Pumas Docstrings

DataFrame(
  pop::Population;
  include_covariates = true,
  include_observations = true,
  include_events = true,
)

Create a DataFrame with the Population information. The data follows Pumas Data Format ready to be imported with read_pumas.

Examples:

julia> events = DosageRegimen(1.0; time = 1.0)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     1.0      1      1.0     1      0.0      0      0.0       0.0     0  NullRoute

julia> pop = [Subject(; id = 1, events), Subject(; id = 2, events)]
Population
  Subjects: 2
  Observations:

julia> DataFrame(pop)
2×12 DataFrame
 Row │ id      time     amt      evid  cmt    rate     duration  ss    ii       route      tad      dosenum
     │ String  Float64  Float64  Int8  Int64  Float64  Float64   Int8  Float64  NCA.Route  Float64  Int64
─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────
   1 │ 1           1.0      1.0     1      1      0.0       0.0     0      0.0  NullRoute      0.0        1
   2 │ 2           1.0      1.0     1      1      0.0       0.0     0      0.0  NullRoute      0.0        1

DataFrame(dr::DosageRegimen)

Create a DataFrame with the information in the dosage regimen. The data follows Pumas Data Format ready to be imported with read_pumas.

Examples:

julia> DataFrame(DosageRegimen(100))
1×10 DataFrame
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1      0.0      0      0.0       0.0     0  NullRoute

julia> DataFrame(DosageRegimen(100; ii = 24, addl = 2))
1×10 DataFrame
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1     24.0      2      0.0       0.0     0  NullRoute

DataFrame(subject::AbstractSubject)

Create a DataFrame with the Subject information. The data follows Pumas Data Format ready to be imported with read_pumas.

Examples:

julia> DataFrame(Subject(; events = DosageRegimen(1.0; time = 1.0)))
1×12 DataFrame
 Row │ id      time     amt      evid  cmt    rate     duration  ss    ii       route      tad      dosenum
     │ String  Float64  Float64  Int8  Int64  Float64  Float64   Int8  Float64  NCA.Route  Float64  Int64
─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────
   1 │ 1           1.0      1.0     1      1      0.0       0.0     0      0.0  NullRoute      0.0        1

julia> DataFrame(
         Subject(;
           id = "AKJ491",
           events = DosageRegimen(1.0; time = 1.0),
           observations = (; glucose = [8.31, 7.709, 5.19]),
           time = [0.3, 0.9, 3.0],
           covariates = (; bloodpressure = [(143, 95.01), (141.3, 94.8), (130.4, 85.1)]),
           covariates_time = [0.0, 0.5, 5.0],
         ),
       )
7×14 DataFrame
 Row │ id      time     evid  glucose      amt        cmt      rate       duration   ss       ii         route       b ⋯
     │ String  Float64  Int8  Float64?     Float64?   Int64?   Float64?   Float64?   Int8?    Float64?   NCA.Route?  T ⋯
─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────────────
   1 │ AKJ491      0.0     2  missing      missing    missing  missing    missing    missing  missing    missing     ( ⋯
   2 │ AKJ491      0.3     0        8.31         0.0  missing        0.0        0.0        0        0.0  NullRoute   (
   3 │ AKJ491      0.5     2  missing      missing    missing  missing    missing    missing  missing    missing     (
   4 │ AKJ491      0.9     0        7.709        0.0  missing        0.0        0.0        0        0.0  NullRoute   (
   5 │ AKJ491      1.0     1  missing            1.0        1        0.0        0.0        0        0.0  NullRoute   ( ⋯
   6 │ AKJ491      3.0     0        5.19         0.0  missing        0.0        0.0        0        0.0  NullRoute   (
   7 │ AKJ491      5.0     2  missing      missing    missing  missing    missing    missing  missing    missing     (
                                                                                                       3 columns omitted

DataFrame(
  fpmi::FittedPumasModelInspection;
  include_covariates = true,
  include_observations = true,
  include_events = true,
  include_icoef = true,
  include_dosecontrol = true,
  include_ebes = true,
  include_dynamics = true,
)

Create a DataFrame with the information in an inspect result fpmi. The data follows Pumas Data Format ready to be imported with read_pumas.

DataFrame(
  obs::SimulatedObservations;
  include_events = true,
  include_covariates = true,
)

Create a DataFrame with the information in a simobs result obs. The data follows Pumas Data Format ready to be imported with read_pumas.

DataFrame(pred::SubjectPrediction)

Create a DataFrame with the information in the subject's prediction pred. The data follows Pumas Data Format ready to be imported with read_pumas.

DataFrame(bts::Union{SimulatedInference, Bootstraps})

Returns a data frame of the parameter estimates.

DataFrame(vpred::Vector{<:SubjectPrediction})

Create a DataFrame with the information in a population prediction vpred. The data follows Pumas Data Format ready to be imported with read_pumas.

DataFrame(vresid::Vector{<:SubjectResidual})

Create a DataFrame with the information in a population weighted residuals vresid. The data follows Pumas Data Format ready to be imported with read_pumas.

MvNormal(pmi::FittedPumasModelInference)

Returns an MvNormal with a mean equal to the optimal parameter values in the fitted model and a covariance equal to the variance-covariance estimate.

MvNormal(model::PumasModel, means::NamedTuple, vcov::NamedTuple)

Returns an MvNormal with mean equal to the vectorized means argument and a covariance equal to the block diagonal matrix of the covariance matrices in the vcov argument. The mean argument should be a named tuple of values in their natural domain. vcov should be a named tuple of numbers (for real parameters) or matrices for vector and matrix parameters. For parameters that are themselves symmetric matrices, e.g:

Ω = [
  Ω₁₁ Ω₁₂ Ω₁₃
  Ω₂₁ Ω₂₂ Ω₂₃
  Ω₃₁ Ω₃₂ Ω₃₃
]

the covariance matrix should be the covariance of the vector: [Ω₁₁, Ω₁₂, Ω₂₂, Ω₁₃, Ω₂₃, Ω₃₃]. Note that because the matrix is symmetric, only the upper triangle is used iterating over the columns first in an outer loop then the over the rows of each column in an inner loop.

Example:

means = (θ1 = 1.0, θ2 = [1.0, 2.0], θ3 = [1.0 0.1; 0.1 1.0])
vcov =
  (θ1 = 0.1, θ2 = [0.5 -0.2; -0.2 0.6], θ3 = [0.5 -0.2 0.1; -0.2 0.6 -0.3; 0.1 -0.3 0.45])
dist = MvNormal(model, means, vcov)

Chains(b::BayesMCMCResults; subject::Union{Nothing, Int} = nothing)

Constructs an MCMCChains.Chains object for the samples of the population or subject-specific parameters. If subject is nothing (the default value) a Chains object of the population parameters' samples is returned. If subject is an integer, the samples of the subject-specific parameters corresponding to the subject with the given index will be returned instead.

NCASubject(subj; group, observations, [kwargs...])

Return an NCASubject constructed from a Subject. The main keyword arguments are

• group, a vector of symbols for covariates to use as grouping variables for the NCA calculations.
• observations, refers to the variable to calculate NCA parameters for.

For other keywords arguments to pass to the NCASubject constructor, see the docstring for the NCASubject constructor that does not take in a Subject but rather builds the NCASubject from time series directly.

AnalyticalPKPDProblem(
  f,
  u0,
  tspan,
  events,
  time,
  p,
  bioav)

An analytical PK(PD) problem.

Fields:

• f: callable that returns the state of the dynamic system
• u0: initial conditions
• tspan: time points that define the intervals in which the problem is defined
• events: events such as dose, reset events, etc
• time: event times
• p: a function that returns pre block evaluated at a time point

AnalyticalPKProblem(pkprob, prob2, sys2)

A problem that is partially an analytical problem that can be evaluated independently of the rest of the system.

Fields:

• pkprob: the analytical part of the problem
• prob2: a problem that represents the rest of the system
• sys2 (optional): the system of prob2 - used for latexification.

BayesMCMC

An instance of the Hamiltonian Monte Carlo (HMC) based No-U-Turn Sampler (NUTS) algorithm. All the options of the algorithm can be passed to the constructor of BayesMCMC as keyword arguments. Example:

alg = BayesMCMC(; nsamples = 2000, nadapts = 1000)

The main options that can be set are:

• target_accept: (default is 0.8) target acceptance ratio for the NUTS algorithm.
• nsamples: (default is 2_000) number of Markov Chain Monte Carlo (MCMC) samples to generate including adaptation samples.
• nadapts: (default is nsamples ÷ 2) number of adaptation steps in the NUTS algorithm.
• nchains: (default is 4) number of MCMC chains to sample.
• ess_per_chain: (default is ∞ ) target effective sample size (ESS) per chain, sampling terminates if the target is reached.
• check_every: (default is ess_per_chain ÷ 5) the number of samples after which the ESS per chain is checked.
• time_limit: (default is ∞) a time limit for sampling in seconds, sampling terminates if the time limit is reached.
• ensemblealg: (default is EnsembleThreads()) can be set to EnsembleSerial() for serial sampling, EnsembleThreads() for multi-threaded sampling, EnsembleDistributed() for multi-processing (aka distributed parallelism) sampling, or EnsembleSplitThreads() for multi-processing over chains, and multi-threading over subjects.
• parallel_chains: (default is true if enough threads/workers exist) can be set to true or false, if set to true the chains will be sampled in parallel using either multi-threading or multi-processing depending on the value of ensemblealg.
• parallel_subjects: (default is true if enough threads/workers exist) can be set to true or false, if set to true the log probability computation will be parallelized over the subjects using either multi-threading or multi-processing depending on the value of ensemblealg.
• rng: the random number generator used.
• diffeq_options: (default is an empty NamedTuple (;), which uses the downstream defaults) a NamedTuple of all the differential equations solver's options.
• constantcoef: (default is an empty NamedTuple (;)) a NamedTuple of the parameters to be fixed during sampling. This can be used to sample from conditional posteriors fixing some parameters to specific values, e.g. constantcoef = (σ = 0.1,) fixes the σ parameter to 0.1 and samples from the posterior of the remaining parameters conditional on σ.
• check_initial: (default is true) a Bool to check if the model's initial parameter values have a valid log-likelihood before begin fitting.

Additionally, there are options to fine-tune the NUTS sampler behavior:

• max_chunk_size: (default is 8) Forward-mode autodiff (currently the only autodiff method available) maximal chunks to use as chunk size. Forward-mode autodiff performs partial derivative evaluation on one "chunk" of the input vector at a time. Each differentiation of a chunk requires a call to the target function as well as additional memory proportional to the square of the chunk's size. Thus, a smaller chunk size makes better use of memory bandwidth at the cost of more calls to the target function, while a larger chunk size reduces calls to the target function at the cost of more memory bandwidth.
• adapt_init_buffer: (default is 0) width of initial fast adaptation interval for the NUTS mass matrix adaptation.
• adapt_term_buffer: (default is the maximum value between nadapts ÷ 20 and 50) width of final fast adaptation interval for the NUTS mass matrix adaptation.
• adapt_window_size: (default is 5) initial width of slow adaptation interval for the NUTS mass matrix adaptation.

For the NUTS mass matrix adaptation, if you want the same behavior as Stan, set the following values:

BayesMCMC(; ..., adapt_init_buffer = 75, adapt_term_buffer = 50, adapt_window_size = 25)

Bootstrap(; rng=Random.default_rng, samples=200, stratify_by=nothing, ensemblealg=EnsembleThreads())

Constructs a Bootstrap object that allows the user to use bootstrapping with the supplies setting in infer function calls. See ?infer and https://docs.pumas.ai/stable/analysis/inference/ for more information about obtaining inference results.

The keyword arguments are as follows:

• rng: used to select a pseudo-random number generator to beused for the random resamples of the datasets.
• samples: controls the number of resampled Populations.
• resamples: the number of model parameter samples from the original sample distribution to re-sample (by weighed re-sampling without replacement). The number of resamples must be less than that of the original sample. An original sample-to-resample ratio of at least 5:1 is suggested.
• stratify_by control the stratification used in the re-sampling scheme. Should be a set to a Symbol with the name of a covariate it is possible to stratify by a covariate with a finite number of possible values (e.g :study for a covariate named study).
• ensemblealg controls the parallelization across fit calls. Each fit is fitted using the ensemblealg that was used in the original fit.

ByObservation(; allsubjects = true)

Using this method, each observation or collection of observations is treated as a single data point. When computing predictive loglikelihoods using this method, the predictive loglikelihood computed is the conditional loglikelihood of one or more observations for one or more subjects.

This method has one option, allsubjects. If allsubjects is set to true (the default value), the ith observation of each subject are all grouped together into a single data point. If allsubjects is set to false, then each observation per subject is its individual data point.

Examples:

Assume there are 2 subjects and 3 observations per subject. When using LeaveK(K = 1) as the splitting method together with ByObservation(allsubjects = false), the training and validation splittings are:

On the other hand if allsubjects is set to true, the training and validation splittings are:

BySubject(; marginal = LaplaceI())

Using this method, each subject is treated as a single data point. The predictive loglikelihood computed for each subject can be either the marginal loglikelihood or conditional loglikelihood.

This method has one option, marginal. If marginal is set to nothing, the predictive loglikelihood computed for each subject is the conditional loglikelihood using 0s for the subject-specific parameters. Otherwise, the predictive loglikelihood computed for each subject is the marginal loglikelihood using marginal as the marginalization algorithm.

The default value of marginal is LaplaceI() which uses the Laplace method to integrate out the subject-specific parameters. Other options include: FOCE(), FO() and Pumas.LLQuad(). Pumas.LLQuad() is a quadrature integration algorithm. To learn more about the options of LLQuad, you can type ?LLQuad in the REPL.

Central1()

An analytical model for a one compartment model with dosing into Central. Equivalent to

Central' = -CL/Vc*Central

where clearance, CL, and volume, Vc, are required to be defined in the @pre block.

Central1Meta1()

An analytical model for a model with a central compartment, Central, and a metabolite, Metabolite. Equivalent to Central' = -(CL+CLfm)/VcCentral Metabolite' = CLfm/VcCentral - CLm/Vm*Metabolite where clearances (CL and CLm) and volumes (Vc and Vm) and formation clearance of metabolite CLfm are required to be defined in the @pre block.

Central1Periph1()

An analytical model for a two-compartment model with a central compartment, Central and a peripheral compartment, Peripheral. Equivalent to

Central'    = -(CL+Q)/Vc*Central + Q/Vp*Peripheral
Peripheral' =       Q/Vc*Central - Q/Vp*Peripheral

where clearance, CL, and volumes, Vc and Vp, and distribution clearance, Q, are required to be defined in the @pre block.

Central1Periph1Meta1()

An analytical model for a two compartment model with a central compartment, Central, with a peripheral compartment, Peripheral, and a metabolite compartment, Metabolite. Equivalent to

Central'     = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
CPeripheral' =            Q/Vc*Central - Q/Vp*CPeripheral
Metabolite'  = -CLm/Vm*Metabolite + CLfm/Vc*Central

where clearances (CL and CLm) and volumes (Vc, Vp and Vm), distribution clearance (Q), and formation clearance of metabolite CLfm are required to be defined in the @pre block.

Central1Periph1Meta1Periph1()

An analytical model for a two compartment model with a central compartment, Central, with a peripheral compartment, Peripheral, and a metabolite compartment, Metabolite, with a peripheral compartment, MPeripheral. Equivalent to

Central'     = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
CPeripheral' =            Q/Vc*Central - Q/Vp*CPeripheral
Metabolite'  = -(CLm+Qm)/Vm*Metabolite + Qm/Vmp*MPeripheral + CLfm/Vc*Central
MPeripheral' =        Qm/Vm*Metabolite - Qm/Vmp*MPeripheral

where clearances (CL and CLm) and volumes (Vc, Vp, Vm and Vmp), distribution clearances (Q and Qm) and formation clearance of metabolite CLfm are required to be defined in the @pre block.

@param x = val
@param x ∈ ConstDomain(val)

Specifies a parameter as a constant.

Constrained

Constrain a Distribution within a Domain. The most common case is an MvNormal constrained within a VectorDomain. The only supported method for Constrained is logpdf. Notice that the result does not represent a probability distribution since the remaining probability mass is not scaled by the mass excluded by the constraints.

