Visual Predictive Check (VPC)

Pumas allows you to generate VPC quantiles for your data and model simulations and utilize the Julia plotting capabilities to generate the relevant VPC plots. This is allowed with the vpc function.

VPC with Simulation

vpc(
  fpm::AbstractFittedPumasModel;
  samples::Integer = 499
  qreg_method = IP(),
  observations::Vector{Symbol} = [keys(fpm.data[1].observations)[1]],
  stratify_by::Vector{Symbol} = Symbol[],
  quantiles::NTuple{3,Float64}=(0.1, 0.5, 0.9),
  level::Real=0.95,
  ensemblealg=EnsembleThreads(),
  bandwidth=nothing,
  maxnumstrats=[6 for i in 1:length(stratify_by)],
  covariates::Vector{Symbol} = [:time],
  smooth::Bool = true,
  rng::AbstractRNG=default_rng(),
  nodes::Dict = Dict{Tuple{},Vector{Float64}}(),
  nnodes::Integer = 0,
  prediction_correction = false,
)

Computes the quantiles for VPC for a FittedPumasModel or FittedPumasEMModel with simulated confidence intervals around the empirical quantiles based on samples simulated populations to be input as a keyword with a default value of 499. The resulting object can then be passed to the vpc_plot function to generate the plots.

Hence, a vpc call with 100 repetitions of simulation would be:

res = fit(model, data, param, FOCE())
vpc_fpm = vpc(res; samples = 100)

The sample simulations are performed in parallel by default using multithreading (ensemblealg = EnsembleThreads()). Stratified resampling of the populations for the bootstrap fits based on the wt covariate is performed as follows:

vpc_fpm_stratwt = vpc(res; stratify_by = [:wt])
# run only 100 simulations
vpc_fpm_stratispm = vpc(res; samples = 100, stratify_by = [:isPM])

Prediction correction can be turned on by specifying prediction_correction = true

vpc_fpm = vpc(res; samples = 100, prediction_correction = true)

The VPC object obtained as the result contains the following fields:

• simulated_quantiles::DataFrame: DataFrame of the simulated quantiles result.
• popvpc::PopVPC: A PopVPC object with the observation quantiles (data_quantiles), data (data), stratification covariate if specified (stratify_by) and the dependent variable (dv).
• level::Float64: The simulation CI's level.

Since the quantiles are stored as DataFrames it is very easy to use the result of vpc and also export it using CSV package to your disk.

vpc_fpm.simulated_quantiles
vpc_fpm.popvpc.data_quantiles

using CSV
CSV.write("sim_quantiles.csv", vpc_fpm.simulated_quantiles)

In Pumas, for models with continuous derived variables (including prediction corrected VPCs), the non-parametric quantile regression approach discussed in Jamsen et al. is used. Discrete and Time to Event models are also supported through the vpc function and can be used without any change in syntax.

Full examples

Let's take a look at a complete example, first we define the model and generate round-trip data.

using Pumas
using PumasUtilities
using DataFrames
using LinearAlgebra

model = @model begin
    @param begin
        tvcl ∈ RealDomain(; lower = 0)
        tvv ∈ RealDomain(; lower = 0)
        pmoncl ∈ RealDomain(; lower = -0.99)
        Ω ∈ PDiagDomain(2)
        σ_prop ∈ RealDomain(; lower = 0)
    end

    @random begin
        η ~ MvNormal(Ω)
    end

    @covariates wt isPM

    @pre begin
        CL = tvcl * (1 + pmoncl * isPM) * (wt / 70)^0.75 * exp(η[1])
        Vc = tvv * (wt / 70) * exp(η[2])
    end

    @dynamics Central1

    @derived begin
        cp = @. 1_000 * (Central / Vc)
        dv ~ @. Normal(cp, sqrt(cp^2 * σ_prop))
    end
end

ev = DosageRegimen(100; time = 0, addl = 2, ii = 24)
s1 = Subject(; id = 1, events = ev, covariates = (isPM = 1, wt = 70))

param = (; tvcl = 4.0, tvv = 70, pmoncl = -0.7, Ω = Diagonal([0.09, 0.09]), σ_prop = 0.04)

choose_covariates() = (; isPM = rand([1, 0]), wt = rand(55:80))
pop_with_covariates =
    map(i -> Subject(; id = i, events = ev, covariates = choose_covariates()), 1:10)
obs = simobs(model, pop_with_covariates, param; obstimes = 0:1:60)
simdf = DataFrame(obs);
data = read_pumas(simdf; time = :time, covariates = [:isPM, :wt])

res = fit(model, data, param, FOCE())

FittedPumasModel

Successful minimization:                      true

Likelihood approximation:                     FOCE
Likelihood Optimizer:                         BFGS
Dynamical system type:                 Closed form

Log-likelihood value:                   -4391.7778
Number of subjects:                             10
Number of parameters:         Fixed      Optimized
                                  0              6
Observation records:         Active        Missing
    dv:                         610              0
    Total:                      610              0

