Calculating Parameter Uncertainty

Author

Patrick Kofod Mogensen

1 Introduction

A typical workflow for fitting a Pumas model and deriving parameter precision typically involves:

  1. Preparing the data and the model.
  2. Checking model-data compatibility.
  3. Obtaining initial parameter estimates.
  4. Fitting the model via a chosen estimation method.
  5. Interpreting the fit results.
  6. Computing parameter precision.
  7. (Optionally) proceeding with more advanced techniques like bootstrapping or SIR for robust uncertainty quantification.

The following sections will walk through these steps using a one-compartment PK model for Warfarin, focusing on the PK aspects only. Exploratory data analysis (EDA), although extremely important, is out of scope here. Readers interested in EDA are encouraged to consult other tutorials.

2 Model and Data

2.1 Model Definition

Below is the PK model, named warfarin_pk_model, defined in Pumas. This model contains:

  • Fixed effects (population parameters): pop_CL, pop_Vc, pop_tabs, pop_lag
  • Inter-individual variability (IIV) components: pk_Ω
  • Residual error model parameters: σ_prop,σ_add
  • Covariates for scaling: FSZCL and FSZV
  • Differential equations describing the PK behavior in the compartments.
using Pumas
using PharmaDatasets
using DataFramesMeta
using PumasUtilities
warfarin_pk_model = @model begin
    @metadata begin
        desc = "Warfarin 1-compartment PK model (PD removed)"
        timeu = u"hr"
    end
    @param begin
        # PK parameters
        """
        Clearance (L/hr)
        """
        pop_CL ∈ RealDomain(lower = 0.0, init = 0.134)
        """
        Central volume (L)
        """
        pop_Vc ∈ RealDomain(lower = 0.0, init = 8.11)
        """
        Absorption lag time (hr)
        """
        pop_tabs ∈ RealDomain(lower = 0.0, init = 0.523)
        """
        Lag time (hr)
        """
        pop_lag ∈ RealDomain(lower = 0.0, init = 0.1)
        # Inter-individual variability
        """
          - ΩCL: Clearance
          - ΩVc: Central volume
          - Ωtabs: Absorption lag time
        """
        pk_Ω ∈ PDiagDomain([0.01, 0.01, 0.01])
        # Residual variability
        """
        σ_prop: Proportional error
        """
        σ_prop ∈ RealDomain(lower = 0.0, init = 0.00752)
        """
        σ_add: Additive error
        """
        σ_add ∈ RealDomain(lower = 0.0, init = 0.0661)
    end
    @random begin
        pk_η ~ MvNormal(pk_Ω)    # mean = 0, covariance = pk_Ω
    end
    @covariates begin
        """
        FSZCL: Clearance scaling factor
        """
        FSZCL
        """
        FSZV: Volume scaling factor
        """
        FSZV
    end
    @pre begin
        CL = FSZCL * pop_CL * exp(pk_η[1])
        Vc = FSZV * pop_Vc * exp(pk_η[2])
        tabs = pop_tabs * exp(pk_η[3])
        Ka = log(2) / tabs
    end
    @dosecontrol begin
        lags = (Depot = pop_lag,)
    end
    @vars begin
        cp := Central / Vc
    end
    @dynamics Depots1Central1

    @derived begin
        """
        Concentration (ng/mL)
        """
        conc ~ @. Normal(cp, sqrt((σ_prop * cp)^2 + σ_add^2))
    end
end
PumasModel
  Parameters: pop_CL, pop_Vc, pop_tabs, pop_lag, pk_Ω, σ_prop, σ_add
  Random effects: pk_η
  Covariates: FSZCL, FSZV
  Dynamical system variables: Depot, Central
  Dynamical system type: Closed form
  Derived: conc
  Observed: conc

2.2 Data Preparation

The Warfarin data used in this tutorial is pulled from PharmaDatasets for demonstration purposes. Note how the code reshapes and prepares the data in “wide” format for reading into Pumas. Only the conc column is treated as observations for the PK model.

warfarin_data = dataset("pumas/warfarin_pumas")

