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: 0.040242910385131836
1 1.499510e+03 9.365700e+01
* time: 1.4167799949645996
2 1.447619e+03 4.714464e+01
* time: 1.488844871520996
3 1.427906e+03 4.439232e+01
* time: 1.5589268207550049
4 1.414326e+03 2.726109e+01
* time: 1.6283228397369385
5 1.387798e+03 1.159019e+01
* time: 1.7087929248809814
6 1.382364e+03 7.060796e+00
* time: 1.8228650093078613
7 1.380839e+03 4.839103e+00
* time: 1.893371820449829
8 1.380281e+03 4.075615e+00
* time: 1.9662859439849854
9 1.379767e+03 3.303901e+00
* time: 2.0538508892059326
10 1.379390e+03 2.856359e+00
* time: 2.165651798248291
11 1.379193e+03 2.650736e+00
* time: 2.2372238636016846
12 1.379036e+03 2.523349e+00
* time: 2.309349775314331
13 1.378830e+03 2.638648e+00
* time: 2.4021639823913574
14 1.378593e+03 3.463990e+00
* time: 2.5112438201904297
15 1.378335e+03 3.471127e+00
* time: 2.6778647899627686
16 1.378143e+03 2.756670e+00
* time: 2.7625479698181152
17 1.378019e+03 2.541343e+00
* time: 2.8422179222106934
18 1.377888e+03 2.163251e+00
* time: 2.936462879180908
19 1.377754e+03 2.571076e+00
* time: 3.018357992172241
20 1.377620e+03 3.370764e+00
* time: 3.1173739433288574
21 1.377413e+03 3.938291e+00
* time: 3.194931983947754
22 1.377094e+03 4.458016e+00
* time: 3.295564889907837
23 1.376674e+03 5.713348e+00
* time: 3.4064459800720215
24 1.375946e+03 5.417530e+00
* time: 3.5360209941864014
25 1.375343e+03 5.862876e+00
* time: 3.6258938312530518
26 1.374689e+03 5.717165e+00
* time: 3.7061758041381836
27 1.374056e+03 4.400490e+00
* time: 3.7888808250427246
28 1.373510e+03 2.191437e+00
* time: 3.9019808769226074
29 1.373277e+03 1.203587e+00
* time: 3.9999279975891113
30 1.373233e+03 1.157761e+00
* time: 4.101386785507202
31 1.373218e+03 8.770728e-01
* time: 4.335424900054932
32 1.373204e+03 8.021952e-01
* time: 4.659853935241699
33 1.373190e+03 6.613857e-01
* time: 4.78418493270874
34 1.373183e+03 7.602394e-01
* time: 4.853357791900635
35 1.373173e+03 8.552154e-01
* time: 4.931084871292114
36 1.373162e+03 6.961928e-01
* time: 5.056737899780273
37 1.373152e+03 3.162546e-01
* time: 5.54597282409668
38 1.373148e+03 1.747381e-01
* time: 5.871567964553833
39 1.373147e+03 1.258699e-01
* time: 5.99270486831665
40 1.373147e+03 1.074908e-01
* time: 6.084695816040039
41 1.373147e+03 6.799619e-02
* time: 6.168896913528442
42 1.373147e+03 1.819329e-02
* time: 6.285821914672852
43 1.373147e+03 1.338880e-02
* time: 6.3634679317474365
44 1.373147e+03 1.370144e-02
* time: 6.441253900527954
45 1.373147e+03 1.315666e-02
* time: 6.5279929637908936
46 1.373147e+03 1.065953e-02
* time: 6.653302907943726
47 1.373147e+03 1.069775e-02
* time: 6.734397888183594
48 1.373147e+03 6.234846e-03
* time: 6.818893909454346
49 1.373147e+03 6.234846e-03
* time: 6.967804908752441
50 1.373147e+03 6.234846e-03
* time: 7.202537775039673FittedPumasModel
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: 9.703636169433594e-5
1 1.436886e+03 9.959639e+01
* time: 0.2816050052642822
2 1.383250e+03 3.318037e+01
* time: 0.3546020984649658
