Hello Team,
I have couple of questions regarding error model representations.
Is dv ~ @. Normal(cp, cp * σ) same as dv ~ @.Normal(cp, cp*exp(σ))?
An exponential error model is represented as dv ~ @. LogNormal(log(cp), σ), how is it different from proportional error model under FOCE
Can expoennalso be represented as dv ~ @. Normal(cp, exp(σ)) in Pumas ?
I would apprciate if you could help me understand how these representations differ under FOCE approximation.
Thank you in advance.
Sorry, for the late response. Did you try ask.pumas.ai for this?
Thank you so much for your response, Vijay. Yes.
To summarize what it says
dv ~@.Normal(cp, cp*σ) and dv ~@.Normal(cp, cp*exp(σ)) are the same model but with different representations. The first is the proportional error model: the observation is Normal, centered at cp, with an SD that scales with cp, so it mimics multiplicative error but being Normal, it stays symmetric and can go negative, unlike the true exponential/LogNormal model.
dv ~ @.Normal(cp, cp*exp(σ)) only reparameterizes the CV through a log link so σ can range freely while the CV stays positive. The model is still proportional, just σ = log(CV) instead of the CV directly.
y ~ @.Normal(cp, exp(σ)) is additive on the natural scale, a constant-SD Normal, where exp only keeps the SD positive. The exponential error model is the multiplicative one, Y = cp·exp(ε), which yields strictly positive, right-skewed observations and is written in Pumas as dv ~ @. LogNormal(log(cp), σ).
This is my understanding. Would appreciate to get more inputs to get more clarity.