MI-210 Essentials of Population PKPD M&S — Pumas Edition
Covariate model building identifies measurable patient factors that explain inter-individual variability in PK/PD parameters:
When a covariate genuinely explains inter-individual variability, incorporating it into the model splits the single population parameter into covariate-defined subgroups, each with its own typical value. The random effect \(\eta_i\) must then only account for the residual difference between the individual’s true parameter and the subgroup-specific population prediction, rather than the overall population mean. This forces \(\eta\) to do less work, producing a tighter distribution and a smaller \(\omega\).
Intuitive example: Suppose males have true CL \(\approx\) 10 L/h and females have true CL \(\approx\) 8 L/h. A model without a sex covariate estimates \(\text{tvCL} \approx 9\) L/h with a large \(\omega_{CL}\) that absorbs the male–female difference. A model that includes sex estimates \(\text{tvCL}_\text{male} = 10\) and \(\text{tvCL}_\text{female} = 8\), and \(\omega_{CL}\) shrinks because within each group the \(\eta\) values cluster more tightly around zero. The individual clearances have not changed — \(\eta\) simply shifts to accommodate the new, more precise population prediction.
Power model (centered on reference value):
\[TVCL = \theta_1 \cdot \left(\frac{CLCR}{80}\right)^{\theta_3}\]
Linear model:
\[TVCL = \theta_1 + \theta_3 \cdot (CLCR - 80)\]
Body weight on \(V\): The apparent volume of distribution is \(V = V_P + \sum_i K_{P,i} \cdot V_{T,i}\), where \(V_{T,i}\) are tissue volumes and \(K_{P,i}\) are tissue-to-plasma partition coefficients. A larger body has greater tissue mass and total body water, so \(V\) scales with body size. Allometric scaling formalizes this: \(V \propto \text{BW}^{1.0}\) and \(CL \propto \text{BW}^{0.75}\), reflecting biological scaling laws derived from fractal geometry and cross-species metabolic rate observations.
Creatinine clearance on \(CL\): For renally eliminated drugs, clearance is proportional to glomerular filtration rate (GFR). Filtration clearance is \(CL_F = f_u \cdot \text{GFR}\), where only unbound drug is filtered. The Cockcroft-Gault equation estimates \(\text{CLcr}\) as a surrogate for GFR, making it a natural covariate for renal clearance.
Hepatic function on \(CL\): For hepatically cleared drugs, the well-stirred model predicts \(CL_H = \frac{Q_H \cdot f_u \cdot CL_{\text{int}}}{Q_H + f_u \cdot CL_{\text{int}}}\), where \(Q_H\) is hepatic blood flow, \(f_u\) is fraction unbound, and \(CL_{\text{int}}\) is intrinsic clearance. Intrinsic clearance depends on hepatic enzyme activity (CYP genotype, enzyme induction/inhibition from DDIs), so covariates reflecting hepatic function or metabolizer status directly modulate \(CL\).
Proportional shift:
\[TVCL = \theta_1 \cdot (1 + \theta_3 \cdot SEX)\]
Separate typical values:
\[TVCL = \theta_1 \cdot (1 - SEX) + \theta_2 \cdot SEX\]
cov_df = CSV.read(joinpath(data_dir, "poppk_wcovs.csv"), DataFrame)
rename!(cov_df,
:ID => :id, :DV => :dv, :AMT => :amt, :TIME => :time,
:II => :ii, :ADDL => :addl, :RATE => :rate,
)
cov_df.evid = ifelse.(cov_df.amt .> 0, 1, 0)
cov_df.cmt = fill(1, nrow(cov_df))
cov_df.dv = ifelse.(cov_df.evid .== 1, missing, cov_df.dv)
pop = read_pumas(cov_df; covariates = [:HT, :WT, :CLCR, :SEX, :AGE])
println("Subjects: ", length(pop))Subjects: 40When \(\eta\)-shrinkage exceeds approximately 30%, individual empirical Bayes estimates (EBEs) regress toward zero and lose their ability to reflect true inter-individual differences. This has two consequences for covariate screening: (1) genuine covariate–parameter relationships may be obscured because the \(\hat{\eta}\) values have collapsed toward the population mean, and (2) spurious patterns may emerge from the shrinkage structure itself rather than from biology. A flat EBE-vs-covariate plot under high shrinkage may reflect information poverty rather than the absence of a real relationship. Always compute and report \(\eta\)-shrinkage before interpreting EBE-covariate diagnostic plots; when shrinkage is substantial, consider alternative screening approaches such as simulation-based diagnostics.