# Transform the data in a single chain of operations
warfarin_data_wide = @chain warfarin_data begin
    @rtransform begin
        # Scaling factors
        :FSZV = :wtbl / 70            # volume scaling
        :FSZCL = (:wtbl / 70)^0.75     # clearance scaling (allometric)
    end
end
330×12 DataFrame
305 rows omitted
Row id time evid amt cmt conc pca wtbl age sex FSZV FSZCL
Int64 Float64 Int64 Float64? Int64? Float64? Float64? Float64 Int64 String1 Float64 Float64
1 1 0.0 1 100.0 1 missing missing 66.7 50 M 0.952857 0.96443
2 1 0.5 0 missing missing 0.0 missing 66.7 50 M 0.952857 0.96443
3 1 1.0 0 missing missing 1.9 missing 66.7 50 M 0.952857 0.96443
4 1 2.0 0 missing missing 3.3 missing 66.7 50 M 0.952857 0.96443
5 1 3.0 0 missing missing 6.6 missing 66.7 50 M 0.952857 0.96443
6 1 6.0 0 missing missing 9.1 missing 66.7 50 M 0.952857 0.96443
7 1 9.0 0 missing missing 10.8 missing 66.7 50 M 0.952857 0.96443
8 1 12.0 0 missing missing 8.6 missing 66.7 50 M 0.952857 0.96443
9 1 24.0 0 missing missing 5.6 44.0 66.7 50 M 0.952857 0.96443
10 1 36.0 0 missing missing 4.0 27.0 66.7 50 M 0.952857 0.96443
11 1 48.0 0 missing missing 2.7 28.0 66.7 50 M 0.952857 0.96443
12 1 72.0 0 missing missing 0.8 31.0 66.7 50 M 0.952857 0.96443
13 1 96.0 0 missing missing missing 60.0 66.7 50 M 0.952857 0.96443
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
319 32 48.0 0 missing missing 6.9 24.0 62.0 21 M 0.885714 0.912999
320 32 72.0 0 missing missing 4.4 23.0 62.0 21 M 0.885714 0.912999
321 32 96.0 0 missing missing 3.5 20.0 62.0 21 M 0.885714 0.912999
322 32 120.0 0 missing missing 2.5 22.0 62.0 21 M 0.885714 0.912999
323 33 0.0 1 100.0 1 missing missing 66.7 50 M 0.952857 0.96443
324 33 0.0 0 missing missing missing 100.0 66.7 50 M 0.952857 0.96443
325 33 24.0 0 missing missing 9.2 49.0 66.7 50 M 0.952857 0.96443
326 33 36.0 0 missing missing 8.5 32.0 66.7 50 M 0.952857 0.96443
327 33 48.0 0 missing missing 6.4 26.0 66.7 50 M 0.952857 0.96443
328 33 72.0 0 missing missing 4.8 22.0 66.7 50 M 0.952857 0.96443
329 33 96.0 0 missing missing 3.1 28.0 66.7 50 M 0.952857 0.96443
330 33 120.0 0 missing missing 2.5 33.0 66.7 50 M 0.952857 0.96443

3 Creating a Pumas Population

Below is the creation of a population object in Pumas using read_pumas. Only the conc data are treated as the observation variable:

pop_pk = read_pumas(
    warfarin_data_wide;
    id = :id,
    time = :time,
    amt = :amt,
    cmt = :cmt,
    evid = :evid,
    covariates = [:sex, :wtbl, :FSZV, :FSZCL],
    observations = [:conc],
)
Population
  Subjects: 32
  Covariates: sex, wtbl, FSZV, FSZCL
  Observations: conc
Note

The same data can contain multiple endpoints or PD observations. In this tutorial, the focus is solely on PK fitting. PKPD modeling on this warfarin dataset will be introduced later.

Also note, that parameter inference can be expensive and for that reason we simplified the model for this tutorial to decrease overall runtime.

3.1 Checking Model-Data Compatibility

Before performing any fit, it is recommended to verify whether the defined model can handle the provided dataset. Pumas offers functions such as loglikelihood and findinfluential for these checks.

3.1.1 The loglikelihood Function

The loglikelihood function computes the log-likelihood of the model given data and parameters. In Pumas, the signature typically looks like:

loglikelihood(model, population, params, approx)

where:

  • model: The Pumas model definition (e.g., warfarin_pk_model).
  • population: A Pumas population object (e.g., pop).
  • params: A named tuple or dictionary containing parameter values.
  • approx: The approximation method to use. Common options include FOCE(), FO(), LaplaceI(), etc.

For example, one might write:

