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 uncertainty based on the asymptotic variance-covariance formulas (robust or not).
  7. (Optionally) proceeding with more advanced techniques like bootstrapping or SIR for robust uncertainty quantification.

In previous tutorials, we already set up the data and performed a fit. We also obtained some parameter uncertainty estimates. In this tutorial, we will go more into depth with parameter uncertainty calculations using different methods. 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 ~ @. CombinedNormal(cp, σ_add, σ_prop)
    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_scales = @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_scales;
    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 Obtaining fit results

Following the examples in previous tutorials, we perform a fit. We need the output of the fit function call to perform inference to obtain parameter uncertainty estimates.

# 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: 0.04807090759277344
     1     2.643772e+03     3.167516e+03
 * time: 2.630375862121582
     2     1.836601e+03     2.118430e+03
 * time: 2.6559369564056396
     3     9.351337e+02     8.722439e+02
 * time: 2.681324005126953
     4     6.402300e+02     4.199225e+02
 * time: 2.7062768936157227
     5     5.103664e+02     1.642121e+02
 * time: 2.7309160232543945
     6     4.760464e+02     5.453749e+01
 * time: 2.7548038959503174
     7     4.703757e+02     3.643518e+01
 * time: 4.733490943908691
     8     4.699019e+02     3.135992e+01
 * time: 4.752802848815918
     9     4.697614e+02     2.953531e+01
 * time: 4.771333932876587
    10     4.693153e+02     2.463233e+01
 * time: 4.7894368171691895
    11     4.685743e+02     2.580427e+01
 * time: 4.8087639808654785
    12     4.675133e+02     3.864937e+01
 * time: 4.828780889511108
    13     4.666775e+02     5.495470e+01
 * time: 4.848706007003784
    14     4.661197e+02     5.692101e+01
 * time: 4.869323015213013
    15     4.656782e+02     4.770992e+01
 * time: 4.888653993606567
    16     4.651802e+02     3.087698e+01
 * time: 4.907728910446167
    17     4.645523e+02     1.184834e+01
 * time: 4.927351951599121
    18     4.641447e+02     1.162249e+01
 * time: 4.948163032531738
    19     4.639978e+02     1.125144e+01
 * time: 4.970925807952881
    20     4.639307e+02     1.156463e+01
 * time: 4.9937310218811035
    21     4.638001e+02     1.312870e+01
 * time: 5.01677393913269
    22     4.635282e+02     1.480920e+01
 * time: 5.038254022598267
    23     4.630353e+02     2.169377e+01
 * time: 5.0582098960876465
    24     4.623847e+02     4.478029e+01
 * time: 5.077876806259155
    25     4.617426e+02     6.468975e+01
 * time: 5.0987138748168945
    26     4.610293e+02     7.776996e+01
 * time: 5.120874881744385
    27     4.597628e+02     8.785260e+01
 * time: 5.141788005828857
    28     4.566753e+02     9.769803e+01
 * time: 5.163262844085693
    29     4.490421e+02     1.008838e+02
 * time: 5.18592381477356
    30     4.391868e+02     9.978816e+01
 * time: 5.209959983825684
    31     4.130704e+02     5.917685e+01
 * time: 5.234653949737549
    32     4.055780e+02     3.852824e+01
 * time: 5.342396974563599
    33     4.023118e+02     3.889618e+01
 * time: 5.367537975311279
    34     4.012516e+02     3.694778e+01
 * time: 5.4082818031311035
    35     4.004391e+02     2.061948e+01
 * time: 5.4308319091796875
    36     3.983040e+02     3.508423e+01
 * time: 5.4529759883880615
    37     3.969705e+02     3.841039e+01
 * time: 5.475142955780029
    38     3.965462e+02     3.738343e+01
 * time: 5.496229887008667
