Oh, it seems like we’ve both made the same mistake 😊
If one SD results in a 10% increase, then I think the relevant effect size should be 110% and 2 SD be (110%)^2 rather than 120%, so that generally the logarithm of the effect would be linear with the test results increase (in SDs). Then, I think it makes more sense to approximate it as a geometric mean of these (all numbers > 0).
I’ve done the wrong calculation earlier, taking the geometric mean of the added percentage which doesn’t make sense, as you say. Correcting this, I got 16.4% increase.
[Note that for small enough effect size this would be very similar, as
I’ve started to draft a formal proof that under reasonable assumptions we would indeed get a linear relationship between the additive test results increase and the log of the effect on income, but accidently submitting too soon got me thinking that I’m spending too much time on this 🤓 If anyone is interested, I will continue with this proof
Oh, it seems like we’ve both made the same mistake 😊
If one SD results in a 10% increase, then I think the relevant effect size should be 110% and 2 SD be (110%)^2 rather than 120%, so that generally the logarithm of the effect would be linear with the test results increase (in SDs). Then, I think it makes more sense to approximate it as a geometric mean of these (all numbers > 0).
I’ve done the wrong calculation earlier, taking the geometric mean of the added percentage which doesn’t make sense, as you say. Correcting this, I got 16.4% increase.
[Note that for small enough effect size this would be very similar, as
∏(1+εi)λi=1+∑iλiεi+O(ε2).]
woops, submitted too early..I’ve started to draft a formal proof that under reasonable assumptions we would indeed get a linear relationship between the additive test results increase and the log of the effect on income, but accidently submitting too soon got me thinking that I’m spending too much time on this 🤓 If anyone is interested, I will continue with this proof