosing the model
Choose the model by the type of outcome.
Model | Outcome | What a coefficient means |
|---|---|---|
Multiple linear regression | One continuous outcome | Change in the conditional mean (average outcome) per one-unit rise in the predictor |
Logistic regression | Binary outcome | Change in log odds per one-unit rise in the predictor |
Multiple means more than one predictor, so logistic regression can also be a multiple regression model.
Adjustment means reading each coefficient with the other included predictors held constant. It does not by itself show causality.
Categorical predictors usually enter as indicator variables, each compared with a stated reference category.
Worked example: linear prediction
Suppose predicted score = 5 + 0.5 x baseline score + 2 x group, with group coded 0 or 1.
Input | Value |
|---|---|
Baseline score | 10 |
Group | 1 |
Baseline contribution = 0.5 x 10 = 5.
Group contribution = 2 x 1 = 2.
Predicted mean score = 5 + 5 + 2 = 12. This is an average, not a guaranteed outcome for any one person.
Interpreting logistic regression
Logistic regression models the log odds, log(p / (1 minus p)), where p is the probability of the outcome. Converting back gives a probability between 0 and 1.
Exponentiating a coefficient gives the odds ratio for a one-unit rise in that predictor, with the other predictors held constant.
Interactions let a predictor's association depend on another variable. One coefficient may then not describe its association for everyone.
Worked example: odds are not risks
Input | Value |
|---|---|
Reference probability | 0.20 |
Odds ratio for the comparison | 2 |
Reference odds = 0.20 / 0.80 = 0.25.
Comparison odds = 0.25 x 2 = 0.50.
Comparison probability = 0.50 / 1.50 = 0.333. The risk ratio is about 1.67, not 2.
Checking the model
Linear models: check the residuals (observed minus predicted values) for dependence, changing variance, outliers and departures from the assumed mean relationship.
Logistic models: predictors need the right functional form on the log-odds scale. Watch for sparse events and complete separation (where a predictor perfectly splits those with and without the outcome).
Multicollinearity (strongly correlated predictors) makes individual coefficients unstable.
Overfitting: an overly complex model fits the development data too closely.