# A named tuple of parameter values
param_vals = (
    pop_CL = 0.134,
    pop_Vc = 8.11,
    pop_tabs = 0.523,
    pop_lag = 0.1,
    pk_Ω = Diagonal([0.01, 0.01, 0.01]),
    σ_prop = 0.00752,
    σ_add = 0.0661,
)
foce_fit = fit(warfarin_pk_model, pop_pk, param_vals, FOCE();)
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter     Function value   Gradient norm 
     0     1.209064e+04     1.489225e+04
 * time: 0.03675198554992676
     1     2.643772e+03     3.167516e+03
 * time: 1.4747381210327148
     2     1.836601e+03     2.118430e+03
 * time: 1.7526400089263916
     3     9.351337e+02     8.722439e+02
 * time: 1.7702059745788574
     4     6.402300e+02     4.199225e+02
 * time: 1.7873950004577637
     5     5.103664e+02     1.642121e+02
 * time: 1.8039441108703613
     6     4.760464e+02     5.453749e+01
 * time: 1.820162057876587
     7     4.703757e+02     3.643518e+01
 * time: 1.8363800048828125
     8     4.699019e+02     3.135992e+01
 * time: 1.8520629405975342
     9     4.697614e+02     2.953531e+01
 * time: 1.8669240474700928
    10     4.693153e+02     2.463233e+01
 * time: 1.8822100162506104
    11     4.685743e+02     2.580427e+01
 * time: 1.8974249362945557
    12     4.675133e+02     3.864937e+01
 * time: 1.9128310680389404
    13     4.666775e+02     5.495470e+01
 * time: 1.9279429912567139
    14     4.661197e+02     5.692101e+01
 * time: 1.943000078201294
    15     4.656782e+02     4.770992e+01
 * time: 1.9578680992126465
    16     4.651802e+02     3.087698e+01
 * time: 1.9735970497131348
    17     4.645523e+02     1.184834e+01
 * time: 1.991873025894165
    18     4.641447e+02     1.162249e+01
 * time: 2.0895190238952637
    19     4.639978e+02     1.125144e+01
 * time: 2.1046030521392822
    20     4.639307e+02     1.156463e+01
 * time: 2.120753049850464
    21     4.638001e+02     1.312870e+01
 * time: 2.1357409954071045
    22     4.635282e+02     1.480920e+01
 * time: 2.1507019996643066
    23     4.630353e+02     2.169377e+01
 * time: 2.1660830974578857
    24     4.623847e+02     4.478029e+01
 * time: 2.1816020011901855
    25     4.617426e+02     6.468975e+01
 * time: 2.1972689628601074
    26     4.610293e+02     7.776996e+01
 * time: 2.2128751277923584
    27     4.597628e+02     8.785260e+01
 * time: 2.2286999225616455
    28     4.566753e+02     9.769803e+01
 * time: 2.245326042175293
    29     4.490421e+02     1.008838e+02
 * time: 2.2656331062316895
    30     4.391868e+02     9.978816e+01
 * time: 2.286550998687744
    31     4.130704e+02     5.917685e+01
 * time: 2.3081319332122803
    32     4.055780e+02     3.852824e+01
 * time: 2.3519699573516846
    33     4.023118e+02     3.889618e+01
 * time: 2.369731903076172
    34     4.012516e+02     3.694778e+01
 * time: 2.3940329551696777
    35     4.004391e+02     2.061948e+01
 * time: 2.4114699363708496
    36     3.983040e+02     3.508423e+01
 * time: 2.4288880825042725
    37     3.969705e+02     3.841039e+01
 * time: 2.4470369815826416
    38     3.965462e+02     3.738343e+01
 * time: 2.464261054992676
    39     3.950409e+02     3.064789e+01
 * time: 2.4817941188812256
    40     3.945750e+02     2.876429e+01
 * time: 2.4992449283599854
    41     3.937725e+02     2.571438e+01
 * time: 2.5170140266418457
    42     3.933955e+02     2.436112e+01
 * time: 2.5357470512390137
    43     3.927564e+02     2.051069e+01
 * time: 2.5753109455108643
    44     3.916020e+02     1.629035e+01
 * time: 2.5926859378814697
    45     3.886991e+02     2.689824e+01
 * time: 2.610496997833252
    46     3.870054e+02     2.298582e+01
 * time: 2.62937593460083
    47     3.853691e+02     2.614992e+01
 * time: 2.6474111080169678
    48     3.841730e+02     2.207557e+01
 * time: 2.665019989013672
    49     3.825113e+02     2.204399e+01
 * time: 2.683074951171875
    50     3.808880e+02     2.444784e+01
 * time: 2.7004339694976807
    51     3.800407e+02     1.250611e+01
 * time: 2.717205047607422
    52     3.798092e+02     1.167926e+01
 * time: 2.7336010932922363
    53     3.797789e+02     1.162382e+01
 * time: 2.7675321102142334
    54     3.797069e+02     1.152441e+01
 * time: 2.7833831310272217
    55     3.794424e+02     1.132717e+01
 * time: 2.799989938735962
    56     3.788131e+02     2.006438e+01
 * time: 2.8164401054382324
    57     3.771525e+02     3.584695e+01
 * time: 2.833604097366333
    58     3.731299e+02     5.697249e+01
 * time: 2.8507890701293945
    59     3.658671e+02     6.542042e+01
 * time: 2.8686940670013428
    60     3.604194e+02     4.036489e+01
 * time: 2.8864059448242188
    61     3.532841e+02     1.574006e+01
 * time: 2.918376922607422
    62     3.520181e+02     1.393300e+01
 * time: 2.935529947280884
    63     3.517984e+02     6.701188e+00
 * time: 2.9518630504608154
    64     3.517541e+02     3.503978e+00
 * time: 2.9681029319763184
    65     3.516436e+02     8.720957e+00
 * time: 2.9857680797576904
    66     3.511845e+02     1.406200e+01
 * time: 3.0082690715789795
    67     3.510647e+02     2.540378e+00
 * time: 3.038135051727295
    68     3.510209e+02     3.157201e+00
 * time: 3.054983139038086
    69     3.509959e+02     3.045642e+00
 * time: 3.0705180168151855
    70     3.509765e+02     2.673143e+00
 * time: 3.0861690044403076
    71     3.509751e+02     2.603975e+00
 * time: 3.1010830402374268
    72     3.509724e+02     2.505719e+00
 * time: 3.117516040802002
    73     3.509666e+02     2.379768e+00
 * time: 3.144541025161743
    74     3.509504e+02     3.572030e+00
 * time: 3.160053014755249
    75     3.509123e+02     6.006350e+00
 * time: 3.175511121749878
    76     3.508288e+02     8.822995e+00
 * time: 3.1909849643707275
    77     3.506944e+02     9.708012e+00
 * time: 3.2066640853881836
    78     3.505767e+02     6.092631e+00
 * time: 3.222661018371582
    79     3.505358e+02     1.734431e+00
 * time: 3.2509260177612305
    80     3.505314e+02     6.749379e-01
 * time: 3.2661449909210205
    81     3.505313e+02     6.721982e-01
 * time: 3.2807891368865967
    82     3.505312e+02     6.699487e-01
 * time: 3.295156955718994
    83     3.505307e+02     6.606824e-01
 * time: 3.3096511363983154
    84     3.505298e+02     6.413909e-01
 * time: 3.3243799209594727
    85     3.505274e+02     9.083363e-01
 * time: 3.351072072982788
    86     3.505222e+02     1.339147e+00
 * time: 3.366420030593872
    87     3.505129e+02     1.608661e+00
 * time: 3.3817520141601562
    88     3.505026e+02     1.293164e+00
 * time: 3.396914005279541
    89     3.504973e+02     5.140504e-01
 * time: 3.4122350215911865
    90     3.504963e+02     6.340189e-02
 * time: 3.4274189472198486
    91     3.504963e+02     3.137914e-03
 * time: 3.454767942428589
    92     3.504963e+02     5.681551e-04
 * time: 3.4686319828033447

