The pye package provides an advanced framework for simultaneous variable selection and prediction within low- and high-dimensional binary classification contexts. Its core methodology focuses on maximizing the penalized Youden index function \(F_{Y=0}(\theta) - F_{Y=1}(\theta) - \Phi(\theta)\) with respect of the parameter vector \(\theta\), where \(F_{Y=y}(\theta)\) represents the distribution function of the feature combination for class \(Y=y\) and \(\Phi(\theta)\) is a sparsity-inducing penalty term. For the Penalized Youden Index Estimator (pye) - https://doi.org/10.1016/j.chemolab.2023.104786 -, \(\theta\) corresponds to the coefficients of a linear combination of biomarkers \(\beta\) together with diagnostic cut-off point \(c\). For the Covariate-Adjusted Youden Index Estimator (covYI), the biomarker score is a single known marker, and \(\theta\) denotes the coefficients of the linear combination of covariate that define the diagnostic cut-off point \(c\). pye is particularly suited for applications in medical diagnostics, where identifying a subset of relevant biomarkers is crucial for effective disease classification. covYI extends this framework by allowing the diagnostic cut-off to depend on patient covariates, thereby enhancing the accuracy of the biomarkers in heterogeneous populations.
The package implements two primary methodologies:
Penalized Youden Index Estimator (pye): The base estimator combining the Youden index function with sparsity-inducing penalization techniques (Lasso, SCAD, MCP) for simultaneous biomarker selection.
Covariate-Adjusted Youden Index Estimator (covYI): An extension of pye that allows the optimal diagnostic cut-off to be a linear function of patient covariates, improving the classification accuracy of the biomarker score.
This section outlines a typical workflow. In practice, the user must
define the specific functions for the gradient (delta_fx)
and the proximal operator (proxx) based on their chosen
penalty (e.g., SCAD, L1) and data structure.
# Load the package
library(pye)
# 1. Simulate data for the example
set.seed(123) # Always good to set a seed for reproducibility in examples
cols <- 200
cols_cov <- 20
max_rho <- 0.2
rows_train <- 200
sim_data <- create_sample_with_covariates(rows_train = rows_train,
cols = cols,
cols_cov = cols_cov,
max_rho = max_rho,
seed = 1)
df <- sim_data$train_df_scaled
X <- sim_data$X
y <- sim_data$y
C <- sim_data$C
regressors_betas <- sim_data$nregressors # True betas for evaluation
regressors_gammas <- sim_data$ncovariates # True gammas for evaluation
# 2. Set cross-validation parameters
penalty <- "SCAD" # Penalty for betas in pye estimation
penalty_g <- "L12" # Penalty for gammas in covYI estimation
trend <- "monotone" # Trend for the KS estimation
alpha <- 0.5
c_function_of_covariates <- TRUE # Use covariates for 'c' estimation
used_cores <- 1 # For this example, no parallelization
max_iter <- 10 # Keep iterations low for a quick example run
n_folds <- 3
# 3. Calibrate lambda_max and lambda_min for betas (pye estimation)
lambda_seq <- create_lambda(n = 4, lmax = 1.5, lmin = 0.1)
lambda_seq <- as.numeric(formatC(lambda_seq, format = "e", digits = 9))
