Deciding Diminishing Returns based on Data (Regression)

[WWGD]

Hi All,
I am thinking of the issue of diminishing returns re linear regression. Can it be determined/decided from the
data itself, or is it decided just from the context? I was thinking of examples like that of grade vs daily study hours or (height )jump length vs year ( winner heights have been increasing.) In the 1st case, say the slope is 0.5 , constant is 23 ,so that every hour studied adds (along the regression line) a half point to the grade . It seems clear that studying 18 hours n a day would not add 9 points, i.e., we hit a diminishing returns at some point. Still, can this diminishing return be deduced from the data itself, or just from common sense/context?
Thanks.

 

[pbuk]

Before you calculate anything (using linear regression or otherwise), always plot your data. If it looks like a straight line across the range, go ahead and fit a straight line. But if the slope reduces as the independent variable increases, you have what you describe as "diminishing returns" which is an example of a non-linear relationship - so don't try and fit a straight line.

 

[RUber]

Like MrAnchovy said, a linear regression will never show diminishing returns.
If you can get data which samples past the point of diminishing returns, you should be able to see it as a "knee" in the curve.

Also, you need some sort of cost associated with what you are putting in. For your example with study hours, the "cost" might be shown by the relationship between sleep and test scores. As soon as sleep is sacrificed to study, the returns would be diminished. But if the sleep cost is non-linear: e.g.
Test score = 40 - .25 * (lost sleep hours)^2, then you could probably still show the value in losing some sleep to studying.

If you tell me I have infinite money and infinite time, there would be no reason to stop putting money and time into something even if the expected return for each additional million dollars was 1/10 the return on the previous million.

If you are stuck with linear regression as your only tool in your tool kit, you can show diminishing returns by dividing the data into segments and running the regression on just the subset of the data. Looking at the slope along each segment, if the slope is decreasing, the you have evidence of the non-linear relationship and the diminishing returns.

Another method would be to transform your data: Y = y^2, Y = y^{1/2}, Y = log(y), etc. If any of these fit the straight line regression, you might be able to make inferences based on the functional relationship to the linear model.