FittedPumasModel

Dynamical system type:                 Closed form

Number of subjects:                             32

Observation records:         Active        Missing
    conc:                       251             47
    Total:                      251             47

Number of parameters:      Constant      Optimized
                                  0              9

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -350.49625

--------------------
           Estimate
--------------------
pop_CL     0.13465
pop_Vc     8.0535
pop_tabs   0.55061
pop_lag    0.87158
pk_Ω₁,₁    0.070642
pk_Ω₂,₂    0.018302
pk_Ω₃,₃    0.91326
σ_prop     0.090096
σ_add      0.39115
--------------------

3.2 Computing Parameter Precision with infer

The infer function in Pumas estimates the uncertainty (precision) of parameter estimates. Depending on the chosen method, infer can provide standard errors, confidence intervals, and correlation matrices.

The signature for infer often looks like:

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

where:

  • fpm::FittedPumasModel: The result of fit (e.g., foce_fit).
  • level: The confidence interval level (e.g., 0.95). The confidence intervals are calculated as the (1-level)/2 and (1+level)/2 quantiles of the estimated parameters
  • rethrow_error: 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.
  • sandwich_estimator: Whether to use the sandwich estimator. If set to true (the default value), the sandwich estimator will be used. If set to false, the standard error will be calculated using the inverse of the Hessian matrix.

An example usage:

inference_results = infer(foce_fit; level = 0.95)
[ Info: Calculating: variance-covariance matrix.
[ Info: Done.

Asymptotic inference results using sandwich estimator

Dynamical system type:                 Closed form

Number of subjects:                             32

Observation records:         Active        Missing
    conc:                       251             47
    Total:                      251             47

Number of parameters:      Constant      Optimized
                                  0              9

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -350.49625

---------------------------------------------------------
           Estimate   SE          95.0% C.I.
---------------------------------------------------------
pop_CL     0.13465    0.0066546   [ 0.12161  ; 0.1477  ]
pop_Vc     8.0535     0.22108     [ 7.6201   ; 8.4868  ]
pop_tabs   0.55061    0.18702     [ 0.18406  ; 0.91717 ]
pop_lag    0.87158    0.056687    [ 0.76048  ; 0.98269 ]
pk_Ω₁,₁    0.070642   0.024577    [ 0.022472 ; 0.11881 ]
pk_Ω₂,₂    0.018302   0.0051549   [ 0.0081988; 0.028406]
pk_Ω₃,₃    0.91326    0.40637     [ 0.11678  ; 1.7097  ]
σ_prop     0.090096   0.014521    [ 0.061636 ; 0.11856 ]
σ_add      0.39115    0.065398    [ 0.26297  ; 0.51932 ]
---------------------------------------------------------

This is the usual asymptotic variance-covariance estimator and we already saw this previous tutorials.

