# How regression coefficients change when shifting the scale of the response variable

I am running a regression with y: a 7 point index ranging from -3 to 3, x: binary indicator (0,1) of second wave of data collection. When I fit this regression, I get the following equation:

y = 0.61 + 0.21x

To comment on the % of change between wave 1 (x = 0) and wave 2 (x = 1) of data collection, I divide 0.21/0.61 = 34%

However, when I shift the scale of y (by adding 3) to range from 0 to 5, I get the following equation:

y = 3.61 + 0.21x

See here, the coefficient of x did not change although the scale of y has shifted. Now the % of change in the average score across the 2 waves is way smaller 0.21/3.61 = 6%

What am I missing here? Why has the percentage of change shifted drastically (from 34% to just 6%? Is it normal that the coefficient of x didn't change although the scale of y has changed?

• Doesn’t this indicate to you that the ratio you’ve computed is meaningless? If you add a billion to $y$ then the percent change will be about a billionth. Compare instead to the width of the range. Jun 2 at 13:06
• You haven't changed the scale of $y:$ you have changed its origin (or, equivalently, center).
– whuber
Jun 2 at 13:43
• @AryaMcCarthy You mean dividing by the standard error? It could be expressed in this way but I was looking for a more intuitive way to express the change in % of y for a report I'm working on to a non specialized audience. This is why I attempted this ratio from the coefficients and got stumbled. Any thoughts?
– Rali
Jun 2 at 14:00
• No, I do mean dividing by the width of the range: 6, whether you’re going from -3 to 3 or 0 to 6. Though using linear regression instead of ordinal regression on Likert data is a broader discussion for another time. Jun 2 at 14:03