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Multiple Linear Regression Calculator
Fit a continuous outcome on several predictors at once, read each coefficient as an adjusted effect, and copy the exact R command behind the result. A full interpretation guide follows below. Validated against R 4.3.3.
In Simple Words
Multiple linear regression estimates how a continuous outcome changes as several predictors change together. Each coefficient is the average change in the outcome for a one-unit rise in that predictor while the others are held fixed — the “adjusted” effect that a simple two-variable comparison cannot give you.
The model also reports R², the share of outcome variance the predictors jointly explain, and a p-value for each coefficient. Together they tell you not just whether a predictor matters, but how much it moves the outcome once the other variables are accounted for.
When to Use Multiple Linear Regression
Reach for multiple linear regression when your outcome is continuous and you want the independent contribution of each predictor after adjusting for the others — for example, modelling blood pressure from age, BMI, and sodium intake at once. It separates a predictor’s own effect from confounding by the variables alongside it, which a one-at-a-time analysis cannot do.
If your outcome is binary, use logistic regression instead; for a single predictor, a simple correlation may be enough. Report each coefficient with its 95% confidence interval, not just its p-value. Unsure which model fits your design? Our interactive Test Selector can point you to it.
Expected: one row per record, y, x1, x2, …
Drop a CSV here or click to choose. Parsed in your browser: nothing is uploaded.
Expected: one row per record, y, x1, x2, …
df <- read.csv(text = "y,x1,x2
{mlr_csv}")
summary(lm(y ~ ., data = df))User guide
When to use. Model a continuous outcome from several predictors simultaneously; read each coefficient as the adjusted effect.
Assumptions. Linearity, independent errors, constant variance, normal residuals, no severe multicollinearity.
Common mistakes. Avoid fitting more predictors than the sample can support; check variance inflation factors.
Worked example
Click Load example data above, then Run Analysis to see a fully worked result with figure, formula and interpretation for this test.
References
- Draper NR, Smith H. Applied Regression Analysis. 3rd ed. 1998.
Worked Examples
Example 1 — Two predictors of an outcome
Suppose a model predicts a quality-of-life score from a treatment dose and a baseline severity rating, fitted as ŷ = 8.4 + 1.9·dose − 0.6·severity. The dose coefficient of 1.9 means that, holding baseline severity fixed, each extra unit of dose is associated with a 1.9-point higher score on average. The negative severity coefficient says higher baseline severity predicts a lower score once dose is accounted for.
Example 2 — Reading R² and significance
If the same model reports R² = 0.62, the two predictors together explain 62% of the variation in the score; the remaining 38% is unexplained. A coefficient is judged statistically meaningful by its p-value and confidence interval — an interval that excludes zero indicates the adjusted effect is distinguishable from no effect at the chosen level.
Assumptions to Check
Linearity and additivity. The outcome is modelled as a straight-line, additive function of the predictors. Curved relationships need transformation or polynomial terms, and genuine interactions must be entered explicitly.
Independent, equal-variance, normal errors. Residuals should be independent, have roughly constant spread across fitted values (homoscedasticity), and be approximately normal. Plot residuals against fitted values and use a normal quantile plot to check.
No severe multicollinearity, adequate sample size. Highly correlated predictors inflate standard errors; screen with variance inflation factors. You also need enough observations per predictor — too many terms for the sample size produces unstable, overfitted coefficients.
Common Mistakes
Overfitting with too many predictors
Adding predictors the sample cannot support inflates R² while making coefficients unstable. Sample size considerations, not a fixed rule of thumb, should guide how many terms to include.
Ignoring multicollinearity
When predictors are strongly correlated, individual coefficients become unreliable and can even flip sign. Check variance inflation factors before interpreting any single effect.
Interpreting coefficients without holding others fixed
Each coefficient is an adjusted effect, conditional on the other predictors. Describing it as a raw, unadjusted relationship misstates what the model estimated.
Skipping residual diagnostics
A high R² does not confirm the model is sound. Non-linearity or non-constant variance can hide behind a good fit; always inspect residual plots before reporting.
Frequently Asked Questions
How is multiple regression different from simple regression?
Simple regression uses one predictor; multiple regression uses several at once and reports each effect adjusted for the rest, separating overlapping influences.
What does an adjusted coefficient mean?
It is the average change in the outcome for a one-unit change in that predictor while every other predictor in the model is held constant.
How many observations do I need?
It depends on the goal — estimating one coefficient precisely versus predicting individual outcomes imply different requirements. Base the count on the standard error you can tolerate, not a blanket subjects-per-variable rule.
Does the result match SPSS, R, and Stata?
Yes. The calculator uses ordinary least squares and shows the exact lm() command so you can reproduce the coefficients, R², and p-values in R.
Related Calculators
References
- Hanley JA. Simple and multiple linear regression: sample size considerations. J Clin Epidemiol. 2017;79:112–119. PMID 27393156.
- Slinker BK, Glantz SA. Multiple linear regression is a useful alternative to traditional analyses of variance. Am J Physiol. 1988;255(3 Pt 2):R353–367. PMID 3046375.
- Draper NR, Smith H. Applied Regression Analysis. 3rd ed. New York: Wiley; 1998.
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