Example

julia> d = Constrained(MvNormal(fill(1.0, 1, 1)), lower = -1, upper = 1)
Constrained{ZeroMeanFullNormal{Tuple{Base.OneTo{Int64}}}, VectorDomain{Vector{Int64}, Vector{Int64}, Vector{Float64}}}(ZeroMeanFullNormal(
dim: 1
μ: Zeros(1)
Σ: [1.0;;]
)
, VectorDomain{Vector{Int64}, Vector{Int64}, Vector{Float64}}([-1], [1], [0.0]))

julia> logpdf(d, [0])
-0.9189385332046728

julia> logpdf(d, [-2])
-Inf

CorrDomain(template)

Return a correlation domain.

template provides both the shape and initial values for the resulting correlation parameter.

• if template is an Int then an identity matrix of size template is returned.
• if template is a square Matrix then the CorrDomain will match its size and will have initial values according to the template elements.

Depots1Central1()

An analytical model for a one compartment model with a central compartment, Central, and a depot, Depot. Equivalent to

Depot'   = -Ka*Depot
Central' =  Ka*Depot - CL/Vc*Central

where absoption rate, Ka, clearance, CL, and volume, Vc, are required to be defined in the @pre block.

Depots1Central1Periph1()

An analytical model for a two-compartment model with a central compartment, Central, a peripheral compartment, Peripheral, and a depot Depot. Equivalent to

Depot'      = -Ka*Depot
Central'    =  Ka*Depot - (CL+Q)/Vc*Central + Q/Vp*Peripheral
Peripheral' =                 Q/Vc*Central - Q/Vp*Peripheral

where absorption rate, Ka, clearance, CL, and volumes, Vc and Vp, and distribution clearance, Q, are required to be defined in the @pre block.

Depots2Central1()

An analytical model for a one compartment model with a central compartment, Central, and two depots, Depot1 and Depot2. Equivalent to

Depot1'  = -Ka1*Depot1
Depot2'  = -Ka2*Depot2
Central' =  Ka1*Depot1 + Ka2*Depot2 - CL/Vc*Central

where absorption rates, Ka1 and Ka2, clearance, CL, and volume, Vc, are required to be defined in the @pre block.

When using this model during simulation or estimation, it is preferred to have 2 dosing rows for each subject in the dataset, where the first dose goes into cmt =1 (or cmt = Depot1) and the second dose goes into cmt=2 (or cmt=Depot2). Central compartment gets cmt=3 or (cmt = Central). e.g.

ev = DosageRegimen([100,100],cmt=[1,2]) s1 = Subject(id=1, events=ev)

DiffEqWrapper

Modifies the diffeq system f to add allow for modifying rates

• f: original diffeq system
• params: ODE parameters
• rates_on: should rates be modified?
• rates: amount to be added to rates

DosageRegimen(
  amt::Numeric;
  time::Numeric = 0,
  cmt::Union{Numeric,Symbol} = 1,
  evid::Numeric = 1,
  ii::Numeric = zero.(time),
  addl::Numeric = 0,
  rate::Numeric = zero.(amt)./oneunit.(time),
  duration::Numeric = zero(amt)./oneunit.(time),
  ss::Numeric = 0,
  route::NCA.Route = NCA.NullRoute,
)

Lazy representation of a series of events.

Examples

julia> DosageRegimen(100; ii = 24, addl = 6)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1     24.0      6      0.0       0.0     0  NullRoute

julia> DosageRegimen(50; ii = 12, addl = 13)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1     50.0     1     12.0     13      0.0       0.0     0  NullRoute

julia> DosageRegimen(200; ii = 24, addl = 2)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    200.0     1     24.0      2      0.0       0.0     0  NullRoute

julia> DosageRegimen(200; ii = 24, addl = 2, route = NCA.IVBolus)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    200.0     1     24.0      2      0.0       0.0     0  IVBolus

You can also create a new DosageRegimen from various existing DosageRegimens:

evs = DosageRegimen(
  regimen1::DosageRegimen,
  regimen2::DosageRegimen;
  offset = nothing
)

offset specifies if regimen2 should start after an offset following the end of the last event in regimen1.

Examples

julia> e1 = DosageRegimen(100; ii = 24, addl = 6)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1     24.0      6      0.0       0.0     0  NullRoute

julia> e2 = DosageRegimen(50; ii = 12, addl = 13)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1     50.0     1     12.0     13      0.0       0.0     0  NullRoute

julia> evs = DosageRegimen(e1, e2)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1     24.0      6      0.0       0.0     0  NullRoute
   2 │     0.0      1     50.0     1     12.0     13      0.0       0.0     0  NullRoute

julia> DosageRegimen(e1, e2; offset = 10)
DosageRegimen
 Row │ time     cmt    amt      evid  ii       addl   rate     duration  ss    route
     │ Float64  Int64  Float64  Int8  Float64  Int64  Float64  Float64   Int8  NCA.Route
─────┼───────────────────────────────────────────────────────────────────────────────────
   1 │     0.0      1    100.0     1     24.0      6      0.0       0.0     0  NullRoute
   2 │   178.0      1     50.0     1     12.0     13      0.0       0.0     0  NullRoute

ExactCrossvalidation(; split_method, split_by, ensemblealg)

Defines an instance of the ExactCrossvalidation algorithm for crossvalidation. In this algorithm, the fitting is re-run while leaving out a subset of the data each time.

The way by which the data is split between training and validation sets is determined using the keyword arguments split_method and split_by. The split_method argument can be of any of the following types:

• LeaveK for leave-K-out cross-validation
• KFold for K-fold cross-validation
• LeaveFutureK for leaving K future points at a time

while split_by can be of any of the following types:

• BySubject for leaving out subjects
• ByObservation for leaving out individual observations per subject

Please refer to the documentation of each of the above types for the parameters of each split method, e.g. typing ?LeaveK or ?BySubject.

Examples:

Assume there are 5 subjects and 10 observations per subject and that ft is the result of the fit function. The following are some of the combinations in which the above inputs can be used:

• Leave-1-observation-out cross-validation, leaving 1 observation for all the subjects at a time. allsubjects = true means that the same observation index is removed for all the subjects, e.g. the 10th observation for all the subjects is used for validation in the first run, then the 9th observation is used for validation in the second run, etc.

      split_method = LeaveK(K = 1)
      split_by = ByObservation(allsubjects = true)
      cv_method = ExactCrossvalidation(; split_method = split_method, split_by = split_by, ensemblealg = EnsembleThreads())
      r = crossvalidate(ft, cv_method)

• Leave-2-future-observations-out cross-validation, leaving 2 future observations per subject such that no less than 4 observations are used in training. allsubjects = false means that only the data of one subject at a time will be split. So in the first run, only observations 1 to 4 of subject 1 are used for training and observations 5 and 6 of subject 1 are used for validation. In the second run, observations 1 to 6 of subject 1 are used for training and observations 7 and 8 are used for validation. In the third run, observations 1 to 8 of subject 1 get used for training and observations 9 and 10 are used for validation. For all 3 runs, all the observations of subjects 2 to 5 are used in training. Then in the fourth to sixth runs, subject 2's data gets split. In the seventh to ninth runs, subject 3's data gets split, etc.

      split_method = LeaveFutureK(K = 2, minimum = 4)
      split_by = ByObservation(allsubjects = false)
      cv_method = ExactCrossvalidation(; split_method = split_method, split_by = split_by, ensemblealg = EnsembleThreads())
      r = crossvalidate(ft, cv_method)

• Leave-1-subject-out cross-validation, marginalizing the subject-specific parameters using quadrature integration when computing predictive likelihoods. Using this method, subjects {1, 2, 3, 4} are used for training in the first run and subject {5} is used for validation. In the second run, subjects {1, 2, 3, 5} are used for training and subject {4} is used for validation, etc. The predictive likelihoods computed are the marginal likelihood of the subject computed using Gaussian quadrature with a maximum of 1000 integration points. Refer to the documentation of LLQuad by typing ?LLQuad for more details on the parameters of the quadrature algorithm. FO(), FOCE() and LaplaceI() can also be used instead.

      split_method = LeaveK(K = 1)
      split_by = BySubject(marginal = LLQuad(imaxiters=1000))
      cv_method = ExactCrossvalidation(; split_method = split_method, split_by = split_by, ensemblealg = EnsembleThreads())
      r = crossvalidate(ft, cv_method)

• Leave-1-future-subject-out cross-validation, where the predictive likelihoods are conditional likelihoods computed using 0s for the subject-specific parameters instead of marginalizing. The minimum number of subjects is 3 so there are only 2 runs. In the first run, the training subjects are {1, 2, 3} and the validation subject is {4}. In the second run, the training subjects are {1, 2, 3, 4} and the validation subject is {5}.

      split_method = LeaveFutureK(K = 1, minimum = 3)
      split_by = BySubject(marginal = nothing)
      cv_method = ExactCrossvalidation(; split_method = split_method, split_by = split_by, ensemblealg = EnsembleThreads())
      r = crossvalidate(ft, cv_method)

FittedPumasEMModel

Contains the results of fitting a PumasEMModel with SAEM.

Fields:

• model: The PumasEMModel that was fit.
• data: The population that was fit.
• μ: Final estimates of @param and @random parameters.
• ω: Final estimates of ω parameters.
• σ: Final estimated standard deviations of residuals.
• opthistory: If approx is SAEM, contains the full history of unconstrained-μ, ω, and, σ estimates across iterations. Otherwise, contains the optimization results.
• approx: Fitting method used.
• η: Random effects from the fit. If approx is SAEM, contains a length(pop) × nη × iteration array of η samples from the smoothing phase. Otherwise, contains orthogonal random effects.
• kwargs: Input keywords stored for use in diagnostics. Includes ensemblealg and diffeq_options.

O ~ C | D

For example: CL ~ 1 + wt | Normal -> Formula{:CL,(1,:wt),:Normal}()

IDR(; direct=true, mode=:stimulation_sigmoid)

Get an indirect response model (IDR) for use as the pharmacodynamic part of model macro generation.

• direct::Bool - toggle whether the drug acts on production (true) or inactivation (false)
• mode::Symbol - pick the mode of drug action. Valid options are: :stimulationlinear, :stimulationemax, :stimulationsigmoid, :stimulationinfinite, :stimulationlogarithmic, :inhibitionemax, :inhibitionsigmoid, :inhibitionhill,

JointMAP

An instance of the maximum a-posteriori (MAP) algorithm where the mode of the joint posterior of the population and subject-specific parameters is found using the BFGS optimization algorithm. All the options of the algorithm can be passed to the constructor of JointMAP as keyword arguments. Example:

alg = JointMAP(ensemblealg = EnsembleThreads())

The main options that can be set are:

• ensemblealg: can be set to EnsembleSerial() for serial sampling, EnsembleThreads() for multi-threaded sampling or EnsembleDistributed() for multi-processing (aka distributed parallelism) sampling
• diffeq_options: a NamedTuple of all the differential equations solver's options

KFold(; K = 5, shuffle = false, rng = nothing)

K-fold splitting method. In this method, the data is split K times into 2 disjoint groups, each time starting from the full data set. The 2 groups are typically called training and validation subsets, where the validation subset has floor(N / K) data points, N being the total number of data points. In the next iteration, the whole data set is then re-split using another disjoint validation subset of floor(N / K) different points, disjoint from the the previous validation subsets. This process is done iteratively until almost each data point has shown up in 1 and only 1 validation subset. If N is divisible by K, each point will show up in 1 and only 1 validation subset. Otherwise, the remainder points will be part of the training subset for all the splittings and will not show up in any validation subset. This method can be used for computing predictive loglikelihoods, in-sample loglikelihoods and doing cross-validation using subsets of out-of-sample data.

The data is typically a vector of some sort, e.g. observations or subjects, and the splittings are order-dependent. Before performing the splittings, you can randomly shuffle the data vector by setting the shuffle keyword argument to true (default is false) getting rid of the sensitivity to the original order of the data. You can additionally pass an optional pseudorandom number generator rng to control the pseudo-randomness for maximum reproducibility.

Example:

Assume the original data is ["A", "B", "C", "D"]. 4-fold splitting without shuffling results in the following splittings of the data:

where each data point showed once and only once in a validation subset.

2-fold splitting without shuffling results in the following splittings:

LLQuad(; iabstol = 0.0, ireltol = 1e-8, imaxiters = 10^10)

Gaussian quadrature integration approximation method for computing the marginal loglikelihood. The integration volume is adaptively subdivided, using a cubature rule due to Genz and Malik (1980), until the estimated error E satisfies E ≤ max(ireltol*norm(I), iabstol) where I is the approximate integrand, i.e. ireltol and iabstol are the relative and absolute tolerances requested, respectively. However, the number of conditional loglikelihood evaluations is not allowed to exceed imaxiters. The default values of iabstol, ireltol and imaxiters are 0, 1e-8 and 10^10 respectively.

Example:

alg = LLQuad(ireltol = 1e-6, imaxiters = 10_000)
Pumas.marginal_nll(model, subject, init_params(model), alg)

LeaveFutureK(; K = 1, minimum = 2)

Leave-future-K-out splitting method. In this method, the data is assumed to be a timeseries. The goal is to split the data into "past" and "future". Using this method, the data is split multiple times into 3 disjoint groups where the third group is discarded, each time starting from the full data set. The first 2 groups are typically called the past/training subset and the future/validation subset, where the future validation subset has K future data points. In the next iteration, the whole data set is then re-split using another disjoint subset of K data points as the future validation subset. This process is done iteratively until the training set has no less than minimum number of points remaining. Using this method, each data point can show up in at most 1 future validation subset. This method can be used for computing predictive loglikelihoods, in-sample loglikelihoods and doing cross-validation using subsets of future out-of-sample data. The default values of K and minimum are 1 and 2 respectively.

Example:

Assume the original data is ["A", "B", "C", "D", "E", "F"]. Leave-1-future-out splitting with minimum = 2 results in the following splittings of the data where the third column is discarded:

Leave-2-future-out splitting with minimum = 2 results in the following splittings:

LeaveK(; K = 5, shuffle = false, rng = nothing)

Leave-K-out splitting method. In this method, the data is split multiple times into 2 disjoint groups, each time starting from the full data set. The 2 groups are typically called training and validation subsets, where the validation subset has K data points. In the next iteration, the whole data set is then re-split using another disjoint subset of K data points as the validation subset. This process is done iteratively until almost each data point has shown up in 1 and only 1 validation subset. This method can be used for computing predictive loglikelihoods, in-sample loglikelihoods and doing cross-validation using subsets of out-of-sample data.

The data is typically a vector of some sort, e.g. observations or subjects, and the splittings are order-dependent. Before performing the splittings, you can randomly shuffle the data vector by setting the shuffle keyword argument to true (default is false) getting rid of the sensitivity to the original order of the data. You can additionally pass an optional pseudorandom number generator rng to control the pseudo-randomness for maximum reproducibility.

Example:

Assume the original data is ["A", "B", "C", "D"]. Leave-1-out splitting without shuffling results in the following splittings of the data:

where each data point showed once and only once in a validation subset.

Leave-2-out splitting without shuffling results in the following splittings:

MarginalMCMC

An instance of the marginal Hamiltonian Monte Carlo (HMC) based No-U-Turn Sampler (NUTS) algorithm where the subject-specific parameters are marginalized and only the population parameters are sampled. All the options of the algorithm can be passed to the constructor of MarginalMCMC as keyword arguments. Example:

alg = MarginalMCMC(marginal_alg = LaplaceI(), nsamples = 2000, nadapts = 1000)

The main options that can be set are:

• marginal_alg: (default is LaplaceI()) the algorithm used to marginalize out the subject-specific parameters, defaults to LaplaceI() but can also be FO() or LaplaceI()
• target_accept: (default is 0.8) target acceptance ratio for the NUTS algorithm
• nsamples: (default is 2000) number of Markov Chain Monte Carlo (MCMC) samples to generate including adaptation samples
• nadapts: (default is nsamples ÷ 2) number of adaptation steps in the NUTS algorithm
• nchains: (default is 4) number of MCMC chains to sample, default value is 4
• ess_per_chain: (default is infinity) target effective sample size (ESS) per chain, sampling terminates if the target is reached
• check_every: (default is ess_per_chain ÷ 5) the number of samples after which the ESS per chain is checked
• time_limit: (default is infinity) a time limit for sampling in seconds, sampling terminates if the time limit is reached
• ensemblealg: (default is EnsembleThreads()) can be set to EnsembleSerial() for serial sampling, EnsembleThreads() for multi-threaded sampling or EnsembleDistributed() for multi-processing (aka distributed parallelism) sampling
• parallel_chains: (default is true if enough threads/workers exist) can be set to true or false, if set to true the chains will be sampled in parallel using either multi-threading or multi-processing depending on the value of ensemblealg
• parallel_subjects: (default is true if enough threads/workers exist) can be set to true or false, if set to true the log probability computation will be parallelized over the subjects using either multi-threading or multi-processing depending on the value of ensemblealg
• rng: the random number generator used
• diffeq_options: (default is an empty named tuple (;), which uses the downstream defaults) a NamedTuple of all the differential equations solver's options
• constantcoef: (default is an empty named tuple (;)) a NamedTuple of the parameters to be fixed during sampling. This can be used to sample from conditional posteriors fixing some parameters to specific values, e.g. constantcoef = (σ = 0.1,) fixes the σ parameter to 0.1 and samples from the posterior of the remaining parameters conditional on σ.