Before formal covariate modeling, screen ETA estimates vs. covariates:
# Base model: 1-cpt IV, BLOCK(2) OMEGA, proportional error
# From 501.ctl
base_model = @model begin
@metadata begin
desc = "Base PopPK Model (no covariates)"
end
@param begin
tvCL ∈ RealDomain(lower = 0.0, init = 13.0)
tvV ∈ RealDomain(lower = 0.0, init = 75.0)
Ω ∈ PSDDomain(2)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates WT CLCR SEX AGE HT
@pre begin
CL = tvCL * exp(η[1])
Vc = tvV * exp(η[2])
end
@dynamics Central1
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, abs(cp) * σ_prop)
end
end┌ Warning: Covariates WT, CLCR, SEX, AGE and HT are not used in the model. └ @ Pumas ~/.julia/packages/Pumas/GZeMg/src/dsl/model_macro.jl:3399
PumasModel
Parameters: tvCL, tvV, Ω, σ_prop
Random effects: η
Covariates: WT, CLCR, SEX, AGE, HT
Dynamical system variables: Central
Dynamical system type: Closed form
Derived: dv
Observed: dvbase_params = (
tvCL = 13.0, tvV = 75.0,
Ω = [0.04 0.02; 0.02 0.04],
σ_prop = 0.2,
)
base_fit = fit(base_model, pop, base_params, Pumas.FOCEI())
base_fit[ 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 7.329348e+02 2.137954e+02 * time: 0.012467145919799805 1 7.311461e+02 1.247906e+02 * time: 0.9712700843811035 2 7.307777e+02 1.862206e+02 * time: 1.0094261169433594 3 7.116558e+02 5.785995e+01 * time: 1.0139260292053223 4 7.089067e+02 1.076801e+02 * time: 1.0181691646575928 5 7.073020e+02 8.481440e+01 * time: 1.0223591327667236 6 7.044493e+02 5.135452e+01 * time: 1.0264620780944824 7 7.004105e+02 9.413825e+00 * time: 1.0909039974212646 8 6.997037e+02 6.836002e+00 * time: 1.0949652194976807 9 6.994446e+02 5.826943e+00 * time: 1.0987060070037842 10 6.993923e+02 8.408957e-01 * time: 1.102431058883667 11 6.993907e+02 3.426738e-01 * time: 1.1056702136993408 12 6.993900e+02 1.631341e-01 * time: 1.1091980934143066 13 6.993897e+02 1.129596e-01 * time: 1.1124200820922852 14 6.993896e+02 3.428446e-02 * time: 1.503749132156372 15 6.993896e+02 4.000807e-03 * time: 1.506422996520996 16 6.993896e+02 5.317351e-04 * time: 1.5086700916290283
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 40
Observation records: Active Missing
dv: 560 0
Total: 560 0
Number of parameters: Constant Optimized
0 6
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -699.38964
-------------------
Estimate
-------------------
tvCL 11.846
tvV 85.0
Ω₁,₁ 0.057303
Ω₂,₁ 0.010948
Ω₂,₂ 0.061453
σ_prop 0.21302
-------------------base_insp = inspect(base_fit)
base_df = DataFrame(base_insp)
# Get one row per subject (first obs row)
subj_df = combine(groupby(@rsubset(base_df, !ismissing(:dv)), :id), first)
fig = Figure(size = (900, 600))
ax1 = Axis(fig[1, 1], xlabel = "CLCR", ylabel = "ETA(CL)")
scatter!(ax1, subj_df.CLCR, subj_df.η₁, color = :navy, markersize = 6)
hlines!(ax1, 0, color = :black)
ax2 = Axis(fig[1, 2], xlabel = "WT", ylabel = "ETA(V)")
scatter!(ax2, subj_df.WT, subj_df.η₂, color = :navy, markersize = 6)
hlines!(ax2, 0, color = :black)
ax3 = Axis(fig[1, 3], xlabel = "AGE", ylabel = "ETA(CL)")
scatter!(ax3, subj_df.AGE, subj_df.η₁, color = :navy, markersize = 6)
hlines!(ax3, 0, color = :black)
ax4 = Axis(fig[2, 1], xlabel = "AGE", ylabel = "ETA(V)")
scatter!(ax4, subj_df.AGE, subj_df.η₂, color = :navy, markersize = 6)
hlines!(ax4, 0, color = :black)
ax5 = Axis(fig[2, 2], xlabel = "SEX", ylabel = "ETA(CL)")
scatter!(ax5, subj_df.SEX, subj_df.η₁, color = :navy, markersize = 6)
hlines!(ax5, 0, color = :black)
ax6 = Axis(fig[2, 3], xlabel = "SEX", ylabel = "ETA(V)")
scatter!(ax6, subj_df.SEX, subj_df.η₂, color = :navy, markersize = 6)
hlines!(ax6, 0, color = :black)
fig[ Info: Calculating predictions. [ Info: Calculating weighted residuals. [ Info: Calculating empirical bayes. [ Info: Evaluating dose control parameters. [ Info: Evaluating individual parameters. [ Info: Done.
