# Fitting Non-Identifiable and Poorly Identifiable Models using Bayesian Inference

Author

Mohamed Tarek

In this tutorial, we will see how to use Bayesian inference in Pumas to fit a non-identifiable (or a poorly identifiable) model by sampling from the full posterior. Unlike maximum likelihood estimation methods which only find point estimates for the model parameters, Bayesian methods can be used to sample from the full posterior of the model parameters making it more robust to identifiability issues.

## 1 Model Identifiability

One of the goals of statistical learning is to identify the underlying parameter values in a parametric model that best fit the observed data. In many practical scenarios, some parameters in a model may not be identifiable, for one of the following problems:

1. The model is over-parameterized with redundant parameters, where a continuum of parameter values, e.g $$0 \leq \theta \leq 1$$, would all give identical model predictions. This tends to happen when a dynamical model has many compartments and associated parameters but only one or a few compartments are observed. Another case where this happens is if there is a typo in the model where a parameter is not used everywhere it should.
2. The model has symmetries such that a discrete set of parameter values, e.g. $$\theta = -1$$ or $$\theta = 1$$, give identical model predictions. This tends to happen with inappropriate use of the “absolute value” function (abs) if used on a function that can be either negative or positive potentially creating 2 separate modes. This problem can sometimes be fixed by using appropriate parameter bounds, e.g. $$\theta \geq 0$$, in the model.
3. The data is insufficient to learn some parameters’ values but more data would have been sufficient.

The first 2 problems are problems in the model describing structural non-identifiability and the third problem is a problem in the data or data-model mismatch describing practical non-identifiability (sometimes called non-estimability).

Type of Identifiability Description
Globally Structurally Identifiable Every set of parameter values $$\theta$$ makes a unique model prediction $$\mu(\theta)$$.
Globally Practically Identifiable Globally structurally identifiable and there is enough data to estimate the data-generating parameter values.
Locally Structurally Identifiable Only dis-connected parameters values $$\{\theta_1, \theta_2, \dots, \theta_m\}$$ can result in the same model prediction $$\mu$$, i.e. $$\mu(\theta_1) = \mu(\theta_2) = \dots = \mu(\theta_m)$$. However, in the neighborhood $$N(\theta)$$ of each set of parameters $$\theta$$, each $$\theta' \in N(\theta)$$ results in a unique model prediction $$\mu(\theta')$$.
Locally Practically Identifiable Locally structurally identifiable and there is enough data to estimate the (potentially non-unique but dis-connected) data-generating parameter values.
Key Points
• Local (or global) practical identifiablity is what we usually want in analyses.
• Practical identifiability (estimability) implies structural identifiability.
Poor Identifiability

When a model may be identifiable in exact arithmetic but its identifiability is sensitive to numerical errors in computations, we will call it poorly identifiable. When the term “poorly identifiable models” is used in the rest of this tutorial, this also includes truly non-identifiable models.

## 2 Example

Let’s consider an example of fitting a poorly identifiable model.

### 2.1 Loading Pumas

First, let’s load Pumas:

using Pumas

### 2.2 Two Compartment Model

Now let’s define a single-subject, 2-compartment model with a depot, central and peripheral compartments and a proportional error model.

model = @model begin
@param begin
θ ∈ VectorDomain(lower = zeros(5))
σ ∈ RealDomain(lower = 0.0)
end
@pre begin
CL = θ[1]
Vc = θ[2]
Ka = θ[3]
Vp = θ[4]
Q = θ[5]
end
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - (CL + Q) / Vc * Central + Q / Vp * Peripheral
Peripheral' = Q / Vc * Central - Q / Vp * Peripheral
end
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, abs(cp) * σ + 1e-6)
end
end
PumasModel
Parameters: θ, σ
Random effects:
Covariates:
Dynamical variables: Depot, Central, Peripheral
Derived: dv
Observed: dv
Subject Model vs Population Model

Note that studying the identifiability of subject models is still relevant when fitting a population version of the model with typical values and random effects. The reason is that when evaluating the log likelihood using the Laplace method or first-order conditional estimation (FOCE), one of the steps involved is finding the empirical Bayes estimates (EBE) which is essentially fitting a subject’s version of the model, fixing population parameters and estimating random effects. Non-identifiability in the EBE estimation can often cause the EBE estimation to fail or to be so numerically unstable that the population parameters’ fit itself fails because the marginal likelihood (and its gradient) computed with Laplace/FOCE relied on incorrect or numerically unstable EBEs.

### 2.3 Parameter Values

Let’s define some parameter values to use for simulation.

params = (θ = [35, 100, 0.5, 210, 30], σ = 0.1)
(θ = [35.0, 100.0, 0.5, 210.0, 30.0],
σ = 0.1,)

### 2.4 Subject Definition

Next we define a subject skeleton with a single bolus dose and no observations.

skeleton = Subject(
id = 1,
time = 0.0:0.5:30.0,
events = DosageRegimen(3000, time = 0.0, cmt = 1),
observations = (; dv = nothing),
)
Subject
ID: 1
Events: 1
Observations: dv: (n=61)

### 2.5 Fisher Information Matrix

The expected Fisher Information Matrix (FIM) is an important diagnostic which can be used to detect local practical identifiability (LPI) given the model and the experiment design [1]. The first order approximation of the expected FIM [2,3] has also been used successfully to analyze the LPI of pharmacometric nonlinear mixed effect (NLME) models [4,5,6].

The positive definiteness (non-singularity) of the expected FIM $$F(\theta)$$ at parameters $$\theta$$ is a sufficient condition for LPI at $$\theta$$. Under more strict assumptions which are more difficult to verify, the positive definiteness of $$F(\theta)$$ is even a necessary condition for LPI [1].

To compute a first-order approximation of the expected FIM, we will use the OptimalDesign package:

using OptimalDesign

times = OptimalDesign.ObsTimes(skeleton.time)
F = OptimalDesign.fim(model, [skeleton], params, [times], FO())
6×6 Symmetric{Float64, Matrix{Float64}}:
1.29489    -0.0538762     13.6934     0.0254882   -0.218694      -2.68028
-0.0538762   0.0188821     -4.38015   -0.00168545   0.0296863     -0.325336
13.6934     -4.38015     1054.71       0.635262    -7.13309       66.7482
0.0254882  -0.00168545     0.635262   0.00215196  -0.00487912    -0.0895271
-0.218694    0.0296863     -7.13309   -0.00487912   0.144604      -0.720992
-2.68028    -0.325336      66.7482    -0.0895271   -0.720992    1667.74

### 2.6 Procedure for Detecting Practical Non-Identifiability

Next, let’s do an eigenvalue decomposition of F to find the smallest eigenvalue. The smallest eigenvalue is always the first one because they are sorted.