NL(pkmodel; hill=false)

Specify nonlinear clearance of a pkmodel for model generation.

pkmodel::Pumas.Explicitmodel - The pharmacokinetic model, e.g. Depots1Central1(). hill::Bool - toggle the use of a Hill coefficient.

ObsAdditive(obsname::Symbol)

Get an additive error model for automatic model macro generation.

• obsname::Symbol - choose what the output should be called. If you'll be fitting the model towards data then this symbol should match the identifier of the observation in your data.

ObsCombined(obsname::Symbol)

Get a combined (proportional and additive) error model for automatic model macro generation.

• obsname::Symbol - choose what the output should be called. If you'll be fitting the model towards data then this symbol should match the identifier of the observation in your data.

ObsProportional(obsname::Symbol)

Get a proportional error model for automatic model macro generation.

• obsname::Symbol - choose what the output should be called. If you'll be fitting the model towards data then this symbol should match the identifier of the observation in your data.

@param x ∈ PDiagDomain(n::Int)
@param x ∈ PDiagDomain(; init::AbstractVector{<:Real})

Specify a parameter as an n-by-n positive semi-definite diagonal matrix by providing either the number n of rows and columns or the initial diagonal (variance) elements init.

If the number n of rows and columns in the matrix is specified, the initial diagonal elements are all set to 1.

@param x ∈ PSDDomain(n::Int)
@param x ∈ PSDDomain(; init=Matrix{Float64}(I, n, n))

Specifies a parameter as a symmetric n-by-n positive semi-definite matrix. The user must either specify n as the number of rows and columns in the matrix or init as the initial value of the parameter. init should be a positive semi- definite Matrix of Float64s.

PSISCrossvalidation(; split_method, split_by, filter_nans = true, pareto_shape_threshold = 0.7)

Defines an instance of the PSISCrossvalidation algorithm for crossvalidation. In this algorithm, the fitting is not re-run but the MCMC samples are re-weighted using importance sampling (IS) while leaving out a subset of the data each time. The weights are then smoothed using a generalized Pareto distribution. The smoothed weights are then used to estimate the predictive likelihood of the out-of-sample data. This algorithm is known as the Pareto-smoothed importance sampling, leave-one-out (PSIS-LOO) crossvalidation method. In Pumas, we generalize the approach to handle any training-validation splitting of the data.

The shape parameters of the Pareto distribution fitted to the importance weights provide a diagnostic for the reliability of estimates of the predictive likelihoods. This happens typically when the samples are too sensitive to the a certain data point or subset removed and used for validation. In some cases NaNs can even result. The filter_nans set to true by default can be used to ignore the cases where the smoothing fails eliminating this run from the cross-validation result. Additionally, any run leading to a shape parameter higher than pareto_shape_threshold (default is 0.7) will also be removed from the cross-validation result. So if there are 100 different training-validation splittings and 9 of those lead to NaN or a shape parameter higher than pareto_shape_threshold, only the remaining 91 will be used for estimating the predictive loglikelihoods.

The way by which the data is split between training and validation sets is determined using the keyword arguments split_method and split_by. The split_method argument can be of any of the following types:

• LeaveK for leave-K-out cross-validation
• KFold for K-fold cross-validation
• LeaveFutureK for leaving K future points at a time

while split_by can be of any of the following types:

• BySubject for leaving out subjects
• ByObservation for leaving out individual observations per subject

Please refer to the documentation of each of the above types for the parameters of each split method, e.g. typing ?LeaveK or ?BySubject.

Examples:

Assume there are 5 subjects and 10 observations per subject and that ft is the result of the fit function. The following are some of the combinations in which the above inputs can be used:

• Leave-1-observation-out cross-validation, leaving 1 observation for all the subjects at a time. allsubjects = true means that the same observation index is removed for all the subjects, e.g. the 10th observation for all the subjects is used for validation in the first run, then the 9th observation is used for validation in the second run, etc.

      split_method = LeaveK(K = 1)
      split_by = ByObservation(allsubjects = true)
      cv_method = PSISCrossvalidation(;
          split_method = split_method, split_by = split_by,
          filter_nans = true, pareto_shape_threshold = 0.7,
      )
      r = crossvalidate(ft, cv_method)

• Leave-2-future-observations-out cross-validation, leaving 2 future observations per subject such that no less than 4 observations are used in training. allsubjects = false means that only the data of one subject at a time will be split. So in the first run, only observations 1 to 4 of subject 1 are used for training and observations 5 and 6 of subject 1 are used for validation. In the second run, observations 1 to 6 of subject 1 are used for training and observations 7 and 8 are used for validation. In the third run, observations 1 to 8 of subject 1 get used for training and observations 9 and 10 are used for validation. For all 3 runs, all the observations of subjects 2 to 5 are used in training. Then in the fourth to sixth runs, subject 2's data gets split. In the seventh to ninth runs, subject 3's data gets split, etc.

      split_method = LeaveFutureK(K = 2, minimum = 4)
      split_by = ByObservation(allsubjects = false)
      cv_method = PSISCrossvalidation(;
          split_method = split_method, split_by = split_by,
          filter_nans = true, pareto_shape_threshold = 0.7,
      )
      r = crossvalidate(ft, cv_method)

• Leave-1-subject-out cross-validation, marginalizing the subject-specific parameters using quadrature integration when computing predictive likelihoods. Using this method, subjects {1, 2, 3, 4} are used for training in the first run and subject {5} is used for validation. In the second run, subjects {1, 2, 3, 5} are used for training and subject {4} is used for validation, etc. The predictive likelihoods computed are the marginal likelihood of the subject computed using Gaussian quadrature with a maximum of 1000 integration points. Refer to the documentation of LLQuad by typing ?LLQuad for more details on the parameters of the quadrature algorithm. FO(), FOCE() and LaplaceI() can also be used instead.

      split_method = LeaveK(K = 1)
      split_by = BySubject(marginal = LLQuad(imaxiters=1000))
      cv_method = PSISCrossvalidation(;
          split_method = split_method, split_by = split_by,
          filter_nans = true, pareto_shape_threshold = 0.7,
      )
      r = crossvalidate(ft, cv_method)

• Leave-1-future-subject-out cross-validation, where the predictive likelihoods are conditional likelihoods computed using 0s for the subject-specific parameters instead of marginalizing. The minimum number of subjects is 3 so there are only 2 runs. In the first run, the training subjects are {1, 2, 3} and the validation subject is {4}. In the second run, the training subjects are {1, 2, 3, 4} and the validation subject is {5}.

      split_method = LeaveFutureK(K = 1, minimum = 3)
      split_by = BySubject(marginal = nothing)
      cv_method = PSISCrossvalidation(;
          split_method = split_method, split_by = split_by,
          filter_nans = true, pareto_shape_threshold = 0.7,
      )
      r = crossvalidate(ft, cv_method)

ParamSet(params::NamedTuple)

Construct a Pumas.jl parameter set.

params should be a NamedTuple that maps a parameter name to a Domain or a Distribution.

Example

ParamSet((;
  tvKa = RealDomain(lower = 0.0),
  tvCL = RealDomain(lower = 0.0),
  tvω = PDiagDomain(2),
))

A Population is an AbstractVector of Subjects.

@param x ∈ RealDomain(;lower=-∞,upper=∞,init=0)

Specifies a parameter as a real value. lower and upper are the respective bounds, init sets the initial value of the parameter and is used for init_params.

SAEM(; iters::Tuple{Int,Int,Int} = (200,100,100), α::Float64 = 0.7, λ::Float64 = 0.975, repeat::Int = 1, verbosity::Int = 0)

• iters: A tuple indicating the number of iterations to run in the exploratory, middle, and smoothing phases of SAEM.
• α: The learning rate for the middle phase. On the kth update, α^k weight is placed on the kth iteration, and 1 - α^k on the previous.
• λ: Maximum standard deviation change rate. A value of 0.95 means that ω_new = clamp(ω_new, ω_old * 0.95, ω_old / 0.95) on each iteration. Slowing the decliine in ωs and σs increases exploration, while limiting the growth prevents excessive flattening of the log likelihood.
• repeat: Indicates how many times to repeat iters. Defaults to 1.
• verbosity: How verbose should it be? 0: silent, 1: print on each repeat, 2: print once per iteration.

SIR(; rng=Random.default_rng, samples, resamples)

Constructs an SIR object that allows the user to use sampling importance re-sampling (SIR) with the supplies settings in infer function calls. See ?infer and https://docs.pumas.ai/stable/analysis/inference/ for more information about obtaining inference results.

The keyword arguments are as follows:

• rng: used to select a pseudo-random number generator to beused within the SIR procedure. Requires loading Random package (using Random)
• samples: the number of model parameter samples simulated from a truncated multivariate normal distribution (proposal distribution) based on the maximum-likelihood estimates and robust variance-covariance matrix. This constitutes the original sample (uncertainty) distribution.
• resamples: the number of model parameter samples from the original sample distribution to re-sample (by weighed re-sampling without replacement). The number of resamples must be less than that of the original sample. An original sample-to-resample ratio of at least 5:1 is suggested.

Both samples and resamples must be supplied, as there are no defaults.

Note that the proposal distribution for the SIR procedure is based on the robust variance-covariance matrix and will therefore fail whenever the computation of the robust variance-covariance matrix fails. Confidence intervals based on SIR respect bounds on the parameters (e.g. the confidence interval for a strictly positive parameter cannot contain negative values).

Subject

The data corresponding to a single subject:

Fields:

• id: identifier
• observations: a named tuple of the dependent variables
• covariates: a named tuple containing the covariates, or nothing.
• events: a DosageRegimen, or nothing.
• time: a vector of time stamps for the observations

When there are time varying covariates, each covariate is interpolated with a common covariate time support. The interpolated values are then used to build a multi-valued interpolant for the complete time support.

From the multi-valued interpolant, certain discontinuities are flagged in order to use that information for the differential equation solvers and to correctly apply the analytical solution per region as applicable.

Constructor

Subject(;id = "1",
         observations = NamedTuple(),
         events = nothing,
         time = observations isa AbstractDataFrame ? observations.time : nothing,
         covariates::Union{Nothing, NamedTuple} = nothing,
         covariates_time = observations isa AbstractDataFrame ? observations.time : nothing,
         covariates_direction = :left)

Subject may be constructed from an <:AbstractDataFrame with the appropriate schema or by providing the arguments directly through separate DataFrames / structures.

Examples:

julia> Subject()
Subject
  ID: 1

julia> Subject(
         id = 20,
         events = DosageRegimen(200, ii = 24, addl = 2),
         covariates = (WT = 14.2, HT = 5.2),
       )
Subject
  ID: 20
  Events: 3
  Covariates: WT, HT

julia> Subject(covariates = (WT = [14.2, 14.7], HT = fill(5.2, 2)), covariates_time = [0, 3])
Subject
  ID: 1
  Covariates: WT, HT

Subject

Constructor

Subject(simsubject::SimulatedObservations)

Roundtrip the result of simobs, i.e. SimulatedObservations to a Subject/Population

Example:

sims = simobs(model, pop, params)

To convert sims to a Population, broadcast as below

Subject.(sims)

Subject(subject::Subject; id::AbstractString, events)

Construct a Subject from an existing subject while replacing information according to the keyword arguments. The possible keyword arguments are

• id, sets the subject identifier to the string id. Defaults to the identifier already in subject.
• events, sets the subject events to the input DosageRegimen. Defaults to the events already present in subject.
• covariates and covariates_times, used as input to add covariate information

Examples:

julia> s1 = Subject(; id = "AKJ491", events = DosageRegimen(1.0; time = 1.0))
Subject
  ID: AKJ491
  Events: 1

julia> Subject(s1; id = "AKJ492")
Subject
  ID: AKJ492
  Events: 1

TimeToEvent{T}

Distribution-like struct to store the hazard and the cumulative hazard in a time-to-event model. The dependent variable in a model that uses TimeToEvent should be a censoring variable that is one when the variable is observed (not censored) zero when the variable is right censored. Currently, no other censoring types are supported.

Example

...
@pre begin
  θeff = θ*DOSE
  λ = λ₀*exp(θeff)
end

@dynamics begin
  Λ' = λ
end

@derived begin
  DV ~ @. TimeToEvent(λ, Λ)
end
...

@param x ∈ VectorDomain(n::Int; lower=-∞,upper=∞,init=0)

Specifies a parameter as a real vector of length n. lower and upper are the respective bounds, init sets the initial value of the parameter and is used for init_params. The keyword arguments can all take either numbers or vectors. If numbers then the same value will be applied to all n vector elements. If a you specify a vector (of length n) then you can adjust the parameters of the VectorDomain elements individually.

ZeroSumDomain(n; init = zeros(n))
ZeroSumDomain(init)
ZeroSumDomain(; init)

Return a zero-sum domain of dimension n. Any n-dimensional vector with a sum of 0 is in this domain. Instead of passing n, an initial value init can be passed instead either as a positional or keyword argument.

isvalid(s::SimulatedObservations) -> Bool

Check if ODE solution of a simulation was successful.

Examples

mdl = @model begin
  @options begin
    checklinear = false # Force numerical integration of ODE
  end
  @param begin
    θkₑ ∈ RealDomain(lower = 0.0)
  end
  @pre begin
    kₑ = θkₑ
    cᵥ = 0.1
    V = 1.0
  end
  @dynamics begin
    Central' = -kₑ * Central
  end
  @derived begin
    μ = Central ./ V
    y ~ @. Gamma(1 / cᵥ^2, abs(μ) * cᵥ^2)
  end
end

sim = simobs(
  mdl,
  Subject(events = DosageRegimen(100), time = range(1.0, 40.0, length = 12)),
  (; θkₑ = 1e20),
)

julia> isvalid(sim)
false

Pumas.truncate(b::BayesMCMCResults; burnin = 0, ratio = 1.0)

Removes burnin samples from the beginning of each chain and then keeps only ratio * 100% of the remaining samples in each chain (i.e. thinning).

sol = solve(
  model::AbstractPumasModel,
  population::Union{Population, Subject},
  param::NamedTuple,
  randeffs::NamedTuple=sample_randeffs(rng, model, param, population);
  ensemblealg=EnsembleSerial(),
  diffeq_options=NamedTuple(),
  rng=default_rng()
)

Solve the model applied to the Subject(s) within population using parameters param and random effects randeffs. By default, the times at which to save the solution are taken to be the observation times for each Subject within population. This can be overridden by supplying a vector or range as saveat to diffeq_options - e.g. diffeq_options=(; saveat=[0., 0.5, 1.]).

Arguments

• model may either be a PumasModel or a PumasEMModel.

• population may either be a Population of Subjects or a single Subject.

• param is parameter set in the form of a NamedTuple, e.g. (; tvCL=1., tvKa=2., tvV=1.).

• randeffs is an optional argument that, if used, should specify the random effects for each subject in population. Such random effects are specified by NamedTuples for PumasModels (e.g. (; tvCL=1., tvKa=2.)) and by Tuples for PumasEMModels (e.g. (1., 2.)). If population is a single Subject (without being enclosed in a vector) then a single such random effect specifier should be passed. If, however, population is a Population of multiple Subjects then randeffs should be a vector containing one such specifier for each Subject. The functions init_randeffs, zero_randeffs, and sample_randeffs are sometimes convenient for generating randeffs:

randeffs = zero_randeffs(model, param, population)
solve(model, population, param, randeffs)

If no randeffs is provided, then random ones are generated according to the distribution in the model.

