Regression output follows the same basic structure whether you’re using Excel, R, SPSS, Python, or any other tool. Once you know what each number means, you can read any of them. The key pieces are the coefficients (which tell you the direction and size of each relationship), the p-values (which tell you whether those relationships are statistically meaningful), and the model fit statistics (which tell you how well the whole equation works). Here’s how to work through each one.
Coefficients: The Core of the Output
The coefficients table is the centerpiece of any regression output. Each row represents one variable you included in the model, plus a row for the intercept (sometimes labeled “constant”). Two columns matter most here: the coefficient estimate itself and its associated p-value.
The intercept is the predicted value of your outcome when every predictor equals zero. Sometimes that’s meaningful (if zero is a realistic value for your predictors), and sometimes it’s just a mathematical anchor with no practical interpretation.
The slope coefficients tell you how much the outcome changes for every one-unit increase in that predictor, holding all other variables constant. A coefficient of 0.389 for a math score predicting a science score means that for every additional point on the math test, the science score goes up by 0.389 points, assuming everything else stays the same. A positive coefficient means the outcome increases as the predictor increases. A negative coefficient means the outcome decreases. If a variable has a coefficient of -0.016, the outcome drops by 0.016 for each one-unit increase in that predictor.
The “holding all other variables constant” part is critical in multiple regression. Each coefficient isolates the effect of one predictor as if the others didn’t change. This is what makes multiple regression more informative than running separate analyses for each variable.
P-Values: Separating Signal From Noise
Next to each coefficient, you’ll see a p-value. This number answers one question: if this predictor actually had no relationship with the outcome, how likely would you be to see a coefficient this large just from random chance in your sample?
The conventional cutoff is 0.05. A p-value below 0.05 means the result is considered statistically significant, and you reject the idea that the predictor has no effect. A p-value of 0.04, for instance, means that if there were truly no relationship, you’d see a result this extreme only about 4% of the time. That’s rare enough that most researchers treat it as evidence of a real effect.
A p-value above 0.05, like 0.14, means you don’t have enough evidence to conclude the predictor matters. That doesn’t prove it has zero effect. It just means your data can’t distinguish its effect from noise. Some fields use stricter thresholds (0.01 or even 0.005) to reduce false positives, so check the conventions in your area.
One important caution: p-values describe probability, not certainty. A significant p-value doesn’t guarantee the effect is real, and a non-significant one doesn’t guarantee it isn’t. They’re a tool for decision-making, not proof.
Standard Error and the T-Statistic
Your output also shows a standard error for each coefficient. This measures how precise that estimate is. A small standard error means the estimate is tightly clustered around the true value. A large one means there’s more uncertainty, and the coefficient could shift substantially with a different sample.
The t-statistic (or t-value) is simply the coefficient divided by its standard error. A large t-value, whether positive or negative, means the coefficient is large relative to its uncertainty. That’s what drives the p-value down. You rarely need to calculate or interpret the t-value directly, but it helps to know what it represents: the ratio of signal to noise for that individual predictor.
R-Squared: How Well the Model Fits
Near the top of most regression output, you’ll find R-squared (R²). This is the proportion of variation in your outcome that the model explains. An R² of 0.72 means your predictors collectively account for 72% of the differences in the outcome variable. The remaining 28% is unexplained, whether from variables you didn’t include, measurement error, or randomness.
What counts as a “good” R² depends entirely on your field. In physics or engineering, you might expect values above 0.90. In social science or psychology, 0.30 can be perfectly respectable because human behavior is inherently noisy.
If your model includes more than one predictor, look at adjusted R-squared instead. Regular R² always increases when you add another variable, even if that variable is useless. Adjusted R² corrects for this by penalizing the addition of predictors that don’t genuinely improve the model. If adjusted R² drops when you add a new variable, that variable is probably not helping. This is your safeguard against overfitting, where a model looks impressive on paper but is really just capturing noise.
The F-Statistic: Does the Model Work at All?
The F-statistic tests whether your model as a whole is statistically significant. While p-values on individual coefficients tell you whether each predictor matters, the F-test asks a bigger question: is this entire set of predictors, taken together, better than using no predictors at all?
The logic is straightforward. The F-test compares how much error your model produces against how much error you’d get from just predicting the average for everyone. A large F-statistic (with a small associated p-value) means your model does meaningfully better than that baseline. If the p-value on the F-test is above 0.05, your model as a whole isn’t explaining the outcome in a statistically significant way, regardless of what any individual coefficient says.
In practice, if you have at least one significant predictor, the overall F-test is usually significant too. But it’s worth checking, especially in models with many weak predictors.
Residuals: Checking Your Assumptions
Most output includes some information about residuals, which are the differences between what your model predicted and what actually happened for each data point. Residuals are your diagnostic tool for checking whether the model is appropriate for your data.
A well-fitting model produces residuals that look random. If you plot residuals against your predicted values (or against individual predictors), you should see a flat, patternless scatter. Specific patterns signal specific problems:
- A curved pattern suggests the relationship isn’t linear, and you may need to transform a variable or add a squared term.
- A fan or cone shape (residuals spreading out as predicted values increase) means the variability in your outcome isn’t constant across the range, a violation called heteroscedasticity.
- Clusters or waves may indicate that your data points aren’t independent, which often happens with time-series data.
If the residuals show no pattern and hover around zero in a roughly even band, your model’s assumptions are holding up.
Watching for Multicollinearity
When two or more predictors are highly correlated with each other, the model has trouble separating their individual effects. This is called multicollinearity, and it inflates standard errors, making coefficients unstable and p-values unreliable.
Many software packages report a Variance Inflation Factor (VIF) for each predictor. A VIF of 1 means no correlation with other predictors. Values above 5 deserve close inspection, and values above 10 almost certainly indicate a serious problem. If you spot high VIF values, you may need to drop one of the correlated predictors or combine them into a single variable.
Reading Logistic Regression Output
If your outcome is binary (yes/no, survived/didn’t, purchased/didn’t), you’re likely running logistic regression instead of linear regression. The output looks similar, but the coefficients mean something different.
In logistic regression, each coefficient represents the change in the log-odds of the outcome for a one-unit increase in the predictor. Since log-odds aren’t intuitive, most people convert them to odds ratios by raising the mathematical constant e to the power of the coefficient. An odds ratio of 1.5 means the odds of the outcome increase by 50% for each one-unit increase in that predictor. An odds ratio below 1 means the odds decrease.
The rest of the output, including p-values, standard errors, and overall model significance, works the same way. The main difference is that logistic regression uses different fit statistics (like the log-likelihood or pseudo R²) instead of the traditional R-squared, since the math behind the model is fundamentally different.
Putting It All Together
When you sit down with regression output, work through it in this order. First, check the F-statistic and its p-value to confirm the model is worth interpreting at all. Next, look at R-squared (or adjusted R-squared) to see how much of the outcome your predictors explain. Then examine each coefficient: its sign tells you the direction, its size tells you the magnitude, and its p-value tells you whether it’s statistically distinguishable from zero. Finally, check residual plots and VIF values to make sure the model isn’t violating assumptions that would make all those numbers unreliable.
The coefficients only make sense in the units of your variables. A coefficient of 5 could be trivial or enormous depending on whether you’re measuring dollars or millions of dollars. Always interpret the size of an effect in context, not just whether it’s significant.