# Interpreting a negative intercept in linear regression

This is my first time of having a negative intercept, so I'm a bit confused.

My line of regression is:

$$\text{starting monthly income} = -7.5 + 0.75\times \text{years of education}.$$

How would I interpret this?

• What does the plot of starting monthly income vs years of education look like? (Is the smallest value of years of education in your data set quite some way from 0? or is the fit near 0 poor?) Nov 20, 2017 at 9:11
• I'd consider working on a logarithmic scale almost by default with a variable like income. Then positive predictions are ensured. (Negative incomes aren't ruled out totally depending on the definition, but the surprise here suggests wages, salaries, etc.) Nov 20, 2017 at 10:34
• Nov 20, 2017 at 13:44

As @whuber comments, this is probably a case of misspecification.

When in doubt, plot. Take a piece of paper and plot your regression line: $y=-7.5+0.75x$, where $y$ is starting income and $x$ is years of education. In R:

xx <- 0:20
plot(xx,-7.5+0.75*xx,lwd=2,type="l")


You see that your model predicts that someone with zero years of education will have a negative starting income of $-7.5$, and each additional year of education will increase starting income by $0.75$.

Now, there are two possibilities:

1. This model may make perfect sense. For instance, you may only have people with 15-20 years of education in your sample, with their starting income varying linearly with education between about 3.75 and 7.5. In this case, you shouldn't predict outside the observed range of [15,20], and your negative intercept won't bother you.

2. This model may not make sense. For instance, you may have people in your sample with few years of education and low - but non-negative - starting income. For instance, your data and the regression line may look like this:

set.seed(1)
education <- sample(x=xx,size=200,replace=TRUE)
income <- pmax(450*education^2+20000+rnorm(length(education),0,10000),0)

(model.linear <- lm(income~education))
plot(education,income,pch=19)
abline(model.linear,col="red",lwd=2)


Here, predicting for someone with 0 or 1 year of education is entirely warranted by your observation, but a negative prediction isn't. (Incidentally, linear predictions will not only underestimate income for low eduction, but also for high education.)

The problem here is that your model is misspecified. The relationship between education and income is not linear.

To model such nonlinearities, I recommend using . (Polynomial fits using poly() are not recommended. See Frank Harrell's textbook Regression Modeling Strategies on why, and on more information on splines.) Here is a spline model and the new fits in green:

library(splines)
(model.spline <- lm(income~ns(education)))
lines(xx,predict(model.squared,newdata=data.frame(education=xx)),col="green",lwd=2)


The fit is much better, and importantly, this model doesn't predict negative values of income for low values of education.

Bottom line:

• When in doubt, plot.
• Always plot your actual data, as well as fits.
• If a linear model doesn't make sense, consider splines.
• First three bottom lines are always good reminders. Otherwise splines are a smart answer; personally I'd stress using a logarithmic link (or some equivalent) before I stressed using splines in guiding students. Nov 20, 2017 at 10:33
• Impressive, nicely illustrated exposition (+1).
– whuber
Nov 20, 2017 at 18:48

all it means is that if someone is 0 years they are "expected" to have a negative income . You should be careful about predicting outside the range of experimentation .

The line of least squares does not necessarily have to go through the origin ( X=0 and Y=0 ) ... Consider the two passengers on the plane before you take your next flight.....

http://autobox.com/dave/OUTSIDE.png

It is always possible to force the equation ti go through the origin BUT it can have serious consequences . https://online.stat.psu.edu/~ajw13/stat501/SpecialTopics/Reg_thru_origin.pdf argues (incorrectly in my opinion) that one should fit an equation that goes through the origin. The consequence of that constraint can be quite arlarming.

• It's certainly possible for people to have zero years of education. Not only that, the dataset might actually include such people.
– whuber
Nov 19, 2017 at 22:25
• very true ... i have corrected my reflection .... Nov 19, 2017 at 22:30