• ensemblealg is a keyword argument that allows you to choose between different modes of parallelization. Options include EnsembleSerial(), EnsembleThreads() and EnsembleDistributed().

• diffeq_options is a keyword argument that takes a NamedTuple of options to pass on to the differential equation solver.

• steadystate_options is a keyword argument that takes a NamedTuple of options to pass on to the steady state solver.

• rng is a keyword argument where you can specify which random number generator to use.

logpdf(d::Pumas.Constrained, x)

Evaluate the logarithm of the probability density of the constrained distribution, d, at x.

• If d is a constrained univariate distribution then x should be a scalar.
• If d is a constrained multivariate distribution then x should be a vector.

Evaluations of x outside of d's constraints returns -Inf.

Note that d itself is not a true distribution since its probability mass is not rescaled to 1.

function GlobalSensitivity.gsa(
  model::PumasModel,
  population::Union{Population,Subject},
  params::NamedTuple,
  method::GlobalSensitivity.GSAMethod,
  vars = [:dv],
  p_range_low::NamedTuple = NamedTuple(),
  p_range_high::NamedTuple = NamedTuple();
  constantcoef::Tuple = (),
  rng::AbstractRNG = Random.default_rng(),
  diffeq_options::NamedTuple = NamedTuple(),
  ensemblealg = EnsembleSerial(),
  samples = 1000,
  kwargs...,
)

Perform a global sensitivity analysis (GSA) of the population mean of the derived variables vars on the parameters of the Pumas model.

Arguments

• model: a PumasModel.
• population: a Population of Subjects or a single Subject.
• params: a named tuple of parameters.
• method: one of the GSA methods from GlobalSensitivity.jl such as Sobol(), Morris(), eFAST(), or RegressionGSA().
• vars: a list of the observed variables (defined in the @observed block) to run GSA on.
• p_range_low & p_range_high: the lower and upper bounds for the parameters you want to run the GSA on. For parameters without lower or upper bounds in p_range_low or p_range_high, respectively, bound(s) in the @param block are used. If bounds for a parameter are neither specified as arguments nor in the @param block, an error is thrown.

Keyword arguments

• constantcoef: A tuple of the names of the parameters that are not to be varied in the GSA.
• rng: The random number generator to use.
• samples: The number of samples to use for the GSA.
• diffeq_options: The options of the differential equation solver, passed on to simobs when computing the derived variables of interest.
• ensemblealg: The mode of parallelization of the simobs calls when computing the derived variables of interest.
• kwargs...: The remaining keyword arguments are passed on to GlobalSensitivity.gsa, as documented in GlobalSensitivity.jl. In particular, evaluation of GSA samples can be parallelized by setting batch = true (default: batch = false).

References

For more details regarding the GSA methods, their options, and keyword arguments please consult the GlobalSensitivity.jl documentation.

cond(pmi::FittedPumasModelInference; correlation = true)

Return the condition number of the correlation matrix (or variance-covariance matrix with correlation = false) stored in pmi. Throws an error if pmi is the result of a call to infer with Pumas.Bootstrap or if the variance-covariance calculation failed.

gewekediag(b::BayesMCMCResults; subject::Union{Nothing,Int} = nothing, first::Real = 0.1, last::Real = 0.5)

Computes the Geweke diagnostic ^[Geweke1991] for each chain outputting a p-value per parameter. If subject is nothing (default), the chains diagnosed are those of the population parameters. If subject is set to an integer index, the chains diagnosed are those of the subject-specific parameters corresponding to the subject with the input index.

The Geweke diagnostic compares the sample means of two disjoint sub-chains X₁ and X₂ of the entire chain using a normal difference of means hypothesis test where the null and alternative hypotheses are defined as:

• H₀: μ₁ = μ₂
• H₁: μ₁ ≂̸ μ₂ where μ₁ and μ₂ are the population means. The first sub-chain X₁ is taken as the first (first * 100)% of the samples in the chain, where first is a keyword argument defaulting to 0.1. The second sub-chain X₂ is taken as the last (last * 100)% of the samples in the chain, where last is a keyword argument defaulting to 0.5.

The test statistic used is: z₀ = x̅₁ - x̅₂ / √(s₁² + s₂²) where x̅₁ and x̅₂ are the sample means of X₁ and X₂ respectively, and s₁ and s₂ are the Markov Chain standard error (MCSE) estimates of X₁ and X₂ respectively. Auto-correlation is assumed within the samples of each individual sub-chain, but the samples in X₁ are assumed to be independent of the samples in X₂. The p-value output is an estimate of P(|z| > |z₀|), where z is a standard normally distributed random variable.

Low p-values indicate one of the following:

• The first and last parts of the chain are sampled from distributions with different means, i.e. non-convergence,
• The need to discard some initial samples as burn-in, or
• The need to run the sampling for longer due to lack of samples or high auto-correlation.

High p-values indicate the inability to conclude that the means of the first and last parts of the chain are different with statistical significance. However, this alone does not guarantee convergence to a fixed posterior distribution because:

• Either the standard deviations or higher moments of X₁ and X₂ may be different, or
• The independence assumption between X₁ and X₂ may not be satisfied when high auto-correlation exists.

heideldiag(b::BayesMCMCResults; subject::Union{Nothing,Int} = nothing, alpha::Real = 0.05, eps::Real = 0.1, start::Integer = 1)

Compute the Heidelberger and Welch diagnostic ^{[Heidelberger1983]} for each chain. If subject is nothing (default), the chains diagnosed are those of the population parameters. If subject is set to an integer index, the chains diagnosed are those of the subject-specific parameters corresponding to the subject with the input index.

The Heidelberger diagnostic attempts to:

• Identify a cutoff point for the initial transient phase for each parameter, after which the samples can be assumed to come from a steady-state posterior distribution. The initial transient phase can be removed as burn-in. The cutoff point for each parameter is given in the burnin column of the output dataframe.
• Estimate the relative confidence interval (confidence interval divided by the mean) for the mean of the steady-state posterior distribution of each parameter, assuming such steady-state distribution exists in the samples. A large confidence interval implies either the lack of convergence to a stationary distribution or lack of samples. Half the relative confidence interval is given in the halfwidth column of the output dataframe. The test column will be true (1) if the halfwidth is less than the input target eps (default is 0.1) and false (0) otherwise. Note that parameters with mean value close to 0 can have erroneous relative confidence intervals because of the division by the mean. The test value can therefore be expected to be false (0) for those parameters without concluding lack of convergence.
• Quantify the extent to which the distribution of the samples is stationary using statistical testing. The returned p-value, shown in the pvalue column of the output dataframe, can be considered a measure of mean stationarity. A p-value lower than the input threshold alpha (default is 0.05) implies lack of stationarity of the mean, i.e. the posterior samples did not converge to a steady-state distribution with a fixed mean.

The Heidelberger diagnostic only tests for the mean of the distribution. Therefore, it can only be used to detect lack of convergence and not to prove convergence. In other words, even if all the numbers seem normal, one cannot conclude that the chain converged to a stationary distribution or that it converged to the true posterior.

rafterydiag(b::BayesMCMCResults; subject::Union{Nothing,Int} = nothing, q = 0.025, r = 0.005, s = 0.95, eps = 0.001)

Compute the Raftery and Lewis diagnostic ^{[Raftery1992]}. This diagnostic is used to determine the number of iterations required to estimate a specified quantile q within a desired degree of accuracy. The diagnostic is designed to determine the number of autocorrelated samples required to estimate a specified quantile $\theta_q$, such that $\Pr(\theta \le \theta_q) = q$, within a desired degree of accuracy. In particular, if $\hat{\theta}_q$ is the estimand and $\Pr(\theta \le \hat{\theta}_q) = \hat{P}_q$ the estimated cumulative probability, then accuracy is specified in terms of r and s, where $\Pr(q - r < \hat{P}_q < q + r) = s$. Thinning may be employed in the calculation of the diagnostic to satisfy its underlying assumptions. However, users may not want to apply the same (or any) thinning when estimating posterior summary statistics because doing so results in a loss of information. Accordingly, sample sizes estimated by the diagnostic tend to be conservative (too large).

Furthermore, the argument r specifies the margin of error for estimated cumulative probabilities and s the probability for the margin of error. eps specifies the tolerance within which the probabilities of transitioning from initial to retained iterations are within the equilibrium probabilities for the chain. This argument determines the number of samples to discard as a burn-in sequence and is typically left at its default value.

Censored(distribution::UnivariateDistribution, lower::Real, upper::Real)

Construct a censored distribution based on distribution and the censoring limits lower and upper.

_convergencedata(obj; metakey="x")

Returns the "timeseries" of optimization as a matrix, with series as columns.

_lpdf(d,x)

The log pdf: this differs from Distributions.logdpf definition in a couple of ways:

• if d is a non-distribution it assumes the Dirac distribution.
• if x is NaN or Missing, it returns 0.
• if d is a NamedTuple of distributions, and x is a NamedTuple of observations, it computes the sum of the observed variables.

_objectivefunctionvalues(obj)

Returns the objective function values during optimization. Must return a Vector{Number}.

_paramnames(obj)

Returns the names of the parameters which convergence is being checked for.

add_t(ex, vars)

Append (t) to all the vars symbols in the expression ex.

applyformatting(inputex, vars; show_t=true, italicize=true)

Apply formatting to the vars of an input expression before latexification.

apply_stat(stat, v::AbstractVector)

Apply the summarizing function stat to the contents of the vector in a way that is equivalent to calling stat on a vector of all the elements that are in the ith position of each element of v. For example, this can be used to create a NamedTuple of the mean across a vector of NamedTuples such that the key k in the output is the mean of all the values belonging to the key k in each element of the vector v.

Example

julia> v = [(a = 3, b = 4), (a = 1, b = 9), (a = 6, b = 0)]
3-element Vector{NamedTuple{(:a, :b), Tuple{Int64, Int64}}}:
 (a = 3, b = 4)
 (a = 1, b = 9)
 (a = 6, b = 0)

julia> apply_stat(mean, v)
(a = 3.3333333333333335,
 b = 4.333333333333333,)

julia> apply_stat(var, v)
(a = 6.333333333333334,
 b = 20.333333333333336,)

julia> apply_stat(std, v)
(a = 2.5166114784235836,
 b = 4.509249752822894,)

build_model_str(pk, pd, obs, dcp)

Generate a string that fully specifies an @model macro call.

• pk - a pharmacokinetic model, for example Depots1Central1()
• pd::Union{Nothing, AbstractPDModule} - a pharmacodynamic model, for example IDR(; direct=true, mode=:stimulation_emax)
• obs::AbstractObservationModule - an observational error model, for example ObsCombined(:MyOutputName)
• dcp::DoseControl - a dose control specification, for example DoseControl(; bioav=true)

censored_latent(
    distribution::UnivariateDistribution,
    lower::Union{Real,Nothing},
    upper::Union{Real,Nothing},
)
censored_latent(
    distribution::UnivariateDistribution;
    lower::Union{Real,Nothing} = nothing,
    upper::Union{Real,Nothing} = nothing,
)

Construct a censored distribution based on distribution and the censoring limits lower and upper. When fitting a model with a censored_latent dependent variable, the log probability of the censored distribution is used in the likelihood calculation. However, when calculating the mean or other summary statistics, the underlying uncensored distribution is used.

conditional_nll(m::AbstractPumasModel, subject::Subject, param, randeffs; diffeq_options)

Compute the conditional negative log-likelihood of model m for subject with parameters param and random effects randeffs. diffeq_options is a NamedTuple of options passed to the ODE solver. Requires that the derived produces distributions.

`cor2cov(ρ::AbstractVector, ω::AbstractVector)`

`cor2cov(ω::AbstractMatrix)`

`cor2cov(ω::AbstractVector)`

Converts correlations and standard deviations to a covariance matrix. There are 3 ways to use the cor2cov function.

The first method has 2 inputs:

• ρ: a vector of correlations which is a vector of the strict lower triangular elements of the correlation matrix of the random variables.
• ω: a vector of standard deviations of the random variables Example:

ω11, ρ12, ω22 = 0.2, 0.1, 0.3

ρ = [ρ12]
ω = [ω11, ω22]
Pumas.cor2cov(ρ, ω)

gives the covariance matrix:

[ ω11^2        ρ12*ω11*ω22;
  ρ12*ω11*ω22  ω22^2      ]

which is equal to:

2×2 Matrix{Float64}:
 0.04   0.006
 0.006  0.09

The second method has a single input matrix ω which is a symmetric matrix whose diagonal is the standard deviations of the random variables and off-diagonals are their correlations. Example:

ω = [
  ω11 ρ12
  ρ12 ω22
]
Pumas.cor2cov(ω)

gives the covariance matrix:

[
  ω11^2 ρ12*ω11*ω22
  ρ12*ω11*ω22 ω22^2
]

which is equal to:

2×2 Matrix{Float64}:
 0.04   0.006
 0.006  0.09

The third method has a single input vector ω which is a vector of the lower triangular elements of the symmetric matrix whose diagonal is the standard deviations of the random variables and off-diagonals are their correlations.

Example:

ω = [ω11, ρ12, ω22]
Pumas.cor2cov(ω)

gives the covariance matrix:

[
  ω11^2 ρ12*ω11*ω22
  ρ12*ω11*ω22 ω22^2
]

which is equal to:

2×2 Matrix{Float64}:
 0.04   0.006
 0.006  0.09

correlation_diagnostic(
  pmi::FittedPumasModelInference;
  high_cor_threshold = 0.7,
  medium_cor_threshold = high_cor_threshold / 2,
)

A function which accepts the output of the infer function as its argument and prints the lists of parameter pairs with high, medium or low correlations. Parameter pairs with high correlation are those with a correlation higher than high_cor_threshold. Parameter pairs with a low correlation are those with a correlation lower than medium_cor_threshold. The remaining parameter pairs have a medium correlation. Assume fpm is the output of fit. The following is an example of how to use correlation_diagnostic:

fpmi = infer(fpm)
correlation_diagnostic(fpmi)

function covariate_select(
    model::AbstractPumasModel,
    population::Population,
    param::NamedTuple,
    approx::LikelihoodApproximation;
    control_param::Union{Tuple,Nothing},
    criterion = aic,
    method::CovariateSelection.T = CovariateSelection.Forward,
    fit_options::NamedTuple = (;),
    verbose::Bool = false,
) -> CovariateSelectionResults

Use a covariate selection procedure to find the best fit of model to population with starting values param and estimation method/likelihood approximation approx according to criterion by restricting parameters to zero for subsets of control_param traversed according to method.

The model should be written such that it includes all relevant covariates, possibly included with multiple different functional forms, such that that restricting on a parameter listed in control_param to zero eliminates the effect of the covariate (see example below).

Currently, the two methods CovariateSelection.Forward (the default) and CovariateSelection.Backward are supported. The method CovariateSelection.Forward starts from the models where all parameters in control_param are restricted to zero. In the next step, a model is fit where all parameters in control_param but one are restricted to zero. If none of these models attain a smaller value of criterion then the model that restricts all parameters to zero is chosen. Otherwise, the model with the lowest criterion is chosen and the procedure continues by fitting all models with two estimated parameters where one of the estimated parameters must be the one corresponding to the best fit in the previous step. New steps are taken as long as the criterion is reduced. If the CovariateSelection.Backward method is chosen then the procedure starts with all parameters in control_param unrestricted and then one parameter is restricted at a time until no there is no further improvement.

Any callable object that returns a real value when a fitted model is passed can be used as criterion. It is always assumed that a smaller value is better. The default value is the Akaike Information Criterion aic but another commonly used criterion the Bayesian Information Criterion bic which puts a higher penalty on the number of parameters compared to aic.

To avoid that logging messages from the calls to fit fills up the output window or default log, all logging messages produced by fit are by default suppressed. This might hide important issues with the model fitting so for it is possible to enable complete logging by passing verbose = true.

