Linear regression with specified slope

Question

I want to fit a linear regression line with a specified slope to a data set. I read this thread about doing the same with an explicit intercept.

0+ suppresses the fitting of the intercept; what is the corresponding trick for the slope? For example, to fit a line with slope 1.5, I tried the following

set.seed(6)
x <- runif(100, -3, 3) 
y <- 2 + x + rnorm(100) 

model1<-lm(y ~ x) 
plot(x,y)
abline(model1,col="red")
abline(coef(model1),1.5,col="dark green")

but second abline function just takes the intercept from model1 and slope 1.5. Whereas I would like the regression line to have slope 1.5, find the best fit to the data points, and then compute intercept from that regression line.

Ine way is to use the offset... `model2 <- lm(y ~ 1 + offset(1.5*x)) ; abline(coef(model2)[1], 1.5, col="red")` (where you still cant just pass the model object to abline) — user20650, Oct 23 '15 at 00:53

cocquemas · Answer 1 · 2015-10-23T01:42:26.077

11

To find the value of the intercept, you don't actually need a regression. Since Y = a + b * X + ϵ, then E[Y - b * X] = E[a] + E[ϵ], and by assumption E[a] = a and E[ϵ] = 0, where E[] is the expectation operator. Therefore, a = E[Y - b * X].

Translated into R, this means the intercept a is:

b1 <- 1.5
a <- mean(y - b1 * x)

This is inspired by the comments to this question.

edited Oct 23 '15 at 01:42

answered Oct 23 '15 at 01:13

cocquemas

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score 9 · Accepted Answer · answered Oct 23 '15 at 00:34

I suppose one approach would be to subtract 1.5*x from y and then fit y using only an intercept term:

mod2 <- lm(I(y-1.5*x)~1)
plot(x, y)
abline(mod2$coefficients, 1.5)

This represents the best linear fit with fixed slope 1.5. Of course, this fit is not very visually appealing because the simulated slope is 1 while the fixed slope is 1.5.

Linear regression with specified slope

2 Answers2