This is tricky as the unobserved effect (leg angle) is correlated with the independent variable (scuff height), but perhaps we can assume a structural model
h_scuff = x * height - cos(knee bend) * y * height
But x and y are related, Your knee is some fraction of your height, and also determines the amount the scuff height varies with angle
h_scuff = x * height - cos(knee bend) * x * height
cosine is probably unrealistically simple but probably fair for now. The key point is the produced scuff mark will be drawn from a distribution centered at knee height, and bounded below by the floor (presumably where your straight leg reaches) and above by 2*knee height. The logit-normal is a typical distribution used in these situations.
So maybe it can be structured
h_scuff = K * height + F * x * height = (1 + F) * x * height
F is the foot factor, a logit-normal random variable representing how high up it is comfortable to rest and K is the fraction of your height up to your knee. At this point I think one can start trying to get rid of the too many free parameters but I realize we still have a key degeneracy, each observation has multiple unknowns (height, knee angle). You were right to call for a Bayesian approach. Here's a stab at it:
height ~ N(μ_h, σ_h²) #height prior
A ~ N(0, σ²) #comfortable angle (who knows)
F = expit(A) # foot factor
K ~ N(0.27, something) # maybe truncated or logit-normal: small variance knee height as proportion of total
h_scuff = 2K · height · F
n1000 | 4 hours ago
This is tricky as the unobserved effect (leg angle) is correlated with the independent variable (scuff height), but perhaps we can assume a structural model
But x and y are related, Your knee is some fraction of your height, and also determines the amount the scuff height varies with angle
cosine is probably unrealistically simple but probably fair for now. The key point is the produced scuff mark will be drawn from a distribution centered at knee height, and bounded below by the floor (presumably where your straight leg reaches) and above by 2*knee height. The logit-normal is a typical distribution used in these situations.
So maybe it can be structured
F is the foot factor, a logit-normal random variable representing how high up it is comfortable to rest and K is the fraction of your height up to your knee. At this point I think one can start trying to get rid of the too many free parameters but I realize we still have a key degeneracy, each observation has multiple unknowns (height, knee angle). You were right to call for a Bayesian approach. Here's a stab at it:
cyplo | 11 hours ago
delightful !
brokebit | 8 hours ago
This made my day. Reminds me of looking at which buttons are worn on a keypad to isolate potential digits in the passcode.