# Interpreting the f ratio in linear regression in r

The last string of my output in r is:

F-statistic: 181.4 on 15 and 2380 DF, p-value: < 2.2e-16

How do I interpret this result?

While I understand that the f value is a ratio that compares the systematic variance to the amount of unsystematic variance i.e. “the ratio of the model to its error” (Fields et al., 2012 p.400), I do not understand how to interpret the string.

## 1 Answer

Mathematically, $F=\frac{\frac{\text{Sum Squares Model}}{\text{df Model}}}{\frac{\text{Sum Squares Error}}{\text{df Error}}}$.

If you think of your data have a certain amount of variation in it, the F-statistic essentially gives you a measure of how much of the variation is explained by the model (per parameter) versus how much of the variation is unexplained (per remaining degrees of freedom). This unexplained variation is your error sums of squares. Through this lens, a higher F-statistic means that your model explains that much more of the variation per parameter than there is error per remaining degree of freedom.

The p-value just gives you an idea of how likely it is that your model would give you these results under the null hypothesis that the model explains none of the variability in your data. The low p-value tells you that the likelihood of getting this fit if the model actually didn't fit your data at all is ridiculously low, leading to the interpretation that at least some part of your model does fit the data.

Hope that helps!

• just so I understand, I will write some of this back to you. Please confirm the validity of the following statements. Since the f-value is high and the p-value is very small, the likelihood of the model not fitting the data is incredibly low. Therefore, at least some part of the model fits the data. – Jacquelyn Renée Sep 9 '15 at 23:54
• pretty much. i might say the probability of the model not explaining any of the data is low, but same general idea. – MeetMrMet Sep 10 '15 at 12:28