---------------------
           Estimate
---------------------
tvcl        2.756
tvv        78.873
pmoncl     -0.64333
Ω₁,₁        0.19806
Ω₂,₂        0.10222
σ_prop      0.039245
---------------------

vpc_fpm = vpc(res);

vpc_plot(vpc_fpm)

vpc_plot(vpc_fpm; axis = (; limits = (0, 24, nothing, nothing)))

Alternatively, using a PumasEMModel:

emmodel = @emmodel begin
    @random begin
        CLdw ~ 1 + isPM | LogNormal # CLdw = μ_CL * exp(μ_CL_isPM * isPM) * exp(η_CL)
        Vcdw ~ 1 | LogNormal # Vcdw = μ_Vc * exp(η_Vc)
    end
    @covariates wt
    @pre begin
        CL = CLdw * (wt / 70)^0.75
        Vc = Vcdw * (wt / 70)
    end
    @dynamics Central1
    @post begin
        cp = 1000 * Central / Vc
    end
    @error begin
        dv ~ ProportionalNormal(cp)
    end
end

emparam = (;
    CLdw = (4.0, -1.2), # -1.2 ≈ log(1 - 0.7) = log(1 - pmoncl)
    Vcdw = 70,
    Ω = (0.09, 0.09),
    σ = (0.04,),
)

emres = fit(emmodel, data, emparam, LaplaceI())

vpc_emfpm = vpc(emres);

Visual Predictive Check
  Type of VPC: Continuous VPC
  Simulated populations: 499
  Subjects in data: 10
  Stratification variable(s): None
  Confidence level: 0.95
  VPC lines: quantiles ([0.1, 0.5, 0.9])

Note that because isPM equals either 0 or 1, pmoncl = exp(μ_CL_isPM) - 1, allowing us to perform a change of variables and express the earlier PumasModel as a PumasEMModel.

vpc_plot(vpc_emfpm)

Let's generate a VPC stratified on isPM covariate with stratify_by. Note that multiple covariates can be passed to stratify_by to get cross stratified VPC plots, in that case the maxnumstrats will have to be adjusted accordingly.

resem_saem = fit(emmodel, data, (; CLdw = (5.0, 0.0), Vcdw = 50.0), SAEM())

vpc_fpmem_saem = vpc(resem_saem; samples = 100);
vpc_plot(vpc_fpmem_saem)

vpc_fpm = vpc(res; stratify_by = [:isPM])
vpc_plot(vpc_fpm)

vpc_saem_fpm = vpc(resem_saem; stratify_by = [:isPM]);
vpc_plot(vpc_saem_fpm)

By specifying the plot and the path the VPC plot can be saved to the disk as below:

p = vpc_plot(vpc_fpm);
CairoMakie.save("obstimesvpc.png", p)

More examples of VPC plots can be found in the plotting page's vpc_plot section.

Prediction corrected VPCs (pcVPCs) can be done with prediction_correction = true:

vpc_fpm = vpc(res; samples = 100, prediction_correction = true);
vpc_plot(vpc_fpm)

VPC after Simulation

It is also possible to run the VPC as a post-processing step on arbitrary simulation output sims from any simobs function call. For more on simulations and how to use simobs, check the Simulation of Pumas Models section of the documentation.

Example

The following is an example of simulating from a bootstrap result and then plotting a VPC using the simulations.

model = @model begin
    @param begin
        tvcl ∈ RealDomain(; lower = 0)
        tvv ∈ RealDomain(; lower = 0)
        pmoncl ∈ RealDomain(; lower = -0.99)
        Ω ∈ PDiagDomain(2)
        σ_prop ∈ RealDomain(; lower = 0)
    end

    @random begin
        η ~ MvNormal(Ω)
    end

    @covariates wt isPM

    @pre begin
        CL = tvcl * (1 + pmoncl * isPM) * (wt / 70)^0.75 * exp(η[1])
        Vc = tvv * (wt / 70) * exp(η[2])
    end

    @dynamics Central1

    @derived begin
        cp = @. 1000 * (Central / Vc)
        dv ~ @. Normal(cp, sqrt(cp^2 * σ_prop))
    end
end

ev = DosageRegimen(100; time = 0, addl = 2, ii = 24)
s1 = Subject(; id = 1, events = ev, covariates = (isPM = 1, wt = 70))

param = (; tvcl = 4.0, tvv = 70, pmoncl = -0.7, Ω = Diagonal([0.09, 0.09]), σ_prop = 0.04)

choose_covariates() = (; isPM = rand([1, 0]), wt = rand(55:80))
pop_with_covariates =
    map(i -> Subject(; id = i, events = ev, covariates = choose_covariates()), 1:10)
obs = simobs(model, pop_with_covariates, param; obstimes = 0:1:60);
simdf = DataFrame(obs);
data = read_pumas(simdf; time = :time, covariates = [:isPM, :wt]);

res = fit(model, data, param, FOCE())
inf = infer(res, Bootstrap(; samples = 100))
sims = simobs(inf; samples = 50)
vpc_sims = vpc(sims);
vpc_plot(vpc_sims)