# Pumas Docstrings

DataFrames.DataFrameType
DataFrame(events::DosageRegimen, expand::Bool = false)

Create a DataFrame with the information in the dosage regimen. If expand, create a DataFrame with the information in the event list (expanded form).

DataFrames.DataFrameMethod
DataFrame(bts::Union{SimulatedInference, Bootstraps})

Returns a data frame of the parameter estimates.

Distributions.MvNormalMethod
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.

Distributions.MvNormalMethod
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)
Pumas.AnalyticalPKPDProblemType
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
• bioav: bioavailability in each compartment
Pumas.AnalyticalPKProblemType

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.
Pumas.CensoredType
Censored(distribution::Distribution, lower::Real, upper::Real)::Censored

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

Pumas.Central1Type
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.

Pumas.Central1Periph1Type
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.

Pumas.Central1Periph1Meta1Type
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.

Pumas.Central1Periph1Meta1Periph1Type
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.

Pumas.ConstrainedType
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: 1-element Zeros{Float64}
Σ: [1.0]
)
, VectorDomain{Vector{Int64}, Vector{Int64}, Vector{Float64}}([-1], , [0.0]))

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

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

Pumas.Depots1Central1Type
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.

Pumas.Depots1Central1Periph1Type
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.

Pumas.Depots2Central1Type
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)

Pumas.DosageRegimenType
DosageRegimen

Lazy representation of a series of Events.

Fields

• data::DataFrame: The tabular representation of a series of Events.

• Signature

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

From various DosageRegimens

• Signature

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

Pumas.PDiagDomainType
@param x ∈ PDiagDomain(n::Int; init=ones(n))

Specifies a parameter as a positive diagonal matrix, with initial diagonal elements specified by init.

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

Specifies a parameter as a symmetric n-by-n positive semi-definite matrix. init sets the initial value of the parameter and should be a positive semi-definite Matrix of Float64s.

Pumas.ParamSetType
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.),
tvCL = RealDomain(lower=0.),
tvω = PDiagDomain(2)
))
Pumas.PumasModelType
PumasModel

A model takes the following arguments (arg [= default]):

• param: a ParamSet detailing the parameters and their domain
• random: a mapping from a named tuple of parameters -> ParamSet
• pre: a mapping from the (params, randeffs, subject) -> ODE parameters.
• dcp: a mapping from the (params, randeffs, subject) -> dose-control parameters.
• init: a mapping (col,t0) -> initial conditions
• prob: a DEProblem describing the dynamics (either exact or analytical)
• derived: the derived variables and error distributions (param, randeffs, data, ode vals) -> sampling dist
• observed = (_, _, _, _, _, _, samples) -> samples: simulated values from the error model and post processing: (param, randeffs, data, ode vals, samples) -> vals
• options::PumasModelOptions = PumasModelOptions(; subject_t0=false): pass additional options, see PumasModelOptions.
• syms::PumasModelSymbols = PumasModelSymbols(): the symbols of all the variables defined inside the model, see PumasModelSymbols.
• sys = nothing: the ModelingToolkit.jl System used to generate prob - if applicable.
• macroexpr::Expr = :(): the expression passed to the @model macro - if applicable.
• desc::Dict{Symbol, String} = Dict{Symbol, String}(): descriptions of the symbols (variables, parameters, etc.) used in the model.
• metadata::Dict{Symbol, Any} = Dict{Symbol, Any}(): include a Dict of arbitrary metadata.
Pumas.RealDomainType
@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.

Pumas.SubjectType
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 vector of Events.
• 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 = nothing,
events = Event[],
time = observations isa AbstractDataFrame ? observations.time : nothing,
event_data = true,
covariates::Union{Nothing, NamedTuple} = nothing,
covariates_time = observations isa AbstractDataFrame ? observations.time : nothing,
covariates_direction = :right)

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

Pumas.SubjectMethod

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)
Pumas.TimeToEventType
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 zero if the variable isn't censored and one if 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
...
Pumas.VectorDomainType
@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.

CommonSolve.solveFunction
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 overriden 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.

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

Distributions.logpdfMethod
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.

GlobalSensitivity.gsaFunction

GlobalSensitivity.gsa(model, population, params, method, vars, prangelow, prangehigh; kwargs...)

Function to perform global sensitivty analysis

The arguments are:

• model: a PumasModel, either defined by the @model DSL or the function-based interface.
• population: a Population.
• params: a named tuple of parameters.
• method: one of the GSAMethods from GlobalSensitivity.jl, Sobol(), Morris(), eFAST(), RegressionGSA().
• vars: a list of the derived variables 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 method specific arguments that are passed with the method constructor you can refer to the GlobalSensitivity.jl documentation.