E = eigen(F)
E.values[1]
9.646842560443467e-5

It is close to 0! This implies that the matrix is very close to being singular. While this is not strictly a proof of local non-identifiability, it is one step towards detecting non-identifiability.

Tip

The non-singularity of the expected FIM is generally only a sufficient (not necessary) condition for local identifiability. So there are some locally identifiable models that have a singular or undefined expected FIM. However, there is a subclass of models satisfying strict assumptions for which the non-singularity of the expected FIM is a necessary condition for local identifiability [1]. In practice, these assumptions are difficult to verify for a general model but if our model happens to satisfy these assumptions, then a singular expected FIM would imply local non-identifiability. In that case, the eigenvector(s) corresponding to the 0 eigenvalue would be useful diagnostics as they point in the directions along which changes in the parameters will have no effect on the log likelihood.

To summarize:

• Being “Non-Singular” Isn’t Always a Must: A non-singular expected FIM always signals that a model is “locally identifiable” (meaning you can find a locally unique solution for its parameters). However, some locally identifiable models can still have singular or undefined expected FIM.
• Sometimes, It’s Absolutely Necessary: There’s a special group of models where a non-singular expected FIM is required to be locally identifiable. It’s hard to know if your model falls into this group.
• Practical Use: Even if your expected FIM is singular, it may be a helpful diagnostic. The eigenvectors of the FIM with 0 eigenvalues show you directions where changing the model’s parameters may not affect the model’s predictions – this may help you pinpoint where the model is fuzzy.

To prove if a model is practically non-identifiable, it suffices to find a single set $$\Theta$$ such that for all parameters $$\theta \in \Theta$$, the log likelihood is unchanged. To find $$\Theta$$ numerically, one can

1. Assume parameter values $$\theta_0$$,
2. Define a criteria for changing the parameters $$\theta$$ from their current values $$\theta_0$$, and then create a candidate set of parameters $$\Theta_c$$,
3. Simulate synthetic data using the parameters $$\theta_0$$,
4. Evaluate the log likelihood $$L(\theta)$$ for all $$\theta \in \Theta_c$$,
5. Evaluate the sensitivity of the log likelihood to local changes within $$\Theta_c$$.

One way to construct a candidate $$\Theta_c$$ is as the set $$\{\theta_0 + \alpha \cdot d : \alpha \in [-\epsilon, \epsilon] \}$$ for a small $$\epsilon > 0$$, where $$d$$ is an eigenvector corresponding to the smallest eigenvalue of the expected FIM, $$F$$. The sensitivity of the log likelihood to changes within $$\Theta_c$$ can then be quantified as the average value of:

$\Bigg( \frac{L(\theta_0 + \alpha \cdot d) - L(\theta_0)}{\alpha} \Bigg)^2$

for all $$\alpha \in [-\epsilon, \epsilon]$$, $$\alpha \neq 0$$. Let’s follow this procedure for the above model assuming $$\theta_0$$ is params.

### 2.7 Simulating Data

Here we simulate a synthetic subject using the skeleton subject we have. We fix the seed of the pseudo-random number generator for reproducibility.

using Random

rng = Random.default_rng()
Random.seed!(rng, 12345)
pop = [Subject(simobs(model, skeleton, params; rng))]
Population
Subjects: 1
Observations: dv

To evaluate the log likelihood of params given pop, we can use the loglikelihood function:

ll0 = loglikelihood(model, pop, params, NaivePooled())
32.82461166030997

Since there are no random effects in this model, we use the NaivePooled() algorithm in Pumas.

### 2.8 Local Sensitivity Analysis of Log Likelihood

The eigenvectors corresponding to the smallest eigenvalue of F give us the directions that are likely to have the largest standard error in the maximum likelihood estimates. These are the most promising directions to test when constructing candidate sets $$\Theta_c$$ for identifiability analysis. To get the eigenvector corresponding to the smallest eigenvalue, you can run:

d = E.vectors[:, 1]
6-element Vector{Float64}:
0.006717009021476735
0.8382731175155917
0.0037560578470219614
-0.5451738773535086
0.004939403956445427
-3.137352344503738e-6

The order of parameters is the same as the order of definition in the model: θ and then σ.

Note that the potential non-identifiability seems to be mostly in the second and fourth parameter, Vc and Vp. More precisely, the above eigenvector implies that simultaneously increasing Vc and decreasing Vp (or vice versa) by the ratios given in d may have little to no effect on the log likelihood.

Let’s try to add α * d[1:end-1] to params.θ (i.e. move params.θ in the direction d[1:end-1]) and evaluate the log likelihood for different step sizes α. To do that, we will first define a function that moves params.θ and call it on different step values α. These choices of α values correspond to a discrete candidate set $$\Theta_c$$.

