`Pumas`

Docstrings

`DataFrames.DataFrame`

— Method```
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:

```jldoctest 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

`DataFrames.DataFrame`

— Method`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))
1×9 DataFrame
Row │ time amt evid cmt rate duration ss ii route
│ Float64 Float64 Int8 Int64 Float64 Float64 Int8 Float64 NCA.Route
─────┼────────────────────────────────────────────────────────────────────────────
1 │ 0.0 100.0 1 1 0.0 0.0 0 0.0 NullRoute
julia> DataFrame(DosageRegimen(100; ii = 24, addl = 2))
3×9 DataFrame
Row │ time amt evid cmt rate duration ss ii route
│ Float64 Float64 Int8 Int64 Float64 Float64 Int8 Float64 NCA.Route
─────┼────────────────────────────────────────────────────────────────────────────
1 │ 0.0 100.0 1 1 0.0 0.0 0 0.0 NullRoute
2 │ 24.0 100.0 1 1 0.0 0.0 0 0.0 NullRoute
3 │ 48.0 100.0 1 1 0.0 0.0 0 0.0 NullRoute
```

`DataFrames.DataFrame`

— Method`DataFrame(subject::AbstractSubject)`

Create a `DataFrame`

with the `Subject`

information. The data follows Pumas Data Format ready to be imported with `read_pumas`

.

Examples:

```jldoctest 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 bloodpressure tad dosenum │ String Float64 Int64 Float64? Float64? Int64? Float64? Float64? Int8? Float64? NCA.Route? Tuple…? Float64 Int64 ─────┼──────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ AKJ491 0.0 2 missing missing missing missing missing missing missing missing (143, 95.01) 0.0 0 2 │ AKJ491 0.3 0 8.31 0.0 missing 0.0 0.0 0 0.0 NullRoute (143, 95.01) 0.3 0 3 │ AKJ491 0.5 2 missing missing missing missing missing missing missing missing (141.3, 94.8) 0.5 0 4 │ AKJ491 0.9 0 7.709 0.0 missing 0.0 0.0 0 0.0 NullRoute (141.3, 94.8) 0.9 0 5 │ AKJ491 1.0 1 missing 1.0 1 0.0 0.0 0 0.0 NullRoute (141.3, 94.8) 0.0 1 6 │ AKJ491 3.0 0 5.19 0.0 missing 0.0 0.0 0 0.0 NullRoute (141.3, 94.8) 2.0 1 7 │ AKJ491 5.0 2 missing missing missing missing missing missing missing missing (130.4, 85.1) 4.0 1

`DataFrames.DataFrame`

— Method```
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`

.

`DataFrames.DataFrame`

— Method```
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`

.

`DataFrames.DataFrame`

— Method`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`

.

`DataFrames.DataFrame`

— Method`DataFrame(bts::Union{SimulatedInference, Bootstraps})`

Returns a data frame of the parameter estimates.

`DataFrames.DataFrame`

— Method`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`

.

`DataFrames.DataFrame`

— Method`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`

.

`Distributions.MvNormal`

— Method`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.MvNormal`

— Method`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)
```

`MCMCChains.Chains`

— Method`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.

`Pumas.AnalyticalPKPDProblem`

— Type```
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

`Pumas.AnalyticalPKProblem`

— TypeAnalyticalPKProblem(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.BayesMCMC`

