Multiple regression analysis is a statistical method that predicts one outcome variable from two or more predictor variables at the same time, telling you how much each predictor contributes while holding the others constant. If you're staring at a dataset with several possible influences on your outcome — and you can't figure out which ones actually matter — this is the test you're looking for.
Key Takeaways
- Multiple regression predicts a single continuous outcome (like exam score or blood pressure) from two or more predictors — for example, predicting salary from years of experience, education, and hours worked together.
- Each predictor gets its own coefficient (b or β) that shows its unique effect while statistically controlling for the other predictors in the model.
- R² tells you how much of the outcome's variation your predictors explain together — an R² of .45 means your model accounts for 45% of the variation.
- You need a continuous outcome variable and roughly 10–20 cases per predictor for stable results.
- StatRyx runs multiple regression and returns an APA 7 write-up automatically, including coefficients, R², and significance tests — no syntax required.
What is multiple regression analysis, in plain English?
Multiple regression is a way to answer the question: "Which things predict my outcome, and how strongly does each one matter on its own?" Instead of testing one relationship at a time, it looks at several predictors simultaneously and untangles their separate effects.
Think of it as an upgrade to simple linear regression. Simple regression uses one predictor (does study time predict exam score?). Multiple regression uses several (do study time, sleep, and prior GPA together predict exam score?) — and crucially, it tells you the effect of each one after accounting for the others. That "holding the others constant" part is what makes it so useful for real research, where nothing happens in isolation.
When should I use multiple regression?
Use multiple regression when your outcome variable is continuous and you have two or more predictor variables you think influence it. Typical questions include:
- Which factors predict patient recovery time — age, BMI, and treatment dose?
- Does job satisfaction predict burnout after controlling for salary and workload?
- How well do income, education, and neighbourhood predict test scores?
Your predictors can be continuous (age, hours, scores) or categorical if you code them as dummy variables (e.g. male/female as 0/1). Your outcome should be continuous. If your outcome is binary (yes/no, survived/died), you need logistic regression instead — a common mix-up worth flagging before you run anything.
How does multiple regression actually work?
Multiple regression fits an equation that looks like this:
Ŷ = b₀ + b₁X₁ + b₂X₂ + … + bₙXₙ
Here's what each piece means without the maths anxiety:
- Ŷ is the predicted outcome value.
- b₀ is the intercept — the predicted outcome when all predictors are zero.
- b₁, b₂… are the unstandardised coefficients. Each one says: "for every one-unit increase in this predictor, the outcome changes by this much, holding the others constant."
- X₁, X₂… are your predictor values.
You'll also see standardised coefficients (β). These put every predictor on the same scale so you can compare them directly — a β of .40 has twice the impact of a β of .20, regardless of the original units. Reviewers love betas because they answer "which predictor is most important?"
What do R² and adjusted R² tell me?
R² is the proportion of variance in the outcome explained by all predictors combined — it ranges from 0 to 1. An R² of .35 means your model explains 35% of why the outcome varies. Adjusted R² is a slightly smaller, more honest version that penalises you for adding predictors that don't earn their keep, so always report it for multiple regression.
A worked example with real numbers
Suppose you study 120 undergraduates and want to predict final exam score (0–100) from three predictors: weekly study hours, average nightly sleep hours, and prior GPA. You run a multiple regression and get:
- Overall model: F(3, 116) = 18.42, p < .001, R² = .32, adjusted R² = .30
- Study hours: b = 1.85, β = .34, p < .001
- Sleep hours: b = 2.10, β = .22, p = .008
- Prior GPA: b = 6.40, β = .28, p = .002
Here's how to read it:
The overall model is significant (p < .001), and together the three predictors explain 32% of the variance in exam scores (R² = .32). That's a meaningful chunk for behavioural data.
Study hours is the strongest predictor (β = .34): each extra hour of weekly study predicts a 1.85-point rise in exam score, holding sleep and GPA constant. Sleep matters too (β = .22, p = .008) — each additional hour of sleep predicts about 2.1 more points. Prior GPA (β = .28) has a large per-unit coefficient (b = 6.40) because GPA moves on a small 0–4 scale, which is exactly why the standardised beta is more honest for comparing importance.
The takeaway a researcher would write: all three factors independently predict performance, with study time being the most influential.
Simple vs multiple regression: key differences
| Feature | Simple Regression | Multiple Regression |
|---|---|---|
| Number of predictors | 1 | 2 or more |
| Question answered | Does X predict Y? | Which of several Xs predict Y, and how much each? |
| Controls for confounds? | No | Yes — isolates each predictor's unique effect |
| Key output | b, R² | b, β for each predictor, R², adjusted R² |
| Minimum sample guide | ~30+ cases | ~10–20 cases per predictor |
What assumptions do I need to check?
Multiple regression rests on a handful of assumptions, and skipping them is where most thesis regressions go wrong:
- Linearity — predictors relate to the outcome in straight-line fashion.
- Independence of residuals — one observation's error doesn't predict another's.
- Homoscedasticity — the spread of residuals stays roughly constant.
- Normality of residuals — the errors are approximately normally distributed.
- No severe multicollinearity — your predictors shouldn't be too highly correlated with each other. Check the VIF (variance inflation factor); a VIF above 5 (some say 10) signals a problem.
Multicollinearity is the sneakiest one: if two predictors measure nearly the same thing, the model can't separate their effects and your coefficients become unstable. StatRyx checks VIF and residual assumptions automatically and flags violations in plain language, so you don't discover a broken model in your viva.
How do I report multiple regression in APA 7?
APA 7 wants the overall model fit and the individual coefficients. A clean write-up of the example above looks like this:
A multiple regression was conducted to predict exam score from study hours, sleep, and prior GPA. The model significantly predicted exam score, F(3, 116) = 18.42, p < .001, and explained 32% of the variance (R² = .32, adjusted R² = .30). Study hours (β = .34, p < .001), sleep (β = .22, p = .008), and prior GPA (β = .28, p = .002) were all significant unique predictors.
Note the APA conventions: test statistics and p are italicised, and p values drop the leading zero (p < .001, not 0.001). If you're weighing which analysis fits your design, our guide on choosing the right statistical test walks through the decision tree.