function moveθ(α)
# unpacking the fields of params to variables with the same names
(; θ, σ) = params
# move θ by step * d[1:end-1]
return (; θ = θ + α * d[1:end-1], σ)
end
# move by both negative and positive steps (excluding α = 0)
αs = vcat(-1e-3 .* (1:10), 1e-3 .* (1:10))
20-element Vector{Float64}:
-0.001
-0.002
-0.003
-0.004
-0.005
-0.006
-0.007
-0.008
-0.009
-0.01
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
newparams = moveθ.(αs)
20-element Vector{NamedTuple{(:θ, :σ), Tuple{Vector{Float64}, Float64}}}:
(θ = [34.99999328299098, 99.99916172688249, 0.499996243942153, 210.00054517387736, 29.999995060596042], σ = 0.1)
(θ = [34.999986565981956, 99.99832345376497, 0.49999248788430595, 210.0010903477547, 29.999990121192088], σ = 0.1)
(θ = [34.999979848972934, 99.99748518064746, 0.4999887318264589, 210.00163552163207, 29.99998518178813], σ = 0.1)
(θ = [34.99997313196391, 99.99664690752994, 0.4999849757686119, 210.00218069550942, 29.999980242384176], σ = 0.1)
(θ = [34.99996641495489, 99.99580863441243, 0.49998121971076487, 210.00272586938678, 29.999975302980218], σ = 0.1)
(θ = [34.99995969794587, 99.9949703612949, 0.49997746365291784, 210.00327104326414, 29.99997036357626], σ = 0.1)
(θ = [34.99995298093685, 99.9941320881774, 0.49997370759507087, 210.00381621714146, 29.999965424172306], σ = 0.1)
(θ = [34.999946263927825, 99.99329381505987, 0.49996995153722384, 210.00436139101882, 29.999960484768348], σ = 0.1)
(θ = [34.99993954691881, 99.99245554194236, 0.4999661954793768, 210.00490656489617, 29.999955545364394], σ = 0.1)
(θ = [34.99993282990979, 99.99161726882484, 0.4999624394215298, 210.00545173877353, 29.999950605960436], σ = 0.1)
(θ = [35.00000671700902, 100.00083827311751, 0.500003756057847, 209.99945482612264, 30.000004939403958], σ = 0.1)
(θ = [35.000013434018044, 100.00167654623503, 0.500007512115694, 209.9989096522453, 30.000009878807912], σ = 0.1)
(θ = [35.000020151027066, 100.00251481935254, 0.500011268173541, 209.99836447836793, 30.00001481821187], σ = 0.1)
(θ = [35.00002686803609, 100.00335309247006, 0.5000150242313881, 209.99781930449058, 30.000019757615824], σ = 0.1)
(θ = [35.00003358504511, 100.00419136558757, 0.5000187802892351, 209.99727413061322, 30.000024697019782], σ = 0.1)
(θ = [35.00004030205413, 100.0050296387051, 0.5000225363470822, 209.99672895673586, 30.00002963642374], σ = 0.1)
(θ = [35.00004701906315, 100.0058679118226, 0.5000262924049291, 209.99618378285854, 30.000034575827694], σ = 0.1)
(θ = [35.000053736072175, 100.00670618494013, 0.5000300484627762, 209.99563860898118, 30.000039515231652], σ = 0.1)
(θ = [35.00006045308119, 100.00754445805764, 0.5000338045206232, 209.99509343510383, 30.000044454635606], σ = 0.1)
(θ = [35.00006717009021, 100.00838273117516, 0.5000375605784703, 209.99454826122647, 30.000049394039564], σ = 0.1)

Now let’s evaluate the log likelihoods of all these parameter sets.

lls = map(newparams) do p
loglikelihood(model, pop, p, NaivePooled())
end
20-element Vector{Float64}:
32.82463258809266
32.824653516776316
32.82467444599981
32.824695375686
32.8247163049834
32.82473723581962
32.82475816602141
32.82477909749674
32.82480002929137
32.824820961111435
32.82459073300738
32.82456980632097
32.8245488798121
32.82452795380517
32.82450702699767
32.824486102815484
32.824465177567234
32.82444425406181
32.824423330072776
32.824402406539846

To compute the average sensitivity within $$\Theta_c$$, we then call:

sens = abs2.((lls .- ll0) ./ αs)
mean(sens)
0.0004379688924138595

Not very sensitive! Therefore, we can conclude that the log likelihood is almost the same inside $$\Theta_c$$. This is a strong sign of non-identifiability, or at least poor identifiability.

Let’s contrast this with a random direction d2:

d2 = normalize(rand(6))
function moveθ2(α)
# unpacking the fields of params to variables with the same names
(; θ, σ) = params
# move θ by step * d2[1:end-1]
return (; θ = θ + α * d2[1:end-1], σ)
end
newparams2 = moveθ2.(αs)
lls2 = map(newparams2) do p
loglikelihood(model, pop, p, NaivePooled())
end
sens2 = abs2.((lls2 .- ll0) ./ αs)
mean(sens2)
1489.899252213142

Notice the difference in sensitivity compared to a random direction.

Note

In practice, it can be difficult to be definitive about practical non-identifiability with numerical tests due to the nature of computation in floating point numbers where numerical errors can accumulate and either:

1. Mask a truly singular matrix by reporting its smallest eigenvalue as very close to 0 but not exactly 0 in floating point numbers, or
2. Make a truly non-singular matrix appear singular because its smallest eigenvalue was close to 0 in exact arithmetic but was computed as exactly 0 in floating point arithmetic.

A small enough average local sensitivity can therefore be taken as numerically equivalent to 0, i.e. local practical non-identifiability. To be more general, we will sometimes use the term poor identifiability to refer to the case when the model is approximately non-identifiable.

## 3 Fitting a Poorly Identifiable Model

### 3.1 Maximum Likelihood Estimation

When using maximum likelihood (ML) estimation to fit a poorly identifiable model, the parameter values you get can be dependent on arbitrary factors such as:

• The initial parameter estimates. Optimization algorithms will typically converge to values close to the initial value which is arbitrary.
• Level of noise in the data. Different levels of noise in the data can cause the optimization algorithm to take different trajectories reaching different optimal parameter values at the end.
• The implementation details of the optimization algorithm. For example, some optimization algorithms implicitly favor parameter values with the smallest norm.

Most of these factors have no statistical significance and can be considered arbitrary in any analysis. Therefore, any insights drawn from poorly identifiable parameter values fitted with ML estimation may be flawed. Luckily, the standard error estimation, if done right, will often reveal signs of poor identifiability. However, common techniques for estimating standard errors can often break down when the model is poorly identifiable. For instance,

1. Asymptotic estimates of standard errors require the model to be locally identifiable,
2. Bootstrap relies on the same arbitrary optimization algorithm for fitting the data resamples so its estimates are as unreliable as the ML estimates. For instance, since all the re-fits in bootstrapping are typically initialized from the ML estimates, there is a high probability that the optimization algorithm will only converge to nearby values, significantly under-estimating the variance in the ML estimates.
3. Sampling importance resampling (SIR) may be able to handle non-identifiable models but it requires a good proposal and its results are sensitive to the proposal used.