— TypeBayesMCMC

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,
)
```

`Pumas.Bootstrap`

— Method`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
`Population`

s. - 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 parallization across`fit`

calls. Each fit is fitted using the`ensemblealg`

that was used in the original`fit`

.

`Pumas.ByObservation`

— Type`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 `i`

th 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:

Training subset | Validation subset |
---|---|

`Sub 1 (obs 1, 2, 3), sub 2 (obs 1, 2)` | `Sub 2 (obs 3)` |

`Sub 1 (obs 1, 2, 3), sub 2 (obs 1, 3)` | `Sub 2 (obs 2)` |

`Sub 1 (obs 1, 2, 3), sub 2 (obs 2, 3)` | `Sub 2 (obs 1)` |

`Sub 1 (obs 1, 2), sub 2 (obs 1, 2, 3)` | `Sub 1 (obs 3)` |

`Sub 1 (obs 1, 3), sub 2 (obs 1, 2, 3)` | `Sub 1 (obs 2)` |

`Sub 1 (obs 2, 3), sub 2 (obs 1, 2, 3)` | `Sub 1 (obs 1)` |

On the other hand if `allsubjects`

is set to `true`

, the training and validation splittings are:

Training subset | Validation subset |
---|---|

`Sub 1 (obs 1, 2), sub 2 (obs 1, 2)` | `Sub 1 (obs 3) and sub 2 (obs 3)` |

`Sub 1 (obs 1, 3), sub 2 (obs 1, 3)` | `Sub 1 (obs 2) and sub 2 (obs 2)` |

`Sub 1 (obs 2, 3), sub 2 (obs 2, 3)` | `Sub 1 (obs 1) and sub 2 (obs 1)` |

`Pumas.BySubject`

— Type`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.

`Pumas.Censored`

— Type`Censored(distribution::Distribution, lower::Real, upper::Real)::Censored`

Construct a censored distribution based on `distribution`

and the censoring limits `lower`

and `upper`

.

`Pumas.Central1`

— Type`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.Central1Meta1`

— Type`Central1Meta1()`

An analytical model for a model with a central compartment, `Central`

, and a metabolite, `Metabolite`

. Equivalent to Central' = -(CL+CLfm)/Vc*Central Metabolite' = CLfm/Vc*Central - 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.

`Pumas.Central1Periph1`

— Type`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.Central1Periph1Meta1`

— Type`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.Central1Periph1Meta1Periph1`

— Type`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.ConstDomain`

— Type```
@param x = val
@param x ∈ ConstDomain(val)
```

Specifies a parameter as a constant.

`Pumas.Constrained`

— Type`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
```

`Pumas.CorrDomain`

— Type`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.Depots1Central1`

— Type`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.Depots1Central1Periph1`

— Type`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.Depots2Central1`

— Type`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.DiffEqWrapper`

— Type`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

`Pumas.DosageRegimen`

— Type`DosageRegimen`

Lazy representation of a series of Events.

**Fields**

`data::DataFrame`

: The tabular representation of a series of`Event`

s.Signature

```
evts = 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)
```

- 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.Event`

— Type`Event`

A single event

Fields:

`amt`

: Dose amount (mass units, e.g. g, mg, etc.)`time`

: Time of dose (time unit, e.g. hours, days, etc.)`evid`

: Event identifier, possible values are`-1`

: End of an infusion dose`0`

: observation (these should have already been removed)`1`

: dose event`2`

: other type event`3`

: reset event (the amounts in each compartment are reset to zero, the on/off status of each compartment is reset to its initial status)`4`

: reset and dose event

`cmt`

: Compartment (an Int or Symbol)`rate`

: The dosing rate:`0`

: instantaneous bolus dose`>0`

: infusion dose (mass/time units, e.g. mg/hr)`-1`

: infusion rate specifed by model`-2`

: infusion duration specified by model

`duration`

: duration of dose, if`dose > 0`

.`ss`

: steady state status`0`

: dose is not a steady state dose.`1`

: dose is a steady state dose, and that 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 that the compartment amounts are to be set to the sum of the steady-state amounts resulting from the given dose plus whatever those amounts would have been at the event time were the steady-state dose not given.

`ii`

: interdose interval (units: time). Time between implied or additional doses.`base_time`

: time at which infusion started`route`

: route of administration to be used in NCA analysis

`Pumas.ExactCrossvalidation`

— Type`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 nineth 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
`0`

s 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)
```