    39     3.950409e+02     3.064789e+01
 * time: 5.518172979354858
    40     3.945750e+02     2.876429e+01
 * time: 5.5390520095825195
    41     3.937725e+02     2.571438e+01
 * time: 5.560366868972778
    42     3.933955e+02     2.436112e+01
 * time: 5.58204984664917
    43     3.927564e+02     2.051069e+01
 * time: 5.602881908416748
    44     3.916020e+02     1.629035e+01
 * time: 5.624118804931641
    45     3.886991e+02     2.689824e+01
 * time: 5.644366979598999
    46     3.870054e+02     2.298582e+01
 * time: 5.663103818893433
    47     3.853691e+02     2.614992e+01
 * time: 5.707810878753662
    48     3.841730e+02     2.207557e+01
 * time: 5.727065801620483
    49     3.825113e+02     2.204399e+01
 * time: 5.746184825897217
    50     3.808880e+02     2.444784e+01
 * time: 5.766003847122192
    51     3.800407e+02     1.250611e+01
 * time: 5.785405874252319
    52     3.798092e+02     1.167926e+01
 * time: 5.804163932800293
    53     3.797789e+02     1.162382e+01
 * time: 5.8227620124816895
    54     3.797069e+02     1.152441e+01
 * time: 5.840788841247559
    55     3.794424e+02     1.132717e+01
 * time: 5.859774827957153
    56     3.788131e+02     2.006438e+01
 * time: 5.878332853317261
    57     3.771525e+02     3.584695e+01
 * time: 5.897320985794067
    58     3.731299e+02     5.697249e+01
 * time: 5.917426824569702
    59     3.658671e+02     6.542042e+01
 * time: 5.937747001647949
    60     3.604194e+02     4.036489e+01
 * time: 5.978269815444946
    61     3.532841e+02     1.574006e+01
 * time: 5.9998250007629395
    62     3.520181e+02     1.393300e+01
 * time: 6.021374940872192
    63     3.517984e+02     6.701188e+00
 * time: 6.041980981826782
    64     3.517541e+02     3.503978e+00
 * time: 6.0625319480896
    65     3.516436e+02     8.720957e+00
 * time: 6.083197832107544
    66     3.511845e+02     1.406200e+01
 * time: 6.107919931411743
    67     3.510647e+02     2.540378e+00
 * time: 6.126168966293335
    68     3.510209e+02     3.157201e+00
 * time: 6.144171953201294
    69     3.509959e+02     3.045642e+00
 * time: 6.161505937576294
    70     3.509765e+02     2.673143e+00
 * time: 6.1968488693237305
    71     3.509751e+02     2.603975e+00
 * time: 6.214148998260498
    72     3.509724e+02     2.505719e+00
 * time: 6.231274843215942
    73     3.509666e+02     2.379768e+00
 * time: 6.248921871185303
    74     3.509504e+02     3.572030e+00
 * time: 6.2667248249053955
    75     3.509123e+02     6.006350e+00
 * time: 6.284411907196045
    76     3.508288e+02     8.822995e+00
 * time: 6.302067995071411
    77     3.506944e+02     9.708012e+00
 * time: 6.319978952407837
    78     3.505767e+02     6.092631e+00
 * time: 6.337718963623047
    79     3.505358e+02     1.734431e+00
 * time: 6.369365930557251
    80     3.505314e+02     6.749379e-01
 * time: 6.387204885482788
    81     3.505313e+02     6.721982e-01
 * time: 6.404048919677734
    82     3.505312e+02     6.699487e-01
 * time: 6.420694828033447
    83     3.505307e+02     6.606824e-01
 * time: 6.437394857406616
    84     3.505298e+02     6.413909e-01
 * time: 6.454203844070435
    85     3.505274e+02     9.083363e-01
 * time: 6.471344947814941
    86     3.505222e+02     1.339147e+00
 * time: 6.488548994064331
    87     3.505129e+02     1.608661e+00
 * time: 6.5224609375
    88     3.505026e+02     1.293164e+00
 * time: 6.5404908657073975
    89     3.504973e+02     5.140504e-01
 * time: 6.558198928833008
    90     3.504963e+02     6.340189e-02
 * time: 6.576372861862183
    91     3.504963e+02     3.137914e-03
 * time: 6.593740940093994
    92     3.504963e+02     5.681551e-04
 * time: 6.6104958057403564