Example: Sensitivity to Initial Estimates

Let’s fit the model to the same subject but with 2 different sets of initial estimates:

params1 = (θ = [35, 100, 0.5, 210, 30], σ = 0.1)
fpm1 = fit(model, pop, params1, NaivePooled())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
0    -3.282461e+01     5.896801e+01
* time: 0.01868605613708496
1    -3.299541e+01     8.371540e+01
* time: 0.3482520580291748
2    -3.440867e+01     3.937303e+01
* time: 0.34865903854370117
3    -3.550285e+01     3.429323e+01
* time: 0.34900999069213867
4    -3.585802e+01     2.078146e+01
* time: 0.3493480682373047
5    -3.605801e+01     4.208231e+00
* time: 0.34969496726989746
6    -3.606867e+01     5.500193e+00
* time: 0.34998297691345215
7    -3.607820e+01     5.143560e-01
* time: 0.35028600692749023
8    -3.607823e+01     1.670014e-01
* time: 0.35057687759399414
9    -3.607823e+01     6.594682e-03
* time: 0.35086894035339355
10    -3.607823e+01     3.931879e-03
* time: 0.3511669635772705
11    -3.607823e+01     5.365673e-02
* time: 0.3514680862426758
12    -3.607823e+01     2.875459e-02
* time: 0.35176801681518555
13    -3.607823e+01     3.163451e-03
* time: 0.3520689010620117
14    -3.607823e+01     1.189333e-04
* time: 0.35236191749572754
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:              NaivePooled
Log-likelihood value:                    36.078234
Number of subjects:                              1
Number of parameters:         Fixed      Optimized
0              6
Observation records:         Active        Missing
dv:                          61              0
Total:                       61              0

-----------------
Estimate
-----------------
θ₁     34.397
θ₂    103.17
θ₃      0.46566
θ₄    204.44
θ₅     26.234
σ       0.092273
-----------------
params2 = (θ = [10.0, 300, 1.0, 10.0, 5], σ = 0.2)
fpm2 = fit(model, pop, params2, NaivePooled())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
0     4.245135e+02     6.802168e+02
* time: 2.193450927734375e-5
1     1.499850e+02     4.520555e+01
* time: 0.0003769397735595703
2     1.436201e+02     4.485781e+01
* time: 0.0006821155548095703
3     1.108786e+02     4.698780e+01
* time: 0.0009810924530029297
4     9.146731e+01     4.889574e+01
* time: 0.001277923583984375
5     8.587677e+01     4.879782e+01
* time: 0.0016400814056396484
6     8.353852e+01     5.068832e+01
* time: 0.0019850730895996094
7     8.061237e+01     4.922992e+01
* time: 0.002337932586669922
8     7.723482e+01     3.939427e+01
* time: 0.002682924270629883
9     7.347395e+01     6.732180e+01
* time: 0.0029840469360351562
10     6.467070e+01     2.181937e+01
* time: 0.003283977508544922
11     5.389941e+01     4.260986e+01
* time: 0.0036420822143554688
12     4.899559e+01     4.746853e+01
* time: 0.003996133804321289
13     4.636064e+01     4.389414e+01
* time: 0.004297018051147461
14     4.311120e+01     3.667380e+01
* time: 0.0046079158782958984
15     3.452695e+01     4.210878e+01
* time: 0.004970073699951172
16     2.786722e+01     4.181505e+01
* time: 0.005335092544555664
17     2.167865e+01     2.502622e+01
* time: 0.0056951045989990234
18     2.102586e+01     3.073394e+01
* time: 0.0059969425201416016
19     2.012361e+01     9.155807e+00
* time: 0.0062999725341796875
20     1.993999e+01     8.931312e+00
* time: 0.0066089630126953125
21     1.899136e+01     3.290293e+01
* time: 0.00690913200378418
22     1.831295e+01     1.555682e+01
* time: 0.007218122482299805
23     1.764099e+01     4.156266e+00
* time: 0.007542133331298828
24     1.734278e+01     4.764555e+00
* time: 0.007853031158447266
25     1.692204e+01     5.545118e+00
* time: 0.008178949356079102
26     1.576408e+01     6.312614e+00
* time: 0.008569955825805664
27     1.433865e+01     8.298700e+00
* time: 0.008966922760009766
28     1.402269e+01     1.935485e+01
* time: 0.009359121322631836
29     1.388981e+01     5.313929e+00
* time: 0.009690999984741211
30     1.380201e+01     6.598755e+00
* time: 0.010023117065429688
31     1.361376e+01     1.819850e+01
* time: 0.010369062423706055
32     1.307080e+01     3.715701e+01
* time: 0.010710954666137695
33     1.084752e+01     4.989789e+01
* time: 0.01105809211730957
34     9.107761e+00     3.568940e+01
* time: 0.011691093444824219
35     8.799303e+00     1.470967e+02
* time: 0.012131929397583008
36     6.685368e+00     4.816807e+01
* time: 0.012501955032348633
37     4.048434e+00     5.757686e+01
* time: 0.01285696029663086
38    -4.570968e+00     4.010359e+01
* time: 0.021431922912597656
39    -6.951249e+00     9.481085e+01
* time: 0.021863937377929688
40    -8.241067e+00     2.924279e+01
* time: 0.022304058074951172
41    -9.267379e+00     1.592312e+01
* time: 0.022752046585083008
42    -9.300539e+00     6.235892e+00
* time: 0.0231170654296875
43    -9.408955e+00     4.087560e+00
* time: 0.023492097854614258
44    -9.420626e+00     5.730458e+00
* time: 0.023858070373535156
45    -9.435635e+00     8.693778e+00
* time: 0.024219989776611328
46    -9.439794e+00     8.483929e+00
* time: 0.024599075317382812
47    -9.443618e+00     7.807829e+00
* time: 0.02496790885925293
48    -9.453743e+00     6.452476e+00
* time: 0.025355100631713867
49    -9.479753e+00     3.781804e+00
* time: 0.025727033615112305
50    -9.545875e+00     5.189641e+00
* time: 0.02609705924987793
51    -9.621161e+00     2.998542e+01
* time: 0.02646613121032715
52    -9.788354e+00     2.377897e+01
* time: 0.026823997497558594
53    -9.957783e+00     2.804857e+01
* time: 0.027240991592407227
54    -1.016832e+01     2.937980e+01
* time: 0.027697086334228516
55    -1.046535e+01     3.264929e+01
* time: 0.02810811996459961
56    -1.071184e+01     7.422024e+00
* time: 0.028469085693359375
57    -1.074836e+01     1.458278e+01
* time: 0.028886079788208008
58    -1.088324e+01     1.552006e+01
* time: 0.0292971134185791
59    -1.094332e+01     1.034929e+01
* time: 0.02965402603149414
60    -1.100763e+01     9.505009e+00
* time: 0.030015945434570312
61    -1.101755e+01     1.637700e+00
* time: 0.03037405014038086
62    -1.101926e+01     9.140574e-01
* time: 0.030726909637451172
63    -1.101993e+01     3.661381e-01
* time: 0.031072139739990234
64    -1.102009e+01     1.784483e-01
* time: 0.031423091888427734
65    -1.102013e+01     2.946407e-02
* time: 0.03178000450134277
66    -1.102013e+01     3.293414e-03
* time: 0.03212594985961914
67    -1.102013e+01     5.169943e-04
* time: 0.0324859619140625
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:              NaivePooled
Log-likelihood value:                    11.020132
Number of subjects:                              1
Number of parameters:         Fixed      Optimized
0              6
Observation records:         Active        Missing
dv:                          61              0
Total:                       61              0