From 510.ctl — power model parameterization with CLCR and AGE on CL, WT and AGE on V:
\[TVCL = \theta_1 \cdot \left(\frac{CLCR}{80}\right)^{\theta_3} \cdot \left(\frac{AGE}{45}\right)^{\theta_4}\]
\[TVV = \theta_2 \cdot \left(\frac{WT}{70}\right)^{\theta_5} \cdot \left(\frac{AGE}{45}\right)^{\theta_6}\]
full_cov_model = @model begin
@metadata begin
desc = "Full Covariate Model (power parameterization)"
end
@param begin
tvCL ∈ RealDomain(lower = 0.0, init = 13.0)
tvV ∈ RealDomain(lower = 0.0, init = 75.0)
dCL_CLCR ∈ RealDomain(init = 0.5) # THETA(3): CL~CLCR power
dCL_AGE ∈ RealDomain(init = -0.3) # THETA(4): CL~AGE power
dV_WT ∈ RealDomain(init = 0.5) # THETA(5): V~WT power
dV_AGE ∈ RealDomain(init = 0.1) # THETA(6): V~AGE power
Ω ∈ PSDDomain(2)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates WT CLCR AGE SEX HT
@pre begin
TVCL = tvCL * (CLCR / 80)^dCL_CLCR * (AGE / 45)^dCL_AGE
CL = TVCL * exp(η[1])
TVV = tvV * (WT / 70)^dV_WT * (AGE / 45)^dV_AGE
Vc = TVV * exp(η[2])
end
@dynamics Central1
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, abs(cp) * σ_prop)
end
end┌ Warning: Covariates SEX and HT are not used in the model. └ @ Pumas ~/.julia/packages/Pumas/GZeMg/src/dsl/model_macro.jl:3399
PumasModel
Parameters: tvCL, tvV, dCL_CLCR, dCL_AGE, dV_WT, dV_AGE, Ω, σ_prop
Random effects: η
Covariates: WT, CLCR, AGE, SEX, HT
Dynamical system variables: Central
Dynamical system type: Closed form
Derived: dv
Observed: dvfull_params = (
tvCL = 13.0, tvV = 75.0,
dCL_CLCR = 0.5, dCL_AGE = -0.3,
dV_WT = 0.5, dV_AGE = 0.1,
Ω = [0.04 0.02; 0.02 0.04],
σ_prop = 0.2,
)
full_fit = fit(full_cov_model, pop, full_params, Pumas.FOCEI())
full_fit[ 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 7.343467e+02 2.109806e+02 * time: 2.3126602172851562e-5 1 7.305659e+02 1.085502e+02 * time: 0.26302218437194824 2 7.205771e+02 1.374909e+02 * time: 0.47431302070617676 3 7.052267e+02 5.024212e+01 * time: 0.48009514808654785 4 7.038430e+02 1.009489e+02 * time: 0.5226230621337891 5 7.007879e+02 5.859850e+01 * time: 0.5270190238952637 6 6.998238e+02 4.166342e+01 * time: 0.531620979309082 7 6.963667e+02 1.073305e+01 * time: 0.5356230735778809 8 6.957381e+02 1.217252e+01 * time: 0.5398869514465332 9 6.954567e+02 4.410505e+00 * time: 0.5589900016784668 10 6.952999e+02 2.106663e+00 * time: 0.5629520416259766 11 6.952374e+02 2.678294e+00 * time: 0.5666561126708984 12 6.951089e+02 2.496807e+00 * time: 0.570357084274292 13 6.949554e+02 3.026525e+00 * time: 0.5740900039672852 14 6.946986e+02 3.345164e+00 * time: 0.581773042678833 15 6.944501e+02 3.343745e+00 * time: 0.5855240821838379 16 6.942799e+02 3.183658e+00 * time: 0.5891799926757812 17 6.941981e+02 2.412127e+00 * time: 0.592799186706543 18 6.941469e+02 1.591244e+00 * time: 0.5993170738220215 19 6.941207e+02 9.072599e-01 * time: 0.6028611660003662 20 6.941138e+02 5.684698e-01 * time: 0.6062710285186768 21 6.941127e+02 3.146168e-01 * time: 0.6095640659332275 22 6.941125e+02 1.474094e-01 * time: 0.6152021884918213 23 6.941123e+02 5.936196e-02 * time: 0.6183240413665771 