LinearAlgebra.condMethod
cond(pmi::FittedPumasModelInference)

Return the condition number of the variance-covariance matrix 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.

Pumas.conditional_nllMethod
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.

Pumas.empirical_bayesMethod
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 Pumas.FO likelihood approximation methods the empirical bayes estimates will be obtained using the Pumas.LaplaceI approximation. If either Pumas.FOCE or Pumas.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 Pumas.LaplaceI approximation.

Pumas.eventnumMethod
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.

Pumas.expectationMethod
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().

Pumas.expectationMethod
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().

Pumas.expectationMethod
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.

Pumas.findinfluentialFunction
findinfluential(fpm::AbstractFittedPumasModel, k::Integer=5)

Return a vector of the k most influencial observations based on the value of (minus) the log-likelihood function.

Pumas.icoefMethod
icoef(fpm::AbstractFittedPumasModel)::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 return a vector of covariate interpolation objects. Each of which can be evaluated at any time point. Each of the covariate interpolation objects can be converted to a DataFrame by calling the DataFrame constructor. Hence, a complete DataFrame of the individual coefficients can be obtained by calling reduce(vcat, DataFrame.(icoef(fpm))).

Pumas.inferMethod
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.

Pumas.inferMethod

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.

Pumas.inferMethod
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.

Pumas.init_paramsFunction
init_params(model::PumasModel)

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

Pumas.init_randeffsFunction
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.

Pumas.inspectMethod
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, and npdes is output.

Pumas.lrtestMethod
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.

Pumas.probstableMethod

probstable(fpm::FittedPumasModel)

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

Pumas.pvalueMethod
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.

Pumas.read_pumasMethod
read_pumas(filepath::String, args...; kwargs...)
df::AbstractDataFrame;
observations=Symbol[:dv],
covariates=Symbol[],
id::Symbol=:id,
time::Symbol=:time,
evid::Union{Nothing,Symbol}=nothing,
amt::Symbol=:amt,
ii::Symbol=:ii,
cmt::Symbol=:cmt,
rate::Symbol=:rate,
ss::Symbol=:ss,
route::Symbol=:route,
mdv::Symbol=nothing,
event_data::Bool=true,
covariates_direction::Symbol=:right,
check::Bool=event_data,
adjust_evid34::Bool=true)

Import NMTRAN-formatted data.

• df : DataFrame contaning the data to be converted to a Vector{<:Subject}
• observations : dependent variables specified by a vector of column names
• covariates : covariates specified by a vector of column names
• id : specifies the ID column of the dataframe
• time : specifies the time column of the dataframe
• evid : specifies the event ID column of the dataframe. See ?Pumas.Event for more details.
• amt : specifies the dose amount column of the dataframe. See ?Pumas.Event for more details.
• addl : specifies the column of the dataframe that indicated the number of repeated dose events. If not specified then the value is zero.
• ii : specifies the dose interval column of the dataframe. See ?Pumas.Event for more details.
• cmt : specifies the compartment column of the dataframe. See ?Pumas.Event for more details.
• rate : specifies the infusion rate column of the dataframe. See ?Pumas.Event for more details.
• ss : specifies the steady state column of the dataframe. See ?Pumas.Event for more details.
• route : specifies the route of administration column of the dataframe. See ?Pumas.Event for more details.
• mdv : specifies the the column of the dataframe indicating if observations are missing.
• event_data : toggles assertions applicable to event data
• covariates_direction : specifies direction of covariate interpolation. Either :left or :right (default)
• check : toggles NMTRAN compliance check of the input data
• adjust_evid34 : 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.
Pumas.sample_randeffsFunction
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.

Pumas.simobsFunction
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.

Pumas.simobsFunction
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 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.

Pumas.simobsFunction
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 multi-variate normal distribution for the parameter values. The optimal parameter values are used 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.

Pumas.simobstteFunction
simobstte(
model::PumasModel,
subject::Union{Subject,Population},
param::NamedTuple,
randeffs::Union{Vector{<:NamedTuple}, NamedTuple, Nothing}=nothing;
minT=0.0,
maxT=nothing,
nT=10,
repeated=false,
rng = 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 function first computes nT values of survival probability from t=minT to maxT and then interpolate with a cubic spline to get a smooth survival funtion. Given a survival funtion, it is possible to simulate from the distribution by using inverse cdf sampling. Instead of sampling a uniform variate to use with the survival probability, we sample an exponential and compare to the cumulative hazard which is 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.

Pumas.tadMethod
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.