# 4. Calibrate tau_max and tau_min for gammas (covYI estimation), if
tau_seq <- create_lambda(n = 4, lmax = 0.15, lmin = 0.05)
tau_seq <- as.numeric(formatC(tau_seq, format = "e", digits = 9))
# 5. Run the cross-validation
pye_cv_result <- pye_KS_compute_cv(
n_folds = n_folds,
df = df,
X = X,
y = y,
C = C,
lambda = lambda_seq,
tau = tau_seq,
trace = 1, # Show final results
alpha = alpha,
penalty = penalty,
regressors_betas = regressors_betas,
regressors_gammas = regressors_gammas,
seed = 1,
used_cores = used_cores,
trend = trend,
max_iter = max_iter,
c_function_of_covariates = c_function_of_covariates,
measure_to_select_lambda = "ccr",
penalty_g = penalty_g,
trend_g = trend,
max_iter_g = max_iter
)
#> Starting CV with the following estimation/convergence parameters:
#> max_iter: 10
#> min_alpha: 1e-10
#> convergence_error: 1e-07
#> stepsizeShrink: 0.8
#> c_function_of_covariates: FALSE
#> ----------------------------------------------------------------
#> | starting with the 1 -th fold for the CV of lambda |
#> ----------------------------------------------------------------
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ;
#> total iters: 10 ; Backtraking iters: 798 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1444 ; youden_index: 0.4252 ; sensitivity: 0.9104 ; specificity: 0.8485 ; geometric_mean: 0.8789 ; fdr: 0.1408 ; mcc: 0.7607 ; auc: 0.9521 ; corrclass: 0.8797 ;
#> TP: 61 ; TN: 56 ; FP: 10 ; FN: 6 ; betas:
#> norm2 norm35 norm51 norm52 norm101 norm102 c
#> -0.028667 -0.003142 0.024869 -0.065990 0.034681 -0.029819 0.002434
#> Estimation time: 0.2831 mins
#>
#>
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ;
#> total iters: 6 ; Backtraking iters: 638 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3781 ; youden_index: 0.7603 ; sensitivity: 0.9552 ; specificity: 0.9394 ; geometric_mean: 0.9473 ; fdr: 0.05882 ; mcc: 0.8948 ; auc: 0.9878 ; corrclass: 0.9474 ;
#> TP: 64 ; TN: 62 ; FP: 4 ; FN: 3 ; betas:
#> norm2 norm12 norm16 norm18 norm23 norm35 norm39
#> -0.0834877 -0.0024775 0.0014795 -0.0001693 -0.0024886 -0.0055629 -0.0139270
#> norm45 norm48 norm51
#> 0.0060359 -0.0003150 0.0702604
#> [ reached 'max' / getOption("max.print") -- omitted 23 entries ]
#> Estimation time: 0.3149 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ;
#> total iters: 10 ; Backtraking iters: 653 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.5969 ; youden_index: 0.9447 ; sensitivity: 0.9851 ; specificity: 1 ; geometric_mean: 0.9925 ; fdr: 0 ; mcc: 0.9851 ; auc: 0.9903 ; corrclass: 0.9925 ;
#> TP: 66 ; TN: 66 ; FP: 0 ; FN: 1 ; betas:
#> norm2 norm5 norm9 norm16 norm17 norm18 norm22
#> -0.1146306 -0.0006052 0.0032371 0.0431150 -0.0016098 -0.0051568 -0.0028814
#> norm23 norm24 norm28
#> -0.0331921 -0.0318302 -0.0122568
#> [ reached 'max' / getOption("max.print") -- omitted 48 entries ]
#> Estimation time: 0.253 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ;
#> total iters: 10 ; Backtraking iters: 364 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.732 ; youden_index: 0.8824 ; sensitivity: 0.9552 ; specificity: 1 ; geometric_mean: 0.9774 ; fdr: 0 ; mcc: 0.9559 ; auc: 0.986 ; corrclass: 0.9774 ;
