Associated with manuscript: Agreement between heuristic shrinkage factor and optimal shrinkage factors in logistic regression for risk prediction: a simulation study across different sample sizes and settings

This supplementary data file contains results for the alternative data generating mechanism, where \(Q_{meas}\) and the C-statistic are independent - see section “Impact of using a different data generating mechanism” from the main manuscript).

1 Plots for results from simulation study 1

We present plots of \(\hat{S}_{VH}\) and \(\hat{S}_{boot}\) plotted against \(S_{opt}\) from simulation study 1 for every combination of the simulation inputs, where the alternative data generating mechanism was used (see section 4 for more details).

Figure S27: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to population-level model performance

Estimator = mean(\(\hat{S}_{VH}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S28: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to apparent model performance

Estimator = mean(\(\hat{S}_{VH}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S29: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{VH}\)) plotted against median(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to population-level model performance

Estimator = median(\(\hat{S}_{VH}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S30: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{VH}\)) plotted against median(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to apparent model performance

Estimator = median(\(\hat{S}_{VH}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S31: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{boot}\)) plotted against mean(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to population-level model performance

Estimator = mean(\(\hat{S}_{boot}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S32: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{boot}\)) plotted against mean(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to apparent model performance

Estimator = mean(\(\hat{S}_{boot}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S33: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{boot}\)) plotted against median(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to population-level model performance

Estimator = median(\(\hat{S}_{boot}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S34: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{boot}\)) plotted against median(\(S_{opt}\)), non-zero covariance in DGM, presented with respect to apparent model performance

Estimator = median(\(\hat{S}_{boot}\))

Covariance structure = non-zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S35: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), zero covariance in DGM, presented with respect to population-level model performance

Estimator = mean(\(\hat{S}_{VH}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S36: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), zero covariance in DGM, presented with respect to apparent model performance

Estimator = mean(\(\hat{S}_{VH}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S37: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{VH}\)) plotted against median(\(S_{opt}\)), zero covariance in DGM, presented with respect to population-level model performance

Estimator = median(\(\hat{S}_{VH}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S38: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{VH}\)) plotted against median(\(S_{opt}\)), zero covariance in DGM, presented with respect to apparent model performance

Estimator = median(\(\hat{S}_{VH}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S39: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{boot}\)) plotted against mean(\(S_{opt}\)), zero covariance in DGM, presented with respect to population-level model performance

Estimator = mean(\(\hat{S}_{boot}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S40: alternative DGM (see section 4 of manuscript); mean(\(\hat{S}_{boot}\)) plotted against mean(\(S_{opt}\)), zero covariance in DGM, presented with respect to apparent model performance

Estimator = mean(\(\hat{S}_{boot}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S41: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{boot}\)) plotted against median(\(S_{opt}\)), zero covariance in DGM, presented with respect to population-level model performance

Estimator = median(\(\hat{S}_{boot}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to population-level model performance

Figure S42: alternative DGM (see section 4 of manuscript); median(\(\hat{S}_{boot}\)) plotted against median(\(S_{opt}\)), zero covariance in DGM, presented with respect to apparent model performance

Estimator = median(\(\hat{S}_{boot}\))

Covariance structure = zero covariance in DGM

By variable = presented with respect to apparent model performance

Figure S43: mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), grouped by mean(\(C_{app}\)), non-zero covariance in DGM

Figure S44: mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), grouped by \(C_{pop}\), non-zero covariance in DGM

Figure S45: mean(\(\hat{S}_VH\)) plotted against mean(\(S_{opt}\)), grouped by mean(\(C_{app}\)), zero covariance in DGM

Figure S46: mean(\(\hat{S}_{VH}\)) plotted against mean(\(S_{opt}\)), grouped by \(C_{pop}\), zero covariance in DGM

2 Simulation study 2

We present a plot of \(N_{sim}\) against \(N_{original}\):

Figure S47: \(N_{sim}\) plotted against \(N_{original}\)