----------------
Estimate
----------------
θ₁     30.963
θ₂    214.26
θ₃     55.46
θ₄    169.43
θ₅     12.071
σ       0.13904
----------------

Now let’s display the estimates of θ side by side:

hcat(coef(fpm1).θ, coef(fpm2).θ)
5×2 Matrix{Float64}:
34.3968     30.9631
103.168     214.264
0.465659   55.4603
204.442     169.429
26.2337     12.0709

Notice the significant difference in the final estimates when changing nothing but the initial estimates. Also note that the 2 sets of coefficients are not in the same neighbourhood and don’t have similar log likelihoods. This is indicative of the existence of multiple dis-connected local optima.

Example: Sensitivity to Noise Level

To demonstrate the sensitivity to noise level, we will re-simulate the synthetic subject from the same pseudo-random number generator and seed but using a higher σ.

rng = Random.default_rng()
Random.seed!(rng, 12345)
newparams = (; θ = params.θ, σ = 0.2)
newpop = [Subject(simobs(model, skeleton, newparams; rng))]
Population
Subjects: 1
Observations: dv

Now let’s do the fit once with pop and once with newpop:

fpm1 = fit(model, pop, params, NaivePooled())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
0    -3.282461e+01     5.896801e+01
* time: 2.288818359375e-5
1    -3.299541e+01     8.371540e+01
* time: 0.0004658699035644531
2    -3.440867e+01     3.937303e+01
* time: 0.0008330345153808594
3    -3.550285e+01     3.429323e+01
* time: 0.0011789798736572266
4    -3.585802e+01     2.078146e+01
* time: 0.0015208721160888672
5    -3.605801e+01     4.208231e+00
* time: 0.001878976821899414
6    -3.606867e+01     5.500193e+00
* time: 0.002173900604248047
7    -3.607820e+01     5.143560e-01
* time: 0.0024678707122802734
8    -3.607823e+01     1.670014e-01
* time: 0.002763032913208008
9    -3.607823e+01     6.594682e-03
* time: 0.003072977066040039
10    -3.607823e+01     3.931879e-03
* time: 0.0033638477325439453
11    -3.607823e+01     5.365673e-02
* time: 0.0036568641662597656
12    -3.607823e+01     2.875459e-02
* time: 0.003957986831665039
13    -3.607823e+01     3.163451e-03
* time: 0.004251003265380859
14    -3.607823e+01     1.189333e-04
* time: 0.0045468807220458984
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:              NaivePooled
Log-likelihood value:                    36.078234
Number of subjects:                              1
Number of parameters:         Fixed      Optimized
0              6
Observation records:         Active        Missing
dv:                          61              0
Total:                       61              0

-----------------
Estimate
-----------------
θ₁     34.397
θ₂    103.17
θ₃      0.46566
θ₄    204.44
θ₅     26.234
σ       0.092273
-----------------
fpm2 = fit(model, newpop, newparams, NaivePooled())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
0     8.763976e+00     2.735808e+01
* time: 1.9073486328125e-5
1     8.405325e+00     2.531358e+01
* time: 0.00045299530029296875
2     6.620382e+00     1.286061e+01
* time: 0.0008070468902587891
3     5.297438e+00     8.779519e+00
* time: 0.001157999038696289
4     4.963913e+00     3.840318e+00
* time: 0.00150299072265625
5     4.930293e+00     1.184668e+01
* time: 0.0018041133880615234
6     4.873975e+00     3.753008e+00
* time: 0.0021109580993652344
7     4.858307e+00     2.308575e+00
* time: 0.002463102340698242
8     4.854947e+00     8.434789e-02
* time: 0.0027561187744140625
9     4.854713e+00     7.173484e-02
* time: 0.0030579566955566406
10     4.853126e+00     8.197275e-01
* time: 0.003358125686645508
11     4.851996e+00     1.269989e+00
* time: 0.003662109375
12     4.850709e+00     1.179594e+00
* time: 0.003957986831665039
13     4.849908e+00     1.616557e-01
* time: 0.004264116287231445
14     4.849724e+00     8.747458e-01
* time: 0.004557132720947266
15     4.849534e+00     1.703991e-01
* time: 0.004845142364501953
16     4.849431e+00     1.385412e-01
* time: 0.005145072937011719
17     4.849400e+00     1.192473e-01
* time: 0.005444049835205078
18     4.849395e+00     8.727150e-03
* time: 0.005731105804443359
19     4.849394e+00     2.408209e-02
* time: 0.006022930145263672
20     4.849394e+00     1.491013e-03
* time: 0.006330013275146484
21     4.849394e+00     1.658892e-04
* time: 0.006623029708862305
FittedPumasModel

Successful minimization:                      true

Likelihood approximation:              NaivePooled
Log-likelihood value:                   -4.8493937
Number of subjects:                              1
Number of parameters:         Fixed      Optimized
0              6
Observation records:         Active        Missing
dv:                          61              0
Total:                       61              0

----------------
Estimate
----------------
θ₁     33.85
θ₂    125.52
θ₃      0.50012
θ₄    192.8
θ₅     22.25
σ       0.18103
----------------

Now let’s display the estimates of θ side by side:

hcat(coef(fpm1).θ, coef(fpm2).θ)
5×2 Matrix{Float64}:
34.3968     33.8499
103.168     125.517
0.465659    0.500121
204.442     192.8
26.2337     22.2497

Given that we have many observations per subject, this level of fluctuation due to a higher noise is a symptom of non-identifiability. Also note the big difference compared to the true data-generating parameter values!

params.θ
5-element Vector{Float64}:
35.0
100.0
0.5
210.0
30.0
Summary

In this section, we have demonstrated beyond reasonable doubt that ML estimation workflows can be unreliable when fitting poorly identifiable models, potentially leading to erroneous conclusions in a study. So if we cannot rely on ML estimation for fitting poorly identifiable models, what can we use? The answer is Bayesian inference.

