How to Interpret Regression Analysis Results: P-values and Coefficients

For simple regression, fitted line plots really bring the math to life

Regression analysis generates an equation to describe the statistical relationship between one or more predictor variables and the response variable. After you use Minitab Statistical Software to fit a regression model, and verify the fit by checking the residual plots, you’ll want to interpret the results. In this post, I’ll show you how to interpret the p-values and coefficients that appear in the output for linear regression analysis.

How do I interpret the p-values in linear regression analysis?

The p-value for each term tests the null hypothesis that the coefficient is equal to zero (no effect). A low p-value (< 0.05) indicates that you can reject the null hypothesis. In other words, a predictor that has a low p-value is likely to be a meaningful addition to your model because changes in the predictor’s value are related to changes in the response variable.

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Comments

Linear or nonlinear regression

Jim, this is a nuanced comment but a critical one. You say "Conversely, a larger (insignificant) p-value suggests that changes in the predictor are not associated with changes in the response." You are missing a key word: linear. The sentence should say "...are not linearly associated with changes in the response." The predictor could have a perfect nonlinear relationship.

Similarly, from your first example you say "Typically, you use the coefficient p-values to determine which terms to keep in the regression model." You should change this to "keep in the linear regression." Also,to your suggestion "In the model above, we should consider removing east" add the caveat "if you are only doing linear regression."

For this reason, plotting the data is the very first thing one should do when analyzing linear relationships.

As you stated, when doing multiple regression (linear or otherwise), you need multidimensional plots. However, some of that can be captured by plotting individually each predictor against the response and each predictor against each other. Perhaps, that's for another article.

Regression

Hi Casey,

You raise very good and valid points. However, keep in mind, I wrote this entire article in the context of linear regression. I mention this in the first paragraph, "In this post, I’ll show you how to interpret the p-values and coefficients that appear in the output for linear regression analysis."

I've also presupposed in the introductory paragraph that the residual plots have passed muster, which the more advanced regression user can interpret as meaning that the relationships are likely to be correctly modeled.

This specific article is an introductory look at what these statistics mean for relatively simple cases. There is definitely plenty of room for additional, and more advanced, articles!

Thanks for reading and commenting!

Jim