using Pumas
using PharmaDatasets
Covariate Selection Methods - Introduction
In pharmacometric workflows, we often have competing models to select from. In this tutorial we will review selection criteria and automated procedures to select the best model out of a set of competing candidate models.
First, we’ll review how to measure model fit, then we’ll cover model selection algorithms.
1 Model Fit Measures
Traditionally in Statistics, model comparison has been done based on a theoretical divergence metric that originates from information theory’s entropy:
\[H = - \operatorname{E}\log(p) = -\sum_i p_i \log(p_i)\]
where \(p_i\) is the probability of occurrence of the \(i\)-th possible value.
Note
We use the \(\log\) scale because it transforms a product of probabilities into a sum, which is both numerically faster and numerically more stable due to the robustness against floating point underflow.
Entropy was the inspiration behind Akaike’s Information Criterion (AIC) (Akaike, 1973):
\[\operatorname{AIC} = -2\log{\hat{\mathcal{L}}} + 2k\]
where \(\hat{\mathcal{L}}\) is the estimated value of the likelihood for a given model and data, and \(k\) is the number of parameters in the model. Generally the likelihood is estimated by maximizing the likelihood function, thus the name maximum likelihood estimation (MLE). The likelihood describes how well the model fits the data, and in certain conditions, can be treated similarly to a probability: higher values means higher plausibility. Hence, models with higher likelihood values demonstrate better fits to the data. Since we are multiplying by a negative value, this means that lower values are preferred.
Note
The \(-2\) was proposed in Akaike’s 1973 original paper to simplify some calculations involving \(\chi^2\) distributions and was kept around since then.
AIC was devised to “punish” model complexity, i.e models that have more parameters to fit to the data. This is why we add \(2\) to the loglikelihood value for every parameter that the model has. Due to the preference of lower AIC values this penalizes models by the number of parameters, while also making it possible to compare models with different complexities.
Building from the AIC, the Bayesian Information Criterion (BIC) (Schwarz, 1978) uses the same idea, but the penalty term is different:
\[\operatorname{BIC} = -2\log{\hat{\mathcal{L}}} + k\log(n)\]
where \(\hat{\mathcal{L}}\) is the estimated value of the likelihood for a given model and data, \(k\) is the model’s number of parameters, and \(n\) is the number of observations. It is called Bayesian because it uses a “Bayesian” argument to derive the punishment term \(k\log(n)\) in the original 1975 paper.
1.1 Example in Pumas
Let’s go over an example of model fit measures in Pumas.
First, let’s import the following packages:
We are going to use the po_sad_1 dataset from PharmaDatasets:
df = dataset("po_sad_1")
first(df, 5)5×14 DataFrame
| Row | id | time | dv | amt | evid | cmt | rate | age | wt | doselevel | isPM | isfed | sex | route |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Int64 | Float64 | Float64? | Float64? | Int64 | Int64? | Float64 | Int64 | Int64 | Int64 | String3 | String3 | String7 | String3 | |
| 1 | 1 | 0.0 | missing | 30.0 | 1 | 1 | 0.0 | 51 | 74 | 30 | no | yes | male | ev |
| 2 | 1 | 0.25 | 35.7636 | missing | 0 | missing | 0.0 | 51 | 74 | 30 | no | yes | male | ev |
| 3 | 1 | 0.5 | 71.9551 | missing | 0 | missing | 0.0 | 51 | 74 | 30 | no | yes | male | ev |
| 4 | 1 | 0.75 | 97.3356 | missing | 0 | missing | 0.0 | 51 | 74 | 30 | no | yes | male | ev |