### 3.2 Bayesian Inference

Using Bayesian methods to sample from the posterior of the parameter estimates is a mathematically sound way to fit non-identifiable models because even non-identifiable models have a well-defined posterior distribution, when their parameters are assigned prior distributions. If multiple parameter values all fit the data well, then all such values will be plausible samples from the posterior distribution, assuming reasonable priors were used.

#### 3.2.1 Model Definition

Let’s see how to minimally change our model above to make it Bayesian using a weakly informative priors “roughly in the ballpark”:

bayes_model = @model begin
@param begin
θ ~ MvLogNormal([log(50), log(150), log(1.0), log(100.0), log(10.0)], I(5))
σ ~ Uniform(0.0, 1.0)
end
@pre begin
CL = θ[1]
Vc = θ[2]
Ka = θ[3]
Vp = θ[4]
Q = θ[5]
end
@dynamics begin
Depot' = -Ka * Depot
Central' = Ka * Depot - (CL + Q) / Vc * Central + Q / Vp * Peripheral
Peripheral' = Q / Vc * Central - Q / Vp * Peripheral
end
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, abs(cp) * σ + 1e-6)
end
end
PumasModel
Parameters: θ, σ
Random effects:
Covariates:
Dynamical variables: Depot, Central, Peripheral
Derived: dv
Observed: dv

#### 3.2.2 Sampling from the Posterior

Now to fit it, we need to pass an instance of BayesMCMC as the algorithm in fit. In this case, we used 4 chains for sampling with 3000 samples per chain out of which the first 1500 samples will be used to adapt the mass matrix and step size of the No-U-Turn sampler (NUTS) used in Pumas. All the chains are also parallelized using multi-threading.

# Setting the pseudo-random number generator's seed for reproducibility
rng = Pumas.default_rng()
Random.seed!(54321)
bayes_alg = BayesMCMC(;
nsamples = 3000,
nchains = 4,
rng,
)
bayes_fpm = fit(bayes_model, pop, params, bayes_alg)
[ Info: Checking the initial parameter values.
[ Info: The initial log probability and its gradient are finite. Check passed.
[ Info: Checking the initial parameter values.
[ Info: The initial log probability and its gradient are finite. Check passed.
[ Info: Checking the initial parameter values.
[ Info: The initial log probability and its gradient are finite. Check passed.
[ Info: Checking the initial parameter values.
[ Info: Checking the initial parameter values.
[ Info: The initial log probability and its gradient are finite. Check passed.
[ Info: The initial log probability and its gradient are finite. Check passed.
Chains MCMC chain (3000×6×4 Array{Float64, 3}):

Iterations        = 1:1:3000
Number of chains  = 4
Samples per chain = 3000
Wall duration     = 30.93 seconds
Compute duration  = 119.37 seconds
parameters        = θ₁, θ₂, θ₃, θ₄, θ₅, σ

Summary Statistics
parameters       mean       std      mcse    ess_bulk    ess_tail      rhat  ⋯
Symbol    Float64   Float64   Float64     Float64     Float64   Float64  ⋯

θ₁    34.2717    0.6362    0.0086   5551.3806   6413.5813    1.0007  ⋯
θ₂   138.8467   34.8978    0.7102   2487.3817   3979.1213    1.0017  ⋯
θ₃     0.6511    0.1843    0.0036   2529.3840   3923.1250    1.0017  ⋯
θ₄   190.0993   18.8466    0.3440   3108.8136   4530.2519    1.0011  ⋯
θ₅    23.2832    3.0216    0.0550   3072.9410   4973.3905    1.0014  ⋯
σ     0.0983    0.0097    0.0001   5752.4801   5753.0447    1.0001  ⋯
1 column omitted

Quantiles
parameters       2.5%      25.0%      50.0%      75.0%      97.5%
Symbol    Float64    Float64    Float64    Float64    Float64

θ₁    33.0120    33.8490    34.2784    34.6904    35.5084
θ₂    71.9391   111.2588   143.1361   166.0002   197.2179
θ₃     0.3491     0.5018     0.6481     0.7779     1.0266
θ₄   158.1985   176.6931   188.1775   201.7879   231.2874
θ₅    17.2545    21.1185    23.4505    25.4717    28.8015
σ     0.0813     0.0916     0.0976     0.1041     0.1195


Now let’s discard the NUTS warmup samples as burn-in:

bayes_fpm_samples = Pumas.discard(bayes_fpm, burnin = 1500)
Chains MCMC chain (1500×6×4 Array{Float64, 3}):

Iterations        = 1:1:1500
Number of chains  = 4
Samples per chain = 1500
Wall duration     = 30.93 seconds
Compute duration  = 119.37 seconds
parameters        = θ₁, θ₂, θ₃, θ₄, θ₅, σ

Summary Statistics
parameters       mean       std      mcse    ess_bulk    ess_tail      rhat  ⋯
Symbol    Float64   Float64   Float64     Float64     Float64   Float64  ⋯

θ₁    34.2756    0.6322    0.0121   2711.9918   3407.4054    1.0004  ⋯
θ₂   139.2094   35.1573    0.9827   1324.9363   2301.4196    1.0008  ⋯
θ₃     0.6537    0.1863    0.0050   1342.8480   2090.4298    1.0009  ⋯
θ₄   189.6751   18.9777    0.4879   1563.5781   2411.6157    1.0012  ⋯
θ₅    23.2415    3.0198    0.0770   1562.4658   2631.9980    1.0023  ⋯
σ     0.0982    0.0097    0.0002   3419.5267   3247.6062    1.0003  ⋯
1 column omitted

Quantiles
parameters       2.5%      25.0%      50.0%      75.0%      97.5%
Symbol    Float64    Float64    Float64    Float64    Float64

