Multiple regression output can look overwhelming at first: tables of coefficients, p-values, F-statistics, and R-squared figures. But each piece answers a specific question, and once you know what question each number addresses, the whole output clicks into place. Here’s how to read it, piece by piece.
Start With the F-Test: Does the Model Work at All?
Before diving into individual variables, check whether your model as a whole is doing anything useful. That’s what the F-test tells you. It compares your regression model against a “no predictors” model, essentially asking: do these variables, taken together, predict the outcome better than just using the average?
If the p-value attached to the F-statistic is below your significance threshold (commonly 0.05), you can conclude that at least some of your predictors, working together, meaningfully explain variation in the outcome. If that p-value is large, the model isn’t providing useful predictions, and interpreting individual coefficients becomes questionable.
One important nuance: it’s possible for the F-test to be significant even when no single predictor reaches significance on its own. This happens when predictors share explanatory power and collectively explain the outcome, but none stands out individually. It’s also possible for the reverse to occur. Always check the F-test first so you know whether the model deserves further interpretation.
R-Squared and Adjusted R-Squared
R-squared tells you the proportion of variation in your outcome that the predictors explain. An R-squared of 0.65 means 65% of the variability in your outcome is accounted for by the model. The remaining 35% is unexplained, left to factors you didn’t measure or random noise.
The problem with plain R-squared is that it always increases when you add more predictors, even useless ones. Adjusted R-squared fixes this by penalizing the addition of variables that don’t genuinely improve the model. It increases only when a new predictor adds real explanatory power, and it decreases when a predictor is just adding noise. If you’re comparing models with different numbers of predictors, adjusted R-squared is the better metric because it balances fit against complexity.
Reading Unstandardized Coefficients
Each predictor in your model gets an unstandardized coefficient (often labeled “B” or “b”). This number tells you the expected change in the outcome for every one-unit increase in that predictor, holding all other predictors constant. That last phrase is critical: the coefficient reflects that variable’s unique contribution after accounting for everything else in the model.
A concrete example makes this easier. In a study of eye pressure, researchers found that a particular corneal measurement had an unstandardized coefficient of -1.17 when predicting intraocular pressure. That means for each one-unit increase in that corneal measurement, eye pressure decreased by 1.17 units, assuming all other variables in the model stayed the same. Another predictor in a different eye study had a coefficient of +0.0154, meaning a one-unit increase in that variable raised the outcome by 0.0154 units.
The sign tells you the direction (positive means the outcome goes up; negative means it goes down), and the size tells you the magnitude. But be careful comparing unstandardized coefficients across predictors, because they depend on how each variable is measured. A coefficient of 0.02 for income measured in dollars is not “smaller” than a coefficient of 3.5 for years of education. They’re in completely different units.
Using Standardized Coefficients to Compare Predictors
When you want to know which predictor matters most, you need standardized coefficients (often labeled “Beta” or “β”). These express each coefficient in standard deviation units: a standardized coefficient of 0.40 means that a one-standard-deviation increase in that predictor is associated with a 0.40-standard-deviation change in the outcome. Because every variable is now on the same scale, you can compare them directly.
This solves what’s sometimes called the “apples and oranges” problem. If you’re predicting sales using both staffing levels (measured in people) and travel budgets (measured in dollars), their unstandardized coefficients can’t be meaningfully compared. But their standardized coefficients can. The predictor with the largest absolute standardized coefficient has the strongest relative association with the outcome. In one textbook example, a standardized coefficient of 0.925 for audience size indicated it was the most influential predictor among three variables, regardless of how the other variables were scaled.
Keep in mind this comparison is rough. Standardized coefficients give you a general sense of relative importance, not a precise ranking, especially when predictors are correlated with each other.
P-Values for Individual Predictors
Each predictor also gets its own p-value (from a t-test), which tells you whether that specific variable contributes to the model beyond what the other predictors already explain. A p-value below 0.05 is the conventional threshold for concluding that a predictor is statistically significant, meaning the association you’re seeing is unlikely to be due to chance alone.
But a few cautions are worth noting. A p-value of 0.05 is a convention, not a magic line. Some researchers argue the bar should be stricter, such as 0.005, to reduce false positives. A non-significant p-value doesn’t mean the variable has no real relationship with the outcome. It may simply mean your sample wasn’t large enough to detect it, or that the variable’s effect overlaps with another predictor already in the model. And a significant p-value doesn’t tell you the effect is large or practically important. Always look at the coefficient’s size alongside its p-value.
Checking for Multicollinearity With VIF
Multicollinearity occurs when two or more predictors are highly correlated with each other. When this happens, the model has trouble separating their individual effects, which inflates the standard errors and makes coefficients unstable. Your overall model fit might look fine, but individual predictors may appear non-significant even when they’re genuinely related to the outcome.
The standard diagnostic is the variance inflation factor (VIF), which most statistical software can calculate for each predictor. The common rules of thumb are:
- VIF below 5: generally not a concern
- VIF between 5 and 10: moderate multicollinearity that deserves closer inspection
- VIF above 10: serious multicollinearity that’s likely distorting your results
If you find high VIF values, your options include removing one of the correlated predictors, combining them into a single variable, or collecting more data. The key is to check VIF before placing too much weight on individual coefficients, because multicollinearity can make a genuinely important predictor look unimportant.
Checking Residual Plots
Regression assumes that the errors (the differences between predicted and actual values) are randomly scattered and roughly equal in spread across all levels of prediction. A residual vs. fitted values plot is the quickest way to check this.
In a well-behaved model, the points in this plot look like a random cloud centered on zero with no obvious pattern. Two common warning signs to watch for: a “fanning” pattern, where residuals are tightly clustered for low predicted values but spread wide for high predicted values, and a “funneling” pattern, which is the reverse. Either pattern signals that the model’s predictions are more accurate in some ranges than others, a violation called heteroscedasticity. If you see fanning or funneling, your standard errors may be unreliable, which means your p-values and confidence intervals could be misleading.
Also watch for curved patterns in the residuals, which suggest a nonlinear relationship that a straight-line model isn’t capturing. In that case, you may need to transform a variable or add a squared term.
Putting It All Together
A practical sequence for interpreting multiple regression output looks like this:
- F-test p-value: Is the model as a whole significant? If not, stop here.
- Adjusted R-squared: How much of the outcome does the model explain?
- Individual p-values: Which specific predictors are significant?
- Unstandardized coefficients: For each significant predictor, what’s the size and direction of the effect in real-world units?
- Standardized coefficients: Which predictor has the strongest relative association?
- VIF values: Are any predictors too correlated to interpret reliably?
- Residual plots: Are the model’s assumptions reasonably met?
When reporting results, you’ll typically present the overall model statistics (F-value, p-value, adjusted R-squared), then a table with each predictor’s coefficient, standard error, confidence interval, and p-value. If you’re writing for an academic audience, standardized coefficients and confidence intervals are expected alongside unstandardized coefficients. The confidence interval is often more informative than the p-value alone, because it shows the plausible range of the true effect rather than just a yes-or-no significance decision.