Multiple regression is a statistical method that estimates how several independent variables relate to one outcome variable at the same time. When you read its output, you focus on three things: the coefficients (the size and direction of each relationship), the p-values (whether each relationship could plausibly be due to chance), and the overall model fit (how much of the outcome the variables together explain). Everything else in the output supports those three numbers.
What Is Multiple Regression Actually Doing?
Multiple regression fits a line (or a curved surface) through your data that best predicts one outcome from several inputs at once. The math chooses the coefficients that make the predicted values as close as possible to the actual values, usually by minimizing the sum of squared errors.
The reason this matters is adjustment. In real life, variables rarely act alone. A simple correlation between coffee and heart rate might really reflect that coffee drinkers also smoke. Multiple regression lets you estimate the coffee effect while holding smoking statistically constant. That is the whole point.
The output is a set of coefficients. Each one tells you how much the outcome is expected to change when that predictor increases by one unit, assuming the other predictors stay the same. That phrase — “assuming the other predictors stay the same” — is doing enormous work. It is also the source of most misinterpretation.
How Do You Read the Coefficients?
A coefficient is a slope. If the coefficient for a predictor is 2.5, the model says the outcome rises by 2.5 units for every one-unit increase in that predictor, holding the others fixed.
The sign tells you direction. Positive means the outcome tends to go up as the predictor goes up. Negative means the opposite. The size tells you the estimated effect per unit of the predictor — but only in the units that predictor is measured in.
This is where people trip. A coefficient of 0.001 for income measured in dollars and a coefficient of 0.5 for age measured in years are not directly comparable. To compare them, you need standardized coefficients, sometimes labeled beta, which express effects in standard deviation units. Even then, comparison is only meaningful if the predictors are on similar conceptual footing.
The intercept is the predicted outcome when every predictor equals zero. Often that scenario is impossible or meaningless — nobody has zero age or zero height. The intercept is usually a mathematical necessity, not a finding.
What Do the P-Values and Confidence Intervals Tell You?
A p-value answers one narrow question: if the true relationship were actually zero, how likely would we be to see a coefficient this large or larger just by chance? A small p-value means the data are unlikely under that “no relationship” assumption.
That is not the same as “the effect is large” or “the effect matters.” A tiny, clinically irrelevant effect can produce a very small p-value if the sample is big enough. This is one of the most common misreadings of regression output.
Confidence intervals are more informative. A 95% confidence interval gives a range of plausible values for the true coefficient. If the interval includes zero, the data are consistent with no relationship. If it runs from, say, 0.2 to 3.1, you have evidence of a positive relationship, but the true size could be anywhere in that range.
Wide intervals mean imprecise estimates. Narrow intervals mean precision — but precision is not the same as importance. A narrow interval around a trivial effect is still a trivial effect.
How Do You Judge the Overall Model Fit?
R-squared is the share of the outcome’s variation the model explains. An R-squared of 0.40 means the predictors together account for about 40% of the variation. The rest comes from variables you did not include, measurement error, and randomness.
What counts as a good R-squared depends entirely on the field. In tightly controlled lab experiments, values above 0.9 are common. In social science and health research predicting human behavior, values of 0.2 to 0.4 can be meaningful. There is no universal cutoff.
Adjusted R-squared penalizes you for adding predictors that do not pull their weight. If adjusted R-squared is much lower than R-squared, your model may be bloated with weak variables. The F-test for the overall model asks whether the predictors together explain more than you would expect by chance.
What Are the Main Ways People Misread Regression Output?
The first mistake is treating correlation as causation. Regression estimates associations. It does not prove that changing one variable will change the outcome. Only a randomized experiment, or very strong causal-inference methods, can support that claim.
The second is ignoring multicollinearity — when predictors are highly correlated with each other. This inflates the standard errors, making coefficients unstable and confidence intervals wide. The model still fits, but individual coefficients become hard to trust. Variance inflation factors above roughly 5 to 10 are a common warning sign.
The third is overinterpreting a coefficient whose confidence interval is wide. A point estimate of 3.0 with an interval from −1 to 7 is not evidence of a positive effect.
The fourth is ignoring the units and scale. A “significant” predictor measured in tiny units can look impressive while representing a trivial real-world change.
The fifth is forgetting that regression assumes a linear relationship. If the true relationship curves, a straight-line model can miss it entirely, or produce a coefficient that averages over a pattern it cannot capture.
What Is the Difference Between Statistical and Practical Significance?
Statistical significance means the data are unlikely under a no-effect assumption. Practical significance means the effect is large enough to matter in the real world.
These can diverge sharply. With a large enough sample, almost any nonzero effect becomes statistically significant. With a small sample, a genuinely important effect might not reach significance.
Always ask two questions separately: Is this likely a real relationship? And is it big enough to care about? The p-value addresses the first. The coefficient size, in real-world units, addresses the second. Neither substitutes for the other.
How Should You Interpret Interaction Terms and Categorical Predictors?
An interaction term means the effect of one predictor depends on the value of another. If the coefficient for an interaction is significant, you cannot interpret the main effects in isolation — the slope of one variable changes depending on the other.
Categorical predictors, like sex or treatment group, are handled through dummy coding. One category becomes the reference group, and each coefficient represents the difference between that category and the reference. The choice of reference group changes the coefficients but not the model’s predictions.
This is a common source of confusion. If a coefficient for “Group B” is negative, it means Group B is lower than the reference group — not lower than some absolute standard.
What Should You Check Before Trusting the Output?
Before interpreting anything, check the assumptions. Regression assumes linearity, independent observations, roughly constant variance of residuals, and normally distributed residuals for inference to hold.
Look at residual plots. Patterns in residuals — curves, funnels, clusters — signal that the model is missing something. Outliers can pull coefficients substantially, especially in small samples.
Missing data matters too. If data are missing in a way related to the outcome, results can be biased. How missing values were handled should always be reported.
A model that violates its assumptions can still produce output. The numbers will look fine. They just may not mean what you think.
Frequently Asked Questions
What does a p-value below 0.05 mean in regression?
It means the observed coefficient would be unlikely if the true effect were zero, assuming the model is correctly specified. It does not prove the effect is large or important.
How do I know which predictor matters most?
Standardized coefficients (beta) let you compare predictors measured in different units. Even then, a larger standardized coefficient does not automatically mean the variable is more important in a causal sense.
Can multiple regression prove that one variable causes another?
No. Regression estimates associations. Establishing causation requires randomized experiments or strong causal-inference designs that address confounding directly.
What is a good R-squared value?
There is no universal threshold. What counts as good depends on the field and the outcome being predicted. In human behavior research, modest values can still be meaningful.
