13 Research methods, statistics, critical review and evidence-based practice13.10 Correlation and regression
13.10.15 Multivariate 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.
What it is
Multivariate multiple regression models several continuous outcomes at the same time, using the same set of predictors.
The two words in the name describe different things:
Multiple means there are several predictors.
Multivariate means there are several outcomes.
Model | Outcomes | Predictors |
|---|---|---|
Simple linear regression | One | One |
Multiple linear regression | One | Several |
Multivariate multiple regression | Several | Several |
For example, a study might model depression score, functioning and quality of life together. The predictors could be treatment group, age and baseline severity.
Each outcome gets its own set of coefficients. The model also takes account of how the residuals of the different outcomes correlate with each other. A residual is the gap between an observed value and the value the model predicts.
What joint analysis adds
A multivariate test can ask whether a predictor is related to the set of outcomes as a whole, rather than testing each outcome on its own.
You still need estimates for each outcome. These show which outcomes differ, in which direction and by how much.
A significant joint test does not mean that every individual outcome has a significant association.
Separate follow-up tests on each outcome bring back the problem of multiple testing. Plan how you will interpret them before the analysis.
Worked example
Suppose a joint model includes treatment group, coded 0 or 1, with the other predictors held fixed.
Outcome | Predicted mean (fitted conditional mean) |
|---|---|
Depression score | 20 minus 3 x group |
Function score | 40 + 5 x group |
For group 1, predicted depression score = 20 minus 3 = 17.
Predicted function score = 40 + 5 = 45.
The two coefficients describe different outcomes, measured in different units. Neither the predictions nor the signs of the coefficients tell you whether the effects are statistically significant.
Appraising a study
Choice of outcomes should follow from the research question. Adding extra correlated measures just to raise the chance of a significant result is poor practice.
Assumptions concern the joint distribution of the errors across outcomes, the right form for the relationship between predictors and mean outcomes, and independence between observational units (for example, individual patients).
Missing outcomes need explicit handling. Analysing only people who have every measure can reduce precision and introduce selection bias.
Causation still depends on the study design and on the assumptions behind any adjustment. Modelling more outcomes does not make a causal claim any stronger.