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 := @. 1000 * (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 variables: Depot, Central, Peripheral
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: 0.021066904067993164
1 1.499510e+03 9.365700e+01
* time: 0.4584949016571045
2 1.447619e+03 4.714464e+01
* time: 0.4771730899810791
3 1.427906e+03 4.439232e+01
* time: 0.49608707427978516
4 1.414326e+03 2.726109e+01
* time: 0.5206539630889893
5 1.387798e+03 1.159019e+01
* time: 0.5929660797119141
6 1.382364e+03 7.060796e+00
* time: 0.610692024230957
7 1.380839e+03 4.839103e+00
* time: 0.6307439804077148
8 1.380281e+03 4.075615e+00
* time: 0.6545388698577881
9 1.379767e+03 3.303901e+00
* time: 0.7008810043334961
10 1.379390e+03 2.856359e+00
* time: 0.7191140651702881
11 1.379193e+03 2.650736e+00
* time: 0.7409839630126953
12 1.379036e+03 2.523349e+00
* time: 0.7769219875335693
13 1.378830e+03 2.638648e+00
* time: 0.7953240871429443
14 1.378593e+03 3.463990e+00
* time: 0.8164880275726318
15 1.378335e+03 3.471127e+00
* time: 0.8400669097900391
16 1.378143e+03 2.756670e+00
* time: 0.8755168914794922
17 1.378019e+03 2.541343e+00
* time: 0.894644021987915
18 1.377888e+03 2.163251e+00
* time: 0.9156858921051025
19 1.377754e+03 2.571076e+00
* time: 0.9398689270019531
20 1.377620e+03 3.370764e+00
* time: 0.9705798625946045
21 1.377413e+03 3.938291e+00
* time: 0.9882309436798096
22 1.377094e+03 4.458016e+00
* time: 1.0111780166625977
23 1.376674e+03 5.713348e+00
* time: 1.0449309349060059
24 1.375946e+03 5.417530e+00
* time: 1.063586950302124
25 1.375343e+03 5.862876e+00
* time: 1.0841259956359863
26 1.374689e+03 5.717165e+00
* time: 1.1095209121704102
27 1.374056e+03 4.400490e+00
* time: 1.1406409740447998
28 1.373510e+03 2.191437e+00
* time: 1.1625609397888184
29 1.373277e+03 1.203587e+00
* time: 1.1894659996032715
30 1.373233e+03 1.157761e+00
* time: 1.2290239334106445
31 1.373218e+03 8.770728e-01
* time: 1.2474348545074463
32 1.373204e+03 8.021952e-01
* time: 1.2703099250793457
33 1.373190e+03 6.613857e-01
* time: 1.3059189319610596
34 1.373183e+03 7.602394e-01
* time: 1.3251218795776367
35 1.373173e+03 8.552154e-01
* time: 1.3448588848114014
36 1.373162e+03 6.961928e-01
* time: 1.3702778816223145
37 1.373152e+03 3.162546e-01
* time: 1.4050400257110596
38 1.373148e+03 1.747381e-01
* time: 1.4235360622406006
39 1.373147e+03 1.258699e-01
* time: 1.4459660053253174
40 1.373147e+03 1.074908e-01
* time: 1.479599952697754
41 1.373147e+03 6.799619e-02
* time: 1.4975519180297852
42 1.373147e+03 1.819329e-02
* time: 1.516563892364502
43 1.373147e+03 1.338880e-02
* time: 1.539396047592163
44 1.373147e+03 1.370144e-02
* time: 1.5748169422149658
45 1.373147e+03 1.315666e-02
* time: 1.5982880592346191
46 1.373147e+03 1.065953e-02
* time: 1.6231789588928223
47 1.373147e+03 1.069775e-02
* time: 1.6480228900909424
48 1.373147e+03 6.234846e-03
* time: 1.6882798671722412
49 1.373147e+03 6.234846e-03
* time: 1.7141470909118652
50 1.373147e+03 6.234846e-03
* time: 1.7596080303192139FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
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 variables: Depot, Central, Peripheral
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: 7.796287536621094e-5