θ₁    33.0335    33.8569    34.2767    34.6907    35.5135
θ₂    71.6983   112.0005   143.9073   166.3824   197.3886
θ₃     0.3484     0.5045     0.6515     0.7826     1.0302
θ₄   157.8467   176.2235   187.5045   201.4310   231.8597
θ₅    17.2540    21.0390    23.4050    25.4315    28.6638
σ     0.0808     0.0914     0.0976     0.1041     0.1190


Notice how Bayesian inference was able to quantify the uncertainty in the non-identifiable parameters (mostly θ₂ which is Vc and θ₄ which is Vp) reflected in the large standard deviation of the marginal posterior of some parameters, std, relative to its mean value, mean. This is consistent with the eigenvector d we used earlier to prove that the model is poorly identifiable which also showed the poor identifiability was mostly prevalent in these 2 parameters.

d
6-element Vector{Float64}:
0.006717009021476735
0.8382731175155917
0.0037560578470219614
-0.5451738773535086
0.004939403956445427
-3.137352344503738e-6

#### 3.2.3 Convergence Diagnostics

By default, we print summary statistics and a few convergence diagnostics: effective samples size (ess_bulk) and $$\hat{R}$$ (rhat). In this case, the diagnostics look reasonable. The rhat is close to 1 and the minimum ess_bulk is around 1000.

Next we show the trace plot of all the parameters

using PumasPlots

trace_plot(bayes_fpm_samples; linkyaxes = false)

It can be seen that the chains are mostly well mixed. Occasional jumps seem to happen which may indicate the presence of another mode in the posterior with a relatively small probability mass.

Now let’s look at the auto-correlation plot:

autocor_plot(bayes_fpm_samples; linkyaxes = false)

Some auto-correlation seems to persist in some chains so let’s try some thinning, by keeping only one out of every 5 samples.

thin_bayes_fpm_samples = Pumas.discard(bayes_fpm, burnin = 1500, ratio = 0.2)

autocor_plot(thin_bayes_fpm_samples; linkyaxes = false)

Looks better!

To Thin Or Not To Thin

In general, thinning is not recommended unless there are extremely high levels of auto-correlation in the samples. This is because the thinned samples will always have less information than the full set of samples before thinning. However, we perform thinning in this tutorial for demonstration purposes.

#### 3.2.4 Posterior Predictive Check

Now let’s do a posterior predictive check by first simulating 1000 scenarios from the posterior distribution of the response including residual noise.

obstimes = 0.0:0.5:30.0
sims = simobs(bayes_fpm_samples; samples = 1000, simulate_error = true, obstimes)
[ Info: Sampling 1000 sample(s) from the posterior predictive distribution of each subject.
Simulated population (Vector{<:Subject})
Simulated subjects: 1000
Simulated variables: dv

then we can do a visual predictive check (VPC) plot using the simulations

vpc_res = vpc(sims; observations = [:dv], ensemblealg = EnsembleThreads())

vpc_plt = vpc_plot(
vpc_res;
simquantile_medians = true,
observations = true,
axis = (xlabel = "Time (h)", ylabel = "Concentration", xticks = 0:2:30),
)
[ Info: Detected 1000 scenarios and 1 subjects in the input simulations. Running VPC.
[ Info: Continuous VPC
Summary

With very few changes in the model and a few lines of code, we were able to obtain samples from the full posterior of the parameters of our poorly identifiable model.

## 4 Uncertainty Propagation, Queries and Decision Making

Just because a model is non-identifiable does not mean that the model is useless or less correct. In fact, more correct models that incorporate more biological processes tend to be non-identifiable because we can only observe/measure very few variables in the model, while simplified models are more likely to be identifiable. Given samples from the posterior of a non-identifiable model, one can do the following:

• Propagate the uncertainty forward to the predictions to get samples from the posterior predictive distribution, instead of relying on a single prediction using the ML estimates.
Value of Uncertainty Quantification

Parameter uncertainty due to structural non-identifiability will by definition have no effect on the model’s predictions when predicting the observed response. However, uncertainty due to practical non-identifiability, or otherwise insufficient data, can have an impact on the model’s predictions. In this case, basing decisions on the full posterior predictive distribution instead of a single prediction from the ML estimates will make the decisions more robust to parameter uncertainty due to insufficient observations and model misspecification.

• Ask probabilistic questions given your data. For example, what’s the probability that the drug effect is $$> 0$$? Or what’s the probability that the new drug A is better than the control drug B after only 3 months of data? Or what’s the probability of satisfying a therapeutic criteria for efficacy and safety given the current dose?
• What-if analysis (aka counter-factual simulation) and dose optimization. For example, you can make predictions assuming the subject is a pediatric using the model parameters’ posterior inferred from an adult’s data. Or you can test different dose levels to select the best dose according to some therapeutic criteria. This can also be done in the non-Bayesian setting.

### 4.1 Probabilistic Questions

In this section, we show how to

1. Estimate the probability that a parameter is more than a specific value, and
2. Estimate the probability that the subject satisfies a desired therapeutic criteria. From there, one can simulate multiple doses and choose the dose that maximizes this probability.

To estimate the probability that CL > 35, we can run:

mean(bayes_fpm_samples) do p
p.θ[1] > 35
end
0.12316666666666666

To estimate the probability that a subject satisfies a desired therapeutic criteria, we first simulate from the posterior predictive distribution (without residual error):

ipreds = simobs(bayes_fpm_samples; samples = 1000, simulate_error = false, obstimes)
[ Info: Sampling 1000 sample(s) from the posterior predictive distribution of each subject.
Simulated population (Vector{<:Subject})
Simulated subjects: 1000
Simulated variables: dv

Next, we can estimate the area-under-curve (auc) and maximum drug concentration (cmax) given the different posterior samples.

