# Linear regression with negative estimated value for intercept

Does a negative value of intercept suggest that the regression line provides poor fit to the data? why? and why not?

• The phrasing of your question implies that you have some reason to think the premise may be true. Why? What prompts such a question? (I ask because I suspect there may be a deeper issue laying beneath this one that should be cleared up -- possibly indicating the unsuitability of a linear regression model for whatever you're thinking about) – Glen_b Sep 4 '18 at 6:03
• If you don't like negative intercepts, just negate the response variable. That will not change the linearity (that is, goodness of fit) of any model but will produce a positive intercept. – whuber Sep 4 '18 at 13:56
• Unless it's a variable where a negative value would be meaningless. – smci Sep 5 '18 at 0:40
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This an example of linear regression fit. The intercept of this fit is negative and it fits well.

No, a negative value of intercept does not suggest that the regression line provides “poor fit to the data”. Why not? Because the intercept is not a measure of fit for a regression. It is a value that represents one element of the model, not the quality of the fit.

The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. WP:EN s.v. Goodness of fit

[...] y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system. WP:EN s.v. y-intercept

See KDG's answer for an illustration.

As others have said, intercept is not a measure of fit at all, so a negative intercept does not indicate a poor fit. But -- you asked if a negative intercept suggests a poor fit. In some contexts, the answer is clearly "yes". If $X$ and $Y$ are physical quantities such that $Y < 0$ makes no physical sense, and furthermore values of $X$ close to 0 are non-outliers which are part of your data set, then a linear model $Y = mX + b$ with $b < 0$ (by a significant amount) won't be a good model for your data. Note that such a model will have large residuals for those $X$ values close to 0, so the poorness of fit will show up in the standard measures.

But in other contexts, a negative intercept is unproblematic even if $Y < 0$ makes no sense. As an example, R's built-in sample dataset trees gives height, volume and girth of 31 black cherry trees. If you do a linear regression of volume as a function of girth (lm(formula = Volume ~ Girth, data = trees)) you get a linear model with a negative intercept. The fit isn't bad (even though a quadratic model would be better) with $R^2 = 0.93$. You could use it to get reasonable predictions of the volume of such a tree in terms of its girth. In this case, the negative intercept simply indicates that it is unreasonable to extrapolate a relationship which (roughly) holds among mature trees to mere saplings. But - you don't need a negative intercept to tell you that such extrapolation is problematic. I wouldn't trust any physical model with $X$ near 0 unless the data itself involved such values.

The intercept value has no relation to goodness-of-fit.

To see if a fit-line is good fit for the data you need to calculate the distance between the fit-line and all the data points (as shown in @KDG's answer).

One metric for calculating this is Root Mean Square Error (RMSE).