Example

model = @model begin
  @param begin
    β0 ∈ RealDomain()
    β1 ∈ RealDomain()
    β2 ∈ RealDomain()
    β3 ∈ RealDomain()
    β4 ∈ RealDomain()
  end
  @covariates x1 x2 x3 x4
  @pre μ = β0 + β1 * x1 + β2 * x2 + β3 * x3 + β4 * x4
  @derived begin
    y ~ Normal.(μ, 0.1)
  end
end

selection = covariate_select(
  model,
  population,
  init_params(model),
  NaivePooled();
  control_param = (:β1, :β2, :β3, :β4),
  method = CovariateSelection.Forward,
  criterion = bic,
)

covariates_interpolant(
  covariates_nt::NamedTuple,
  covariates_times::NamedTuple,
  id::AbstractString;
  [covariates_direction::Symbol]) -> time_grid, interpolant

Creates an interpolation of the elements of covariates_nt and covariates_times using the a piecewise constant interpolation scheme. The two NamedTuples should have the same keys. The values of covariates_nt and covariates_times should contain the interpolation values and time point respectively. The argument id is a string indicating the subject ID and the argument is only used for providing useful error messages.

Returns a callable interpolation struct as well as the common time grid for the covariate observations. This is safe for values which are not time-varying and it also allows for mixing subjects with multiple measurements and subjects with a single measurement. The default is left-sided constant interpolation.

covariates_interpolant(
  covariates::Vector{Symbol},
  data::AbstractDataFrame,
  time::Symbol,
  id::AbstractString;
  [covariates_direction::Symbol]) -> time_grid, interpolant

Creates an interpolation of the columns of data listed in the covariates vector at the time points in the time column of data using the a piecewise constant interpolation scheme. The argument id is a string indicating the subject ID and the argument is only used for providing useful error messages.

Returns a callable interpolation struct as well as the common time grid for the covariate observations. This is safe for values which are not time-varying and it also allows for mixing subjects with multiple measurements and subjects with a single measurement. The default is left-sided constant interpolation.

crossvalidate(b::BayesMCMCResults, cv_method::ExactCrossvalidation)

Runs cross-validation by re-running the MCMC sampling while leaving out some data points each time. The first argument b is the output of the MCMC fit. The choice of how the data is split in each run is specified in the cv_method argument which is an instance of ExactCrossvalidation. Please refer to the documentation of ExactCrossvalidation by typing ?ExactCrossvalidation for details on the parameters of the method.

crossvalidate(b::BayesMCMCResults, cv_method::PSISCrossvalidation)

Performs cross-validation using the Pareto smoothed importance sampling (PSIS) approach by re-weighing the samples while leaving out some data points each time. The first argument b is the output of the MCMC fit. The choice of how the data is split in each run is specified in the cv_method argument which is an instance of PSISCrossvalidation.

discard(b::BayesMCMCResults; burnin = 0, ratio = 1.0)

Removes burnin samples from the beginning of each chain and then keeps only ratio * 100% of the remaining samples in each chain (i.e. thinning).

dosecontrol(fpm::AbstractFittedPumasModel)::Vector{ConstantInterpolationStructArray}

Return the dosecontrol parameters from fpm. The dosecontrol parameters are the variables bioav, lags, duration, rate if they're defined in the @dosecontrol block in the @model macro. The function returns a vector of covariate interpolation objects. Each of them can be evaluated at an arbitrary time point, but the interpolant is piecewise constant between the event times. Note, that in the model solution, subject simulation, and parameter fitting the dose control parameters can be time-varying, but they only affect the dosing according to their value at the beginning of each dose event. This means that the dose control parameters are only relevevant at these time points. Each of the covariate interpolation objects can be converted to a DataFrame by calling the DataFrame constructor on each element, or a DataFrame with all subjects can be obtained by calling the DataFrame constructor on the vector output by this function. This DataFrame will only contain the times that correspond to the start of each dose event for the reason discussed above.

dosenum(t, events) -> # of doses for each time

Creates an array that matches t in length which denotes the number of events that have occurred prior to the current point in time. If t is a scalar, outputs the number of events before that time point. Only dose events (evid 1 and 4) are counted.

eiwres(model::PumasModel, subject::Subject, param::NamedTuple, nsim::Integer)

Calculate the Expected Simulation based Individual Weighted Residuals (EIWRES).

elpd(r::CVResult)

Computes an estimate of the expected log predictive density (ELPD) for an out-of-sample data point/set using the result r of crossvalidate. The output is a scalar value estimating the model's mean predictive accuracy on out-of-sample data.

em_to_m_params(m::PumasEMModel, params::NamedTuple)

Given that params is a NamedTuple of coefficients used as inits for fitting m or returned by coef(fpm::FittedPumasEMModel), then em_to_m_params will return a NamedTuple of coefficients usable as inits for fitting or performing diagnostics on a PumasModel created by calling convert(PumasModel, m::PumasEMModel).

empirical_bayes(
  fpm::FittedPumasModel,
  data::Union{Subject, Population};
  diffeq_options=NamedTuple()
)

Return empirical bayes estimates for a fitted model, fpm, when applied to data.

empirical_bayes(fpm::Pumas.FittedPumasModel)
empirical_bayes(fpm::Pumas.FittedPumasEMModel)
empirical_bayes(insp::Pumas.FittedPumasModelInspection)

Return sampled random effects or empirical bayes estimates from a fit or model inspection. If the model was estimated with the FO likelihood approximation methods the empirical bayes estimates will be obtained using the LaplaceI approximation. If either FOCE or LaplaceI was used the final empirical bayes estimates will be returned. If Pumas.SAEM was used to fit the empirical bayes estimates are obtained using the LaplaceI approximation.

empirical_bayes(
  model::PumasModel,
  data::Union{AbstractSubject, Population},
  param::NamedTuple,
  approx::LikelihoodApproximation;
  diffeq_options=NamedTuple()
)

Return sampled random effects or empirical bayes estimates. The empirical bayes estimates will be obtained using the LaplaceI approximation regardless of the value of the approx argument except for NaivePooled where an empty NamedTuple is returned.

empirical_bayes_dist(fpm::AbstractFittedPumasModel)

Estimate the distribution of random effects (typically a Normal approximation of the empirical Bayes posterior) for the subjects in a FittedPumasModel. The result is returned as a vector of MvNormals.

epredict(fpm::AbstractFittedPumasModel; nsim::Integer, rng::AbstractRNG=default_rng(), ensemblealg::::SciMLBase.EnsembleAlgorithm=EnsembleThreads())

Calculate the expected simulation based population predictions.

The keyword nsim controls the number of times a subject is simulated and the keyword rng can be used to provide a custom random number generator. The parallelization can be controlled using the ensemblealg keyword that defaults to EnsembleThreads().

eventnum(t, events) -> # of doses for each time

Creates an array that matches t in length which denotes the number of events that have occurred prior to the current point in time. If t is a scalar, outputs the number of events before that time point. The following events are counted: 1 (dose), 3 (reset), 4 (reset and dose).

expectation(g, ::MonteCarloExpectation, model::PumasModel, subject::Subject, dist::MvNormal; imaxiters = 10000, diffeq_options::NamedTuple = NamedTuple(), rng::AbstractRNG = Pumas.default_rng())

Computes the expected predictions for subject with respect to the population parameters and the subject-specific random effects using Monte Carlo integration. dist is a multivariate normal distribution used to sample the population parameters e.g. dist = MvNormal(pmi) where pmi is the output of infer. Sampling is done using rejection sampling to ensure the parameter values are in the parameters' domains. imaxiters is the number of Monte Carlo samples used. diffeq_options can be used to customize the options of the differential equation solver used. rng is the random number generator used during the sampling, which defaults to Pumas.default_rng().

expectation(g, ::MonteCarloExpectation, model::PumasModel, subject::Subject, param_dists::NamedTuple; imaxiters = 10000, diffeq_options::NamedTuple = NamedTuple(), rng::AbstractRNG = Pumas.default_rng())

Computes the expected predictions for subject with respect to the population parameters and the subject-specific random effects using Monte Carlo integration. param_dists should be a named tuple of distributions for the population parameters. imaxiters is the number of Monte Carlo samples to use. diffeq_options can be used to customize the options of the differential equation solver used. rng is the random number generator used in the sampling, which defaults to Pumas.default_rng().

expectation(g, quant::QuadratureExpectation, model::PumasModel, subject::Subject, param_dists::NamedTuple; ireltol=1e-3, iabstol=1e-3, imaxiters=10000, diffeq_options::NamedTuple = NamedTuple())

Computes the expected predictions for subject with respect to the population parameters using quadrature methods. param_dists should be a named tuple of distributions for the population parameters. Currently, random effects are not supported. ireltol and iabstol are the tolerances for the integration. The integration algorithm will terminate if the tolerance or imaxiters is reached, whichever is first. diffeq_options can be used to customize the options of the differential equation solver.

findinfluential(m::AbstractPumasModel,
                data::AbstractPopulation,
                param::NamedTuple,
                approx::LikelihoodApproximation;
                diffeq_options::NamedTuple = NamedTuple())
findinfluential(fpm::AbstractFittedPumasModel, 
                diffeq_options::NamedTuple = NamedTuple())

Return a vector of the negative log-likelihood values.

It can be used before fitting a model in order to find the subjects that contributes the most the log-likelihood value or to identify subjects for which the evaluation of the log-likelihood fails, e.g. due to incorrect data. The latter will show up as NaNs.

It can also be used on a fitted model to identify the subjects with maximum contribution to the log-likelihood.

A DataFrame method is defined on the resulting object for easy viewing and plotting of the results.

fi = findinfluential(model, pop, parameters, FOCE())
fi_df = DataFrame(fi)

If the above results contain NaNs the id column can be useful in finding problematic data points. To visualize the negative log-likelihood values across subjects a bar plot can be produced as follows

using AlgebraOfGraphics
draw(
  data(fi_df) * mapping(:nll) * visual(BarPlot);
  axis = (;
    xlabel = "id",
    ylabel = "negative log-likelihood",
    xticks = 1:length(pop),
    xtickformat = x -> fi_df.id,
  ),
)

or a histogram to see the distribution:

using AlgebraOfGraphics
draw(
  data(fi_df) *
  mapping(:nll) *
  histogram() *
  visual(BarPlot);
  axis = (; xlabel = "negative loglikelihood"),

)

or a density plot if preferred over a histogram:

using AlgebraOfGraphics
draw(
  data(fi_df) *
  mapping(:nll) *
  density() *
  visual(BarPlot);
  axis = (; xlabel = "negative loglikelihood"),
)

get_model_combinations()

Get all valid arguments to build_model_str or build_model_expr.

Returns a tuple (pks, pds, obs, dcps) where pks is a vector of all possible PK specifications, and so on, for the model generation.

Example usage:

for args in Iterators.product(get_model_combinations())
  obs = make_obs(args[3], args[1], args[2])
  build_model_str(args[1], args[2], obs, args[4])
end

to create all model strings (~15k different ones, 1M lines of code).

Useful for testing and possibly for automatic building, etc.

getblock(blockname, model::AbstractPumasModel) -> array of expressions for a PumasModel macro block.

The blockname should match the name of the blocks used for model creation with the @model or @emmodel macro. Only the blocks that were explicitly used in model creation can be used. Valid such blocknames are :covariates, :derived, :dynamics, :dosecontrol, :init, :observations, :options, :param, :post, :pre, :random, :reactions and :vars. Only models that are defined using the @model or @emmodel macro are supported.

icoef(model::PumasModel, subject::Subject, param::NamedTuple, approx::LikelihoodApproximation = LaplaceI(); obstimes, diffeq_options::NamedTuple = NamedTuple())::Vector{ConstantInterpolationStructArray}

Return the individual coefficients for the input arguments. The individual coefficients are the variables defined in the @pre block in the @model macro. The empirical bayes estimates given the input param will be calculated if the input subjects have observations. If any subject does not have any observations, the mode of the domain-transformed prior will be used. The function returns a vector of covariate interpolation objects defined on obstimes. If no obstimes are given, the time points used to define the covariates will be used. Each of them can be evaluated at any time point, but the interpolant is piecewise constant between the obstimes. Each of the covariate interpolation objects can be converted to a DataFrame by calling the DataFrame constructor on each element, or a DataFrame with all subjects can be obtained by calling the DataFrame constructor on the vector output by this function.

icoef(fpm::AbstractFittedPumasModel; obstimes)::Vector{ConstantInterpolationStructArray}

Return the individual coefficients from fpm. The individual coefficients are the variables defined in the @pre block in the @model macro. The function returns a vector of covariate interpolation objects defined on obstimes. If no obstimes are given, the time points used to define the covariates will be used. Each of them can be evaluated at any time point, but the interpolant is piecewise constant between the obstimes. Each of the covariate interpolation objects can be converted to a DataFrame by calling the DataFrame constructor on each element, or a DataFrame with all subjects can be obtained by calling the DataFrame constructor on the vector output by this function.

icoef(model::PumasModel, population::Population, param::NamedTuple; kwargs...)::Vector{ConstantInterpolationStructArray}

Return the individual coefficients for the input arguments. The individual coefficients are the variables defined in the @pre block in the @model macro. The empirical bayes estimates given the input param will be calculated if the input subjects have observations. If any subject does not have any observations, the mode of the domain-transformed prior will be used. The function returns a vector of covariate interpolation objects defined on obstimes. If no obstimes are given, the time points used to define the covariates will be used. Each of them can be evaluated at any time point, but the interpolant is piecewise constant between the obstimes. Each of the covariate interpolation objects can be converted to a DataFrame by calling the DataFrame constructor on each element, or a DataFrame with all subjects can be obtained by calling the DataFrame constructor on the vector output by this function.

infer(fpm::FittedPumasModel; level=0.95, rethrow_error::Bool=false, sandwich_estimator::Bool=true) -> FittedPumasModelInference

Compute the vcov matrix and return a struct used for inference based on the fitted model fpm. The confidence intervals are calculated as the (1-level)/2 and (1+level)/2 quantiles of the estimated parameters. sandwich_estimator is a boolean that switches on or off the sandwich estimator. If rethrow_error is false (the default value), no error will be thrown if the variance-covariance matrix estimator fails. If it is true, an error will be thrown if the estimator fails.

infer(fpm::FittedPumasModel, bts::Pumas.Bootstrap; level=0.95)

Perform bootstrapping by resampling the Subjects from the Population stored in fpm. The confidence intervals are calculated as the (1-level)/2 and (1+level)/2 quantiles of the estimated parameters. The number of samples used in the bootstrapping is bts.samples. bts.ensemblealg specifies the ensemblealg used here. If ensemblealg is EnsembleSerial(), a single thread will be used. If it is EnsembleThreads() (the default value), multiple threads will be used. See the documentation of Bootstrap for more details on constructing an instance of Bootstrap.

infer(fpm::FittedPumasModel, alg::MarginalMCMC; level=0.95)

Infer the uncertainty of the parameters of the fitted model fpm by sampling from the marginal likelihood with the MCMC algorithm alg.

The confidence intervals are calculated as the (1-level)/2 and (1+level)/2` quantiles of the sampled parameters.

infer(fpm::FittedPumasModel, sir::SIR; level=0.95, ensemblealg = EnsembleThreads()) -> FittedPumasModelInference

Perform sampling importance re-sampling for the Subjects from the Population stored in fpm. The confidence intervals are calculated as the (1-level)/2 and (1+level)/2quantiles of the estimated parameters.ensemblealgcan beEnsembleThreads()(the default value) to use multi-threading orEnsembleSerial()` to use a single thread.

init_params(model::PumasModel)

Create a parameter set populated with the initial parameters of PumasModel.

init_randeffs(model::AbstractPumasModel, param::NamedTuple, [, pop::Population])

Create an object with random effects locked to their mean values.

The optional argument pop takes a Population and changes the output to a vector of such random effects with one element for each subject within the population.

inspect(fpm::AbstractFittedPumasModel; wres_approx::LikelihoodApproximation, nsim::Int, rng)

Output a summary of the model predictions, residuals, Empirical Bayes estimates, and NPDEs (when requested).

Called on a fit output and allows the keyword argument wres_approx for approximation method to be used in residual calculation. The default value is the approximation method used for the marginal likelihood calculation in the fit that produced fpm. The keyword nsim controls the number of times each subject is simulated for npde computation. A FittedPumasModelInspection object with pred, wres, ebes, ewres, and npdes is output.

lrtest(fpm_0::AbstractFittedPumasModel, fpm_A::AbstractFittedPumasModel)::LikelihoodRatioTest

Compute the likelihood ratio test statistic of the null hypothesis defined by fpm_0 against the the alternative hypothesis defined by fpm_A. The pvalue function can be used for extracting the p-value based on the asymptotic Χ²(k) distribution of the test statistic.

lrtest(loglikelihood_0::Real, loglikelihood_A::Real, dof_0::Integer, dof_A::Integer)::LikelihoodRatioTest

Compute the likelihood ratio test using number.

marginal_nll(model, subject, param, approx, ...)
marginal_nll(model, population, param, approx, ...)

