Clarification on combined residual error

1

Hi … i have a basic doubt here, seeking some clarification.

I learned to use combined error in NONMEM in perticular way, which is
F(1+EPS1) + EPS2
I am trying to understand how pumas translate this
Pumas additive error :

Normal(μ, σ)

which i understand as : F + EPS_add

Pumas proportinal error

Normal(μ, μ*σ)

which i understand as : F * EPS_prop

I tried to write combined error from these two informations , which will be equvivalnet to : F(1+EPS1) + EPS2

my understanding of code is

Normal(μ, (σ_add + sqrt((μ*σ_prop)^2)))
or
Normal(μ, (sqrt(σ_add^2) + sqrt((μ*σ_prop)^2)))

or if i want variance for aditive eror
Normal(μ, ((σ_add^2) + sqrt((μ*σ_prop)^2)))

but it is given as

Normal(μ, sqrt(σ_add^2 + (μ*σ_prop)^2))

https://docs.pumas.ai/stable/model_components/error_models/

Can someone help me to understand this

Thanyou

2

It is related to how you derive the distribution of a sum of 2 independent normally distributed random variables (see e.g. https://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables). Basically, the variance of the sum is the sum of the variances. If σ_add and μ*σ_prop are the standard deviations for the additive and proportional error components respectively, then σ_add^2 and (μ*σ_prop)^2 are the respective variances. Take the square root of the sum to get the standard deviation for the combined error.

2 Likes
3

Thanks for answering. i have one more doubt here
if i want SD, formula will be

σ_add +  μ*σ_prop

similarly for variance , it should be

σ_add^2  +   (μ*(σ_prop)^2)

why should i square the μ as well?

Thanks

4

Please see How to define error - #7 by andreasnoack which explains the relationship between the formuations.