24 6.941123e+02 3.644993e-02 * time: 0.6212880611419678 25 6.941123e+02 1.119091e-02 * time: 0.6266829967498779 26 6.941123e+02 9.393166e-03 * time: 0.6295230388641357 27 6.941123e+02 1.301718e-02 * time: 0.6324219703674316 28 6.941123e+02 3.038450e-02 * time: 0.6376190185546875 29 6.941123e+02 4.816357e-02 * time: 0.6406090259552002 30 6.941122e+02 4.946959e-02 * time: 0.6434280872344971 31 6.941122e+02 2.676456e-02 * time: 0.6464600563049316 32 6.941122e+02 5.802006e-03 * time: 0.6518580913543701 33 6.941122e+02 2.525897e-04 * time: 0.6545531749725342
FittedPumasModel
Dynamical system type: Closed form
Number of subjects: 40
Observation records: Active Missing
dv: 560 0
Total: 560 0
Number of parameters: Constant Optimized
0 10
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -694.11224
---------------------
Estimate
---------------------
tvCL 12.462
tvV 91.672
dCL_CLCR 0.21568
dCL_AGE 0.55065
dV_WT 0.40743
dV_AGE -0.45537
Ω₁,₁ 0.049235
Ω₂,₁ 0.003531
Ω₂,₂ 0.050735
σ_prop 0.21309
---------------------full_infer = infer(full_fit)
full_infer[ Info: Calculating: variance-covariance matrix. [ Info: Done.
Asymptotic inference results using sandwich estimator
Dynamical system type: Closed form
Number of subjects: 40
Observation records: Active Missing
dv: 560 0
Total: 560 0
Number of parameters: Constant Optimized
0 10
Likelihood approximation: FOCE
Likelihood optimizer: BFGS
Termination Reason: GradientNorm
Log-likelihood value: -694.11224
------------------------------------------------------------
Estimate SE 95.0% C.I.
------------------------------------------------------------
tvCL 12.462 0.51434 [ 11.454 ; 13.47 ]
tvV 91.672 4.4503 [ 82.949 ; 100.39 ]
dCL_CLCR 0.21568 0.092126 [ 0.035111; 0.39624 ]
dCL_AGE 0.55065 0.49402 [ -0.41761 ; 1.5189 ]
dV_WT 0.40743 0.15347 [ 0.10663 ; 0.70824 ]
dV_AGE -0.45537 0.73577 [ -1.8975 ; 0.9867 ]
Ω₁,₁ 0.049235 0.011635 [ 0.026431; 0.072038]
Ω₂,₁ 0.003531 0.0070028 [ -0.010194; 0.017256]
Ω₂,₂ 0.050735 0.012472 [ 0.026289; 0.07518 ]
σ_prop 0.21309 0.0093414 [ 0.19478 ; 0.2314 ]
------------------------------------------------------------println("Covariate Model Comparison")
println("=" ^ 50)
println("Base Model AIC: ", round(aic(base_fit), digits = 1))
println("Full Cov AIC: ", round(aic(full_fit), digits = 1))
delta_ofv = -2 * loglikelihood(base_fit) - (-2 * loglikelihood(full_fit))
println("\nDelta OFV: ", round(delta_ofv, digits = 2), " (4 df)")
println("Chi-sq critical (p=0.05, 4df): 9.49")
println("Chi-sq critical (p=0.005, 4df): 14.86")Covariate Model Comparison
==================================================
Base Model AIC: 1410.8
Full Cov AIC: 1408.2
Delta OFV: 10.55 (4 df)
Chi-sq critical (p=0.05, 4df): 9.49
Chi-sq critical (p=0.005, 4df): 14.86Pumas provides covariate_select() for automated forward addition / backward elimination. The key design principle: define all covariate effects in the model upfront, then use control_param to specify which parameters “turn on/off” each effect. Setting a control parameter to zero removes its covariate effect.