Compute the marginal negative loglikelihood of a subject or dataset, using the integral approximation approx.

See also loglikelihood.

npde(fpm::AbstractFittedPumasModel; nsim::Integer)

Calculate the normalised prediction distribution errors (NPDE).

The keyword nsim controls the number of times each subject is simulated.

penalized_conditional_nll(m::AbstractPumasModel, subject::Subject, param::NamedTuple, randeffs::NamedTuple; diffeq_options)

Compute the penalized conditional negative log-likelihood. This is the same as conditional_nll, except that it incorporates the penalty from the prior distribution of the random effects. This penalty is equal up to a constant to the log-probability-density of the given randeffs under the prior distribution.

Here param can be either a NamedTuple or a vector (representing a transformation of the random effects to Cartesian space).

postprocess(obs::SimulatedObservations)
postprocess(func::Function, obs::SimulatedObservations)
postprocess(obs::SimulatedPopulation; stat = identity)
postprocess(func::Function, obs::SimulatedPopulation; stat = identity)

Post-processes the output of simobs. The output of a simobs call stores the simulated observations but also all the intermediate values computed such as the parameters used, individual coefficients, dose control parameters, covariates, differential equation solution, etc. There are a number of post-processing operations you can do on the simulation output to compute various queries and summary statistics based on the simulations.

The postprocess function is a powerful tool that allows you to make various queries using the simulation results. There are multiple ways to use the postprocess function. The first way to use the postprocess function is to extract all the information stored in the simulation result in the form of a vector of named tuples. Each named tuple has all the intermediate values evaluated when simulating 1 run. Let sims be the output of any simobs operation.

generated = Pumas.postprocess(sims)

generated is a vector of simulation runs. Hence, generated[i] is the result from the ith simulation run. Time-dependent variables are stored as a vector with 1 element for each time point where there is an observation. Alternatively, if obstimes was set instead when calling simobs, the time-dependent variables will be evaluated at the time points in obstimes instead.

The second way to use postprocess is by passing in a post-processing function. The post-processing function can be used to:

• Transform the simulated quantities, or
• Compare the simulated quantities to the observations.

We use the do syntax here which is short for passing in a function as the first argument to postprocess. The do syntax to pass in a post-processing function is:

Pumas.postprocess(sims) do gen, obs
  ...
end

where gen is the named tuple of all generated quantities from 1 simulation run and obs is the named tuple of observations. For instance to query the ratio of simulated observations dv that are higher than the observed quantity dv at the respective time points, you can use:

Pumas.postprocess(sims) do gen, obs
  sum(gen.dv .> obs.dv) / length(gen.dv)
end

gen.dv is the simulated vector of dv whose length is the same as the number of observation time points. obs.dv is the observation vector dv. gen.dv .> obs.dv returns a vector of true/false, with one element for each time point. The sum of this vector gives the number of time points where the simulation was higher than the observation. Dividing by the number of time points gives the ratio. When using postprocess in this way, the output is always a vector of the query results. In the query function body, you can choose to use only gen or only obs but the header must always have both gen and obs:

Pumas.postprocess(sims) do gen, obs
  ...
end

The third way to use the postprocess function is to compute summary statistics of the simulated quantities or of functions of the simulated quantities. Summary statistics can be computed by passing a statistic function as the stat keyword argument. For example in order to estimate the probability that a simulated value is higher than an observation, you can use:

Pumas.postprocess(sims, stat = mean) do gen, obs
  gen.dv .> obs.dv
end

This function will do 2 things:

• Concatenate the query results (e.g. gen.dv .> obs.dv) from all the simulation runs into a single vector.
• Compute the mean value of the combined vector.

Alternatively, you can use the mean function to do the same thing without using the keyword argument. The following will call the postprocess function under the hood.

mean(sims) do gen, obs
  gen.dv .> obs.dv
end

The result of this operation will be a scalar equal to the mean value of the concatenated vector of queries.

In order to get the probability that the simulated quantity is higher than the observation for each time point, you can call the mean function externally as such:

generated = Pumas.postprocess(sims) do gen, obs
  gen.dv .> obs.dv
end
mean(generated)

This returns a vector of probabilities of the same length as the number of time points without concatenating all the queries together.

To compute a summary statistic of all the generated quantity, you can also call:

Pumas.postprocess(sims, stat = mean)

without specifying a post-processing function or for short:

mean(sims)

Besides mean, you can also use any of the following summary statistic functions in the same way:

• std for element-wise standard deviation
• var for element-wise variance
• cor for correlation between multiple quantities
• cov for covariance between multiple quantities

These functions can be passed in as the stat keyword argument to postprocess or they can be used in the short form, e.g.:

generated = Pumas.postprocess(sims, stat = std) do gen, obs
  ...
end
std(sims) do gen, obs
  ...
end
std(sims)

generated = Pumas.postprocess(sims, stat = var) do gen, obs
  ...
end
var(sims) do gen, obs
  ...
end
var(sims)

The cor and cov statistics are unique in that they require a post-processing function that outputs a vector. For example, to estimate the correlation between the CL and Vc parameters in a 1-compartment model, you can use any of the following:

Pumas.postprocess(s, stat = cor) do gen, obs
  [gen.CL[1], gen.Vc[1]]
end
cor(s) do gen, obs
  [gen.CL[1], gen.Vc[1]]
end

Note that gen.CL is a vector of simulated CL values for all the time points. But since the value is constant across time, we can use the first element gen.CL[1] only. cov can be used instead of cor to compute the covariance matrix. The output of this operation is either a correlation or a covariance matrix.

probstable(fpm::FittedPumasModel)

Return a DataFrame with outcome probabilities of all discrete dependent variables.

pvalue(t::LikelihoodRatioTest)::Real

Compute the p-value of the likelihood ratio test t based on the asymptotic Χ²(k) distribution of the test statistic.

qradd!(Q::AbstractMatrix, R::UpperTriangular, v::AbstractVector)

Replace the right-most column of F = Q * R with v by updating Q and R, and return the updated entry R[end, end] indicating the norm of the projection of the new column to the new orthogonal vector Q[:, end].

If v is mutable, this implementation modifies vector v as well. Only Q and R are valid on exit.

qrdelete!(Q, R)

Delete the left-most column of F = Q * R by updating Q and R.

Only Q[:, begin:(end-1)] and R[begin:(end-1), begin:(end-1)] are valid on exit.

read_pumas(filepath::AbstractString; missingstring = ["", ".", "NA"], kwargs...)
read_pumas(df::AbstractDataFrame; kwargs...)

Import NMTRAN-formatted data. You can either pass a CSV file path or a data frame as the first and only positional argument.

Keyword Arguments

• observations (default: [:dv]): a vector of column names of dependent variables.

• covariates (default: Symbol[]): a vector of column names of covariates.

• id::Symbol (default: :id): the name of the column with the IDs of the individuals. Each individual should have a unique integer or string.

• time::Symbol (default: :time): the name of the column with the time corresponding to the row. Time should be unique per ID, i.e., there should be no duplicate time values for a given subject.

• evid::Union{Symbol,Nothing} (default: nothing): the name of the column with event IDs, or nothing. Possible event IDs are:

  • 0: observation
  • 1: dose event
  • 2: other type event
  • 3: reset event (the amounts in each compartment are reset to zero and the on/off status of each compartment is reset to its initial status)
  • 4: reset and dose event

  The event ID defaults to 0 if the dose amount is 0 or missing, and 1 otherwise.

• amt::Symbol (default: :amt): the name of the column of dose amounts. If the event ID is specified and non-zero, the dose amount should be non-zero. The default dose amount is 0.

• addl::Symbol (default: :addl): the name of the column that indicates the number of repeated dose events. The number of additional doses defaults to 0.

• ii (default: :ii): the name of the column of inter-dose intervals. When the number of additional doses is specified and non-zero, this is the time to the next dose. For steady-state events with multiple infusions or bolus doses, this is the time between implied doses. The default inter-dose interval is 0. It is required to be non-zero for steady-state events with multiple infusions or bolus doses, and it is required to be zero for steady-state events with constant infusion.

• cmt::Symbol (default: :cmt): the name of the column with the compartment to be dosed. Compartments can be specified by integers, strings and Symbols. The default compartment is 1.

• rate::Symbol (default: :rate): the name of the column with the rate of administration. A rate=-2 allows the rate to be determined by Dose Control Parameters (DCP). Defaults to 0. Possible values are:

  • 0: instantaneous bolus dose
  • >0: infusion dose administered at a constant rate for a duration equal to amt/rate
  • -2: infusion rate or duration specified by the dose control parameters (see @dosecontrol)

• ss::Symbol (default: :ss): the name of the column that indicates whether a dose is a steady-state dose. A steady-state dose is defined as the result of having applied the same dose from the infinite past. Possible values of the steady-state indicator are:

  • 0: dose is not a steady state dose.
  • 1: dose is a steady state dose, and the compartment amounts are to be reset to the steady-state amounts resulting from the given dose. Compartment amounts resulting from prior dose event records are "zeroed out", and infusions in progress or pending additional doses are cancelled.
  • 2: dose is a steady state dose and the compartment amounts are to be set to the sum of the steady-state amounts resulting from the given dose plus the amounts at the event time were the steady-state dose not given.

  The default value is 0.

• route::Symbol: the name of the column that specifies the route of administration.

• mdv::Union{Symbol,Nothing} (default: nothing): the name of the column that indicates if observations are missing, or nothing.

• event_data::Bool (default: true): toggles assertions applicable to event data. More specifically, checks if the following columns are present in the DataFrame, either as the default values or as user-defined values: :id, :time, and :amt. If no :evid column is present, then a warning will be thrown, and :evid is set to 1 when :amt values are >0 or not missing, or :evid is set to 0 when :amt values are missing and observations are not missing. Otherwise, read_pumas will throw an error.

• covariates_direction::Symbol (default: :left): the direction of covariate interpolation. Either :left (Last Observation Carried Forward, LOCF) (default), or :right (Next Observation Carried Backward, NOCB). Notice, that for models with occasion variables it is important to use :left for the correct behavior of the interpolation.

• check::Bool (default: event_data): toggles NMTRAN compliance check of the input data. More specifically, checks if the following columns are present in the DataFrame, either as the default values or as user-defined values: :id, :time, :amt, :cmt, :evid, :addl, :ii, :ss, and :route. Additional checks are:

  • all variables in observations are numeric, i.e. Integer or AbstractFloat.
  • :amt column is numeric, i.e. Integer or AbstractFloat.
  • :cmt column is either a positive Integer, an AbstractString, or a Symbol.
  • :amt values must be missing or 0 when evid = 0; or >=0 otherwise.
  • all variables in observation are missing when evid = 1.
  • :ii column must be present if :ss is specified or present.
  • :ii values must be missing or 0 when evid = 0.
  • :ii column must be >0 if :addl values are >0, and vice-versa.
  • :addl column, if present, must be >=0 when evid = 1.
  • :evid values must be !=0 when :amt values are >0, or :addl and :ii values are >0.

• adjust_evid34::Bool (default: true): toggles adjustment of time vector for reset events (evid = 3 and evid = 4). If true (the default) then the time of the previous event is added to the time on record to ensure that the time vector is monotonically increasing.

If a column does not exist, its values are imputed to be the default values.

remove_italics(ex, vars)

Ensure that the variables specified in vars are not italicized during latexification of ex.

sample_params(model::PumasModel; rng = Pumas.default_rng())

Create a parameter set populated with values for the parameters of a PumasModel. Random values will be drawn from the prior distributions for parameters that have a prior. Otherwise, a deterministicitic value will be assigned to the parameter similar to init_params.

sample_randeffs([rng::AbstractRNG=Random.default_rng(),] model::AbstractPumasModel, param::NamedTuple [, pop::Population])

Generate a random set of random effects for model model, using parameters param. Optionally, a random number generator object rng can be passed as the first argument.

The optional argument pop takes a Population and changes the output to a vector of such random effects with one element for each subject within the population.

simobs(pmi::FittedPumasModelInference, population::Population = pmi.fpm.data, randeffs::Union{Nothing, AbstractVector{<:NamedTuple}} = nothing; samples::Int, rng = default_rng(), kwargs...,)

Simulates observations from the fitted model using a truncated multivariate normal distribution for the parameter values when a covariance estimate is available and using non-parametric sampling for simulated inference and bootstrapping. When a covariance estimate is available, the optimal parameter values are used as the mean and the variance-covariance estimate is used as the covariance matrix. Rejection sampling is used to avoid parameter values outside the parameter domains. Each sample uses a different parameter value. samples is the number of samples to sample. population is the population of subjects which defaults to the population associated the fitted model. randeffs can be set to a vector of named tuples, one for each sample. If randeffs is not specified (the default behaviour), it will be sampled from its distribution.

simobs(fpm::FittedPumasModel, [population::Population,] vcov::AbstractMatrix, randeffs::Union{Nothing, AbstractVector{<:NamedTuple}} = nothing; samples::Int, rng = default_rng(), kwargs...)

Simulates observations from the fitted model using a truncated multi-variate normal distribution for the parameter values. The optimal parameter values are used for the mean and the user supplied variance-covariance (vcov) is used as the covariance matrix. Rejection sampling is used to avoid parameter values outside the parameter domains. Each sample uses a different parameter value. samples is the number of samples to sample. population is the population of subjects which defaults to the population associated the fitted model so it's optional to pass. randeffs can be set to a vector of named tuples, one for each sample. If randeffs is not specified (the default behaviour), it will be sampled from its distribution.

simobs(
  model::AbstractPumasModel,
  population::Union{Subject, Population},
  param,
  randeffs=sample_randeffs(model, param, population);
  obstimes=nothing,
  ensemblealg=EnsembleSerial(),
  diffeq_options=NamedTuple(),
  rng=Random.default_rng(),
  simulate_error=Val(true),
)

Simulate random observations from model for population with parameters param at obstimes (by default, use the times of the existing observations for each subject). If no randeffs is provided, then random ones are generated according to the distribution in the model.

Arguments

• model may either be a PumasModel or a PumasEMModel.

• population may either be a Population of Subjects or a single Subject.

• param should be either a single parameter set, in the form of a NamedTuple, or a vector of such parameter sets. If a vector then each of the parameter sets in that vector will be applied in turn. Example: (; tvCL=1., tvKa=2., tvV=1.)

• randeffs is an optional argument that, if used, should specify the random effects for each subject in population. Such random effects are specified by NamedTuples for PumasModels (e.g. (; tvCL=1., tvKa=2.)) and by Tuples for PumasEMModels (e.g. (1., 2.)). If population is a single Subject (without being enclosed in a vector) then a single such random effect specifier should be passed. If, however, population is a Population of multiple Subjects then randeffs should be a vector containing one such specifier for each Subject. The functions init_randeffs, zero_randeffs, and sample_randeffs are sometimes convenient for generating randeffs:

randeffs = zero_randeffs(model, param, population)
solve(model, population, param, randeffs)

If no randeffs is provided, then random ones are generated according to the distribution in the model.

• obstimes is a keyword argument where you can pass a vector of times at which to simulate observations. The default, nothing, ensures the use of the existing observation times for each Subject.

• ensemblealg is a keyword argument that allows you to choose between different modes of parallelization. Options include EnsembleSerial(), EnsembleThreads() and EnsembleDistributed().

• diffeq_options is a keyword argument that takes a NamedTuple of options to pass on to the differential equation solver.

• rng is a keyword argument where you can specify which random number generator to use.