To get a matrix representation of this, use vcov()

vcov(inference_results)
9×9 Symmetric{Float64, Matrix{Float64}}:
  4.42841e-5    0.000217445   0.000302094  …   3.99916e-6    0.00019736
  0.000217445   0.0488775     0.00571323      -0.000846166  -0.0056657
  0.000302094   0.00571323    0.0349767        0.000227818   0.00412692
 -7.40855e-5   -0.00207014   -0.00450616       0.000458813   0.000494683
  0.000120614   5.09406e-5    0.00164596      -9.1424e-5     0.000734901
  2.90008e-7    0.000292148  -0.000131446  …   3.99746e-6    1.80866e-5
 -0.000263152  -0.023877     -0.0275659        0.00328879    0.0126135
  3.99916e-6   -0.000846166   0.000227818      0.000210856   0.000518153
  0.00019736   -0.0056657     0.00412692       0.000518153   0.00427687

and to get the condition number of the correlation matrix implied by vcov use

cond(inference_results)
50.11623683487897

Some users request the condition number of the covariance matrix itself and even if the use is often misguided it can be calculated as well.

cond(inference_results; correlation = false)
13082.75373321667

It is also possible to calculate the correlation matrix from the vcov output using the cov2cor function

cor_from_cov = cov2cor(vcov(inference_results))
9×9 Symmetric{Float64, Matrix{Float64}}:
  1.0          0.147799     0.242733  …  -0.0973098   0.0413859   0.453494
  0.147799     1.0          0.138178     -0.265766   -0.263578   -0.391865
  0.242733     0.138178     1.0          -0.362707    0.083889    0.337422
 -0.196394    -0.165183    -0.425047      0.555027    0.557394    0.133439
  0.737483     0.00937536   0.358102     -0.28125    -0.25618     0.45724
  0.00845409   0.256348    -0.136345  …   0.315212    0.0534038   0.0536508
 -0.0973098   -0.265766    -0.362707      1.0         0.557335    0.47462
  0.0413859   -0.263578     0.083889      0.557335    1.0         0.545635
  0.453494    -0.391865     0.337422      0.47462     0.545635    1.0

And we see that the cond call above matches the condition number of the correlation matrix

cond(cor_from_cov)
50.1162368348788

3.2.1 Failure of the asymptotic variance-covariance matrix

It is well-known that the asymptotic variance-covariance matrix can sometimes fail to compute. This can happen for a variety of reasons including:

  1. There are parameters very close to a bound (often 0)
  2. The parameter vector does not represent a local minimum (optimization failed)
  3. The parameter vector does represent a local minimum but it’s not the global solution

The first one is often easy to check. The problematic parameters can be ones than have lower or upper bounds set. Often this will be a variance of standard deviation that has moved very close to the lower boundary. Consider removing the associated random effect if the problematic parameter is a variance in its specification or error model component if a combined additive and proportional error model is used and a standard deviation is close to zero.

It is also possible that the parameters do not represent a local minimum. In other words, they come from a failed fit. In this case, it can often be hard to perform the associated calculations in a stable way, but most importantly the results would not be interpretable even if they can be calculated in this case. The formulas are only valid for parameters that represent the actual (maximum likelihood) estimates. Please try to restart the optimization at different starting points in this case.

If you have reasons to believe that the model should indeed be a combined error model or if the random effect should be present it is also possible that the model converged to a local minimum that is not the global minimum. If the optimization happened to move to a region of the parameter state space that is hard to get out of you will often have to restart the fit at different parameter values. It is not possible to verify if the minimum is global in general, but it is always advised to try out more than one set of initial parameters when fitting models.

3.2.2 Bootstrap

Sometimes it is appropriate to use a different method to calculate estimate the uncertainty of the estimated parameters. Bootstrapping is a very popular approach that is simply but can often come a quite significant computational cost. Researchers often perform a bootstrapping step if their computational budget allows it or if the asymptotic variance-covariance estimator fails. Bootstrapping is advantageous because it does not rely on any invertability of matrices and it cannot produce negative variance confidence intervals because the resampled estimator respects the bounds of the model.

The signature for bootstrapping in infer looks as follows.

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

This does not help much before also looking at the interface for Bootstrap itself.

