2. Scale shifting should always lead to attenuation (if the underlying relationship is negative and convex, as stated in the piece)
Your linear probability function doesn’t satisfy convexity. But, this seems more realistic, given the plots from Oswald/Kaiser look less than-linear, and probabilities are bounded (whilst happiness is not).
Again consider:
P(h)=1/h
T=1: LS = h ⇒ P(h) =1/LS
T=2: LS = h-5 ⇔ h = LS+5 ⇒ P(h) = 1/(LS+5)
Overall, I think the fact that the relationship stays the same is some weak evidence against shifting – not stretching. FWIW, in the quality-of-life literature, shifting occurs but little stretching.
Thanks for this example, Geoffrey. Hm, that’s interesting! This has gotten a bit more complicated than I thought.
It seems:
Surprisingly, scale stretching could lead to attenuation or magnification depending on the underlying relationship (which is unobserved)
Let h be latent happiness; let LS be reported happiness.
Your example:
P(h)=0.40−h/100
t=1,h≡LS=>dP/dLS=−1/100
t=2,h=2LS=>dP/dh∗dh/dLS=2∗−1/100=−1/50
So yes, the gradient gets steeper.
Consider another function. (This is also decreasing in h)
P(h)=1/h
t=1,h=LS=>dP/dLS=dh/dLS=dP/dh∗dh/dLS=−1/h2∗1=−1/(LS2)
t=2,h=2LS=>dp/dLS=dP/dh∗dh/dLS=−1/h2∗2=−2/(4LS2)=−1/(2LS2)
i.e., the gradient gets flatter.
2. Scale shifting should always lead to attenuation (if the underlying relationship is negative and convex, as stated in the piece)
Your linear probability function doesn’t satisfy convexity. But, this seems more realistic, given the plots from Oswald/Kaiser look less than-linear, and probabilities are bounded (whilst happiness is not).
Again consider:
P(h)=1/h
T=1: LS = h ⇒ P(h) =1/LS
T=2: LS = h-5 ⇔ h = LS+5 ⇒ P(h) = 1/(LS+5)
Overall, I think the fact that the relationship stays the same is some weak evidence against shifting – not stretching. FWIW, in the quality-of-life literature, shifting occurs but little stretching.
Interesting! I think my intuition going into this has always been stretching so that’s something I could rethink