• simulate_error is a keyword argument that allows you to choose whether a sample (Val(true) or true) or the mean (Val(false) or false) of the predictive distribution's error model will be returned for each subject.

simobs(b::BayesMCMCResults; subject = nothing, samples::Int = 10, simulate_error = false, rng::AbstractRNG = default_rng())

Simulates model predictions using the posterior samples for the population and subject-specific parameters. The predictions are either for the subject with index subject or the whole population if no index is input. samples is the number of simulated predictions returned per subject. If simulate_error is false, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng is the pseudo-random number generator used in the sampling.

simobs(model::PumasModel, population::Population; samples::Int = 10, simulate_error = true, rng::AbstractRNG = default_rng())

Simulates model predictions for population using the prior distributions for the population and subject-specific parameters. samples is the number of simulated predictions returned per subject. If simulate_error is false, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng is the pseudo-random number generator used in the sampling.

simobs(model::PumasModel, subject::Subject; samples::Int = 10, simulate_error = true, rng::AbstractRNG = default_rng())

Simulates model predictions for subject using the prior distributions for the population and subject-specific parameters. samples is the number of simulated predictions returned. If simulate_error is false, the mean of the predictive distribution's error model will be returned, otherwise a sample from the predictive distribution's error model will be returned for each subject. rng is the pseudo-random number generator used in the sampling.

simobstte(
  model::PumasModel,
  subject::Union{Subject,Population},
  param::NamedTuple,
  randeffs::Union{Vector{<:NamedTuple}, NamedTuple, Nothing}=nothing;
  minT::Real=0.0,
  maxT::Real,
  nT::Integer=10,
  repeated::Bool=false,
  rng::AbstractRNG = default_rng())

Simulate observations from a time-to-event model, i.e. one with a TimeToEvent dependent variable and return either a new Subject or Population with random event time stamps in the time vector. The observations vector contains the censoring information (see TimeToEvent for more details on the data coding).

The simobstte function first computes nT values of the survival probability from t=minT to maxT and then interpolates with a cubic spline to get a smooth survival function. The user must supply maxT. Given a survival function, it is possible to simulate from the distribution by using inverse cdf sampling. Instead of sampling a uniform variate to be compared with the survival function, we sample an exponential variate and compare with cumulative hazard. The two approaches are equivalent. The Roots package is then used for computing the root. If repeated=true then new event times are drawn until maxT has been reached.

strip_t(ex, vars)

Strip the (t) from the vars in the expression ex.

Works both when (t) is appended using expressions, :(x(t)), or symbols, Symbol("x(t)").

successful_fits(vfpm::Vector{<:AbstractFittedPumasModel)
successful_fits(fpm::FittedPumasModelInference{<:Any, <:Bootstraps})
successful_fits(fpm::Bootstraps)

Returns a vector of all successful fits stored in a vector of fits or the results of Bootstrap inference. Can be compared to the input number of samples or length of vector to calculate the fraction of successful fits.

tad(t, events) -> time after most recent dose

Converts absolute time t (scalar or array) to relative time after the most recent dose. If t is earlier than all dose times, then the (negative) difference between t and the first dose time is returned instead. If no dose events exist, t is returned unmodified.

truncated_latent(
    distribution::UnivariateDistribution,
    lower::Union{Real,Nothing},
    upper::Union{Real,Nothing},
)
truncated_latent(
    distribution::UnivariateDistribution;
    lower::Union{Real,Nothing} = nothing,
    upper::Union{Real,Nothing} = nothing,
)

Construct a truncated distribution based on distribution and the truncation limits lower and upper. When fitting a model with a truncated_latent dependent variable, the log probability of the truncated distribution is used in the likelihood calculation. However, when calculating the mean or other summary statistics, the underlying untruncated distribution is used.

variables(dynobj)

Get a list of variable names for a model or dynamics object.

vech(A; diag = true)

Converts the symmetric matrix A to a vector of lower triangular elements. If diag is set to true, the vector will include the diagonal elements of A. If diag is false, the vector will only include the strict lower triangle and the diagonal elements of A will be ignored. For example when diag is true:

A = [
  1.0 0.1
  0.1 2.0
]
Pumas.vech(A, diag = true)

gives

3-element Vector{Float64}:
 1.0
 0.1
 2.0

And when diag is false:

Pumas.vech(A, diag = false)

gives

1-element Vector{Float64}:
 0.1

vechinv(x; diag = nothing, lower = nothing, lower = true)

Converts a vector of lower or upper triangular elements (traversed column by column) to a symmetric matrix. If diag is set to nothing, the vector is assumed to include diagonal elements. Otherwise, the vector is assumed to only include the strict triangle and the diagonal elements of the output matrix will be set to the value of diag. The keyword argument lower can be used to specify whether the vector is that of lower triangular elements or upper triangular elements. For example when lower is true and diag is not set:

Pumas.vechinv([1.0, -0.1, 2.0], lower = true)

gives

2×2 Matrix{Float64}:
  1.0  -0.1
 -0.1   2.0

And when diag is 1:

Pumas.vechinv([-0.1], diag = 1)

gives

2×2 Matrix{Float64}:
  1.0  -0.1
 -0.1   1.0

vpc(
  sims;
  qreg_method = IP(),
  observations::Array{Symbol}, # defaults to all observations
  stratify_by::Array{Symbol} = Symbol[],
  quantiles::NTuple{3,Float64} = (0.1, 0.5, 0.9),
  level::Real = 0.95,
  ensemblealg = EnsembleThreads(),
  ensemblealg = EnsembleSerial(),
  bandwidth::Real = 2,
  maxnumstrats::Array{<:Integer} = [6 for i = 1:length(stratify_by)],
  covariates::Array{Symbol} = [:time],
  smooth::Bool = true,
)

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.

The following keyword arguments are supported:

• samples: The number of simulated populations to generate, defaults to 499
• quantiles::NTuple{3,Float64}: A three-tuple of the quantiles for which the quantiles will be computed. The default is (0.1, 0.5, 0.9) which computes the 10th, 50th and 90th percentile.
• level::Real: Probability level to use for the simulated confidence intervals. The default is 0.95.
• observations::Vector{Symbol}: The name of the dependent variable to use for the VPCs. The default is the first dependent variable in the Population.
• stratify_by: The covariates to be used for stratification. Takes an Vector of the Symbols of the stratification covariates.
• ensemblealg: This is the parallelization choice.
• bandwidth: The kernel bandwidth in the quantile regression. If you are seeing NaNs or an error, increasing the bandwidth should help in most cases. With higher values of the bandwidth you will get more smoothened plots of the quantiles so it's a good idea to check with your data to pick the right bandwidth. For a DiscreteVPC, the default value is nothing to indicate automatic bandwidth tuning. For a CensoredVPC, the bandwidth should be given as a tuple where the first value applies to the ContinuousVPC and the second to the DiscreteVPC for the censoring variable. For example, (4.0, nothing) applies a bandwidth of 4.0 to the ContinuousVPC and automatically tunes the DiscreteVPC. Automatic tuning is only applicable to DiscreteVPCs at this moment.
• maxnumstrats: The maximum number of strata for each covariate. Takes an Vector with the number of strata for the corresponding covariate, passed in stratify_by. It defaults to 6 for each of the covariates.
• covariates: The independent variable for VPC, defaults to [:time].
• smooth: In case of discrete VPC used to determine whether to interpolate the dependent variable if independent variable is continuous, defaults to true.
• rng: A random number generator, uses the default_rng from Random as default.
• nodes: A Dict specifying the local regression nodes. If stratify_by argument has been specified, the dict should have tuples of values as keys corresponding to each stratum. Otherwise, the key should be an empty tuple. The values of the Dict should be the nodes of the local regressions. If the nodes argument is not passed, at most nnodes quantiles is used from each stratum.
• nnodes: An integer specifying the number of points to use for the local regressions for each stratum. The default value is 11 but the actual number of nodes will not exceed the number of unique covariate values in the stratum.
• qreg_method: Defaults to IP(). For most users the method used in quantile regression is not going to be of concern, but if you see large run times switching qreg_method to IP(true) should help in improving the performance with a tradeoff in the accuracy of the fitting.
• prediction_correction: Default to false. Set to true to enable prediction correction that multiplies all observations with the ratio between the mean prediction and each individuals population prediction.

wresiduals(fpm::AbstractFittedPumasModel, approx::LikelihoodApproximation; nsim=nothing)

Calculate the individual and population weighted residual.

Takes a fit result, an approximation method for the marginal likelihood calculation which defaults to the method used in the fit and the number of simulations with the keyword argument nsim. If nsim is specified only the Expected Simulation based Individual Weighted Residuals (EIWRES) is included in the output as individual residual and population residual is not computed. Using the FO approximation method corresponds to the WRES and while FOCE(I) corresponds to CWRES. The output is a SubjectResidual object that stores the population (wres) and individual (iwres) residuals along with the subject and approximation method (approx).

zero_randeffs(model::AbstractPumasModel, param::NamedTuple)
zero_randeffs(model::AbstractPumasModel, pop::Population, param::NamedTuple)

Create an object to signify that the random effects should be zero.

The optional argument pop takes a Population and changes the output to a vector of such random effects with one element for each subject within the population.

ηshrinkage(fpm::AbstractFittedPumasModel)

Calculate the η-shrinkage.

Takes the result of a fit as the only input argument. A named tuple of the random effects and corresponding η-shrinkage values is output.

ϵshrinkage(fpm::AbstractFittedPumasModel)

Calculate the ϵ-shrinkage.

Takes the result of a fit as the only input argument. A named tuple of derived variables and corresponding ϵ-shrinkage values is output.

aic(fpm::AbstractFittedPumasModel)

Calculate the Akaike information criterion (AIC) of the fitted Pumas model fpm.

bic(fpm::AbstractFittedPumasModel)

Calculate the Bayesian information criterion (BIC) of the fitted Pumas model fpm.

coeftable(pmi::FittedPumasModelInference; level = pmi.level) -> DataFrame

Construct a DataFrame of parameter names, estimates, standard error, and confidence interval with confidence level level. The default value of level is the one that was passed in infer.

coeftable(fpm::FittedPumasModel) -> DataFrame

Construct a DataFrame of parameter names and estimates from fpm.

coeftable(vfpm::Vector{<:FittedPumasModel}) -> DataFrame

Construct a DataFrame of parameter names and estimates and their standard deviation from vector of fitted single-subject models vfpm.

fit(fpm::AbstractFittedPumasModel; kwargs...)

Convenience method to start a new fit where an old one left off.

This uses both the fitted parameters and the fitted random effects from fpm as the starting points for the new fit.

Distributions.fit(
  model::PumasModel,
  data::Population,
  param::NamedTuple,
  alg::BayesMCMC,
)

Samples from the posterior distribution of the model's population and subject-specific parameters given the observations in data using the Hamiltonian Monte Carlo (HMC) based No-U-Turn Sampler (NUTS) algorithm. The MCMC sampler is initialized starting from param for the population parameters. The subject-specific parameters are also apprioriately initialized. The Bayesian inference algorithm's options are passed in the alg struct. Run ?BayesMCMC for more details on the options that can be set.

fit(
  model::PumasModel,
  population::Population,
  param::NamedTuple,
  approx::Union{LikelihoodApproximation, MAP};
  optim_alg = Optim.BFGS(linesearch = Optim.LineSearches.BackTracking(), initial_stepnorm = 1.0),
  optim_options::NamedTuple = NamedTuple(),
  optimize_fn = nothing,
  constantcoef::Tuple = (),
  ensemblealg::SciMLBase.EnsembleAlgorithm = EnsembleThreads(),
  checkidentification = true,
  diffeq_options = NamedTuple(),
  verbose = true,
  ignore_numerical_error = true,
  )

Fit the Pumas model model to the dataset population with starting values param using the estimation method approx. Currently supported values for the approx argument are FO, FOCE, LaplaceI, NaivePooled, and BayesMCMC. See the online documentation for more details about the different methods.

To control the optimization procedure for finding population parameters (fixed effects), use the optim_alg and optim_options keyword arguments. In previous versions of Pumas the argument optimize_fn was used, but is now discouraged and will be removed in a later version of Pumas. These options control the optimization all approx methods except BayesMCMC. The default optimization function uses the quasi-Newton routine BFGS method from the Optim package. It can be changed by setting the optim_alg to an algorithm implemented in Optim.jl as long as it does not use second order derivatives. Optimization specific options can be passed in using the optim_options keyword and has to be a NamedTuple with names and values that match the Optim.Options type. For example, the optimization trace can be disabled and the algorithm can be changed to L-BFGS by passing optim_alg=Optim.LBFGS(), optim_options = ;(show_trace=false) to fit. See Optim for more defails.

It is possible to fix one or more parameters of the fit by passing a Tuple of Symbols as the constantcoef argument with elements corresponding to the names of the fixed parameters, e.g. constantcoef=(:σ,).

When models include an @random block and fitting with NaivePooled is requested, it is required that the user sets all variability parameters to zero with constantcoef such that these can be ignored in the optimization, e.g. constantcoef = (:Ω,) while overwriting the corresponding values in param with (; init_params..., Ω = zeros(3, 3)).

Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. Currently, the only supported options are:

• EnsembleThreads(): the default. Accelerate fits by using multiple threads.
• EnsembleSerial(): fit using a single thread.
• EnsembleDistributed(): fit by using multiple worker processes.

The fit function will check if any gradients and throw an exception if any of the elements are exactly zero unless checkidentification is set to false.

Further keyword arguments can be passed via the diffeq_options argument. This allows for passing arguments to the differential equations solver such as alg, abstol, and reltol. The default values for these are AutoVern7(Rodas5P(autodiff=true)), 1e-12, and 1e-8 respectively. See the DifferentialEquations.jl documentation for more details.

The keyword verbose controls if info statements about initial evaluations of the loglikelihood function and gradient should be printed or not. Defaults to true.

Since the numerical optimization routine can sometimes visit extreme regions of the parameter space during it's exploration we automatically handle the situation where the input parameters result in errors when solving the dynamical system. This allows the algorithm to recover and continue in many situations that would have otherwise stalled early. Sometimes, it is useful to turn this error handling off when debugging a model fit, and this can be done by setting ignore_numerical_error = false.

fit(
  model::PumasEMModel,
  population::Population,
  param::NamedTuple,
  approx::Union{SAEM,LaplaceI};
  ensemblealg = EnsembleThreads(),
  rng = default_rng()
)

Fit the PumasEMModel to the dataset population using the initial values specified in param. The supported methods for the approx argument are currently SAEM and LaplaceI. See the online documentation for more details about these two methods.

When fitting with SAEM the variance terms of the random effects (Ω) and the dispersion parameters for the error model (σ) are initialized to the identity matrix or 1 as appropriate. They may also be specified. With SAEM, it is recommended to choose an init larger than the true values to facilitate exploration of the parameter space and avoid getting trapped in local optima early. Currently LaplaceI fits with a PumasEMModel require Ω to be diagonal.

Options for ensemblealg are EnsembleSerial() and EnsembleThreads() (the default). The fit will be parallel if ensemblealg == EnsembleThreads(). SAEM imposes the additional requirement for threaded fits that either the rng used supports Future.randjump or that rng be a vector of rngs, one per thread. The default_rng() supports randjump().

loglikelihood(fpm::AbstractFittedPumasModel)

Compute the loglikelihood of a fitted Pumas model.

loglikelihood(b::BayesMCMCResults; split_method, split_by)

Computes a 3D array of the in-sample loglikelihoods of all the data points/subsets using the output of the MCMC fit function. The output is a 3D array of size (ndata, nsamples, nchains) where ndata is the number of data points, nsamples is the number of MCMC samples per chain and nchains is the number of chains.

The way by which the data is split into points/subsets is determined using the split_method and the split_by keyword arguments. The split_method argument can be of any of the following types:

• LeaveK for a leave-K-out splitting
• KFold for a K-fold splitting
• LeaveFutureK for leaving K future points at a time while split_by can be of any of the following types:
• BySubject for leaving out subjects
• ByObservation for leaving out individual observations per subject

Please refer to the documentation of each of the above types for the parameters of each split method, e.g. by typing ?LeaveK or ?BySubject.