`Pumas.ExplicitModel`

— Method```
(m::ExplicitModel)(
Δt::Real,
amounts::AbstractVector,
doses::AbstractVector,
pre,
rates::AbstractVector)
```

Solve the model for a given time step `Δt`

.

The initial/previous time point is associated with the initial system state `amounts`

, bolus `doses`

and the start of an infusion with a rates going into compartments given by `rates`

. For each evaluation, `pre`

will be evaluated at the current time point.

`Pumas.FittedPumasEMModel`

— TypeFittedPumasEMModel

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`

.

`Pumas.Formula`

— TypeO ~ C | D

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

`Pumas.IDR`

— Method`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: :stimulation
*linear, :stimulation*emax, :stimulation*sigmoid, :stimulation*infinite, :stimulation*logarithmic, :inhibition*emax, :inhibition*sigmoid, :inhibition*hill,

`Pumas.JointMAP`

— TypeJointMAP

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

`Pumas.KFold`

— Type`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:

Training subset | Validation subset |
---|---|

`["A", "B", "C"]` | `["D"]` |

`["A", "B", "D"]` | `["C"]` |

`["A", "C", "D"]` | `["B"]` |

`["B", "C", "D"]` | `["A"]` |

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

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

Training subset | Validation subset |
---|---|

`["A", "B"]` | `["C", "D"]` |

`["C", "D"]` | `["A", "B"]` |

`Pumas.LLQuad`

— Type`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)
```

`Pumas.LeaveFutureK`

— Type`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:

Past training subset | Future validation subset | Discarded subset |
---|---|---|

`["A", "B", "C", "D", "E"]` | `["F"]` | `[]` |

`["A", "B", "C", "D"]` | `["E"]` | `["F"]` |

`["A", "B", "C"]` | `["D"]` | `["E", "F"]` |

`["A", "B"]` | `["C"]` | `["D", "E", "F"]` |

Leave-2-future-out splitting with `minimum = 2`

results in the following splittings:

Past training subset | Future validation subset | Discarded subset |
---|---|---|

`["A", "B", "C", "D"]` | `["E", "F"]` | `[]` |

`["A", "B"]` | `["C", "D"]` | `["E", "F"]` |

`Pumas.LeaveK`

— Type`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:

Training subset | Validation subset |
---|---|

`["A", "B", "C"]` | `["D"]` |

`["A", "B", "D"]` | `["C"]` |

`["A", "C", "D"]` | `["B"]` |

`["B", "C", "D"]` | `["A"]` |

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

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

Training subset | Validation subset |
---|---|

`["A", "B"]` | `["C", "D"]` |

`["C", "D"]` | `["A", "B"]` |

`Pumas.MarginalMCMC`

— TypeMarginalMCMC

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`

.

`Pumas.NL`

— Type`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.

`Pumas.ObsAdditive`

— Type`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.

`Pumas.ObsCombined`

— Type`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.

`Pumas.ObsProportional`

— Type`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.

`Pumas.PDiagDomain`

— Type`@param x ∈ PDiagDomain(n::Int; init=ones(n))`

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

.

`Pumas.PSDDomain`

— Type```
@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 `Float64`

s.

`Pumas.PSISCrossvalidation`

— Type`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 `NaN`

s 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 nineth 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
`0`

s 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)
```

`Pumas.ParamSet`

— Type`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.Population`

— TypeA `Population`

is an `AbstractVector`

of `Subject`

s.

`Pumas.PumasEMModel`

— Type`PumasEMModel{Σ,NRE,C,PRR,DCR,DR,PR}`

Parameters: Σ - Covariance structure; (2,1) indicates a matrix with `2x2`

and `1x1`

diagonal blocks. NRE - The number of formulas without an associated random effect. C - Covariates defined in `@covariates`

block PRR - Parameters defined in `@pre`

block DCR - Parameters defined in `@dosecontrol`

block DR - Parameters defined in `@dynamics`

block POR - Parameters defined in `@post`

block

Currently supported fitting methods are `SAEM()`

and `LaplaceI()`

.

`Pumas.PumasModel`

— Type`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.RealDomain`

— Type`@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.SAEM`

— TypeSAEM(; 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`k`

th update,`α^k`

weight is placed on the`k`

th 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`ω`

s and`σ`

s 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.

`Pumas.SIR`

— Method`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).

`Pumas.Subject`

— Type`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`Event`

s.`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 = 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 = :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
```

`Pumas.Subject`

— MethodSubject

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.Subject`

— MethodSubject(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:

```jldoctest 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

`Pumas.TimeToEvent`

— Type`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
...
```

`Pumas.VectorDomain`

— Type`@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.

`Pumas.ZeroSumDomain`

— Type```
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.