| 5 | 1 | 1.0 | 128.919 | missing | 0 | missing | 0.0 | 51 | 74 | 30 | no | yes | male | ev |
This is an oral dosing (route = "ev") NMTRAN-formatted dataset. It has 18 subjects, each with 1 dosing event (evid = 1) and 18 measurement events (evid = 0); and the following covariates:
age: age in years (continuous)wt: weight in kg (continuous)doselevel: dosing amount, either30,60or90milligrams (categorical)isPM: subject is a poor metabolizer (binary)isfed: subject is fed (binary)sex: subject sex (binary)
Let’s parse df into a Population with read_pumas:
population =
read_pumas(df; observations = [:dv], covariates = [:wt, :isPM, :isfed], route = :route)Population
Subjects: 18
Covariates: wt, isPM, isfed
Observations: dvLet’s create a 2-compartment oral absorption base model with no covariate effects:
base_model = @model begin
@metadata begin
desc = "base model"
timeu = u"hr"
end
@param begin
"""
Clearance (L/hr)
"""
tvcl ∈ RealDomain(; lower = 0)
"""
Central Volume (L)
"""
tvvc ∈ RealDomain(; lower = 0)
"""
Peripheral Volume (L)
"""
tvvp ∈ RealDomain(; lower = 0)
"""
Distributional Clearance (L/hr)
"""
tvq ∈ RealDomain(; lower = 0)
"""
Absorption rate constant (1/h)
"""
tvka ∈ RealDomain(; lower = 0)
"""
- ΩCL
- ΩVc
- ΩKa
- ΩVp
- ΩQ
"""
Ω ∈ PDiagDomain(5)
"""
Proportional RUV (SD scale)
"""
σₚ ∈ RealDomain(; lower = 0)
end
@random begin
η ~ MvNormal(Ω)
end
@pre begin
CL = tvcl * exp(η[1])
Vc = tvvc * exp(η[2])
Ka = tvka * exp(η[3])
Q = tvq * exp(η[4])
Vp = tvvp * exp(η[5])
end
@dynamics Depots1Central1Periph1
@derived begin
cp := @. 1_000 * (Central / Vc)
"""
Drug Concentration (ng/mL)
"""
dv ~ @. Normal(cp, cp * σₚ)
end
endPumasModel
Parameters: tvcl, tvvc, tvvp, tvq, tvka, Ω, σₚ
Random effects: η
Covariates:
Dynamical system variables: Depot, Central, Peripheral
Dynamical system type: Closed form
Derived: dv
Observed: dvLet’s go over the model.
In the @metadata block we are adding a model description and adding information regarding the time units (hours).
Next, we define the model’s parameters in @param while also prepending them with a string that serves as an annotation for the parameter description. This is helpful for post-processing, since Pumas can use the description instead of the parameter name in tables and figures.
Our ηs are defined in the @random block and are sampled from a multivariate normal distribution with mean 0 and a positive-diagonal covariance matrix Ω. We have 5 ηs, one for each PK typical value (also known as θs).
We proceed by defining the individual PK parameters in the @pre block. Each typical value is incremented by the subject’s ηs in a non-linear exponential transformation. This is done to enforce that all individual PK parameters are constrained to being positive. This also has a side effect that the individual PK parameters will be log-normally distributed.
We use the aliased short notation Depots1Central1Periph1 for the ODE system in the @dynamics. This is equivalent to having the following equations:
Depot' = -Ka * Depot
Central' = Ka * Depot - (CL + Q) / Vc * Central + Q / Vp * Peripheral
Peripheral' = Q / Vc * Central - Q / Vp * PeripheralNote that, in order for Depots1Central1Periph1 work correctly, we need to define Ka, CL, Q, Vc, and Vp in the @pre block.
Finally, in the @derived block we define our error model (or likelihood for the statistically-minded). Here we are using a proportional error model with the Gaussian/normal likelihood. Note that Normal is parameterized with mean and standard deviation, not with variance. That’s why we name our proportional error parameter as σₚ and not σ²ₚ.