using NCA

nca_params = postprocess(ipreds) do gen, obs
pk_auc = NCA.auc(gen.dv, obstimes)
pk_cmax = NCA.cmax(gen.dv, obstimes)
(; pk_auc, pk_cmax)
end
1000-element Vector{NamedTuple{(:pk_auc, :pk_cmax), Tuple{Float64, Float64}}}:
(pk_auc = 86.79995342676538, pk_cmax = 9.94421432561801)
(pk_auc = 83.38689796617597, pk_cmax = 8.788207729010148)
(pk_auc = 88.5271122946146, pk_cmax = 9.488139509630434)
(pk_auc = 87.9674297127427, pk_cmax = 9.890777164505783)
(pk_auc = 85.77240685046621, pk_cmax = 9.523649964910653)
(pk_auc = 86.06018457319989, pk_cmax = 9.613001384427145)
(pk_auc = 83.74037672933108, pk_cmax = 8.852464924602755)
(pk_auc = 86.01736367307765, pk_cmax = 9.118261858823686)
(pk_auc = 86.71837359267508, pk_cmax = 9.652371954177086)
(pk_auc = 83.47963749754102, pk_cmax = 9.132655312024102)
⋮
(pk_auc = 87.0566631838652, pk_cmax = 9.94801197531114)
(pk_auc = 86.12882451521197, pk_cmax = 9.922821569667523)
(pk_auc = 86.75688939436282, pk_cmax = 10.131306750758936)
(pk_auc = 88.71064226092466, pk_cmax = 10.343798917904172)
(pk_auc = 86.80809048079591, pk_cmax = 9.919681690841498)
(pk_auc = 88.27828636451963, pk_cmax = 10.171349052461661)
(pk_auc = 87.23316154030888, pk_cmax = 9.428904174393166)
(pk_auc = 86.71054114072048, pk_cmax = 9.727517911197543)
(pk_auc = 87.82867553282719, pk_cmax = 9.906530246201767)

Finally, we can estimate the probability of satisfying a therapeutic criteria.

prob = mean(nca_params) do p
p.pk_auc > 90 && p.pk_cmax < 15.0
end
0.036

To compute the probability of efficacy and safety separately, one can instead run:

prob1 = mean(nca_params) do p
p.pk_auc > 90
end
prob2 = mean(nca_params) do p
p.pk_cmax < 15.0
end
prob1, prob2
(0.036, 1.0)

### 4.2 Counter-factual Analysis and Dose Optimization

After developing a model, one may be interested in simulating scenarios, e.g. different covariates or doses, that have not been observed in the data while reusing the same posterior distribution of the parameters learnt from the data. This can be used to select a new dose that maximizes the probabilities of efficacy and safety simultaneously given the previously observed data.

In Pumas, you can do this by defining a new subject that includes the new covariates or dose and then passing that to simobs. First, let’s define a new skeleton subject that represents our counter-factual scenario where a dose of 3200 was administered instead of the 3000 used in the observed case:

cf_skeleton = Subject(
id = 1,
time = 0.0:0.5:30.0,
events = DosageRegimen(3200, time = 0.0, cmt = 1),
observations = (; dv = nothing),
)
Subject
ID: 1
Events: 1
Observations: dv: (n=61)

To simulate from the posterior predictive distribution of this new subject using the posterior of the parameters, you can run:

cf_ipreds = simobs(
thin_bayes_fpm_samples,
cf_skeleton;
samples = 1000,
simulate_error = false,
obstimes,
subject = 1,
)
[ Info: Simulating 1000 sample(s) from the posterior predictive distribution of subject 1 using the dose and covariates in the input subject.
Simulated population (Vector{<:Subject})
Simulated subjects: 1000
Simulated variables: dv

We can then re-evaluate the probability of satisfying a therapeutic criteria:

cf_nca_params = postprocess(cf_ipreds) do gen, obs
pk_auc = NCA.auc(gen.dv, obstimes)
pk_cmax = NCA.cmax(gen.dv, obstimes)
(; pk_auc, pk_cmax)
end
cf_prob = mean(cf_nca_params) do p
p.pk_auc > 90 && p.pk_cmax < 15.0
end
0.953

We can see that the probability increased. To understand why, let’s look at the probabilities of the auc and cmax criteria separately:

cf_prob1 = mean(cf_nca_params) do p
p.pk_auc > 90
end
cf_prob2 = mean(cf_nca_params) do p
p.pk_cmax < 15.0
end
cf_prob1, cf_prob2
(0.953, 1.0)

Contrast this to the old dose’s probabilities:

prob1, prob2
(0.036, 1.0)

Note that this example was not particularly interesting because of the dense sampling which despite of it, the uncertainty in the parameters was still high. This implies that the uncertainty was largely due to structural identifiability issues in the model. Since uncertainty in parameters due to structural non-identifiability does not affect the model predictions, the posterior predictive distribution was much more concentrated than the parameters’ posterior.

To estimate the mean and standard deviation of the predictions at each point in time we can run:

μs = mean(ipreds) do gen, obs
gen.dv
end
σs = std(ipreds) do gen, obs
gen.dv
end
61-element Vector{Float64}:
0.0
0.32438572540826127
0.391690840310383
0.3705877679087534
0.33321472795209905
0.3012131018142511
0.2748534589253588
0.25023300009189414
0.22545319723985358
0.20073948617016002
⋮
0.014668790979526194
0.014999231755638758
0.01536309729582706
0.015748645680335207
0.016145850688921524
0.016546344358783056
0.016943276833101463
0.017331138995852258
0.01770557704397631

To get the average relative standard deviations (ignoring the first prediction which is 0), we run:

mean(σs[2:end] ./ μs[2:end])
0.025175698760351927

So the predictions are not very sensitive to the parameter uncertainty.

## 5 Summary

In this tutorial, we have seen how to test for model non-identifiability using the Fisher information matrix and sensitivity analysis. We have shown that maximum likelihood estimation is unreliable when fitting poorly identifiable models. And we have seen how one can use Bayesian inference to: 1) fit non-identifiable or poorly identifiable models to data, 2) ask probabilistic questions of the model, and 3) simulate counter-factual scenarios.

## 6 References

1. Thomas J. Rothenberg. Identification in parametric models. Econometrica, 1971.
2. F. Mentre, A Mallet, and D. Baccar. Optimal design in random-effects regression models. Biometrika, 1997.
3. S. Retout and F Mentre. Further development of the fisher information matrix in nonlinear mixed-effects models with evaluation in population pharmacokinetics. Journal of biopharmaceutical statistics, 2003.
4. V. Shivva, K. Korell, I. Tucker, and S. Duffull. An approach for identifiability of population pharmacokinetic-pharmacodynamic models. CPT Pharmacometrics & Systems Pharmacology, 2013.
5. Stephen Dufful, A workflow for resolving model internal consistency in use-reuse settings (aka repairing unstable models). PAGANZ, 2024.
6. Dan Wright. The identifiability of a turnover model for allopurinol urate-lowering effect. PAGANZ, 2024.