#> TP: 64 ; TN: 66 ; FP: 0 ; FN: 3 ; betas:
#> norm2 norm16 norm18 norm23 norm24 norm33 norm35 norm36
#> -0.183942 0.013181 -0.034843 -0.018366 -0.010160 -0.024587 -0.014625 0.058445
#> norm39 norm48
#> -0.007670 -0.009777
#> [ reached 'max' / getOption("max.print") -- omitted 28 entries ]
#> Estimation time: 0.1132 mins
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
#> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1251 ; youden_index: 0.4287 ; sensitivity: 0.9118 ; specificity: 0.7879 ; geometric_mean: 0.8476 ; fdr: 0.1842 ; mcc: 0.706 ; auc: 0.9287 ; corrclass: 0.8507
#> TP: 31 ; TN: 26 ; FP: 7 ; FN: 3 ; betas:
#> norm2 norm35 norm51 norm52 norm101 norm102 c
#> -0.028667 -0.003142 0.024869 -0.065990 0.034681 -0.029819 0.002434
#> Cross-validation time: 0.2831 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
#> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.219 ; youden_index: 0.4349 ; sensitivity: 0.8529 ; specificity: 0.8182 ; geometric_mean: 0.8354 ; fdr: 0.1714 ; mcc: 0.6717 ; auc: 0.9349 ; corrclass: 0.8358
#> TP: 29 ; TN: 27 ; FP: 6 ; FN: 5 ; betas:
#> norm2 norm12 norm16 norm18 norm23 norm35 norm39
#> -8.349e-02 -2.477e-03 1.480e-03 -1.693e-04 -2.489e-03 -5.563e-03 -1.393e-02
#> norm45 norm48 norm51 norm52 norm60 norm66 norm78
#> 6.036e-03 -3.150e-04 7.026e-02 -1.272e-01 8.754e-03 -6.666e-04 -4.795e-03
#> norm98 norm99 norm101 norm102 norm103 norm107 norm118
#> 5.411e-03 -1.746e-02 9.307e-02 -9.918e-02 1.797e-04 -5.460e-03 -2.026e-02
#> norm122 norm131 norm139 norm141 norm150 norm155 norm161
#> 7.857e-03 9.453e-03 -2.590e-02 -5.859e-04 -1.075e-03 -5.640e-04 4.895e-03
#> norm170 norm177 norm193 norm197 c
#> 2.545e-04 5.668e-03 3.508e-03 1.465e-14 1.074e-02
#> Cross-validation time: 0.3149 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
#> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1986 ; youden_index: 0.3654 ; sensitivity: 0.8529 ; specificity: 0.7576 ; geometric_mean: 0.8038 ; fdr: 0.2162 ; mcc: 0.6138 ; auc: 0.8654 ; corrclass: 0.806
#> TP: 29 ; TN: 25 ; FP: 8 ; FN: 5 ; betas:
#> norm2 norm5 norm9 norm16 norm17 norm18 norm22
#> -1.146e-01 -6.052e-04 3.237e-03 4.312e-02 -1.610e-03 -5.157e-03 -2.881e-03
#> norm23 norm24 norm28 norm33 norm35 norm36 norm38
#> -3.319e-02 -3.183e-02 -1.226e-02 -4.374e-03 -2.868e-02 3.012e-02 3.311e-03
#> norm39 norm48 norm51 norm52 norm55 norm56 norm57
#> -2.704e-03 -1.934e-02 8.884e-02 -1.551e-01 -5.938e-03 4.229e-02 -2.972e-03
#> norm60 norm64 norm66 norm69 norm78 norm98 norm99
#> 2.970e-02 -1.330e-03 -1.884e-02 -4.437e-03 -3.688e-02 4.168e-03 -5.819e-02
#> norm101 norm102 norm103 norm107 norm113 norm117 norm118
#> 8.004e-02 -1.101e-01 3.458e-02 -1.518e-02 2.411e-04 -5.445e-04 -3.529e-02
#> norm122 norm123 norm131 norm132 norm135 norm139 norm141
#> 2.523e-02 -6.490e-04 1.657e-02 -1.709e-03 8.427e-05 -6.095e-02 -3.342e-02
#> norm144 norm147 norm148 norm150 norm155 norm158 norm161
#> -2.348e-02 1.581e-05 -1.454e-03 -3.283e-03 -2.430e-02 -3.952e-02 2.446e-02
#> norm170 norm171 norm177 norm188 norm193 norm194 norm196
#> 1.290e-03 1.950e-03 3.425e-02 -6.419e-03 1.719e-03 -2.443e-02 1.518e-02
#> norm200 c
#> -8.214e-03 9.015e-03
