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 ~ @. ProportionalNormal(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: 0.0431978702545166 1 1.499510e+03 9.365700e+01 * time: 2.400017023086548 2 1.447619e+03 4.714464e+01 * time: 2.451387882232666 3 1.427906e+03 4.439232e+01 * time: 2.4984378814697266 4 1.414326e+03 2.726109e+01 * time: 2.545409917831421 5 1.387798e+03 1.159019e+01 * time: 2.6449568271636963 6 1.382364e+03 7.060796e+00 * time: 2.691849946975708 7 1.380839e+03 4.839103e+00 * time: 2.7386419773101807 8 1.380281e+03 4.075615e+00 * time: 2.8184540271759033 9 1.379767e+03 3.303901e+00 * time: 2.864485025405884 10 1.379390e+03 2.856359e+00 * time: 2.911728858947754 11 1.379193e+03 2.650736e+00 * time: 2.956501007080078 12 1.379036e+03 2.523349e+00 * time: 3.0115108489990234 13 1.378830e+03 2.638648e+00 * time: 3.0559439659118652 14 1.378593e+03 3.463990e+00 * time: 3.110844850540161 15 1.378335e+03 3.471127e+00 * time: 3.1590700149536133 16 1.378143e+03 2.756670e+00 * time: 3.22012996673584 17 1.378019e+03 2.541343e+00 * time: 3.2662229537963867 18 1.377888e+03 2.163251e+00 * time: 3.3207130432128906 19 1.377754e+03 2.571076e+00 * time: 3.366581916809082 20 1.377620e+03 3.370764e+00 * time: 3.4227869510650635 21 1.377413e+03 3.938291e+00 * time: 3.469061851501465 22 1.377094e+03 4.458016e+00 * time: 3.52512788772583 23 1.376674e+03 5.713348e+00 * time: 3.5810680389404297 24 1.375946e+03 5.417530e+00 * time: 3.6310129165649414 25 1.375343e+03 5.862876e+00 * time: 3.689424991607666 26 1.374689e+03 5.717165e+00 * time: 3.747764825820923 27 1.374056e+03 4.400490e+00 * time: 3.7992348670959473 28 1.373510e+03 2.191437e+00 * time: 3.860785961151123 29 1.373277e+03 1.203587e+00 * time: 3.915057897567749 30 1.373233e+03 1.157761e+00 * time: 3.9769680500030518 31 1.373218e+03 8.770728e-01 * time: 4.034328937530518 32 1.373204e+03 8.021952e-01 * time: 4.083461046218872 33 1.373190e+03 6.613857e-01 * time: 4.142199993133545 34 1.373183e+03 7.602394e-01 * time: 4.198353052139282 35 1.373173e+03 8.552154e-01 * time: 4.246068954467773 36 1.373162e+03 6.961928e-01 * time: 4.303383827209473 37 1.373152e+03 3.162546e-01 * time: 4.352378845214844 38 1.373148e+03 1.747381e-01 * time: 4.411082029342651 39 1.373147e+03 1.258699e-01 * time: 4.466372966766357 40 1.373147e+03 1.074908e-01 * time: 4.513504981994629 41 1.373147e+03 6.799619e-02 * time: 4.569118022918701 42 1.373147e+03 1.819329e-02 * time: 4.615039825439453 43 1.373147e+03 1.338880e-02 * time: 4.668769836425781 44 1.373147e+03 1.370144e-02 * time: 4.721163988113403 45 1.373147e+03 1.315666e-02 * time: 4.7642600536346436 46 1.373147e+03 1.065953e-02 * time: 4.816534042358398 47 1.373147e+03 1.069775e-02 * time: 4.858845949172974 48 1.373147e+03 6.234846e-03 * time: 4.912385940551758 49 1.373147e+03 6.234846e-03 * time: 4.9834959506988525 50 1.373147e+03 6.234846e-03 * time: 5.073554039001465
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 18
Observation records: Active Missing
dv: 270 0
Total: 270 0
Number of parameters: Constant Optimized
0 11
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoObjectiveChange
Log-likelihood value: -1373.1468
-----------------