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

Bootstrap accepts a random number generator rng, the number of resampled datasets to produce samples, if sampling should be stratified according to the covariates in stratify_by, and finally the ensemble algorithm to control parallelization across fits. On the JuliaHub platform this can be used together with distributed computing to perform many resampled estimations in a short time.

bootstrap_results = infer(foce_fit, Bootstrap(samples = 50); level = 0.95)
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Warning: Terminated early due to NaN in gradient.
└ @ Optim ~/run/_work/PumasTutorials.jl/PumasTutorials.jl/custom_julia_depot/packages/Optim/7krni/src/multivariate/optimize/optimize.jl:117
┌ Info: Bootstrap inference finished.
│   Total resampled fits = 50
│   Success rate = 1.0
└   Unique resampled populations = 50

Bootstrap inference results

Dynamical system type:                 Closed form

Number of subjects:                             32

Observation records:         Active        Missing
    conc:                       251             47
    Total:                      251             47

Number of parameters:      Constant      Optimized
                                  0              9

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -350.49625

---------------------------------------------------------
           Estimate   SE          95.0% C.I.
---------------------------------------------------------
pop_CL     0.13465    0.0090551   [ 0.12428  ; 0.14912 ]
pop_Vc     8.0535     0.26504     [ 7.6274   ; 8.5159  ]
pop_tabs   0.55061    0.22486     [ 0.21093  ; 1.085   ]
pop_lag    0.87158    0.14698     [ 0.58479  ; 1.2821  ]
pk_Ω₁,₁    0.070642   561220.0    [ 0.028144 ; 0.11522 ]
pk_Ω₂,₂    0.018302   0.004948    [ 0.0095219; 0.027023]
pk_Ω₃,₃    0.91326    0.79202     [ 0.083747 ; 3.0749  ]
σ_prop     0.090096   0.014425    [ 0.070673 ; 0.11906 ]
σ_add      0.39115    0.10914     [ 0.036122 ; 0.47577 ]
---------------------------------------------------------
Successful fits: 50 out of 50
Unique resampled populations: 50
No stratification.

Again, we can calculate a covariance matrix based on the samples with vcov

vcov(bootstrap_results)
9×9 Symmetric{Float64, Matrix{Float64}}:
     8.19945e-5      -0.000336711  …    -3.50055e-5        0.000565455
    -0.000336711      0.0702444         -0.000894007      -0.0176072
     0.000558086      0.0178959         -0.000197232       0.00387331
    -0.000111521     -0.00466242         0.000512402       0.000357513
 -2951.78         56078.7             2709.21         -27779.4
    -8.40975e-6       0.000347322  …     1.4446e-5        -7.80706e-5
    -0.000171646     -0.0827463          0.00520728        0.0267877
    -3.50055e-5      -0.000894007        0.000208079       2.8332e-5
     0.000565455     -0.0176072          2.8332e-5         0.0119106

and we can even get a DataFrame that includes all the estimated parameters from the sampled population fits

DataFrame(bootstrap_results.vcov)
50×9 DataFrame
25 rows omitted
Row pop_CL pop_Vc pop_tabs pop_lag pk_Ω₁,₁ pk_Ω₂,₂ pk_Ω₃,₃ σ_prop σ_add
Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64
1 0.13323 8.20728 0.791375 0.561339 0.0823593 0.0136195 0.318462 0.100694 0.232389
2 0.135569 8.44134 0.92093 0.84089 0.0878481 0.0223436 0.434904 0.0793278 0.379577
3 0.133214 8.15451 0.209688 0.946414 0.0548924 0.0191834 1.43885 0.0821152 0.303382
4 0.146243 8.09525 0.680989 0.848553 0.0948266 0.0112764 0.664869 0.0992746 0.379791
5 0.124129 7.73684 0.441653 0.842237 0.0444963 0.0117525 1.15752 0.0921815 0.397297
6 0.12592 8.25691 0.635714 0.836178 0.0550814 0.014361 0.458466 0.0753128 0.298527
7 0.126899 7.78716 0.439159 0.972769 0.0385816 0.0224259 1.9789 0.114202 0.420057
8 0.130988 8.07576 0.311589 0.98284 0.0282472 0.0253044 3.4725 0.112748 0.294359
9 0.136263 8.37749 0.454351 0.870117 0.0382458 0.0208256 0.727583 0.095338 0.240551
10 0.134155 7.54576 0.279517 0.973944 0.0564424 0.0197273 3.07544 0.108975 0.469758
11 0.130906 8.2373 0.564182 0.862791 0.0450941 0.0180138 0.364845 0.0973565 0.213472
12 0.143213 8.47163 0.552519 0.889574 0.0876238 0.0155491 0.39629 0.061615 0.338408
13 0.0971723 8.77843 0.675515 0.710536 3.96844e6 0.025277 0.0204085 0.126042 1.10943e-82
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
39 0.140619 8.4742 1.09293 0.787401 0.0878795 0.0235782 0.393446 0.0779579 0.383531
40 0.128871 8.43283 0.21521 0.91707 0.0429032 0.0176422 0.668267 0.0780908 0.186376
41 0.125697 7.91828 0.480461 0.919592 0.0336388 0.0149118 0.664353 0.0994008 0.316135
42 0.140551 7.78494 0.549585 0.928 0.0883875 0.0147415 0.898622 0.10708 0.424709
43 0.144365 8.08573 0.5153 0.87089 0.0971503 0.014082 0.488065 0.0844205 0.398055
44 0.133604 8.38586 0.260778 0.972484 0.050185 0.0255931 2.89587 0.101091 0.290101
45 0.127517 7.91961 0.428351 0.976971 0.0371348 0.0272688 3.07325 0.119089 0.441089
46 0.124923 8.51307 0.640704 0.855135 0.0286727 0.0176239 0.342263 0.0818859 0.160004
47 0.135456 8.18322 0.742447 0.904681 0.0676346 0.0180688 0.496355 0.0966218 0.440981
48 0.135845 7.69767 0.536743 0.915432 0.0840448 0.0181668 0.948525 0.0969929 0.424724
49 0.126358 8.19263 0.682 0.841587 0.0470215 0.0145047 1.02964 0.0787489 0.358142
50 0.128161 7.76568 0.436558 0.546353 0.0614043 0.0164034 3.73374e-7 0.0736364 0.307656