`Base.isvalid`

— Method`isvalid(s::SimulatedObservations) -> Bool`

Check if ODE solution of a simulation was successful.

**Examples**

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

`Base.truncate`

— Method`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).

`CommonSolve.solve`

— Function```
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`Subject`

s 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`NamedTuple`

s for`PumasModels`

(e.g.`(; tvCL=1., tvKa=2.)`

) and by`Tuples`

for`PumasEMModel`

s (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`Subject`

s 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.

`Distributions.logpdf`

— Method`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.gsa`

— Functionfunction GlobalSensitivity.gsa( m::PumasModel, pop::Population, params::NamedTuple, method::GlobalSensitivity.GSAMethod, vars = [:dv], p*range*low = NamedTuple(), p*range*high = NamedTuple(); constantcoef::NTuple{N,Symbol} = (), rng::AbstractRNG = default*rng(), diffeq*options::NamedTuple = NamedTuple(), N = nothing, n = nothing, samples = 1000, 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`GSAMethod`

s 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, if not specified the bounds from`@param`

block are used.

Keyword arguments are:

`constantcoef`

: The parameters that are not to be varied in the GSA, passed as a Tuple of Symbols.`rng`

: The random number generator to use.`samples`

: The number of samples to use for the GSA.`diffeq_options`

: The options to pass to the differential equation solver, passed on to the`simobs`

called inside`gsa`

.

For method specific arguments that are passed with the method constructor you can refer to the GlobalSensitivity.jl documentation.

`LinearAlgebra.cond`

— Method`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.

`MCMCDiagnosticTools.gewekediag`

— Method`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.

`MCMCDiagnosticTools.heideldiag`

— Method`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.

`MCMCDiagnosticTools.rafterydiag`

— Method`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.

`Pumas._convergencedata`

— Method`_convergencedata(obj; metakey="x")`

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

This must return parameter data in the same order that `_paramnames`

returns names.

`Pumas._lpdf`

— Method`_lpdf(d,x)`

The log pdf: this differs from `Distributions.logdpf`

definintion 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.

`Pumas._objectivefunctionvalues`

— Method`_objectivefunctionvalues(obj)`

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

.

`Pumas._paramnames`

— Method`_paramnames(obj)`

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

This must return parameter names in the same order that `_convergencedata`

returns data.

`Pumas.add_t`

— Methodadd_t(ex, vars)

Append `(t)`

to all the `vars`

symbols in the expression `ex`

.

`Pumas.apply_formatting`

— Methodapply*formatting(input*ex, vars; show_t=true, italicize=true)

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

`Pumas.apply_stat`

— Method`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 `i`

th position of each element of `v`

. For example, this can be used to create a `NamedTuple`

of the mean across a vector of `NamedTuple`

s 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,)
```

`Pumas.build_model_str`

— Method`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)`

`Pumas.conditional_nll`

— Method`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.cor2cov`

— Method`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
```

`Pumas.correlation_diagnostic`

— Method```
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)
```

`Pumas.covariate_select`

— Method```
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 imposing zero restrictions on 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 a zero restriction on a parameter listed in `control_param`

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 with all zero restrictions is chosen. Otherwise, the model with the lowest `criterion`

is chosen and the procedure continues by fitting all models with two free parameters where one of the free 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 penality 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,
)
```

`Pumas.covariates_interpolant`

— Method```
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.

`Pumas.covariates_interpolant`

— Method```
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.

`Pumas.crossvalidate`

— Method`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.

`Pumas.crossvalidate`

— 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`

.

`Pumas.discard`

— Method`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).

`Pumas.dosecontrol`

— Method`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 containt the times that correspond to the start of each dose event for the reason discussed above.

`Pumas.dosenum`

— Method`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.

`Pumas.eiwres`

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

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

`Pumas.elpd`

— Method`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.

`Pumas.em_to_m_params`

— Method`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)`

.

`Pumas.empirical_bayes`

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

Return empirical bayes estimates for a fitted model, `fpm`

, when applied to `data`

.

`Pumas.empirical_bayes`

— Method```
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.

`Pumas.empirical_bayes`

— Method```
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.