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 also known as the robust variance-covariance 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 calculated using finite difference derivatives of the gradient calculated using automatic differentiation.

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)
50.11623683496925

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)
13082.753733216321

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)
50.11623683496949

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)
[ Info: Bootstrap inference finished.

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    0.007318   [ 0.12393  ; 0.14864 ]
pop_Vc     8.0535     0.18886    [ 7.7182   ; 8.4181  ]
pop_tabs   0.55061    0.20547    [ 0.20703  ; 0.97832 ]
pop_lag    0.87158    0.16753    [ 0.5563   ; 1.3962  ]
pk_Ω₁,₁    0.070642   0.024112   [ 0.028097 ; 0.12167 ]
pk_Ω₂,₂    0.018302   0.004648   [ 0.0093965; 0.026484]
pk_Ω₃,₃    0.91326    0.45909    [ 0.28885  ; 2.2622  ]
σ_prop     0.090096   0.014031   [ 0.065768 ; 0.11794 ]
σ_add      0.39115    0.083113   [ 0.23252  ; 0.497   ]
--------------------------------------------------------
Unique resampled populations: 50 out of 50
Stratification by .

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

vcov(bootstrap_results)
9×9 Symmetric{Float64, Matrix{Float64}}:
  5.35534e-5    0.000526588   0.000363393  …   3.51342e-5    0.000232887
  0.000526588   0.035669      0.012749         0.000167912  -0.000590455
  0.000363393   0.012749      0.0422168        0.000707704   0.00491784
 -8.87407e-5   -0.000693103  -0.011693        -5.54334e-5    0.00597319
  0.000128811   0.00054429    0.001967         1.17787e-5    0.000644754
 -5.87486e-6    8.79343e-5   -0.000294543  …  -9.29461e-6   -1.54616e-5
 -0.000242651  -0.0315627    -0.0489178        0.000704611   0.0097512
  3.51342e-5    0.000167912   0.000707704      0.000196866   0.000307252
  0.000232887  -0.000590455   0.00491784       0.000307252   0.0069078

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 0.138605 7.8957 0.61819 0.899563 0.0793847 0.00794878 0.889524 0.095197 0.473749
2 0.141255 8.43289 1.08284 0.805332 0.0966564 0.0141378 0.288559 0.0996487 0.389441
3 0.140112 8.14507 0.866356 0.699606 0.130808 0.0106341 0.584682 0.0774065 0.36524
4 0.145865 8.22682 0.722411 0.804394 0.0876616 0.0101384 0.449542 0.0982301 0.327512
5 0.125549 8.08315 0.213336 0.88386 0.0351899 0.0199125 0.581266 0.0790613 0.226018
6 0.140798 7.929 0.67995 0.498887 0.123383 0.0267636 0.545571 0.0868177 0.319068
7 0.133774 8.22447 0.750283 0.920569 0.0441059 0.016859 0.653901 0.104632 0.379911
8 0.134981 8.05638 0.660891 0.726493 0.072265 0.01523 0.94635 0.0712545 0.317572
9 0.124975 7.70236 0.475108 0.85103 0.0474784 0.0179118 1.25672 0.0902731 0.382067
10 0.134226 7.80776 0.823588 0.964136 0.0754916 0.016448 1.232 0.118263 0.471111
11 0.125236 8.10652 0.613488 0.89103 0.0246407 0.0270565 1.14984 0.111691 0.320553
12 0.140294 7.89533 0.502416 0.875518 0.0941851 0.018318 0.462664 0.0794753 0.35946
13 0.133306 8.16105 0.498093 0.840694 0.0623879 0.0201541 0.643516 0.0714039 0.320308
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
39 0.128703 8.14686 0.836465 0.847671 0.0730447 0.0142105 1.11243 0.084295 0.434388
40 0.148653 8.11767 0.818758 0.880443 0.0905497 0.00923782 0.895422 0.116814 0.477267
41 0.142026 7.87599 0.651159 0.785371 0.105251 0.0191469 1.00333 0.0848797 0.431012
42 0.141387 8.25926 0.705095 0.939312 0.0590468 0.0186882 0.789904 0.113852 0.436806
43 0.140998 8.20895 0.86227 1.44247 0.0836423 0.0226791 0.799693 0.081046 0.701528
44 0.145075 7.98025 0.965633 0.542868 0.115756 0.00994323 0.405128 0.105754 0.254936
45 0.139512 8.36727 0.504903 0.821997 0.0814494 0.025519 0.683388 0.0616799 0.344199
46 0.135102 7.94908 0.12582 1.40393 0.0639436 0.0200853 2.70475 0.0823103 0.363263
47 0.14688 7.85921 0.430771 0.837858 0.1065 0.0103214 1.09176 0.0993948 0.457964
48 0.122249 7.79551 0.47798 0.931226 0.0260374 0.0166578 1.16476 0.0861424 0.320469
49 0.131743 8.09641 0.873679 0.8026 0.0797947 0.0137281 0.374015 0.0885568 0.322999
50 0.131349 7.77275 0.583034 0.835038 0.076252 0.0234827 1.0178 0.0997629 0.438348