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 ~ @. ProportionalNormal(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: 1.3828277587890625e-5 1 1.436886e+03 9.959639e+01 * time: 0.9919579029083252 2 1.383250e+03 3.318037e+01 * time: 1.0381968021392822 3 1.372961e+03 2.525341e+01 * time: 1.0835928916931152 4 1.365242e+03 2.081002e+01 * time: 1.1282308101654053 5 1.350200e+03 1.667386e+01 * time: 1.2084159851074219 6 1.346374e+03 9.195785e+00 * time: 1.253931999206543 7 1.344738e+03 8.614309e+00 * time: 1.2977240085601807 8 1.343902e+03 4.950745e+00 * time: 1.3531699180603027 9 1.343662e+03 1.478699e+00 * time: 1.3973538875579834 10 1.343626e+03 9.575005e-01 * time: 1.4409418106079102 11 1.343609e+03 8.509968e-01 * time: 1.4955949783325195 12 1.343589e+03 7.964671e-01 * time: 1.5388939380645752 13 1.343567e+03 8.202459e-01 * time: 1.5923669338226318 14 1.343550e+03 8.133359e-01 * time: 1.63578200340271 15 1.343542e+03 6.865506e-01 * time: 1.6882319450378418 16 1.343538e+03 3.869567e-01 * time: 1.7404289245605469 17 1.343534e+03 2.805019e-01 * time: 1.7833359241485596 18 1.343531e+03 3.271442e-01 * time: 1.835453987121582 19 1.343529e+03 4.584302e-01 * time: 1.878220796585083 20 1.343527e+03 3.951940e-01 * time: 1.93436598777771 21 1.343525e+03 1.928385e-01 * time: 2.003405809402466 22 1.343524e+03 1.958575e-01 * time: 2.0476009845733643 23 1.343523e+03 2.008844e-01 * time: 2.1009058952331543 24 1.343522e+03 1.636364e-01 * time: 2.144014835357666 25 1.343522e+03 1.041929e-01 * time: 2.196622848510742 26 1.343521e+03 7.417497e-02 * time: 2.2391929626464844 27 1.343521e+03 7.297961e-02 * time: 2.2912797927856445 28 1.343521e+03 8.109591e-02 * time: 2.342733860015869 29 1.343520e+03 7.067080e-02 * time: 2.3850820064544678 30 1.343520e+03 5.088025e-02 * time: 2.436757802963257 31 1.343520e+03 4.980085e-02 * time: 2.4812488555908203 32 1.343520e+03 4.778940e-02 * time: 2.533667802810669 33 1.343520e+03 5.667067e-02 * time: 2.5852508544921875 34 1.343520e+03 5.825591e-02 * time: 2.629962921142578 35 1.343519e+03 5.354660e-02 * time: 2.680583953857422 36 1.343519e+03 5.300792e-02 * time: 2.7223379611968994 37 1.343519e+03 4.011720e-02 * time: 2.7733888626098633 38 1.343519e+03 3.606197e-02 * time: 2.8149449825286865 39 1.343519e+03 3.546034e-02 * time: 2.8659818172454834 40 1.343519e+03 3.525307e-02 * time: 2.9155077934265137 41 1.343519e+03 3.468091e-02 * time: 2.956763982772827 42 1.343519e+03 3.313732e-02 * time: 3.008697986602783 43 1.343518e+03 4.524162e-02 * time: 3.0505998134613037 44 1.343518e+03 5.769309e-02 * time: 3.1015048027038574 45 1.343518e+03 5.716613e-02 * time: 3.1435189247131348 46 1.343517e+03 4.600797e-02 * time: 3.195077896118164 47 1.343517e+03 3.221948e-02 * time: 3.2452259063720703 48 1.343517e+03 2.610758e-02 * time: 3.2862648963928223 49 1.343517e+03 2.120270e-02 * time: 3.337394952774048 50 1.343517e+03 1.887916e-02 * time: 3.3787198066711426 51 1.343517e+03 1.229271e-02 * time: 3.4291329383850098 52 1.343517e+03 4.778802e-03 * time: 3.473356008529663 53 1.343517e+03 2.158460e-03 * time: 3.5236868858337402 54 1.343517e+03 2.158460e-03 * time: 3.593242883682251 55 1.343517e+03 2.158460e-03 * time: 3.663424015045166 56 1.343517e+03 2.158460e-03 * time: 3.7352898120880127
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 18
Observation records: Active Missing
dv: 270 0
Total: 270 0
Number of parameters: Constant Optimized
0 13
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: NoObjectiveChange
Log-likelihood value: -1343.5173
------------------
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.