Let’s now define a initial set of parameter estimates to fit our model:
iparams = (;
tvka = 0.4,
tvcl = 4.0,
tvvc = 70.0,
tvq = 4.0,
tvvp = 50.0,
Ω = Diagonal(fill(0.04, 5)),
σₚ = 0.1,
)(tvka = 0.4,
tvcl = 4.0,
tvvc = 70.0,
tvq = 4.0,
tvvp = 50.0,
Ω = [0.04 0.0 … 0.0 0.0; 0.0 0.04 … 0.0 0.0; … ; 0.0 0.0 … 0.04 0.0; 0.0 0.0 … 0.0 0.04],
σₚ = 0.1,)We call the fit function to estimate the parameters of the model:
base_fit = fit(base_model, population, iparams, 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 1.630402e+03 2.604358e+02
* time: 0.019212961196899414
1 1.499510e+03 9.365700e+01
* time: 0.3771688938140869
2 1.447619e+03 4.714464e+01
* time: 0.4136660099029541
3 1.427906e+03 4.439232e+01
* time: 0.43051695823669434
4 1.414326e+03 2.726109e+01
* time: 0.46167993545532227
5 1.387798e+03 1.159019e+01
* time: 0.4904749393463135
6 1.382364e+03 7.060796e+00
* time: 0.5070619583129883
7 1.380839e+03 4.839103e+00
* time: 0.5291860103607178
8 1.380281e+03 4.075615e+00
* time: 0.5453250408172607
9 1.379767e+03 3.303901e+00
* time: 0.5671589374542236
10 1.379390e+03 2.856359e+00
* time: 0.5829870700836182
11 1.379193e+03 2.650736e+00
* time: 0.6046218872070312
12 1.379036e+03 2.523349e+00
* time: 0.620046854019165
13 1.378830e+03 2.638648e+00
* time: 0.6416778564453125
14 1.378593e+03 3.463990e+00
* time: 0.6575629711151123
15 1.378335e+03 3.471127e+00
* time: 0.6793630123138428
16 1.378143e+03 2.756670e+00
* time: 0.6956138610839844
17 1.378019e+03 2.541343e+00
* time: 0.717810869216919
18 1.377888e+03 2.163251e+00
* time: 0.7343258857727051
19 1.377754e+03 2.571076e+00
* time: 0.7560968399047852
20 1.377620e+03 3.370764e+00
* time: 0.7779998779296875
21 1.377413e+03 3.938291e+00
* time: 0.7946150302886963
22 1.377094e+03 4.458016e+00
* time: 0.8168649673461914
23 1.376674e+03 5.713348e+00
* time: 0.8337528705596924
24 1.375946e+03 5.417530e+00
* time: 0.8568329811096191
25 1.375343e+03 5.862876e+00
* time: 0.8743579387664795
26 1.374689e+03 5.717165e+00
* time: 0.8976559638977051
27 1.374056e+03 4.400490e+00
* time: 0.915618896484375
28 1.373510e+03 2.191437e+00
* time: 0.939857006072998
29 1.373277e+03 1.203587e+00
* time: 0.9642598628997803
30 1.373233e+03 1.157761e+00
* time: 0.9823899269104004
31 1.373218e+03 8.770728e-01
* time: 1.0051820278167725
32 1.373204e+03 8.021952e-01
* time: 1.0228400230407715
33 1.373190e+03 6.613857e-01
* time: 1.0469269752502441
34 1.373183e+03 7.602394e-01
* time: 1.06427001953125
35 1.373173e+03 8.552154e-01
* time: 1.0872828960418701
36 1.373162e+03 6.961928e-01
* time: 1.110374927520752
37 1.373152e+03 3.162546e-01
* time: 1.1283788681030273
38 1.373148e+03 1.747381e-01
* time: 1.1514759063720703
39 1.373147e+03 1.258699e-01
* time: 1.1679868698120117
40 1.373147e+03 1.074908e-01
* time: 1.1902110576629639
41 1.373147e+03 6.799619e-02
* time: 1.2066938877105713
42 1.373147e+03 1.819329e-02
* time: 1.2289319038391113
43 1.373147e+03 1.338880e-02
* time: 1.2446088790893555
44 1.373147e+03 1.370144e-02
* time: 1.2661468982696533
45 1.373147e+03 1.315666e-02
* time: 1.2815539836883545
46 1.373147e+03 1.065953e-02
* time: 1.3027808666229248
47 1.373147e+03 1.069775e-02
* time: 1.3179759979248047
48 1.373147e+03 6.234846e-03
* time: 1.3389978408813477
49 1.373147e+03 6.234846e-03
* time: 1.3684320449829102
50 1.373147e+03 6.234846e-03
* time: 1.4051878452301025FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: -1373.1468
Number of subjects: 18
Number of parameters: Fixed Optimized
0 11
Observation records: Active Missing
dv: 270 0
Total: 270 0
-------------------
Estimate
-------------------
tvcl 2.8344
tvvc 77.801
tvvp 48.754
tvq 3.9789
tvka 1.028
Ω₁,₁ 0.2638
Ω₂,₂ 0.2288
Ω₃,₃ 0.40047
Ω₄,₄ 0.37968
Ω₅,₅ 0.21495
σₚ 0.097805
-------------------Now we are ready to showcase model fit measures. All of these functions should take a result from fit and output a real number.
Let’s start with aic and bic which are included in Pumas:
aic(base_fit)2768.2935804173985bic(base_fit)2807.876221966381We are also free to create our own functions if we want to use something different than aic or bic.
