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: 0.0256350040435791
1 1.499510e+03 9.365700e+01
* time: 0.5525038242340088
2 1.447619e+03 4.714464e+01
* time: 0.608814001083374
3 1.427906e+03 4.439232e+01
* time: 0.6387197971343994
4 1.414326e+03 2.726109e+01
* time: 0.6889159679412842
5 1.387798e+03 1.159019e+01
* time: 0.7198519706726074
6 1.382364e+03 7.060796e+00
* time: 0.7655198574066162
7 1.380839e+03 4.839103e+00
* time: 0.7954869270324707
8 1.380281e+03 4.075615e+00
* time: 0.8332569599151611
9 1.379767e+03 3.303901e+00
* time: 0.8631279468536377
10 1.379390e+03 2.856359e+00
* time: 0.8992037773132324
11 1.379193e+03 2.650736e+00
* time: 0.9349758625030518
12 1.379036e+03 2.523349e+00
* time: 0.96343994140625
13 1.378830e+03 2.638648e+00
* time: 0.9986958503723145
14 1.378593e+03 3.463990e+00
* time: 1.0279159545898438
15 1.378335e+03 3.471127e+00
* time: 1.0640449523925781
16 1.378143e+03 2.756670e+00
* time: 1.0935959815979004
17 1.378019e+03 2.541343e+00
* time: 1.13028883934021
18 1.377888e+03 2.163251e+00
* time: 1.1597719192504883
19 1.377754e+03 2.571076e+00
* time: 1.1965768337249756
20 1.377620e+03 3.370764e+00
* time: 1.22586989402771
21 1.377413e+03 3.938291e+00
* time: 1.2633228302001953
22 1.377094e+03 4.458016e+00
* time: 1.2941539287567139
23 1.376674e+03 5.713348e+00
* time: 1.332401990890503
24 1.375946e+03 5.417530e+00
* time: 1.3716039657592773
25 1.375343e+03 5.862876e+00
* time: 1.4040858745574951
26 1.374689e+03 5.717165e+00
* time: 1.4432268142700195
27 1.374056e+03 4.400490e+00
* time: 1.4767279624938965
28 1.373510e+03 2.191437e+00
* time: 1.5190207958221436
29 1.373277e+03 1.203587e+00
* time: 1.5539929866790771
30 1.373233e+03 1.157761e+00
* time: 1.5946807861328125
31 1.373218e+03 8.770728e-01
* time: 1.632908821105957
32 1.373204e+03 8.021952e-01
* time: 1.665618896484375
33 1.373190e+03 6.613857e-01
* time: 1.704927921295166
34 1.373183e+03 7.602394e-01
* time: 1.7359898090362549
35 1.373173e+03 8.552154e-01
* time: 1.774021863937378
36 1.373162e+03 6.961928e-01
* time: 1.805851936340332
37 1.373152e+03 3.162546e-01
* time: 1.8466768264770508
38 1.373148e+03 1.747381e-01
* time: 1.8785569667816162
39 1.373147e+03 1.258699e-01
* time: 1.9156317710876465
40 1.373147e+03 1.074908e-01
* time: 1.9528110027313232
41 1.373147e+03 6.799619e-02
* time: 1.9831678867340088
42 1.373147e+03 1.819329e-02
* time: 2.0198628902435303
43 1.373147e+03 1.338880e-02
* time: 2.048513889312744
44 1.373147e+03 1.370144e-02
* time: 2.0829968452453613
45 1.373147e+03 1.315666e-02
* time: 2.110224962234497
46 1.373147e+03 1.065953e-02
* time: 2.1445279121398926
47 1.373147e+03 1.069775e-02
* time: 2.1712119579315186
48 1.373147e+03 6.234846e-03
* time: 2.2115838527679443
49 1.373147e+03 6.234846e-03
* time: 2.2549779415130615
50 1.373147e+03 6.234846e-03
* time: 2.3182058334350586FittedPumasModel
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: 7.009506225585938e-5
1 1.436886e+03 9.959639e+01
* time: 0.0768890380859375
2 1.383250e+03 3.318037e+01
* time: 0.10787320137023926