This is very useful for histogram plotting of parameter distributions.

3.2.3 Sampling Importance Re-sampling

Pumas has support for inference through Sampling Importance Re-sampling through the SIR() input to infer. The signature for SIR in infer looks as follows.

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

This performs sampling importance re-sampling for the population in fpm. The confidence intervals are calculated as the (1-level)/2 and (1+level)/2 quantiles of the sampled parameters. ensemblealg can be EnsembleThreads() (the default value) to use multi-threading or EnsembleSerial() to use a single thread.

The signature for the SIR specification is

SIR(; rng, samples, resamples)

SIR accepts a random number generator rng, the number of samples from the proposal, samples, can be set and to complete the specification the resample has to be set. It is suggested that samples is at least 5 times larger than resamples in practice to have sufficient samples to resample from.

sir_results = infer(foce_fit, SIR(samples = 1000, resamples = 200); level = 0.95)
[ Info: Calculating: variance-covariance matrix.
[ Info: Done.
[ Info: Running SIR.
[ Info: Resampling.

Simulated inference results

Dynamical system type:                 Closed form

Number of subjects:                             32

Observation records:         Active        Missing
    conc:                       251             47
    Total:                      251             47

Number of parameters:      Constant      Optimized
                                  0              9

Likelihood approximation:                     FOCE
Likelihood optimizer:                         BFGS

Termination Reason:                   GradientNorm
Log-likelihood value:                   -350.49625

--------------------------------------------------------
           Estimate   SE          95.0% C.I.
--------------------------------------------------------
pop_CL     0.13465    0.0055232   [ 0.12578 ; 0.14599 ]
pop_Vc     8.0535     0.21377     [ 7.6726  ; 8.4502  ]
pop_tabs   0.55061    0.15057     [ 0.29822 ; 0.89115 ]
pop_lag    0.87158    0.042053    [ 0.78352 ; 0.94544 ]
pk_Ω₁,₁    0.070642   0.016553    [ 0.045459; 0.10976 ]
pk_Ω₂,₂    0.018302   0.0048786   [ 0.010063; 0.028944]
pk_Ω₃,₃    0.91326    0.29955     [ 0.50762 ; 1.6117  ]
σ_prop     0.090096   0.0085777   [ 0.074035; 0.10566 ]
σ_add      0.39115    0.036412    [ 0.33811 ; 0.47912 ]
--------------------------------------------------------

Notice, that SIR bases its first samples number of samples from a truncated multivariate normal distribution with mean of the maximum likelihood population level parameters and covariance matrix that is the asymptotic matrix calculated by infer(fpm). This means that to use SIR the matrix is question has to be successfully calculated by infer(fpm) under the hood.