Here’s an example of a function that takes a fitted Pumas model, m, and outputs the -2LL (minus 2 times log-likelihood) without the constant. This is a model fit measure commonly used by NONMEM users and is is known as OFV: Objective Function Value. Hence, we will name the function ofv:
ofv(m) = (-2 * loglikelihood(m)) - (nobs(m) * log(2π))ofv (generic function with 1 method)We can use it on our base_fit model fit result:
ofv(base_fit)2250.06677248687542 Likelihood Ratio Tests
A likelihood-ratio test (LRT) is a statistical hypothesis test used in the field of statistics and probability theory to compare two statistical models and determine which one provides a better fit to a given set of observed data. It is particularly useful in the context of maximum likelihood estimation (MLE) and is commonly used for hypothesis testing in parametric statistical modeling.
The basic idea behind the likelihood ratio test is to compare the likelihoods of two competing models:
Null Hypothesis (\(H_0\)): This is the model that you want to test against. It represents a specific set of parameter values or restrictions on the model.
Alternative Hypothesis (\(H_a\)): This is the alternative model, often a more complex one or the one you want to support.
The test statistic is calculated as the ratio of the likelihood under the alternative model (\(H_a\)) to the likelihood under the null model (\(H_0\)). Mathematically, it can be expressed as:
\[\operatorname{LRT} = - 2 \log \left( \frac{\mathcal{L}(H_0)}{\mathcal{L}(H_a)} \right)\]
where:
- \(\operatorname{LRT}\): likelihood ratio test statistic
- \(\mathcal{L}(H_0)\): likelihood under \(H_0\), the likelihood of the data under the null hypothesis
- \(\mathcal{L}(H_a)\): likelihood under \(H_a\), the likelihood of the data under the alternative hypothesis
The LRT statistic follows a \(\chi^2\) (chi-squared) distribution with degrees of freedom equal to the difference in the number of parameters between the two models (i.e., the degrees of freedom is the number of additional parameters in the alternative model). In practice, you compare the LRT statistic to \(\chi^2\) distribution to determine whether the alternative model is a significantly better fit to the data than the null model.
The key idea is that if the p-value derived from the LRT statistic is lower than your desired \(\alpha\) (the type-1 error rate, commonly set to \(0.05\)), you would reject the null hypothesis in favor of the alternative hypothesis, indicating that the alternative model provides a better fit to the data.
Note
The likelihood-ratio test requires that the models be nested, i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former’s parameters.
This is generally the case when performing LRT in a covariate selection context. However, be mindful of not violating this assumption when performing LRT.
2.1 Example in Pumas
Pumas provides us with the lrtest function to perform LRT. It takes 2 positional arguments as competing models:
- Model under \(H_0\) (i.e. the model with less parameters)
- Model under \(H_a\) (i.e. the model with more parameters)
Let’s define a covariate model that takes wt into consideration for all the clearance and volume PK parameters:
covariate_model = @model begin
@metadata begin
desc = "covariate model that uses weight covariate information"
timeu = u"hr"
end
@param begin
"""
Clearance (L/hr)
"""
tvcl ∈ RealDomain(; lower = 0)
"""
Central Volume (L)
"""
tvvc ∈ RealDomain(; lower = 0)
"""
Peripheral Volume (L)
"""
tvvp ∈ RealDomain(; lower = 0)
"""
Distributional Clearance (L/hr)
"""
tvq ∈ RealDomain(; lower = 0)
"""
Absorption rate constant (h-1)
"""
tvka ∈ RealDomain(; lower = 0)
"""
Power exponent on weight for Clearance # new
"""
dwtcl ∈ RealDomain() # new
"""
Power exponent on weight for Distributional Clearance # new
"""
dwtq ∈ RealDomain() # new
"""
- ΩCL