#> Cross-validation time: 0.253 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 1 ;
#> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.4726 ; youden_index: 0.4002 ; sensitivity: 0.9118 ; specificity: 0.7879 ; geometric_mean: 0.8476 ; fdr: 0.1842 ; mcc: 0.706 ; auc: 0.9002 ; corrclass: 0.8507
#> TP: 31 ; TN: 26 ; FP: 7 ; FN: 3 ; betas:
#> norm2 norm16 norm18 norm23 norm24 norm33 norm35 norm36
#> -0.183942 0.013181 -0.034843 -0.018366 -0.010160 -0.024587 -0.014625 0.058445
#> norm39 norm48 norm51 norm52 norm56 norm60 norm64 norm66
#> -0.007670 -0.009777 0.177061 -0.254797 0.014226 0.032208 -0.048231 -0.008406
#> norm69 norm78 norm81 norm98 norm99 norm101 norm102 norm103
#> -0.012087 -0.003914 -0.006369 0.010994 -0.052027 0.104744 -0.174681 0.020391
#> norm107 norm118 norm122 norm130 norm131 norm134 norm139 norm141
#> -0.005471 -0.045037 0.053654 0.010044 0.035643 -0.006386 -0.051197 -0.015690
#> norm150 norm155 norm158 norm176 norm177 c
#> -0.003824 -0.010137 -0.031014 -0.002137 0.016758 0.005168
#> Cross-validation time: 0.1132 ; Number of iterations: 10
#>
#>
#> ----------------------------------------------------------------
#> | starting with the 2 -th fold for the CV of lambda |
#> ----------------------------------------------------------------
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ;
#> total iters: 10 ; Backtraking iters: 1236 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1533 ; youden_index: 0.4168 ; sensitivity: 0.9104 ; specificity: 0.8182 ; geometric_mean: 0.8631 ; fdr: 0.1644 ; mcc: 0.7321 ; auc: 0.9414 ; corrclass: 0.8647 ;
#> TP: 61 ; TN: 54 ; FP: 12 ; FN: 6 ; betas:
#> norm2 norm51 norm52 norm101 norm102 norm142 c
#> -0.0004153 0.0239183 -0.0613325 0.0436339 -0.0382461 -0.0080857 -0.0000515
#> Estimation time: 0.426 mins
#>
#>
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ;
#> total iters: 7 ; Backtraking iters: 1124 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3894 ; youden_index: 0.7422 ; sensitivity: 0.9403 ; specificity: 0.9242 ; geometric_mean: 0.9322 ; fdr: 0.07353 ; mcc: 0.8647 ; auc: 0.9916 ; corrclass: 0.9323 ;
#> TP: 63 ; TN: 61 ; FP: 5 ; FN: 4 ; betas:
#> norm2 norm9 norm14 norm26 norm33 norm35 norm39
#> -0.0221927 0.0094674 -0.0043182 0.0079288 -0.0002049 -0.0029455 -0.0003129
#> norm50 norm51 norm52
#> -0.0010401 0.0553048 -0.1180324
#> [ reached 'max' / getOption("max.print") -- omitted 21 entries ]
#> Estimation time: 0.4533 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ;
#> total iters: 10 ; Backtraking iters: 428 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.6411 ; youden_index: 0.9376 ; sensitivity: 1 ; specificity: 1 ; geometric_mean: 1 ; fdr: 0 ; mcc: 1 ; auc: 1 ; corrclass: 1 ;
#> TP: 67 ; TN: 66 ; FP: 0 ; FN: 0 ; betas:
#> norm2 norm7 norm9 norm14 norm17 norm23 norm26
#> -0.0645211 0.0003921 0.0321560 -0.0106883 -0.0045308 -0.0002037 0.0313995
#> norm27 norm28 norm33
#> 0.0031757 -0.0042952 -0.0077072
#> [ reached 'max' / getOption("max.print") -- omitted 54 entries ]
#> Estimation time: 0.1171 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ;