`Pumas.empirical_bayes_dist`

— Method`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 `MvNormal`

s.

`Pumas.epredict`

— Method`epredict(fpm::AbstractFittedPumasModel; nsim::Integer, rng::AbstractRNG=default_rng())`

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.

`Pumas.eventnum`

— Method`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 (end of infusions), 1 (dose), 3 (reset), 4 (reset and dose).

`Pumas.expectation`

— Method`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.expectation`

— Method`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.expectation`

— Method`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.findinfluential`

— Method```
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 `fit`

ting 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 `NaN`

s.

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"),
)
```

`Pumas.get_model_combinations`

— Method`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.

`Pumas.getblock`

— Method`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.

`Pumas.icoef`

— Method`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.

`Pumas.infer`

— Method`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.infer`

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

Perform bootstrapping by resampling the `Subject`

s 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.infer`

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

Perform sampling importance re-sampling for the `Subject`

s 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.`

ensemblealg`can be`

EnsembleThreads()`(the default value) to use multi-threading or`

EnsembleSerial()` to use a single thread.

`Pumas.init_params`

— Function`init_params(model::PumasModel)`

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

.

`Pumas.init_randeffs`

— Function`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.inspect`

— Method`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.lrtest`

— Method`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.lrtest`

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

Compute the likelihood ratio test using number.

`Pumas.marginal_nll`

— Function```
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`

.

`Pumas.npde`

— Method`npde(fpm::AbstractFittedPumasModel; nsim::Integer)`

Calculate the normalised prediction distribution errors (NPDE).

The keyword `nsim`

controls the number of times each subject is simulated.

`Pumas.penalized_conditional_nll`

— Method`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.

Here `param`

can be either a `NamedTuple`

or a vector (representing a transformation of the random effects to Cartesian space).

`Pumas.postprocess`

— Method```
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 `i`

th 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.

`Pumas.probstable`

— Methodprobstable(fpm::FittedPumasModel)

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

`Pumas.pvalue`

— Method`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.qradd!`

— Method`qradd!(Q::AbstractMatrix, R::AbstractMatrix, 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.

`Pumas.qrdelete!`

— Method`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.

`Pumas.read_pumas`

— Method```
read_pumas(filepath::String; missingstring = ["", ".", "NA"], kwargs...)
read_pumas(
df::AbstractDataFrame;
observations=Symbol[:dv],
covariates=Symbol[],
id::Symbol=:id,
time::Symbol=:time,
evid::Union{Nothing,Symbol}=nothing,
amt::Symbol=:amt,
addl::Symbol=:addl,
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=:left,
check::Bool=event_data,
adjust_evid34::Bool=true
)
```

Import NMTRAN-formatted data. You can either pass a CSV file path as a `String`

or a `DataFrame`

as the first and only positional argument.

`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. 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`

: specifies direction of covariate interpolation. Either`:left`

(**L**ast**O**bservation**C**arried**F**orward, LOCF) (default), or`:right`

(**N**ext**O**bservation**C**arried**B**ackward, NOCB). Notice, that for model with occasion variables it is important to use`:left`

for the correct behavior of the interpolation.`check`

: 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`

.

- all variables in
`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.remove_italics`

— Methodremove_italics(ex, vars)

Ensure that the variables specified in `vars`

are not italicized during latexification of `ex`

.

`Pumas.sample_params`

— Method`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 determinsitic value will be assigned to the parameter similar to `init_params`

.

`Pumas.sample_randeffs`

— Function`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.simobs`

— Function`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.

`Pumas.simobs`

— Function`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.simobs`

— Method`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.

`Pumas.simobs`

— Method`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.

`Pumas.simobs`

— Method`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.

`Pumas.simobs`

— Method```
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`Subject`

s 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`NamedTuple`

s for`PumasModels`

(e.g.`(; tvCL=1., tvKa=2.)`

) and by`Tuples`

for`PumasEMModel`

s (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`Subject`

s 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.simobstte`

— Function```
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 funtion. The user must supply `maxT`

. Given a survival funtion, 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.

`Pumas.strip_t`

— Methodstrip_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)")`

.

