Model a mix of correlated and uncorrelated random effects

How-to
1

Hello Team,

I was trying to replicate an interesting scenario where we are required to model correlation
between CL and Vc, but there is no correlation with Ka. Based on the known truth, this
was the intended covariance structure:

          V        CL       Ka
 V  [ 0.1844   0.0369    0      ]
 CL [ 0.0369   0.2089    0      ]
 Ka [ 0         0         0.1844 ]

## How the other two tools write it

NONMEM — two $OMEGA records, one per block:

$OMEGA BLOCK(2)
0.1844   ; IIV_V
0.0368   ; IIV_CL_IIV_V
0.2089   ; IIV_CL
$OMEGA 0.1844 ; IIV_KA

Monolix :

V  = {distribution=logNormal, typical=V_pop,  sd=omega_V}
Cl = {distribution=logNormal, typical=Cl_pop, sd=omega_Cl}
ka = {distribution=logNormal, typical=ka_pop, sd=omega_ka}
correlation = {level=id, r(V, Cl)=corr_V_Cl}

We name only the pair that correlates. Anything unnamed is uncorrelated.

## What I have working in Pumas

Two @param entries and two @random draws:

@param begin
    tvvc   ∈ RealDomain(; lower = 0.0, init = 628.0)
    tvcl   ∈ RealDomain(; lower = 0.0, init = 19.2)
    tvka   ∈ RealDomain(; lower = 0.0, init = 0.75)
    Ωvc_cl ∈ PSDDomain(; init = [0.1844 0.0369; 0.0369 0.2089])
    Ωka    ∈ PDiagDomain(; init = [0.1844])
    σ      ∈ RealDomain(; lower = 0.0, init = 0.2)
end

@random begin
    ηvc_cl ~ MvNormal(Ωvc_cl)
    ηka    ~ MvNormal(Ωka)
end

@pre begin
    Vc = tvvc * exp(ηvc_cl[1])
    CL = tvcl * exp(ηvc_cl[2])
    Ka = tvka * exp(ηka[1])
end

My question is whether it is the intended way of coding it.

Thanks in advance!

2

Yes, that is the way. However, a single scalar omega could be written just as a Normal. See below

@param begin
    tvvc   ∈ RealDomain(; lower = 0.0, init = 628.0)
    tvcl   ∈ RealDomain(; lower = 0.0, init = 19.2)
    tvka   ∈ RealDomain(; lower = 0.0, init = 0.75)
    Ωvc_cl ∈ PSDDomain(; init = [0.1844 0.0369; 0.0369 0.2089])
    omegaka    ∈ RealDomain(; init = sqrt(0.1844))
    σ      ∈ RealDomain(; lower = 0.0, init = 0.2)
end

@random begin
    ηvc_cl ~ MvNormal(Ωvc_cl)
    ηka    ~ Normal(omegaka)
end

This section in the Pumas Tutorials gives you an example - Between Subject Variability

Best,

Vijay

1 Like
3

Thanks for the clarification!