Pumas.vpcMethod
vpc(fpm::AbstractFittedPumasModel;
samples::Integer = 499
qreg_method = IP(),
observations::Array{Symbol} = [keys(fpm.data.observations)],
stratify_by::Array{Symbol} = Symbol[],
quantiles::NTuple{3,Float64}=(0.1, 0.5, 0.9),
level::Real=0.95,
ensemblealg=EnsembleSerial(),
bandwidth=2,
maxnumstrats=[6 for i in 1:length(stratify_by)],
covariates::Array{Symbol} = [:time],
smooth::Bool = true,
rng::AbstractRNG=default_rng(),
obstimes::AbstractVector = [])

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::Array{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 array of the Symbols of the stratification covariates.
• ensemblealg: This is passed to the simobs call for the samples simulations. For more description check the docs for simobs.
• 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 the right bandwidth.
• maxnumstrats: The maximum number of strata for each covariate. Takes an array 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.
• obstimes: The times for simulation in case of continuous VPC, same as obstimes in simobs. Defaults to union of all subject's unique times in the data where observation is not missing.
• 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.
Pumas.wresidualsFunction
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).

Pumas.zero_randeffsFunction
zero_randeffs(model::AbstractPumasModel, param::NamedTuple [, pop::Population])

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.

Pumas.ηshrinkageMethod
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.

Pumas.ϵshrinkageMethod
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.

StatsBase.aicMethod
aic(fpm::AbstractFittedPumasModel)

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

StatsBase.bicMethod
bic(fpm::AbstractFittedPumasModel)

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

StatsBase.coeftableMethod
coeftable(pmi::FittedPumasModelInference) -> DataFrame

Construct a DataFrame of parameter names, estimates, standard error, and confidence interval from a pmi.

StatsBase.coeftableMethod
coeftable(fpm::FittedPumasModel) -> DataFrame

Construct a DataFrame of parameter names and estimates from fpm.

StatsBase.coeftableMethod
coeftable(cfpm::Vector{<:FittedPumasModel}) -> DataFrame

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

StatsBase.fitMethod
fit(
model::PumasEMModel,
population::Population,
param::NamedTuple,
approx::Union{SAEM,LaplaceI};
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 reccomended 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().

StatsBase.fitMethod
fit(
model::PumasModel,
population::Population,
param::NamedTuple,
approx::Union{LikelihoodApproximation, MAP};
optimize_fn = DefaultOptimizeFN(),
constantcoef::NamedTuple = NamedTuple(),
omegas::Tuple = tuple(),
ensemblealg::DiffEqBase.EnsembleAlgorithm = EnsembleSerial(),
checkidentification=true,
diffeq_options))

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.

The argument optimize_fn is used for optimizing the objective function for all approx methods except BayesMCMC. The default optimization function uses the quasi-Newton routine BFGS method from the Optim package. Optimization specific arguments can be passed to DefaultOptimizeFN, e.g. the optimization trace can be disabled and the algorithm can be changed to L-BFGS by passing optimize_fn=DefaultOptimizeFN(Optim.LBFGS(); show_trace=false) to fit. The positional argument is a zero or first or method from Optim and the keywords are used to the available options in Optim. See Optim for more defails.

It is possible to fix one or more parameters of the fit by passing a NamedTuple as the constantcoef argument with keys and values corresponding to the names and values of the fixed parameters, e.g. constantcoef=(σ=0.1,).

When models include an @random block and fitting with NaivePooled is requested, it is required that the user supplies the names of the parameters of the random effects as the omegas argument such that these can be ignored in the optimization, e.g. omegas=(Ω,).

Parallelization of the optimization is supported for most estimation methods via the ensemble interface of DifferentialEquations.jl. The default is EnsembleSerial(). Currently, the only supported parallelization for model fitting is EnsembleThreads().

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(Rodas5(autodiff=true)), 1e-12, and 1e-8 respectively. See the DifferentialEquations.jl documentation for more details.

StatsBase.predictMethod
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.

StatsBase.predictMethod
predict(
model::AbstractPumasModel,
population::Union{Subject,Population},
param::NamedTuple;
[obstimes::AbstractVector,
diffeq_options::NamedTuple]
)::Union{SubjectPrediction,Vector{SubjectPrediction}}

Compute population and individual predictions for either the single subject or vector of subjects (Population) population based on model and the population parameters param.

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.

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.

StatsBase.predictMethod
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.

StatsBase.stderrorMethod
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.

StatsBase.vcovMethod
vcov(f::AbstractFittedPumasModel) -> Matrix

Compute the covariance matrix of the population parameters

Pumas.@emmodelMacro
@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 evalauted 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.
Pumas.@modelMacro
@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 recomended 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.
Pumas.@ncaMacro
 @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
...`