3 1.372961e+03 2.525341e+01
* time: 0.41500401496887207
4 1.365242e+03 2.081002e+01
* time: 0.4851231575012207
5 1.350200e+03 1.667386e+01
* time: 0.5676741600036621
6 1.346374e+03 9.195785e+00
* time: 0.6971111297607422
7 1.344738e+03 8.614309e+00
* time: 0.7563571929931641
8 1.343902e+03 4.950745e+00
* time: 0.8160359859466553
9 1.343662e+03 1.478699e+00
* time: 0.8939940929412842
10 1.343626e+03 9.575005e-01
* time: 0.9750401973724365
11 1.343609e+03 8.509968e-01
* time: 1.069136142730713
12 1.343589e+03 7.964671e-01
* time: 1.1240291595458984
13 1.343567e+03 8.202459e-01
* time: 1.1890740394592285
14 1.343550e+03 8.133359e-01
* time: 1.2690300941467285
15 1.343542e+03 6.865506e-01
* time: 1.3377130031585693
16 1.343538e+03 3.869567e-01
* time: 1.4197230339050293
17 1.343534e+03 2.805019e-01
* time: 1.4709761142730713
18 1.343531e+03 3.271442e-01
* time: 1.5270140171051025
19 1.343529e+03 4.584302e-01
* time: 1.5971651077270508
20 1.343527e+03 3.951940e-01
* time: 1.665959119796753
21 1.343525e+03 1.928385e-01
* time: 1.743624210357666
22 1.343524e+03 1.958575e-01
* time: 1.7941360473632812
23 1.343523e+03 2.008844e-01
* time: 1.8476510047912598
24 1.343522e+03 1.636364e-01
* time: 1.9175779819488525
25 1.343522e+03 1.041929e-01
* time: 1.9873590469360352
26 1.343521e+03 7.417497e-02
* time: 2.0684590339660645
27 1.343521e+03 7.297961e-02
* time: 2.1181700229644775
28 1.343521e+03 8.109591e-02
* time: 2.170884132385254
29 1.343520e+03 7.067080e-02
* time: 2.247303009033203
30 1.343520e+03 5.088025e-02
* time: 2.3210561275482178
31 1.343520e+03 4.980085e-02
* time: 2.397836208343506
32 1.343520e+03 4.778940e-02
* time: 2.4450020790100098
33 1.343520e+03 5.667067e-02
* time: 2.4944710731506348
34 1.343520e+03 5.825591e-02
* time: 2.559128999710083
35 1.343519e+03 5.354660e-02
* time: 2.6239540576934814
36 1.343519e+03 5.300792e-02
* time: 2.7063241004943848
37 1.343519e+03 4.011720e-02
* time: 2.7527761459350586
38 1.343519e+03 3.606197e-02
* time: 2.7994441986083984
39 1.343519e+03 3.546034e-02
* time: 2.8602640628814697
40 1.343519e+03 3.525307e-02
* time: 2.9246320724487305
41 1.343519e+03 3.468091e-02
* time: 3.0076990127563477
42 1.343519e+03 3.313732e-02
* time: 3.0582430362701416
43 1.343518e+03 4.524162e-02
* time: 3.1043920516967773
44 1.343518e+03 5.769309e-02
* time: 3.1581051349639893
45 1.343518e+03 5.716613e-02
* time: 3.223306179046631
46 1.343517e+03 4.600797e-02
* time: 3.3101701736450195
47 1.343517e+03 3.221948e-02
* time: 3.3645169734954834
48 1.343517e+03 2.610758e-02
* time: 3.4124221801757812
49 1.343517e+03 2.120270e-02
* time: 3.464189052581787
50 1.343517e+03 1.887916e-02
* time: 3.523365020751953
51 1.343517e+03 1.229271e-02
* time: 3.608626127243042
52 1.343517e+03 4.778802e-03
* time: 3.661943197250366
53 1.343517e+03 2.158460e-03
* time: 3.713374137878418
54 1.343517e+03 2.158460e-03
* time: 3.79180908203125
55 1.343517e+03 2.158460e-03
* time: 3.9187190532684326FittedPumasModel
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.