Reference Sheet

Key Points

You can parse a NM-TRAN-formatted DataFrame into a Population with read_pumas
A Population is just a collection (Vector) of Subjects
You can slice and index Populations to get another Population subset or a single Subject
You can reconstruct a NM-TRAN-formatted DataFrame from a Population/Subject with the DataFrame constructor
To define a model in Pumas, you use the @model macro along with the model blocks:
- @metadata for model metadata such as description and time units
- @param for the population parameters (i.e. typical values or fixed effects)
- @random for the subject-specific parameters (η or random effects)
- @covariates for subject covariates
- @pre for pre computations such as individual coefficients or any other statistical transformation
- @dynamics for the model dynamics, either as an analytical solution or a system of ordinary differential equations
- @derived for derived variables and error model
In the @model you can have two types of assignments:
- Deterministic assignments with =
- Probabilistic assignments with ~
The fit function is very flexible, it has 4 positional arguments:
1. model: which model to fit
2. population: which population to fit
3. initial_parameters: a NamedTuple of initial parameter estimates
4. estimation_method: which estimation method to use; for maximum likelihood: FOCE, NaivePooled, and LaplaceI are the most common
Additionally, the fit function has the following most used keyword arguments:
- constantcoef: if you want to set any parameter value to a constant value, similar to FIX in NONMEM
- omegas: a tuple with the value of the "omegas" in the @param block, needed for NaivePooled estimation method
All results from the fit function can be converted to a:
- NamedTuple with coef
- DataFrame with coeftable
You can extract individual coefficients with the icoef function, if you want in a DataFrame format use the DataFrame constructor on the result
The infer function can be used to generate confidence intervals using:
- Variance-covariance matrix (default)
- Bootstrap with a second argument Pumas.Bootstrap()
- Sampling importance resampling (SIR) with a second argument Pumas.SIR()
All results from the infer function can be converted to a DataFrame with coeftable

Summary of Basic Commands

Action	Command	Observations
Parse data into a `Population`	`read_pumas`	NM-TRAN-formatted `DataFrame`s
Index or slice a `Population`	`pop[1]` or `pop[1:10]`
Reconstruct data from a `Population`	`DataFrame(pop)`	NM-TRAN-formatted `DataFrame`s
Reconstruct data from a index or slice `Population`	`DataFrame(pop[1])` or `DataFrame(pop[1:10])`	NM-TRAN-formatted `DataFrame`s
Define a model	`@model`
Define model metadata	`@metadata`
Define the population parameters of a model	`@param`
Define the subject-specific parameters of a model	`@random`
Define the subject covariates	`@covariates`
Define individual coefficients, precomputations or any statistical transformation	`@pre`
Define model dynamics	`@dynamics`
Define model derived variables and error model	`@derived`
Fit a model using `FOCE()`	`fit(model, population, initial_parameters, FOCE())`	`initial_parameters` is a `NamedTuple` of parameter name and values
Fit a model using `NaivePooled`	`fit(model, population, initial_parameters, NaivePooled(); omegas=(:Ω,))`	`omegas` is a keyword argument that should be a tuple specifying the variable name where the Ωs are defined in the model
Fit a model using `FOCE()` with fixed parameter values	`fit(model, population, initial_parameters, FOCE(); constantcoef=(; parameter=value))`	`constantcoef` is a keyword argument that should be a `NamedTuple` specifying the parameter name along with the value to be fixed
Get model fit coefficients as a `NamedTuple`	`coef(fit_result)`
Get model fit coefficients as a `DataFrame`	`coeftable(fit_result)`
Get model individual parameters as a `DataFrame`	`DataFrame(icoef(fit_result))`
Calculate confidence intervals using the Variance-covariance matrix	`infer(fit_result)`	Uses the sandwich estimator by default
Calculate confidence intervals using bootstrap	`infer(fit_result, Pumas.Bootstrap())`	Perform 200 samples by default
Calculate confidence intervals using sampling importance resampling (SIR)	`infer(fit_result, SIR())`	User needs to specify the number of samples and resamples, i.e. there aren't default values
Get model confidence intervals as a `DataFrame`	`coeftable(infer_result)`

Glossary

NM-TRAN: Official NONMEM dataset format. Check Pumas documentation on Data Representation for more information.
DataFrame: A tabular data format from the package DataFrames.jl. It is the standard tabular data format in Julia and is present in the Pumas app.
Population: Pumas' representation of a collection of Subjects. Generally parsed from NM-TRAN-formatted DataFrames.
Vector: Contiguous data structure that allows ordering, indexing, looping, slicing, and shape-destructing operations, i.e. grow or shrink. Most used to group elements into a collection.
Subject: Pumas' representation of a subject that has covariates, time, events, observations, and any other relevant information.
@model: how users define models in Pumas, it allows for several other blocks inside. The syntax is similar to NONMEM model specification, but with higher flexibility and expressiveness.
@metadata: @model block with additional details such as model description and model time units.
@param: @model block for the population parameters. Analogous to NONMEM's $THETA, $OMEGA, and $SIGMA.
@random: @model block for the subject-specific parameters, also known as η or random effects.
@covariates: @model block for subject covariates.
@pre: @model block for pre computations such as individual coefficients or any other statistical transformation. Analogous to NONMEM's $PK.
@dynamics: @model block for the model dynamics, either as an analytical solution or a system of ordinary differential equations. Analogous to NONMEM's $DES.
@derived: @model block for derived variables and error model. Analogous to NONMEM's $ERROR.
Deterministic assignments: assignments that will always return the same value. The standard assignment operator in programming languages, e.g. x=1 or y="hello".
Probabilistic assignments: An assignment operator that instead of returning the same value, samples a random value under a distribution. For example, x ~ Normal(0, 1) will generate a new value sampled from a normal distribution with mean 0 and standard deviation 1 every time it is executed.
Model: Mathematical representation of the underlying process regarding a certain phenomena.
Fit: Condition the data into the model and infer the model's parameter values by an estimation method.
FOCE: Estimation method originally from NONMEM, it means First Order Conditional Effects. Please refer to the Pumas documentation for more details.
Naive Pooled: Estimation method that ignores subject-specific parameters relying only on population parameters. Please refer to the Pumas documentation for more details.
Laplace: Estimation method that uses Laplacian approximation under the hood. Please refer to the Pumas documentation for more details.
Ω: The covariance matrix of the subject-specific parameters. Commonly referred to as the "Omega" matrix.
η: The individual subject-specific parameters. Commonly referred to as "etas". Generally a vector for each subject, e.g. η = [η₁, η₂].
icoef: Individual coefficients, also known as subject-specific parameters.
Confidence intervals: Commonly used procedure to measure uncertainty on parameter estimates in maximum likelihood estimation methods.
Variance-covariance matrix: A symmetric matrix with NxN dimensions where N is the number of parameters, the diagonals are the parameter variances, and the symmetric off-diagonal elements are the covariance between the parameter in the ith row and the parameter in the j row.
Sandwich estimator: Commonly used procedure to generate a Variance-covariance matrix from pharmacometrics model fit.
Bootstrap: Alternative way to calculate confidence intervals by fitting the model to N bootstrapped samples, that are generated by sampling with replacement the original sample a new sample with the same size as the original.
Sampling importance resampling (SIR): Alternative way to calculate confidence intervals that rely on the importance sampling procedure.

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