Starts with base model (all covariate effects zeroed out), iteratively adds the single most impactful covariate:
# Step 1: Define model with ALL candidate covariate effects pre-coded
full_cov_model = @model begin
@param begin
tvCL ∈ RealDomain(lower = 0.0, init = 4.0)
tvV ∈ RealDomain(lower = 0.0, init = 70.0)
dWTCL ∈ RealDomain(init = 0.0) # WT effect on CL (power)
dWTV ∈ RealDomain(init = 0.0) # WT effect on V (power)
dCLCR_CL ∈ RealDomain(init = 0.0) # CLCR effect on CL (power)
dAGE_CL ∈ RealDomain(init = 0.0) # AGE effect on CL (power)
dSEX_CL ∈ RealDomain(init = 0.0) # SEX effect on CL (proportional shift)
Ω ∈ PSDDomain(2)
σ_prop ∈ RealDomain(lower = 0.0, init = 0.2)
end
@random begin
η ~ MvNormal(Ω)
end
@covariates WT CLCR AGE SEX
@pre begin
CL = tvCL * (WT / 70)^dWTCL * (CLCR / 80)^dCLCR_CL *
(AGE / 45)^dAGE_CL * (1 + dSEX_CL * (SEX == 1)) * exp(η[1])
Vc = tvV * (WT / 70)^dWTV * exp(η[2])
end
@dynamics Central1
@derived begin
cp := @. Central / Vc
dv ~ @. Normal(cp, abs(cp) * σ_prop)
end
end
# Step 2: Forward selection — control_param lists the covariate effect parameters
covar_result = covariate_select(
full_cov_model, pop, init_params, Pumas.FOCEI();
control_param = (:dWTCL, :dWTV, :dCLCR_CL, :dAGE_CL, :dSEX_CL),
method = CovariateSelection.Forward,
criterion = aic,
)
# Access results
covar_result.best_model # final selected model
covar_result.fits # all intermediate fitsStarts with full model (all effects active), iteratively removes the least impactful:
covar_result_be = covariate_select(
full_cov_model, pop, full_params, Pumas.FOCEI();
control_param = (:dWTCL, :dWTV, :dCLCR_CL, :dAGE_CL, :dSEX_CL),
method = CovariateSelection.Backward,
criterion = bic, # BIC penalizes complexity more than AIC
)Stepwise Covariate Modeling (SCM): The most widely used approach, consisting of forward addition followed by backward elimination within the full nonlinear mixed-effects model. Forward steps add covariates using a liberal threshold (e.g., \(p < 0.05\)); backward steps remove covariates using a stricter threshold (e.g., \(p < 0.001\)). Each step constitutes a hypothesis test, raising concerns about alpha spending from multiple comparisons — in practice, corrections such as Bonferroni are rarely applied, which is a known limitation.
Generalized Additive Modeling (GAM): A non-parametric, EBE-based screening method that operates outside the PK model. Individual \(\hat{\eta}\) values are regressed against candidate covariates using simple linear models, and the best-fitting covariate is selected iteratively. GAM is computationally efficient but inherits the shrinkage bias of EBEs: when \(\eta\)-shrinkage is substantial, GAM results may be unreliable.
Full Fixed-Effects Model (FFM): All biologically plausible covariates are included simultaneously from the outset, and no stepwise hypothesis testing is performed. Covariate importance is judged by the clinical significance of effect magnitudes (e.g., whether a covariate shifts exposure by \(\geq\) 20%) rather than by \(p\)-values. FFM avoids the multiple-testing problem inherent in SCM but requires careful prior identification and de-correlation of candidate covariates using domain knowledge.
