Covariate Selection Methods - Introduction

Authors

Jose Storopoli

Andreas Noack

Joel Owen

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:

using Pumas
using PharmaDatasets

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, either 30, 60 or 90 milligrams (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: dv

Let’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
end
PumasModel
  Parameters: tvcl, tvvc, tvvp, tvq, tvka, Ω, σₚ
  Random effects: η
  Covariates: 
  Dynamical variables: Depot, Central, Peripheral
  Derived: dv
  Observed: dv

Let’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 * Peripheral

Note 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: 0.019262075424194336
     1     1.499510e+03     9.365700e+01
 * time: 0.3825111389160156
     2     1.447619e+03     4.714464e+01
 * time: 0.42285919189453125
     3     1.427906e+03     4.439232e+01
 * time: 0.4408080577850342
     4     1.414326e+03     2.726109e+01
 * time: 0.4711310863494873
     5     1.387798e+03     1.159019e+01
 * time: 0.48942017555236816
     6     1.382364e+03     7.060796e+00
 * time: 0.5128810405731201
     7     1.380839e+03     4.839103e+00
 * time: 0.5305690765380859
     8     1.380281e+03     4.075615e+00
 * time: 0.5534660816192627
     9     1.379767e+03     3.303901e+00
 * time: 0.5712330341339111
    10     1.379390e+03     2.856359e+00
 * time: 0.5939550399780273
    11     1.379193e+03     2.650736e+00
 * time: 0.6156289577484131
    12     1.379036e+03     2.523349e+00
 * time: 0.632591962814331
    13     1.378830e+03     2.638648e+00
 * time: 0.6541311740875244
    14     1.378593e+03     3.463990e+00
 * time: 0.6710929870605469
    15     1.378335e+03     3.471127e+00
 * time: 0.6934189796447754
    16     1.378143e+03     2.756670e+00
 * time: 0.7108991146087646
    17     1.378019e+03     2.541343e+00
 * time: 0.7340610027313232
    18     1.377888e+03     2.163251e+00
 * time: 0.7514331340789795
    19     1.377754e+03     2.571076e+00
 * time: 0.7740380764007568
    20     1.377620e+03     3.370764e+00
 * time: 0.7957761287689209
    21     1.377413e+03     3.938291e+00
 * time: 0.8136889934539795
    22     1.377094e+03     4.458016e+00
 * time: 0.8372941017150879
    23     1.376674e+03     5.713348e+00
 * time: 0.8553481101989746
    24     1.375946e+03     5.417530e+00
 * time: 0.8797471523284912
    25     1.375343e+03     5.862876e+00
 * time: 0.8985521793365479
    26     1.374689e+03     5.717165e+00
 * time: 0.9231390953063965
    27     1.374056e+03     4.400490e+00
 * time: 0.9471249580383301
    28     1.373510e+03     2.191437e+00
 * time: 0.9672830104827881
    29     1.373277e+03     1.203587e+00
 * time: 0.9944219589233398
    30     1.373233e+03     1.157761e+00
 * time: 1.013942003250122
    31     1.373218e+03     8.770728e-01
 * time: 1.0381159782409668
    32     1.373204e+03     8.021952e-01
 * time: 1.0617749691009521
    33     1.373190e+03     6.613857e-01
 * time: 1.081106185913086
    34     1.373183e+03     7.602394e-01
 * time: 1.1048600673675537
    35     1.373173e+03     8.552154e-01
 * time: 1.123277187347412
    36     1.373162e+03     6.961928e-01
 * time: 1.1476850509643555
    37     1.373152e+03     3.162546e-01
 * time: 1.1666109561920166
    38     1.373148e+03     1.747381e-01
 * time: 1.1901881694793701
    39     1.373147e+03     1.258699e-01
 * time: 1.2132911682128906
    40     1.373147e+03     1.074908e-01
 * time: 1.2312140464782715
    41     1.373147e+03     6.799619e-02
 * time: 1.2544581890106201
    42     1.373147e+03     1.819329e-02
 * time: 1.2721481323242188
    43     1.373147e+03     1.338880e-02
 * time: 1.294600009918213
    44     1.373147e+03     1.370144e-02
 * time: 1.3110980987548828
    45     1.373147e+03     1.315666e-02
 * time: 1.3322641849517822
    46     1.373147e+03     1.065953e-02
 * time: 1.348663091659546
    47     1.373147e+03     1.069775e-02
 * time: 1.3697171211242676
    48     1.373147e+03     6.234846e-03
 * time: 1.3860671520233154
    49     1.373147e+03     6.234846e-03
 * time: 1.4151251316070557
    50     1.373147e+03     6.234846e-03
 * time: 1.4486780166625977
FittedPumasModel

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)
2768.2935804173985
bic(base_fit)
2807.876221966381

We 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)
2250.0667724868754

2 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:

  1. 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.

  2. 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:

  1. Model under \(H_0\) (i.e. the model with less parameters)
  2. 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
end
PumasModel
  Parameters: tvcl, tvvc, tvvp, tvq, tvka, dwtcl, dwtq, Ω, σₚ
  Random effects: η
  Covariates: wt
  Dynamical variables: Depot, Central, Peripheral
  Derived: dv
  Observed: dv

This is almost the same model as before. However, we are adding a few tweaks (commented with # new):

  1. wt in the new @covariates block
  2. allometric scaling based on wt for the individual PK parameters CL, Q, Vc and Vp
  3. new parameters in @param for the exponent of the power function of wt on both individual clearance PK parameters CL and Q

Since covariate_model has two new parameters in the @param block, we need to add them to the initial set of parameter estimates. We can do this by creating a new NamedTuple that builts upon the last one iparams, while also adding initial values for dwtcl and dwtq:

iparams_covariate = (; iparams..., dwtcl = 0.75, dwtq = 0.75)
(tvka = 0.4,
 tvcl = 4.0,
 tvvc = 70.0,
 tvq = 4.0,
 tvvp = 50.0,
 Ω = [0.04 0.0 … 0.0 0.0; 0.0 0.04 … 0.0 0.0; … ; 0.0 0.0 … 0.04 0.0; 0.0 0.0 … 0.0 0.04],
 σₚ = 0.1,
 dwtcl = 0.75,
 dwtq = 0.75,)
Tip

We are using Julia’s splatting ... operator to expand inline the iparams NamedTuple.