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    0.0053461   [ 0.12577 ; 0.14651]
pop_Vc     8.0535     0.20686     [ 7.6622  ; 8.5006 ]
pop_tabs   0.55061    0.15169     [ 0.27332 ; 0.87778]
pop_lag    0.87158    0.041225    [ 0.77382 ; 0.93899]
pk_Ω₁,₁    0.070642   0.016142    [ 0.045643; 0.10825]
pk_Ω₂,₂    0.018302   0.0051396   [ 0.010675; 0.02967]
pk_Ω₃,₃    0.91326    0.32089     [ 0.42914 ; 1.6287 ]
σ_prop     0.090096   0.0076047   [ 0.076376; 0.10732]
σ_add      0.39115    0.039066    [ 0.34067 ; 0.48667]
-------------------------------------------------------

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}}:
  2.8581e-5    0.00029673   -9.09871e-7   …  -9.07816e-7    3.75947e-5
  0.00029673   0.0427923     0.00303384      -0.00013039   -0.00261001
 -9.09871e-7   0.00303384    0.0230104        0.000171038   0.000395179
  6.63875e-6  -0.000554134  -0.00263481       2.63091e-5    9.07884e-5
  3.92151e-5   0.000153061   0.000117752     -3.20371e-5    0.000312731
  3.2684e-6    0.00016708   -7.52297e-5   …  -1.10472e-5    2.10414e-5
 -3.24527e-5  -0.00254273   -0.0269288       -6.94775e-5    0.00318881
 -9.07816e-7  -0.00013039    0.000171038      5.78319e-5   -2.29718e-5
  3.75947e-5  -0.00261001    0.000395179     -2.29718e-5    0.00152612

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 0.148834 8.02681 0.677182 0.894045 0.0618706 0.0196805 0.833632 0.0923761 0.379526
2 0.123407 8.11705 0.544741 0.827331 0.0713491 0.0313455 1.55015 0.0849969 0.395499
3 0.129893 7.68071 0.540395 0.878731 0.0810069 0.0244153 0.720865 0.0753865 0.468424
4 0.132654 8.02556 0.393569 0.815684 0.0841987 0.0163122 1.08254 0.0889067 0.405191
5 0.131583 8.36759 0.598634 0.810375 0.0648606 0.0218532 1.34706 0.0862424 0.374545
6 0.142828 7.9624 0.764774 0.86322 0.0677335 0.0219139 0.840477 0.090357 0.39977
7 0.132105 7.89105 0.310026 0.852319 0.0938946 0.0329353 1.64907 0.0763842 0.44548
8 0.140689 8.12315 0.62155 0.884604 0.0835961 0.0201084 1.01081 0.0767262 0.431156
9 0.135067 8.00154 0.586279 0.807686 0.102373 0.0188927 0.649483 0.0857812 0.34447
10 0.12577 8.24548 0.604978 0.82272 0.0598739 0.0192316 0.904508 0.0900032 0.341017
11 0.135399 7.8939 0.421548 0.881832 0.073266 0.0331707 1.15673 0.0823352 0.400908
12 0.139302 8.16877 0.456889 0.912965 0.0670079 0.0229405 1.41614 0.0974649 0.385054
13 0.127224 7.95361 0.687725 0.831963 0.0653026 0.0149402 0.803249 0.100121 0.355874
⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮
189 0.1385 8.1888 0.845646 0.73552 0.0829002 0.022071 0.714751 0.0908622 0.395658
190 0.132445 8.01448 0.51379 0.896139 0.0816485 0.0135399 0.65619 0.087299 0.374343
191 0.136933 8.0237 0.253909 0.911829 0.0953016 0.0172296 1.29648 0.0696516 0.398116
192 0.132463 8.0635 0.631412 0.906857 0.0683555 0.0171024 0.66143 0.0907139 0.412705
193 0.137453 8.14856 0.37214 0.773924 0.10695 0.0224389 1.40659 0.0775836 0.433066
194 0.127269 8.02639 0.582777 0.862158 0.0731455 0.0154684 1.08289 0.0950091 0.40565
195 0.144326 7.88323 0.465608 0.897779 0.079922 0.0168162 1.21281 0.0984038 0.449549
196 0.132659 8.01706 0.59734 0.860069 0.0863569 0.021399 1.18842 0.0895817 0.451507
197 0.134792 8.19744 0.657815 0.898409 0.0658468 0.0169043 0.674749 0.0886008 0.380711
198 0.136712 8.18057 0.729274 0.829358 0.0998849 0.0240643 0.817538 0.072843 0.450667
199 0.140301 8.16379 0.578488 0.830088 0.0758107 0.0159641 1.02873 0.0776839 0.408784
200 0.13507 7.43913 0.547349 0.852778 0.069478 0.0143328 1.43076 0.0892834 0.455747

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.