The methods for vcov and DataFrame(sir_results.vcov) that we saw for Bootstrap also applies here

vcov(sir_results)
9×9 Symmetric{Float64, Matrix{Float64}}:
  3.05057e-5    0.00017376    3.16741e-5   …  -1.08916e-6    3.89227e-5
  0.00017376    0.0456974     0.00910612       3.82711e-7   -0.002038
  3.16741e-5    0.00910612    0.0226718        9.46417e-5    0.000396189
 -1.51387e-5   -0.000610178  -0.00206591       2.29116e-5    0.000309257
  4.8686e-5     0.000280504   0.000245383     -5.19439e-5    0.000251573
 -2.90886e-6    0.000189743   4.79939e-5   …   7.45685e-7    8.72613e-6
 -0.000366786  -0.00720379   -0.0174358        0.000261838   0.00316718
 -1.08916e-6    3.82711e-7    9.46417e-5       7.35768e-5    2.44909e-5
  3.89227e-5   -0.002038      0.000396189      2.44909e-5    0.00132585

and

DataFrame(sir_results.vcov)
200×9 DataFrame
175 rows omitted
Row pop_CL pop_Vc pop_tabs pop_lag pk_Ω₁,₁ pk_Ω₂,₂ pk_Ω₃,₃ σ_prop σ_add
Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64 Float64
1 0.135756 8.18513 0.457613 0.862083 0.0676902 0.0162232 0.904281 0.0978225 0.369858
2 0.127934 8.42382 0.952258 0.774151 0.0795748 0.0221921 0.513181 0.0886247 0.346316
3 0.127604 8.08186 0.641844 0.93448 0.083258 0.0276672 1.52666 0.0910196 0.490284
4 0.145313 8.2139 0.566593 0.85655 0.0744819 0.0291473 0.935252 0.0888907 0.351748
5 0.126793 7.97982 0.419822 0.743315 0.0623115 0.018158 1.12384 0.0913996 0.389919
6 0.135247 7.95475 0.609286 0.896347 0.0525052 0.0225094 1.39235 0.0825916 0.360687
7 0.126216 7.73592 0.530339 0.89951 0.0732214 0.0207412 1.3877 0.0737389 0.462568
8 0.139774 8.23812 0.530381 0.910542 0.090941 0.017896 1.45225 0.0895194 0.435131
9 0.13722 8.16592 0.621373 0.853793 0.0759836 0.0242841 0.618417 0.0849662 0.338282
10 0.141234 7.66084 0.596256 0.88745 0.103689 0.01412 0.637426 0.0820216 0.381504
11 0.144751 7.87291 0.497025 0.87971 0.0737385 0.018409 1.12892 0.0864012 0.415998
12 0.129264 7.97477 0.522049 0.869903 0.0670709 0.00955507 0.571405 0.0896627 0.406787
13 0.138135 7.82616 0.527258 0.867346 0.0738438 0.0171755 0.750477 0.102415 0.36373
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
189 0.137869 8.19329 0.299244 0.945433 0.0801471 0.0172385 0.799525 0.0948349 0.382412
190 0.131889 7.76434 0.233769 0.86404 0.0485432 0.0195741 1.79979 0.101398 0.415065
191 0.139349 8.23077 0.338273 0.91902 0.1022 0.0267065 1.07595 0.0723795 0.39755
192 0.128974 8.43291 0.699697 0.899561 0.0809745 0.0251594 0.843238 0.0754693 0.370244
193 0.143735 7.95856 0.624922 0.896066 0.0918681 0.0124163 0.626936 0.0887354 0.454116
194 0.132335 7.9153 0.455948 0.911961 0.0513684 0.0209757 1.27935 0.0910339 0.410635
195 0.141326 8.16733 0.656509 0.878783 0.0713747 0.0233366 1.2275 0.100199 0.421034
196 0.121969 8.01575 0.546848 0.918101 0.0591082 0.0100144 0.668178 0.0836779 0.373118
197 0.135153 7.91359 0.718934 0.806777 0.0720071 0.00968338 0.986095 0.105033 0.450284
198 0.136877 7.79438 0.522326 0.865159 0.0891616 0.0128566 0.901763 0.0854831 0.440135
199 0.143798 8.01544 0.548749 0.899228 0.084835 0.0167247 0.608901 0.090777 0.373111
200 0.139225 7.96604 0.35164 0.870921 0.0849718 0.0160526 0.834434 0.0928043 0.392189

3.2.4 Marginal MCMC

An alternative to Bootstrap and SIR is to simply use the MarginalMCMC sampler which is a Hamiltonian Monte Carlo (HMC) No-U-Turn Sampler (NUTS) that will sample from the marginal loglikelihood. This means that individual effects are marginalized out and then we sample the population level parameters. This does not resample populations like Bootstrap so inference may be more stable if many resampled populations lead to extreme estimates and it differs from SIR in that it does not need the asymptotic covariance matrix to be calculated and sampled from.

This method requires slightly more understanding from the user when setting the options that can be found through the docstring of MarginalMCMC. Some knowledge of Bayesian inference is advised.

inference_results = infer(foce_fit, MarginalMCMC(); level = 0.95)

As sampling based inference can be computationally intensive we exclude the actual invocation of this method from this tutorial.

4 Concluding Remarks

This tutorial showcased a typical Pumas workflow for parameter inference in Pumas models. We showed the different methods supported by Pumas for calculating parameter uncertainty.