#> total iters: 10 ; Backtraking iters: 357 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.7421 ; youden_index: 0.9452 ; sensitivity: 1 ; specificity: 0.9697 ; geometric_mean: 0.9847 ; fdr: 0.02899 ; mcc: 0.9704 ; auc: 0.9943 ; corrclass: 0.985 ;
#> TP: 67 ; TN: 64 ; FP: 2 ; FN: 0 ; betas:
#> norm2 norm9 norm14 norm17 norm22 norm26 norm28
#> -0.0645914 0.0598891 -0.0674312 -0.0416586 0.0007512 0.0631425 -0.0340830
#> norm32 norm35 norm36
#> -0.0079048 -0.0315867 0.0050180
#> [ reached 'max' / getOption("max.print") -- omitted 45 entries ]
#> Estimation time: 0.1109 mins
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
#> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.102 ; youden_index: 0.4082 ; sensitivity: 0.9412 ; specificity: 0.8182 ; geometric_mean: 0.8775 ; fdr: 0.1579 ; mcc: 0.7662 ; auc: 0.9082 ; corrclass: 0.8806
#> TP: 32 ; TN: 27 ; FP: 6 ; FN: 2 ; betas:
#> norm2 norm51 norm52 norm101 norm102 norm142 c
#> -0.0004153 0.0239183 -0.0613325 0.0436339 -0.0382461 -0.0080857 -0.0000515
#> Cross-validation time: 0.426 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
#> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1794 ; youden_index: 0.3913 ; sensitivity: 0.9118 ; specificity: 0.8182 ; geometric_mean: 0.8637 ; fdr: 0.1622 ; mcc: 0.7339 ; auc: 0.8913 ; corrclass: 0.8657
#> TP: 31 ; TN: 27 ; FP: 6 ; FN: 3 ; betas:
#> norm2 norm9 norm14 norm26 norm33 norm35 norm39
#> -0.0221927 0.0094674 -0.0043182 0.0079288 -0.0002049 -0.0029455 -0.0003129
#> norm50 norm51 norm52 norm61 norm71 norm72 norm74
#> -0.0010401 0.0553048 -0.1180324 0.0001712 0.0189608 -0.0086096 -0.0011047
#> norm75 norm79 norm82 norm83 norm101 norm102 norm103
#> -0.0003637 -0.0084191 0.0005923 0.0066332 0.1199288 -0.0834833 0.0087865
#> norm107 norm131 norm142 norm152 norm165 norm171 norm188
#> -0.0096822 0.0114343 -0.0483673 -0.0001453 0.0077519 0.0145624 -0.0089935
#> norm192 norm196 c
#> 0.0001891 -0.0001453 0.0070873
#> Cross-validation time: 0.4533 ; Number of iterations: 7
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
#> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2358 ; youden_index: 0.3663 ; sensitivity: 0.8235 ; specificity: 0.697 ; geometric_mean: 0.7576 ; fdr: 0.2632 ; mcc: 0.5252 ; auc: 0.8663 ; corrclass: 0.7612
#> TP: 28 ; TN: 23 ; FP: 10 ; FN: 6 ; betas:
#> norm2 norm7 norm9 norm14 norm17 norm23 norm26
#> -6.452e-02 3.921e-04 3.216e-02 -1.069e-02 -4.531e-03 -2.037e-04 3.140e-02
#> norm27 norm28 norm33 norm35 norm36 norm43 norm47
#> 3.176e-03 -4.295e-03 -7.707e-03 -2.345e-02 1.202e-02 -5.128e-03 2.393e-02
#> norm50 norm51 norm52 norm60 norm61 norm71 norm72
#> -2.582e-03 8.086e-02 -1.556e-01 2.066e-02 1.962e-03 2.832e-02 -3.919e-02
#> norm74 norm75 norm77 norm79 norm82 norm83 norm92
#> -2.223e-02 -2.900e-03 8.845e-03 -3.774e-02 4.273e-03 1.898e-02 1.073e-03
#> norm100 norm101 norm102 norm103 norm106 norm107 norm108
#> -1.753e-04 1.700e-01 -1.172e-01 1.883e-02 -1.393e-02 -2.934e-02 1.042e-03
#> norm113 norm118 norm119 norm120 norm124 norm125 norm126
#> 2.273e-04 -1.929e-03 -5.546e-03 -3.346e-04 -5.538e-04 1.265e-03 -3.133e-03
#> norm131 norm135 norm142 norm149 norm151 norm158 norm159
#> 9.129e-04 -6.387e-05 -5.643e-02 -3.018e-03 3.461e-04 7.187e-04 -7.313e-04
#> norm160 norm165 norm166 norm167 norm170 norm171 norm175