| Criterion | Formula | Favors |
|---|---|---|
| AIC | \(-2LL + 2k\) | Predictive accuracy; may overfit with large n |
| BIC | \(-2LL + k \cdot \log(n)\) | Parsimony; stronger penalty for large datasets |
| LRT | \(\Delta OFV \sim \chi^2(\Delta df)\) | Nested model comparison only; requires p-value cutoff |
NONMEM vs Pumas Covariate Workflow:
covariate_select() automates forward/backward selection, or use manual AIC/LRT comparison as shown aboveCovariates that change during the study (e.g., weight, creatinine clearance, concomitant medications) require special handling:
# In the dataset: include covariate values at each observation/event time
# Pumas interpolates between recorded values using piece-wise constant interpolation:
# :left → LOCF (Last Observation Carried Forward) — Pumas default
# :right → NOCB (Next Observation Carried Backward) — NONMEM default
pop = read_pumas(df; covariates = [:WT, :CLCR], covariates_direction = :left)
# The covariate is automatically available as a time-varying quantity in @preNONMEM defaults to NOCB (right-continuous interpolation), while Pumas defaults to LOCF (left-continuous). When comparing results between the two, explicitly set covariates_direction = :right in Pumas for NONMEM equivalence.
| Strategy | When to Use | Pumas Implementation |
|---|---|---|
| Median imputation | Standard approach for sporadic missingness | @rtransform :WT = coalesce(:WT, 70.0) before read_pumas |
| Group-wise imputation | When covariate differs by subgroup | groupby(:SEX) → @transform :WT = coalesce.(:WT, median(skipmissing(:WT))) |
| Complete case | When missingness is minimal | dropmissing(df, [:WT, :CLCR]) |
| Flag + sensitivity | When imputation impact is uncertain | Add :WT_IMPUTED column, run with/without imputed subjects |
full_insp = inspect(full_fit)
full_df = DataFrame(full_insp)
obs = @rsubset(full_df, !ismissing(:dv))
fig = Figure(size = (800, 600))
ax1 = Axis(fig[1, 1], xlabel = "PRED", ylabel = "DV", title = "DV vs PRED")
scatter!(ax1, obs.dv_pred, obs.dv, color = :navy, markersize = 4, alpha = 0.5)
ablines!(ax1, 0, 1, color = :black)
ax2 = Axis(fig[1, 2], xlabel = "IPRED", ylabel = "DV", title = "DV vs IPRED")
scatter!(ax2, obs.dv_ipred, obs.dv, color = :navy, markersize = 4, alpha = 0.5)
ablines!(ax2, 0, 1, color = :black)
ax3 = Axis(fig[2, 1], xlabel = "PRED", ylabel = "CWRES", title = "CWRES vs PRED")
scatter!(ax3, obs.dv_pred, obs.dv_wres, color = :navy, markersize = 4, alpha = 0.5)
hlines!(ax3, 0, color = :black)
ax4 = Axis(fig[2, 2], xlabel = "Time", ylabel = "CWRES", title = "CWRES vs TIME")
scatter!(ax4, obs.time, obs.dv_wres, color = :navy, markersize = 4, alpha = 0.5)
hlines!(ax4, 0, color = :black)
fig[ Info: Calculating predictions. [ Info: Calculating weighted residuals. [ Info: Calculating empirical bayes. [ Info: Evaluating dose control parameters. [ Info: Evaluating individual parameters. [ Info: Done.
| Topic | Tutorial | Key Additions |
|---|---|---|
| Lecture: Covariate Fundamentals | PKPD W7.5 | Covariate types, functional forms, parameterization strategies |
| Lecture: Covariate Diagnostics & Methodology | PKPD W8 | EBE screening, η-shrinkage, SCM/GAM/FFM comparison |
| Lecture: Volume of Distribution | PK W4 | Why body weight affects V — tissue mass, body water, protein binding |
| Lecture: Renal Elimination | PK W7b | Why CLcr predicts CL — GFR, filtration, secretion, reabsorption |
| Covariate model introduction | Covariates | Allometric scaling, hepatic/renal CL breakdown, time-varying covariates, categorical effects |
| AIC/BIC/LRT theory | Covariate Selection Intro | Mathematical derivations, penalty comparison, custom OFV functions |
| Forward selection workflow | Forward Selection | covariate_select() API, control_param patterns, custom criteria |
| Backward elimination workflow | Backward Elimination | Full backward workflow, BIC criterion, result interpretation |
| Mixed stepwise (SCM) | Mixed Selection | Combined forward + backward in one pipeline |
| Cloud batch covariate search | Batch Job | Running covariate_select on JuliaHub for large populations |