I think rescaling could make it steeper or flatter, depending on the particular rescaling. Consider that there is nothing that requires the rescaling to be a linear transformation of the original scale (like you’ve written in your example). A rescaling that compresses the life satisfaction scores that were initially 0-5 into the range 0-3, while leaving the life satisfaction score of 8-10 unaffected will have a different effect on the slope than if we disproportionately compress the top end of life satisfaction scores.
Sorry if I expressed this poorly—it’s quite late :)
Hi Zachary, yeah, see the other comment I just wrote. I think stretching could plausibly magnify or attenuate the relationship, whilst shifting likely wouldn’t.
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.
I think rescaling could make it steeper or flatter, depending on the particular rescaling. Consider that there is nothing that requires the rescaling to be a linear transformation of the original scale (like you’ve written in your example). A rescaling that compresses the life satisfaction scores that were initially 0-5 into the range 0-3, while leaving the life satisfaction score of 8-10 unaffected will have a different effect on the slope than if we disproportionately compress the top end of life satisfaction scores.
Sorry if I expressed this poorly—it’s quite late :)
Hi Zachary, yeah, see the other comment I just wrote. I think stretching could plausibly magnify or attenuate the relationship, whilst shifting likely wouldn’t.
While I agree in principle, I think the evidence is that the happiness scale doesn’t compress at one end. There’s a bunch of evidence that people use happiness scales linearly. I refer to Michael Plant’s report (pp20-22 ish): https://​​wellbeing.hmc.ox.ac.uk/​​wp-content/​​uploads/​​2024/​​02/​​2401-WP-A-Happy-Probability-DOI.pdf
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