`Pumas.tad`

— Method`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.variables`

— Functionvariables(dynobj)

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

`Pumas.vech`

— Methodvech(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
```

`Pumas.vechinv`

— Methodvechinv(x; diag = nothing, lower = nothing, lower = true)

Converts a vector of lower or upper triangular elements (travesed 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
```

`Pumas.vpc`

— Method```
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=EnsembleSerial(),
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`Symbol`

s 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`NaN`

s 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 regresions. 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.

`Pumas.wresiduals`

— Function`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_randeffs`

— Function```
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.

`Pumas.ηshrinkage`

— Method`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.ϵshrinkage`

— Method`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.

`StatsAPI.aic`

— Method`aic(fpm::AbstractFittedPumasModel)`

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

.

`StatsAPI.bic`

— Method`bic(fpm::AbstractFittedPumasModel)`

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

.

`StatsAPI.coeftable`

— Method`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`

.

`StatsAPI.coeftable`

— Method`coeftable(fpm::FittedPumasModel) -> DataFrame`

Construct a DataFrame of parameter names and estimates from `fpm`

.

`StatsAPI.coeftable`

— Method`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`

.

`StatsAPI.fit`

— Method`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`

.

`StatsAPI.fit`

— MethodDistributions.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.

`StatsAPI.fit`

— Method```
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())
```

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 varibility 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(Rodas5(autodiff=true))`

, `1e-12`

, and `1e-8`

respectively. See the DifferentialEquations.jl documentation for more details.

`StatsAPI.fit`

— Method```
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 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()`

.

`StatsAPI.loglikelihood`

— Method`loglikelihood(fpm::AbstractFittedPumasModel)`

Compute the loglikelihood of a fitted Pumas model.

`StatsAPI.loglikelihood`

— Method`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 nineth 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
`0`

s 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)
```

`StatsAPI.loglikelihood`

— Method`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 `j`

th sample of the `k`

th chain.

`StatsAPI.predict`

— Method```
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.

`StatsAPI.predict`

— Method```
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.

`StatsAPI.stderror`

— Method`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.

`StatsAPI.vcov`

— Method`vcov(f::AbstractFittedPumasModel) -> AbstractMatrix`

Compute the covariance matrix of the population parameters

`Pumas.@cache`

— Macro@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
```

`Pumas.@covariance`

— Macro@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`

.

`Pumas.@covariates`

— Macro@covariates

Defines the model's covariates, e.g. `wt`

. Must be used in an `@model`

block. For example:

```
@model begin
@covariates wt
end
```

`Pumas.@delay`

— Macro```
@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 Piotrovskii (1987)

```
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 + 1)), 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
```

`Pumas.@derived`

— Macro@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
```

`Pumas.@dosecontrol`

— Macro@dosecontrol

defines dose control parameters as a function of fixed and random effects. Options include `bioav`

, `duration`

, `lag`

, 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
```

`Pumas.@dynamics`

— Macro@dynamics

Defines the dynamic system. Must be used in an `@model`

block. For example:

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

`Pumas.@emmodel`

— Macro`@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.@error`

— Macro@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
```

`Pumas.@init`

— Macro@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
```

`Pumas.@model`

— Macro`@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.@nca`

— Macro` @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
...
```

`Pumas.@observed`

— Macro@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
```

`Pumas.@options`

— Macro@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`Array`

s (`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
```

`Pumas.@param`

— Macro@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 `PumasEMModel`

s, 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.

`Pumas.@post`

— Macro@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
```

`Pumas.@pre`

— Macro@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
```

`Pumas.@random`

— Macro@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 `PumasEMModel`

s, the `@random`

block is equivalent to the `@param`

s 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.

`Pumas.@vars`

— Macro@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
```

- Geweke1991Geweke, J. F. (1991). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments (No. 148). Federal Reserve Bank of Minneapolis.
- Heidelberger1983Heidelberger, P., & Welch, P. D. (1983). Simulation run length control in the presence of an initial transient. Operations Research, 31(6), 1109-1144.
- Raftery1992A L Raftery and S Lewis. Bayesian Statistics, chapter How Many Iterations in the Gibbs Sampler? Volume 4. Oxford University Press, New York, 1992.