#> 6.255e-03 1.889e-02 3.316e-03 -1.896e-03 -4.601e-04 1.911e-02 -1.701e-02
#> norm180 norm184 norm188 norm189 norm192 norm195 norm196
#> 8.102e-03 4.528e-03 -1.803e-02 1.839e-03 2.262e-02 -2.896e-03 -2.573e-03
#> c
#> 1.484e-02
#> Cross-validation time: 0.1171 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 2 ;
#> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2474 ; youden_index: 0.3289 ; sensitivity: 0.7353 ; specificity: 0.6667 ; geometric_mean: 0.7001 ; fdr: 0.3056 ; mcc: 0.403 ; auc: 0.8289 ; corrclass: 0.7015
#> TP: 25 ; TN: 22 ; FP: 11 ; FN: 9 ; betas:
#> norm2 norm9 norm14 norm17 norm22 norm26 norm28
#> -0.0645914 0.0598891 -0.0674312 -0.0416586 0.0007512 0.0631425 -0.0340830
#> norm32 norm35 norm36 norm47 norm50 norm51 norm52
#> -0.0079048 -0.0315867 0.0050180 0.0085638 -0.0488746 0.1158676 -0.3073828
#> norm53 norm62 norm69 norm71 norm72 norm74 norm76
#> -0.0171630 -0.0323436 0.0114990 0.0042339 -0.0895150 -0.0155223 0.0050926
#> norm79 norm82 norm83 norm92 norm100 norm101 norm102
#> -0.0641022 0.0031302 0.0061992 0.0045952 -0.0006344 0.3353582 -0.1677424
#> norm103 norm104 norm107 norm115 norm118 norm119 norm124
#> 0.0452843 0.0030895 -0.0590861 0.0008549 -0.0138541 -0.0266964 -0.0002408
#> norm126 norm129 norm130 norm131 norm135 norm142 norm146
#> -0.0025368 0.0016493 0.0061323 0.0297875 -0.0450681 -0.1276665 -0.0096072
#> norm148 norm158 norm160 norm165 norm166 norm171 norm174
#> -0.0075802 0.0046764 0.0114635 0.0553625 0.0051047 0.0115194 0.0028345
#> norm180 norm184 norm188 norm192 norm195 c
#> 0.0069444 0.0352305 -0.0739338 0.0191150 -0.0042548 -0.0070457
#> Cross-validation time: 0.1109 ; Number of iterations: 10
#>
#>
#> ----------------------------------------------------------------
#> | starting with the 3 -th fold for the CV of lambda |
#> ----------------------------------------------------------------
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ;
#> total iters: 10 ; Backtraking iters: 906 ; lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1595 ; youden_index: 0.4309 ; sensitivity: 0.9706 ; specificity: 0.8333 ; geometric_mean: 0.8993 ; fdr: 0.1429 ; mcc: 0.8129 ; auc: 0.934 ; corrclass: 0.903 ;
#> TP: 66 ; TN: 55 ; FP: 11 ; FN: 2 ; betas:
#> norm51 norm52 norm98 norm101 norm102 c
#> 0.02201 -0.06112 0.01045 0.03708 -0.05029 -0.01111
#> Estimation time: 0.3829 mins
#>
#>
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ;
#> total iters: 6 ; Backtraking iters: 619 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.394 ; youden_index: 0.7675 ; sensitivity: 0.9706 ; specificity: 0.9242 ; geometric_mean: 0.9471 ; fdr: 0.07042 ; mcc: 0.8963 ; auc: 0.9828 ; corrclass: 0.9478 ;
#> TP: 66 ; TN: 61 ; FP: 5 ; FN: 2 ; betas:
#> norm2 norm7 norm8 norm10 norm12 norm33 norm36
#> -8.924e-03 1.760e-04 3.634e-03 -1.820e-03 -3.012e-02 -2.632e-03 9.833e-04
#> norm38 norm45 norm51
#> 5.766e-05 2.123e-04 5.859e-02
#> [ reached 'max' / getOption("max.print") -- omitted 22 entries ]
#> Estimation time: 0.308 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ;