- ΩVc
- ΩKa
- ΩVp
- ΩQ
"""
Ω ∈ PDiagDomain(5)
"""
Proportional RUV (SD scale)
"""
σₚ ∈ RealDomain(; lower = 0)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates begin
"""
Weight (kg) # new
"""
wt # new
end
@pre begin
CL = tvcl * exp(η[1]) * (wt / 70)^dwtcl # new
Vc = tvvc * exp(η[2]) * (wt / 70) # new
Ka = tvka * exp(η[3])
Q = tvq * exp(η[4]) * (wt / 70)^dwtq # new
Vp = tvvp * exp(η[5]) * (wt / 70) # new
end
@dynamics Depots1Central1Periph1
@derived begin
cp := @. 1000 * (Central / Vc)
"""
Drug Concentration (ng/mL)
"""
dv ~ @. Normal(cp, cp * σₚ)
end
endPumasModel
Parameters: tvcl, tvvc, tvvp, tvq, tvka, dwtcl, dwtq, Ω, σₚ
Random effects: η
Covariates: wt
Dynamical system variables: Depot, Central, Peripheral
Dynamical system type: Closed form
Derived: dv
Observed: dvThis is almost the same model as before. However, we are adding a few tweaks (commented with # new):
wtin the new@covariatesblock- allometric scaling based on
wtfor the individual PK parametersCL,Q,VcandVp - new parameters in
@paramfor the exponent of the power function ofwton both individual clearance PK parametersCLandQ
Since covariate_model has two new parameters in the @param block, we need to add them to the initial set of parameter estimates. We can do this by creating a new NamedTuple that builts upon the last one iparams, while also adding initial values for dwtcl and dwtq:
iparams_covariate = (; iparams..., dwtcl = 0.75, dwtq = 0.75)(tvka = 0.4,
tvcl = 4.0,
tvvc = 70.0,
tvq = 4.0,
tvvp = 50.0,
Ω = [0.04 0.0 … 0.0 0.0; 0.0 0.04 … 0.0 0.0; … ; 0.0 0.0 … 0.04 0.0; 0.0 0.0 … 0.0 0.04],
σₚ = 0.1,
dwtcl = 0.75,
dwtq = 0.75,)
Tip
We are using Julia’s splatting ... operator to expand inline the iparams NamedTuple.
Now we fit our covariate_model:
covariate_fit = fit(covariate_model, population, iparams_covariate, 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 1.555051e+03 2.584685e+02
* time: 6.794929504394531e-5
1 1.436886e+03 9.959639e+01
* time: 0.018018007278442383
2 1.383250e+03 3.318037e+01
* time: 0.04724001884460449
3 1.372961e+03 2.525341e+01
* time: 0.06327390670776367
4 1.365242e+03 2.081002e+01
* time: 0.09148287773132324
5 1.350200e+03 1.667386e+01
* time: 0.10741281509399414
6 1.346374e+03 9.195785e+00
* time: 0.12876391410827637
7 1.344738e+03 8.614309e+00
* time: 0.1439969539642334
8 1.343902e+03 4.950745e+00
* time: 0.16510581970214844
9 1.343662e+03 1.478699e+00
* time: 0.1802690029144287
10 1.343626e+03 9.575005e-01
* time: 0.2010178565979004
11 1.343609e+03 8.509968e-01
* time: 0.21589183807373047
12 1.343589e+03 7.964671e-01
* time: 0.23654484748840332
13 1.343567e+03 8.202459e-01
* time: 0.25145387649536133
14 1.343550e+03 8.133359e-01
* time: 0.27205491065979004
15 1.343542e+03 6.865506e-01
* time: 0.2868068218231201
16 1.343538e+03 3.869567e-01
* time: 0.30745983123779297
17 1.343534e+03 2.805019e-01
* time: 0.32227492332458496
18 1.343531e+03 3.271442e-01
* time: 0.34326887130737305
19 1.343529e+03 4.584302e-01
* time: 0.3579540252685547
20 1.343527e+03 3.951940e-01
* time: 0.37822890281677246
21 1.343525e+03 1.928385e-01
* time: 0.39289283752441406
22 1.343524e+03 1.958575e-01
* time: 0.4133338928222656
23 1.343523e+03 2.008844e-01
* time: 0.42783594131469727
24 1.343522e+03 1.636364e-01
* time: 0.4480478763580322
25 1.343522e+03 1.041929e-01
* time: 0.46254587173461914
26 1.343521e+03 7.417497e-02
* time: 0.4828488826751709
27 1.343521e+03 7.297961e-02
* time: 0.49725890159606934
28 1.343521e+03 8.109591e-02
* time: 0.5173909664154053
29 1.343520e+03 7.067080e-02
* time: 0.5315268039703369
30 1.343520e+03 5.088025e-02
* time: 0.551414966583252
31 1.343520e+03 4.980085e-02
* time: 0.5657808780670166
32 1.343520e+03 4.778940e-02
* time: 0.5857248306274414
33 1.343520e+03 5.667067e-02
* time: 0.5999288558959961
34 1.343520e+03 5.825591e-02
* time: 0.6197519302368164
35 1.343519e+03 5.354660e-02