Examples:

Assume there are 5 subjects and 10 observations per subject. The following are some of the combinations in which the above inputs can be used:

• Leave-1-observation-out splitting, splitting 1 observation for all the subjects at a time. allsubjects = true means that the same observation index is removed for all the subjects, e.g. the 10th observation for all the subjects is used as a single subset, then the 9th observation is used for all the subjects is used as another subset, etc.

      split_method = LeaveK(K = 1)
      split_by = ByObservation(allsubjects = true)
      loglikelihood(b; split_method = split_method, split_by = split_by)

• Leave-2-future-observations-out splitting, splitting 2 future observations per subject such that no less than 4 past observations are remaining. allsubjects = false means that only the data of one subject at a time will be split. So observations 5 and 6 of subject 1 are the first data subset. The second subset is observations 7 and 8 of subject 1. The third subset is observations 9 and 10 of subject 1, etc. Then in the fourth to sixth runs, subject 2's data gets split. In the seventh to ninth runs, subject 3's data gets split, etc.

      split_method = LeaveFutureK(K = 2, minimum = 4)
      split_by = ByObservation(allsubjects = false)
      loglikelihood(b; split_method = split_method, split_by = split_by)

• Leave-1-subject-out splitting, marginalizing the subject-specific parameters using quadrature integration when computing the in-sample likelihoods. Using this method, each subject is a data point. The in-sample likelihoods computed are the marginal likelihoods of the subjects computed using Gaussian quadrature with a maximum of 1000 integration points. Refer to the documentation of LLQuad by typing ?LLQuad for more details on the parameters of the quadrature algorithm. FO(), FOCE() and LaplaceI() can also be used instead.

      split_method = LeaveK(K = 1)
      split_by = BySubject(marginal = LLQuad(imaxiters = 1000))
      loglikelihood(b; split_method = split_method, split_by = split_by)

• Leave-1-future-subject-out splitting, where the in-sample likelihoods are conditional likelihoods computed using 0s for the subject-specific parameters instead of marginalizing. In this method, each subject after the third one is considered a data point, i.e. the 4th and 5th subjects are the only 2 points for which the in-sample likelihood is computed.

      split_method = LeaveFutureK(K = 1, minimum = 3)
      split_by = BySubject(marginal = nothing)
      loglikelihood(b; split_method = split_method, split_by = split_by)

loglikelihood(r::BayesianCVResult)

Computes the out-of-sample loglikelihoods from the result r of crossvalidate. The output is vector of matrices, e.g. loglik where loglik[i] is the loglikelihoods matrix of data point/set i which was not used in training. loglik[i][j, k] is the loglikelihood using the parameter values from the jth sample of the kth chain.

predict(
  model::AbstractPumasModel,
  subject::Subject,
  param::NamedTuple,
  [randeffs,];
  [obstimes::AbstractVector,
  diffeq_options::NamedTuple]
)::Union{SubjectPrediction,Vector{SubjectPrediction}}

Compute population and individual predictions for the single subject based on model and the population parameters param. A NamedTuple of random effects, randeffs, can be omitted or provided by the user. If they are omitted, they will be estimated from the data in the subject.

If the optional obstimes argument is passed then the time points in obstimes are used for the predictions. Otherwise, the time points of the observations of the subject are used for the predictions.

The function allows for extra keyword arguments to be passed on to the differential equations solver through the diffeq_options keyword. See the online documentation for more details.

predict(
  fpm::AbstractFittedPumasModel,
  [population::Union{Subject,Population}];
  [obstimes::AbstractVector]
)::Union{SubjectPrediction,Vector{SubjectPrediction}}

Compute population and individual predictions for the fitted model fpm. By default, the predictions are computed for the estimation data but the predictions can also be computed for user supplied data by passing either a single subject or a vector of subjects (Population) as the population argument.

If the optional obstimes argument is passed then the time points in obstimes are used for the predictions. Otherwise, the time points of the observations for each subject in the population are used for the predictions.

Any optional keyword arguments used when fitting fpm are reused when computing the predictions.

stderror(f::AbstractFittedPumasModel) -> NamedTuple

Compute the standard errors of the population parameters and return the result as a NamedTuple matching the NamedTuple of population parameters.

vcov(f::AbstractFittedPumasModel) -> AbstractMatrix

Compute the covariance matrix of the population parameters

@cache

Defines the model's cache variables. Must be used in an @model block. For example:

@model begin
  @cache begin
    onoff = 1.0
  end
end

@covariance

Allows specifying a block-diagonal covariance structure for the Ω parameters in an @emmodel. The default is fully diagonal.

For example, to specify that the Ω is block diagonal with 1x1, 2x2, and 2x2 blocks:

@emmodel begin
  @random begin
    Ka ~ 1 | LogNormal
    CL ~ 1 | LogNormal
    Vc ~ 1 | LogNormal
    Q ~ 1 | LogNormal
    Vp ~ 1 | LogNormal
  end
  @covariance 1 2 2
end

This would mean that both CL and Vc may be correlated with each other, and Q and Vp may also be, but all other pairwise correlations are zero. @covariance 5 would result in a dense matrix.

Only diagonal is currently supported when a PumasEMModel with LaplaceI.

@covariates

Defines the model's covariates, e.g. wt. Must be used in an @model block. For example:

@model begin
  @covariates wt
end

@delay(
    distribution::Distribution,
    cmt,
) -> Real

Compute the distributed delay contribution for distribution for all dose amounts in administered to cmt up until the current time point. This macro should be used in the @pre block to compute an absorption rate explicitly added to the ODE in the @dynamics block.

Examples

Weibull delay

The model below implements the Weibull absorption model of Hartmann, D., Gysel, D., Dubach, U. C., & Forgo, I. (1990)

mdl_weibull = @model begin
  @param begin
    θCb ∈ RealDomain(lower = 0.0)
    θVc ∈ RealDomain(lower = 0.0)
    θVm ∈ RealDomain(lower = 10.0)
    θKm ∈ RealDomain(lower = 0.0)

    θA ∈ RealDomain(lower = 0.0)
    θB ∈ RealDomain(lower = 0.0)

    θf ∈ RealDomain(lower = 0.0)

    σ ∈ RealDomain(lower = 0.0)
  end

  @pre begin
    Baseline = θCb * θVc
    Vc = θVc
    VVm = Vc * θVm
    VKm = Vc * θKm

    λ = θA^(-1 / θB)
    B = θB

    f = θf

    dose_density = @delay(Weibull(B, λ), Central)
  end

  @dosecontrol begin
    bioav = (; Central = 0.0)
  end

  @init begin
    Central = Baseline
  end

  @dynamics begin
    Central' =
      VVm * (Baseline / (VKm + Baseline) - Central / (VKm + Central)) +
      f * Vc * dose_density
  end

  @derived begin
    μ := @. Central ./ Vc
    y ~ @. Gamma(1 / σ^2, μ * σ^2)
  end
end

Gamma delay

The model below below uses the the delay function to implement a gamma distributed delay demonstrated in Savic, Jonker, Kerbusch, and Karlsson (2007).

mdl_gamma_delay = @model begin
  @param begin
    θn ∈ RealDomain(lower = 0.0)
    θmat ∈ RealDomain(lower = 0.0)
    θCL ∈ RealDomain(lower = 0.0)
    θVc ∈ RealDomain(lower = 0.0)

    σ ∈ RealDomain(lower = 0.0)
  end

  @pre begin
    kₑ = θCL / θVc
    Vc = θVc
    dose_density = @delay(Gamma(θn, θmat / θn), Central)
  end

  @dosecontrol begin
    bioav = (; Central = 0.0)
  end

  @dynamics begin
    Central' = dose_density - kₑ * Central
  end

  @derived begin
    μ := Central ./ Vc
    y ~ @. Gamma(1 / σ^2, μ * σ^2)
  end
end

@derived

Defines the error model of the dependent variables. Must be used in an @model block. For example:

@model begin
  @derived begin
    dv ~ @. Normal(1000 * Central / Vc, 1000 * Central / Vc * σ)
  end
end

@dosecontrol

Define dose control parameters as a function of fixed and random effects. Options include bioav, duration, lags, and rate. Must be used in an @model block. For example:

@model begin
  @dosecontrol begin
    bioav = (Depot1 = max(0, θ[5]), Depot2 = clamp(1 - θ[5], 0.0, 1.0), Central = 1)
    lags = (Depot1 = 0, Depot2 = max(0.1, θ[6]), Central = 0)
  end
end

@dynamics

Defines the dynamic system by specifying differential equations. Must be used in an @model block.

Examples

A @dynamics block with a system of differential equations:

@model begin
  @dynamics begin
    Depot1' = -ka * Depot1
    Depot2' = ka * Depot1 - ka * Depot2
    Central' = ka * Depot2 - CL / Vc * Central
  end
end

A @dynamics block with a differential equation model:

@model begin
  @dynamics Depots1Central1
end

@emmodel

Define a PumasEMModel. It may have the following blocks:

• @param and @random: Defines fixed and random effects, e.g.

@random begin
  CL ~ 1 + wt | LogNormal
  θbioav ~ 1 | LogitNormal
end

Distributions specify the parameter's support. Only Normal (-∞,∞) LogNormal (0,∞), and LogitNormal (0,1) are currently supported.

These define 1 unconstrained parameter per term in the formula, for example CL = exp(μ + wt * β + η) where μ, β ∈ (-∞,∞) and wt is a covariate constant with respect to time. η is a random effect. The @param block is equivalent, with the exception that there is no random effect η. These variables are accessible in @pre, @dosecontrol, @dynamics, @post, and @error blocks.

• @covariates: Covariates available in the @pre, @dosecontrol, @dynamics, and @post blocks.
• @pre: Block evaluated before the @dynamics.
• @dosecontrol: Block specifying dose control parameters. Function of @param and @random variables only.
• @init: return initial conditions for dynamical variables.
• @dynamics: Dynamics, equivalent to the functionality in the @model macro.
• @post: Block evaluated after the @dynamics and before the @error.
• @error: Block describing the error model. Dispersion parameters are implicit. Y ~ ProportionalNormal(μ) indicates that Y has a Normal distribution with mean μ. Y must be observed subject data, while μ can can be defined in the @param, @random, @pre, @dynamics, or @post blocks.

@error

Defines the error model of the dependent variables in a PumasEMModel. Distributions are parameterized by the mean only, the dispersion parameter(s), if any, are left implicit. See the documentation for details on the available error models. The mean parameter(s) must be (a) symbol(s). Use an earlier block, e.g. @post to define any necessary transformations.

Must be used in an @emmodel block. For example:

@emmodel begin
  @random begin
    CL ~ 1 | LogNormal
    Vc ~ 1 | LogNormal
  end
  @dynamics Central1
  @post begin
    cp = Central / Vc # cp is calculated here
  end
  @error begin
    dv ~ CombinedNormal(cp) # cp is the mean of the error distribution
  end
end

@init

Defines initial conditions of the dynamic system. Must be used in an @model block. For example:

@model begin
  @init begin
    Depot1 = 0.0
    Depot2 = 0.0
    Central = 0.0
  end
end

@metadata

Allows including additional details of a model as metadata. These take the form of key/value pairs defined with = as follows, where the values can take on <:Any value:

@model begin
  @metadata begin
    desc = "A short description of the model."
    timeu = u"hr"
  end
end

In the above example, a textual description of the model is provided by the desc key. It must be written in plaintext and should preferably not take up more than a single line. The timeu key is assigned a Unitful value to describe the units used in the model. At present desc and timeu are the only metadata keys that are used.

@model

Defines a Pumas model, with the following possible blocks:

• @param: defines the model's fixed effects and corresponding domains, e.g. tvcl ∈ RealDomain(lower = 0).
• @random: defines the model's random effects and corresponding distribution, e.g. η ~ MvNormal(Ω).
• @covariates: Names of covariates used in the model.
• @pre: pre-processes variables for the dynamic system and statistical specification.
• @dosecontrol: defines dose control parameters as a function of fixed and random effects.
• @vars: define variables usable in other blocks. It is recommended to define them in @pre instead if not a function of dynamic variables.
• @init: defines initial conditions of the dynamic system.
• @dynamics: defines the dynamic system.
• @derived: defines the statistical model of the dependent variables.
• @observed: lists model information to be stored in the solution.
• @options specifies model options, including checklinear, inplace, and subject_t0.

 @nca

Defines a macro that can be used in the observed block to compute NCA metrics of the computed model solutions.

Example

...
@derived begin
  cp := Central/Vc
  DV ~ @. Normal(cp, σ*cp)
end

@observed begin
  nca = @nca cp
  auc = NCA.auc(nca, interval = (0,4))
  cmax = NCA.cmax(nca)
end
...

@observed

Lists model information to be stored in the solution. Must be used in an @model block. For example:

@model begin
  @observed begin
    conc = @. 1000 * Central / Vc
  end
end

@options

Specifies model options, including the following:

• checklinear::Bool: determines whether the solver should check if the system defined in the @dynamics block is linear. If the system is linear, setting this option to true (default) enables calculating the solution through matrix exponentials. If it is set to false or time (t) appears in the @pre block, this optimization is disabled. This option can be useful when the matrix exponential solver is not superior to general numerical integrators or for debugging purposes.

• inplace::Bool: controls whether the solver uses mutable operations on Arrays (true), which can improve performance for large systems. By default, it is set to true for systems containing more than 5 dynamic variables, and set to false otherwise.

• subject_t0::Bool: determines the starting time point from which the solver integrates the system. By default, all systems are solved from time 0 to the last time point in the subject, making subject_t0 false. If subject_t0 is set to true, the solver will begin integration from the first time point found within the subject.

Must be used in an @model block. For example (defaults are shown below):

@model begin
  @options begin
    checklinear = true
    inplace = length(odevars) < 5
    subject_t0 = false
  end
end

@param

Defines the model's fixed effects and corresponding domains, e.g. tvcl ∈ RealDomain(lower = 0). Must be used in an @model block. For example:

@model begin
  @param begin
    tvcl ∈ RealDomain(lower = 0)
    tvv ∈ RealDomain(lower = 0)
    Ω ∈ PDiagDomain(2)
    σ_prop ∈ RealDomain(lower = 0, init = 0.04)
  end
end

For PumasEMModels, the @param block uses a formula syntax for covariate relationships. A distribution from the normal family is used to indicate domain (currently only Normal, LogNormal, and LogitNormal are supported):

@emmodel begin
  @param begin
    CL ~ 1 + wt | LogNormal
    Vc ~ 1 | Normal
  end
end

this gives CL = exp(log(θ.CL[1]) + θ.CL[2] * wt) and VC = θ.VC, where θ is the parameter named tuple. If we only have 1 in the formula, then it is a scalar. If we have covariates, we have a tuple of parameters for variable.

@post

Allows for post-processing dynamical variables, to define any variables needed in the @error block.

Must be used in an @emmodel block. For example:

@emmodel begin
  @random begin
    CL ~ 1 | LogNormal
    Vc ~ 1 | LogNormal
  end
  @dynamics Central1
  @post begin
    cp = Central / Vc
  end
  @error begin
    dv ~ CombinedNormal(cp)
  end
end

@pre

Pre-processes variables for the dynamic system and statistical specification. Must be used in an @model block. For example:

@model begin
  @params begin
    tvcl ∈ RealDomain(lower = 0)
    tvv ∈ RealDomain(lower = 0)
    ωCL ∈ RealDomain(lower = 0)
    ωV ∈ RealDomain(lower = 0)
    σ_prop ∈ RealDomain(lower = 0, init = 0.04)
  end
  @random begin
    ηCL ~ Normal(ωCL)
    ηV ~ Normal(ωV)
  end
  @covariates wt
  @pre begin
    CL = tvcl * (wt / 70)^0.75 * exp(ηCL)
    Vc = tvv * (wt / 70) * exp(ηV)
    ka = CL / Vc
    Q = Vc
  end
end

@random

defines the model's random effects and corresponding distribution, e.g. η ~ MvNormal(Ω). Must be used in an @model block. For example:

@model begin
  @random begin
    η ~ MvNormal(Ω)
  end
end

For PumasEMModels, the @random block is equivalent to the @params block, except that each variable defined here has an additional random effect. For example:

@emmodel begin
  @random begin
    CL ~ 1 + wt | LogNormal
    Vc ~ 1 | Normal
  end
end

this gives CL = exp(log(θ.CL[1]) + θ.CL[2] * wt + η_CL) and VC = θ.VC + η_VC, where θ is the parameter named tuple. If we only have 1 in the formula, then it is a scalar. If we have covariates, we have a tuple of parameters for variable.

@reactions

Defines the dynamic system by specifying chemical reactions with the Catalyst.jl DSL. Must be used in an @model block.

Example

@model begin
  @reactions begin
    ka, Depot1 --> Depot2
    ka, Depot2 --> Central
    CL / Vc, Central --> 0
  end
end

This is equivalent to

@model begin
  @dynamics begin
    Depot1' = -ka * Depot1
    Depot2' = ka * Depot1 - ka * Depot2
    Central' = ka * Depot2 - CL / Vc * Central
  end
end

@vars

Define variables usable in other blocks. It is recommended to define them in @pre instead if not a function of dynamic variables. Must be used in an @model block. For example:

@model begin
  @vars begin
    conc = Central / Vc
  end
end