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

[rahma ali]

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?

[Richard Williams]

Talking about percent change doesn’t make much sense unless Zero really means Zero for a variable. So, for example, it is legitimate to say that six feet is 20% more than five feet. But, it is not legitimate to say that 60 degrees Centigrade is 20% more than 50 degrees Centigrade, because zero doesn’t mean the absence of heat; somebody just decided that 0 would equal the temperature at which water freezes and 100 = Temperature at which water boils. Those may be logical choices but still arbitrary. In Fahrenheit the corresponding temperatures are 32 and 212.

Also the coefficient for x should not change if all you do is add or subtract a constant to y. If you graph it the graph will be the same except everything is shifted three units to the left.