1 1.436886e+03 9.959639e+01
* time: 0.05277109146118164
2 1.383250e+03 3.318037e+01
* time: 0.07068014144897461
3 1.372961e+03 2.525341e+01
* time: 0.0888071060180664
4 1.365242e+03 2.081002e+01
* time: 0.11189103126525879
5 1.350200e+03 1.667386e+01
* time: 0.1663820743560791
6 1.346374e+03 9.195785e+00
* time: 0.1834731101989746
7 1.344738e+03 8.614309e+00
* time: 0.20300507545471191
8 1.343902e+03 4.950745e+00
* time: 0.22629904747009277
9 1.343662e+03 1.478699e+00
* time: 0.26311206817626953
10 1.343626e+03 9.575005e-01
* time: 0.27896618843078613
11 1.343609e+03 8.509968e-01
* time: 0.29868102073669434
12 1.343589e+03 7.964671e-01
* time: 0.3307809829711914
13 1.343567e+03 8.202459e-01
* time: 0.34679317474365234
14 1.343550e+03 8.133359e-01
* time: 0.36248016357421875
15 1.343542e+03 6.865506e-01
* time: 0.3824479579925537
16 1.343538e+03 3.869567e-01
* time: 0.41402196884155273
17 1.343534e+03 2.805019e-01
* time: 0.4298889636993408
18 1.343531e+03 3.271442e-01
* time: 0.4454970359802246
19 1.343529e+03 4.584302e-01
* time: 0.4662330150604248
20 1.343527e+03 3.951940e-01
* time: 0.4963700771331787
21 1.343525e+03 1.928385e-01
* time: 0.5121190547943115
22 1.343524e+03 1.958575e-01
* time: 0.5282301902770996
23 1.343523e+03 2.008844e-01
* time: 0.5487761497497559
24 1.343522e+03 1.636364e-01
* time: 0.5782961845397949
25 1.343522e+03 1.041929e-01
* time: 0.5937831401824951
26 1.343521e+03 7.417497e-02
* time: 0.6101059913635254
27 1.343521e+03 7.297961e-02
* time: 0.6305670738220215
28 1.343521e+03 8.109591e-02
* time: 0.6596550941467285
29 1.343520e+03 7.067080e-02
* time: 0.674839973449707
30 1.343520e+03 5.088025e-02
* time: 0.6931421756744385
31 1.343520e+03 4.980085e-02
* time: 0.713670015335083
32 1.343520e+03 4.778940e-02
* time: 0.7457261085510254
33 1.343520e+03 5.667067e-02
* time: 0.7609071731567383
34 1.343520e+03 5.825591e-02
* time: 0.777076005935669
35 1.343519e+03 5.354660e-02
* time: 0.797173023223877
36 1.343519e+03 5.300792e-02
* time: 0.826287031173706
37 1.343519e+03 4.011720e-02
* time: 0.8416121006011963
38 1.343519e+03 3.606197e-02
* time: 0.8579649925231934
39 1.343519e+03 3.546034e-02
* time: 0.8780810832977295
40 1.343519e+03 3.525307e-02
* time: 0.9066531658172607
41 1.343519e+03 3.468091e-02
* time: 0.9216461181640625
42 1.343519e+03 3.313732e-02
* time: 0.9378111362457275
43 1.343518e+03 4.524162e-02
* time: 0.9579789638519287
44 1.343518e+03 5.769309e-02
* time: 0.987511157989502
45 1.343518e+03 5.716613e-02
* time: 1.0028300285339355
46 1.343517e+03 4.600797e-02
* time: 1.018996000289917
47 1.343517e+03 3.221948e-02
* time: 1.0391440391540527
48 1.343517e+03 2.610758e-02
* time: 1.0682811737060547
49 1.343517e+03 2.120270e-02
* time: 1.0832781791687012
50 1.343517e+03 1.887916e-02
* time: 1.099147081375122
51 1.343517e+03 1.229271e-02
* time: 1.1192879676818848
52 1.343517e+03 4.778802e-03
* time: 1.14857816696167
53 1.343517e+03 2.158460e-03
* time: 1.1634421348571777
54 1.343517e+03 2.158460e-03
* time: 1.1860601902008057
55 1.343517e+03 2.158460e-03
* time: 1.2272720336914062
56 1.343517e+03 2.158460e-03
* time: 1.2546041011810303FittedPumasModel
Successful minimization: true
Likelihood approximation: FOCE
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.