* time: 0.634023904800415
36 1.343519e+03 5.300792e-02
* time: 0.6537508964538574
37 1.343519e+03 4.011720e-02
* time: 0.6681828498840332
38 1.343519e+03 3.606197e-02
* time: 0.6881649494171143
39 1.343519e+03 3.546034e-02
* time: 0.7024168968200684
40 1.343519e+03 3.525307e-02
* time: 0.7212250232696533
41 1.343519e+03 3.468091e-02
* time: 0.735374927520752
42 1.343519e+03 3.313732e-02
* time: 0.7544009685516357
43 1.343518e+03 4.524162e-02
* time: 0.7688698768615723
44 1.343518e+03 5.769309e-02
* time: 0.7882528305053711
45 1.343518e+03 5.716613e-02
* time: 0.8028240203857422
46 1.343517e+03 4.600797e-02
* time: 0.8225250244140625
47 1.343517e+03 3.221948e-02
* time: 0.8369159698486328
48 1.343517e+03 2.610758e-02
* time: 0.8557069301605225
49 1.343517e+03 2.120270e-02
* time: 0.8699707984924316
50 1.343517e+03 1.887916e-02
* time: 0.8841710090637207
51 1.343517e+03 1.229271e-02
* time: 0.9041240215301514
52 1.343517e+03 4.778802e-03
* time: 0.9181668758392334
53 1.343517e+03 2.158460e-03
* time: 0.9376258850097656
54 1.343517e+03 2.158460e-03
* time: 0.9636399745941162
55 1.343517e+03 2.158460e-03
* time: 0.9853479862213135FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
Likelihood Optimizer: BFGS
Dynamical system type: Closed form
Log-likelihood value: -1343.5173
Number of subjects: 18
Number of parameters: Fixed Optimized
0 13
Observation records: Active Missing
dv: 270 0
Total: 270 0
--------------------
Estimate
--------------------
tvcl 2.7287
tvvc 70.681
tvvp 47.396
tvq 4.0573
tvka 0.98725
dwtcl 0.58351
dwtq 1.176
Ω₁,₁ 0.21435
Ω₂,₂ 0.050415
Ω₃,₃ 0.42468
Ω₄,₄ 0.040356
Ω₅,₅ 0.045987
σₚ 0.097904
--------------------Now we are ready to perform LRT with lrtest:
mytest = lrtest(base_fit, covariate_fit)Statistic: 59.3
Degrees of freedom: 2
P-value: 0.0The degrees of freedom of the underlying \(\chi^2\) distribution is \(2\), i.e. we have two additional parameters in the model under \(H_a\); and the test statistic is \(59.3\).
The \(p\)-value corresponding for the test statistic and degree of freedom is very close to \(0\). It prints as 0.0, but we can access the value with the pvalue function:
pvalue(mytest)1.3554737256701043e-13This indicates strong evidence against the base_model (i.e. model under \(H_0\)) and in favor of the covariate_model (i.e. model under \(H_a\)).
3 Model Selection Algorithms
There are several model selection techniques that take into account covariate selection. In the statistical literature, the reader can check Thayer (1990), and for the pharmacometric context, the reader can check Hutmacher & Kowalski (2015) and Jonsson & Karlsson (1998).
Pumas currently only implements the Stepwise Covariate Model (SCM). SCM, also known as stepwise procedures, is a model building strategy that is used to identify the best covariate model for a given dataset by a series of iterations (Hutmacher & Kowalski, 2015). Broadly, there are two main types of SCM:
- Forward Selection (FS)
- Backward Elimination (BE)
We will be covering these in detail in a new set of tutorials, please check tutorials.pumas.ai.
4 References
Akaike, H. (1973). Information theory and the extension of the maximum likelihood principle. Proceedings of the Second International Symposium on Information Theory.
Hutmacher, M. M., & Kowalski, K. G. (2015). Covariate selection in pharmacometric analyses: a review of methods. British journal of clinical pharmacology, 79(1), 132–147. https://doi.org/10.1111/bcp.12451
Jonsson, E. N., & Karlsson, M. O. (1998). Automated covariate model building within NONMEM. Pharmaceutical research, 15(9), 1463–1468. https://doi.org/10.1023/a:1011970125687
Schwarz, Gideon E. (1978). Estimating the dimension of a model. Annals of Statistics, 6 (2): 461–464, doi:10.1214/aos/1176344136.
Thayer, J. D. (1990). Implementing Variable Selection Techniques in Regression. ERIC.