3 1.372961e+03 2.525341e+01
* time: 0.15416407585144043
4 1.365242e+03 2.081002e+01
* time: 0.19887804985046387
5 1.350200e+03 1.667386e+01
* time: 0.2284400463104248
6 1.346374e+03 9.195785e+00
* time: 0.2642841339111328
7 1.344738e+03 8.614309e+00
* time: 0.29253101348876953
8 1.343902e+03 4.950745e+00
* time: 0.3272430896759033
9 1.343662e+03 1.478699e+00
* time: 0.3552391529083252
10 1.343626e+03 9.575005e-01
* time: 0.3893721103668213
11 1.343609e+03 8.509968e-01
* time: 0.4177851676940918
12 1.343589e+03 7.964671e-01
* time: 0.4516720771789551
13 1.343567e+03 8.202459e-01
* time: 0.4792931079864502
14 1.343550e+03 8.133359e-01
* time: 0.5135500431060791
15 1.343542e+03 6.865506e-01
* time: 0.5410001277923584
16 1.343538e+03 3.869567e-01
* time: 0.5749030113220215
17 1.343534e+03 2.805019e-01
* time: 0.6023211479187012
18 1.343531e+03 3.271442e-01
* time: 0.6361720561981201
19 1.343529e+03 4.584302e-01
* time: 0.6633820533752441
20 1.343527e+03 3.951940e-01
* time: 0.6965391635894775
21 1.343525e+03 1.928385e-01
* time: 0.7234930992126465
22 1.343524e+03 1.958575e-01
* time: 0.7569329738616943
23 1.343523e+03 2.008844e-01
* time: 0.7838151454925537
24 1.343522e+03 1.636364e-01
* time: 0.8171169757843018
25 1.343522e+03 1.041929e-01
* time: 0.8439559936523438
26 1.343521e+03 7.417497e-02
* time: 0.8776090145111084
27 1.343521e+03 7.297961e-02
* time: 0.905493974685669
28 1.343521e+03 8.109591e-02
* time: 0.9383101463317871
29 1.343520e+03 7.067080e-02
* time: 0.9646949768066406
30 1.343520e+03 5.088025e-02
* time: 0.9976310729980469
31 1.343520e+03 4.980085e-02
* time: 1.0241010189056396
32 1.343520e+03 4.778940e-02
* time: 1.0567371845245361
33 1.343520e+03 5.667067e-02
* time: 1.083010196685791
34 1.343520e+03 5.825591e-02
* time: 1.1085700988769531
35 1.343519e+03 5.354660e-02
* time: 1.1416091918945312
36 1.343519e+03 5.300792e-02
* time: 1.167375087738037
37 1.343519e+03 4.011720e-02
* time: 1.2010581493377686
38 1.343519e+03 3.606197e-02
* time: 1.2266511917114258
39 1.343519e+03 3.546034e-02
* time: 1.259639024734497
40 1.343519e+03 3.525307e-02
* time: 1.2846851348876953
41 1.343519e+03 3.468091e-02
* time: 1.318108081817627
42 1.343519e+03 3.313732e-02
* time: 1.3436081409454346
43 1.343518e+03 4.524162e-02
* time: 1.3766260147094727
44 1.343518e+03 5.769309e-02
* time: 1.4022960662841797
45 1.343518e+03 5.716613e-02
* time: 1.4368071556091309
46 1.343517e+03 4.600797e-02
* time: 1.4625630378723145
47 1.343517e+03 3.221948e-02
* time: 1.4955010414123535
48 1.343517e+03 2.610758e-02
* time: 1.5206501483917236
49 1.343517e+03 2.120270e-02
* time: 1.5535030364990234
50 1.343517e+03 1.887916e-02
* time: 1.5793371200561523
51 1.343517e+03 1.229271e-02
* time: 1.6123430728912354
52 1.343517e+03 4.778802e-03
* time: 1.6382269859313965
53 1.343517e+03 2.158460e-03
* time: 1.6708261966705322
54 1.343517e+03 2.158460e-03
* time: 1.7105541229248047
55 1.343517e+03 2.158460e-03
* time: 1.759639024734497FittedPumasModel
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.