Now we fit our covariate_model:

covariate_fit = fit(covariate_model, population, iparams_covariate, FOCE())
[ Info: Checking the initial parameter values.
[ Info: The initial negative log likelihood and its gradient are finite. Check passed.
Iter     Function value   Gradient norm 
     0     1.555051e+03     2.584685e+02
 * time: 6.699562072753906e-5
     1     1.436886e+03     9.959639e+01
 * time: 0.01912093162536621
     2     1.383250e+03     3.318037e+01
 * time: 0.05005478858947754
     3     1.372961e+03     2.525341e+01
 * time: 0.06690287590026855
     4     1.365242e+03     2.081002e+01
 * time: 0.09607481956481934
     5     1.350200e+03     1.667386e+01
 * time: 0.11301589012145996
     6     1.346374e+03     9.195785e+00
 * time: 0.13460779190063477
     7     1.344738e+03     8.614309e+00
 * time: 0.15095782279968262
     8     1.343902e+03     4.950745e+00
 * time: 0.1721630096435547
     9     1.343662e+03     1.478699e+00
 * time: 0.18846392631530762
    10     1.343626e+03     9.575005e-01
 * time: 0.209885835647583
    11     1.343609e+03     8.509968e-01
 * time: 0.22597193717956543
    12     1.343589e+03     7.964671e-01
 * time: 0.24733591079711914
    13     1.343567e+03     8.202459e-01
 * time: 0.26822686195373535
    14     1.343550e+03     8.133359e-01
 * time: 0.2846639156341553
    15     1.343542e+03     6.865506e-01
 * time: 0.3054788112640381
    16     1.343538e+03     3.869567e-01
 * time: 0.32187390327453613
    17     1.343534e+03     2.805019e-01
 * time: 0.3427557945251465
    18     1.343531e+03     3.271442e-01
 * time: 0.3588719367980957
    19     1.343529e+03     4.584302e-01
 * time: 0.3793330192565918
    20     1.343527e+03     3.951940e-01
 * time: 0.395129919052124
    21     1.343525e+03     1.928385e-01
 * time: 0.41553497314453125
    22     1.343524e+03     1.958575e-01
 * time: 0.43143391609191895
    23     1.343523e+03     2.008844e-01
 * time: 0.45183897018432617
    24     1.343522e+03     1.636364e-01
 * time: 0.46746301651000977
    25     1.343522e+03     1.041929e-01
 * time: 0.4879188537597656
    26     1.343521e+03     7.417497e-02
 * time: 0.5036628246307373
    27     1.343521e+03     7.297961e-02
 * time: 0.5239439010620117
    28     1.343521e+03     8.109591e-02
 * time: 0.5394430160522461
    29     1.343520e+03     7.067080e-02
 * time: 0.5596129894256592
    30     1.343520e+03     5.088025e-02
 * time: 0.5751769542694092
    31     1.343520e+03     4.980085e-02
 * time: 0.5955708026885986
    32     1.343520e+03     4.778940e-02
 * time: 0.611198902130127
    33     1.343520e+03     5.667067e-02
 * time: 0.6316440105438232
    34     1.343520e+03     5.825591e-02
 * time: 0.647367000579834
    35     1.343519e+03     5.354660e-02
 * time: 0.6681888103485107
    36     1.343519e+03     5.300792e-02
 * time: 0.6838879585266113
    37     1.343519e+03     4.011720e-02
 * time: 0.7044088840484619
    38     1.343519e+03     3.606197e-02
 * time: 0.7199928760528564
    39     1.343519e+03     3.546034e-02
 * time: 0.7402529716491699
    40     1.343519e+03     3.525307e-02
 * time: 0.7554399967193604
    41     1.343519e+03     3.468091e-02
 * time: 0.7758100032806396
    42     1.343519e+03     3.313732e-02
 * time: 0.7912368774414062
    43     1.343518e+03     4.524162e-02
 * time: 0.8117139339447021
    44     1.343518e+03     5.769309e-02
 * time: 0.827387809753418
    45     1.343518e+03     5.716613e-02
 * time: 0.847815990447998
    46     1.343517e+03     4.600797e-02
 * time: 0.8634707927703857
    47     1.343517e+03     3.221948e-02
 * time: 0.8838319778442383
    48     1.343517e+03     2.610758e-02
 * time: 0.8991489410400391
    49     1.343517e+03     2.120270e-02
 * time: 0.9193739891052246
    50     1.343517e+03     1.887916e-02
 * time: 0.9348440170288086
    51     1.343517e+03     1.229271e-02
 * time: 0.9552040100097656
    52     1.343517e+03     4.778802e-03
 * time: 0.9708089828491211
    53     1.343517e+03     2.158460e-03
 * time: 0.9912419319152832
    54     1.343517e+03     2.158460e-03
 * time: 1.0130889415740967
    55     1.343517e+03     2.158460e-03
 * time: 1.0419409275054932
    56     1.343517e+03     2.158460e-03
 * time: 1.0726759433746338
FittedPumasModel

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:               0.0

The 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-13

This 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:

  1. Forward Selection (FS)
  2. 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.