The most common way to tell if a variable is significant in regression is to check its p-value: if the p-value is less than 0.05, the variable is typically considered statistically significant. But that single number doesn’t tell the whole story. Understanding what’s behind it, and what can go wrong, helps you avoid mistakes that lead to wrong conclusions.
What the P-Value Actually Tells You
Every variable in a regression model gets tested against a “null hypothesis,” which assumes the variable has no real effect on the outcome. The p-value represents the probability of seeing a result as extreme as yours if that null hypothesis were true. So a p-value of 0.04 means that if the variable truly had no effect, you’d still get a result this large about 4% of the time just by chance.
The standard cutoff is 0.05 (5%). If a variable’s p-value falls below 0.05, you reject the null hypothesis and call the variable statistically significant. This threshold has a loose mathematical basis: roughly 5% of a normal distribution falls more than two standard deviations from the mean, representing the “outlying” zone. Some fields set the bar higher. A threshold of 0.01 is common in more rigorous contexts, and some researchers have argued for moving to 0.005 to reduce false positives.
The key thing to remember: a p-value of 0.06 doesn’t mean a variable has no effect. And a p-value of 0.03 doesn’t guarantee it does. These are probabilities, not proof. The 0.05 line is a convention, not a law of nature.
How to Read a Regression Output Table
Most statistical software produces a table with four key columns for each variable. Here’s what each one means:
- Coefficient (Estimate): The estimated effect of the variable on the outcome. If the coefficient for “years of experience” is 2,500 in a salary regression, it means each additional year is associated with $2,500 more in salary, holding other variables constant.
- Standard Error: A measure of how uncertain that estimate is. Smaller standard errors mean more precise estimates.
- T-statistic: The coefficient divided by its standard error. This ratio tells you how many standard errors the coefficient is away from zero. Larger absolute values (roughly above 2) suggest significance.
- P-value: Derived from the t-statistic. This is the column most people look at first. Values below your chosen threshold (usually 0.05) indicate significance.
The t-statistic is the engine behind the p-value. A coefficient of 2,500 with a standard error of 800 gives a t-statistic of about 3.1, which translates to a very small p-value. That same coefficient of 2,500 with a standard error of 2,000 gives a t-statistic of 1.25, which would not be significant. The size of the effect matters, but so does how precisely you can estimate it.
Using Confidence Intervals Instead
Confidence intervals offer another way to assess significance, and many statisticians prefer them because they’re more informative. A 95% confidence interval gives you a range of plausible values for the coefficient. If that range does not include zero, the variable is significant at the 0.05 level. If it crosses zero, you can’t rule out the possibility that the variable has no effect.
For example, if the 95% confidence interval for “years of experience” runs from 900 to 4,100, zero isn’t in that range, and the variable is significant. If the interval runs from -500 to 5,500, it crosses zero, so you can’t call it significant. The advantage of confidence intervals is they also show you the range of realistic effect sizes, not just a yes-or-no answer.
Individual Variables vs. the Overall Model
Regression output includes two different types of significance tests, and confusing them is a common mistake. The t-test (and its p-value) evaluates each variable individually. The F-test evaluates whether all the variables in your model, taken together, explain a meaningful amount of variation in the outcome.
It’s possible for the overall F-test to be significant while none of the individual t-tests are. This happens when variables collectively predict the outcome but no single one stands out on its own. The reverse can also occur, though it’s less common. In simple regression with just one predictor, the p-value from the t-test and the F-test will be identical. The distinction only matters when you have multiple variables.
When interpreting your results, check the F-test first to confirm your model as a whole is meaningful. Then look at individual p-values to identify which specific variables are contributing.
When Multicollinearity Hides Significance
Sometimes a variable that genuinely affects the outcome shows up as insignificant. One of the most common reasons is multicollinearity, which happens when two or more predictor variables are highly correlated with each other.
Here’s the mechanism: when predictors are correlated, the model struggles to separate their individual effects. This inflates the standard error of each coefficient. Since the t-statistic is the coefficient divided by its standard error, a larger standard error shrinks the t-statistic and pushes the p-value upward. The result is wider confidence intervals and variables that appear insignificant even when they have real predictive power. In one documented example, multicollinearity inflated a standard error to 32.7 and produced a 95% confidence interval stretching from -104 to 29, far too wide to detect any effect.
If you suspect multicollinearity, look at the correlations between your predictors. A variance inflation factor (VIF) above 5 or 10 is a common warning sign. Possible fixes include removing one of the correlated variables, combining them into a single measure, or using a technique like ridge regression that handles correlated predictors more gracefully.
Sample Size Matters More Than You Think
Small samples make it harder to detect real effects. With fewer data points, your estimates are less precise, standard errors are larger, and p-values tend to be higher. A variable that would be clearly significant with 200 observations might show a p-value of 0.15 with only 30. This doesn’t mean the effect isn’t real; it means you don’t have enough data to detect it reliably.
On the flip side, very large samples can make trivially small effects statistically significant. With 10,000 observations, even a tiny relationship between two variables can produce a p-value well below 0.05. This is where the distinction between statistical and practical significance becomes critical. As a general guideline, samples of 40 or more provide reasonably stable estimates of how well your model fits the data.
Statistical Significance vs. Practical Significance
A variable can be statistically significant but practically meaningless. The classic example comes from a study of over 22,000 people that found aspirin reduced heart attacks with a p-value below 0.00001. That’s extraordinarily significant in statistical terms. But the actual risk difference was only 0.77%, and aspirin explained just 0.1% of the variation in outcomes. The effect was real but tiny.
This is why effect size matters alongside p-values. Effect size measures how large the impact actually is, not just whether it exists. One common framework classifies standardized effect sizes of 0.2 as small, 0.5 as medium, and 0.8 or above as large. A medium effect (0.5) is one a careful observer would notice in practice. In regression specifically, you can look at the coefficient itself: does the predicted change in the outcome actually matter for real-world decisions? A variable that raises test scores by 0.3 points on a 100-point scale might be significant with enough data, but nobody would redesign a curriculum around it.
When reporting or interpreting regression results, both pieces are essential. The p-value tells you whether an effect likely exists. The coefficient and effect size tell you whether it’s worth caring about.
A Practical Checklist
When you’re staring at regression output and need to evaluate a variable, work through these steps:
- Check the p-value: Is it below your chosen threshold (usually 0.05)? If yes, the variable is statistically significant.
- Look at the confidence interval: Does the 95% interval exclude zero? This confirms the p-value result and shows the plausible range of the effect.
- Examine the coefficient: Is the effect large enough to matter in your context? A significant but negligible coefficient may not be worth acting on.
- Check for multicollinearity: Are correlated predictors inflating standard errors and hiding real effects?
- Consider your sample size: With small samples, non-significance might reflect low statistical power rather than the absence of an effect.
- Verify the overall model: Is the F-test significant? If the model as a whole doesn’t explain the outcome, individual variable significance is harder to trust.
No single number definitively proves a variable matters. The p-value is the starting point, but the coefficient size, confidence interval, sample context, and potential multicollinearity all shape the full picture.