#> total iters: 10 ; Backtraking iters: 384 ; lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.6339 ; youden_index: 0.9067 ; sensitivity: 0.9853 ; specificity: 0.9848 ; geometric_mean: 0.9851 ; fdr: 0.01471 ; mcc: 0.9701 ; auc: 0.9993 ; corrclass: 0.9851 ;
#> TP: 67 ; TN: 65 ; FP: 1 ; FN: 1 ; betas:
#> norm1 norm2 norm8 norm10 norm12 norm25 norm27 norm32
#> 0.009573 -0.019962 0.002220 -0.003928 -0.045978 0.018931 -0.004585 -0.012847
#> norm33 norm36
#> -0.030224 0.018665
#> [ reached 'max' / getOption("max.print") -- omitted 27 entries ]
#> Estimation time: 0.1114 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ;
#> total iters: 10 ; Backtraking iters: 345 ; lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.7302 ; youden_index: 0.9938 ; sensitivity: 1 ; specificity: 1 ; geometric_mean: 1 ; fdr: 0 ; mcc: 1 ; auc: 1 ; corrclass: 1 ;
#> TP: 68 ; TN: 66 ; FP: 0 ; FN: 0 ; betas:
#> norm2 norm5 norm7 norm8 norm10 norm12 norm15
#> -0.0429020 -0.0002879 0.0013894 0.0009275 -0.0530969 -0.0856319 0.0009420
#> norm17 norm19 norm20
#> -0.0018709 0.0002089 0.0003063
#> [ reached 'max' / getOption("max.print") -- omitted 87 entries ]
#> Estimation time: 0.1143 mins
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
#> lambda: 1.5 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1056 ; youden_index: 0.4128 ; sensitivity: 0.8788 ; specificity: 0.6364 ; geometric_mean: 0.7478 ; fdr: 0.2927 ; mcc: 0.531 ; auc: 0.9128 ; corrclass: 0.7576
#> TP: 29 ; TN: 21 ; FP: 12 ; FN: 4 ; betas:
#> norm51 norm52 norm98 norm101 norm102 c
#> 0.02201 -0.06112 0.01045 0.03708 -0.05029 -0.01111
#> Cross-validation time: 0.3829 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
#> lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2052 ; youden_index: 0.4229 ; sensitivity: 0.8485 ; specificity: 0.7576 ; geometric_mean: 0.8017 ; fdr: 0.2222 ; mcc: 0.6086 ; auc: 0.9229 ; corrclass: 0.803
#> TP: 28 ; TN: 25 ; FP: 8 ; FN: 5 ; betas:
#> norm2 norm7 norm8 norm10 norm12 norm33 norm36
#> -8.924e-03 1.760e-04 3.634e-03 -1.820e-03 -3.012e-02 -2.632e-03 9.833e-04
#> norm38 norm45 norm51 norm52 norm64 norm66 norm79
#> 5.766e-05 2.123e-04 5.859e-02 -1.341e-01 -1.386e-03 -1.653e-04 -5.358e-03
#> norm81 norm98 norm101 norm102 norm104 norm107 norm139
#> -3.771e-02 3.610e-02 1.091e-01 -9.620e-02 -2.750e-03 -1.400e-02 -2.828e-03
#> norm141 norm149 norm162 norm165 norm171 norm182 norm183
#> -1.449e-03 -1.951e-03 2.269e-03 1.886e-02 3.859e-03 4.895e-03 -3.751e-03
#> norm188 norm195 norm197 c
#> -2.126e-02 -4.267e-03 4.607e-03 1.371e-02
#> Cross-validation time: 0.308 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
#> lambda: 0.2466 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3042 ; youden_index: 0.4219 ; sensitivity: 0.8182 ; specificity: 0.7879 ; geometric_mean: 0.8029 ; fdr: 0.2059 ; mcc: 0.6063 ; auc: 0.9219 ; corrclass: 0.803
#> TP: 27 ; TN: 26 ; FP: 7 ; FN: 6 ; betas:
#> norm1 norm2 norm8 norm10 norm12 norm25 norm27 norm32
#> 0.009573 -0.019962 0.002220 -0.003928 -0.045978 0.018931 -0.004585 -0.012847
#> norm33 norm36 norm37 norm38 norm45 norm49 norm50 norm51
#> -0.030224 0.018665 0.035816 0.014334 0.002386 -0.015544 -0.020119 0.101471
#> norm52 norm60 norm79 norm81 norm98 norm100 norm101 norm102
#> -0.185250 0.004312 -0.018793 -0.044783 0.047704 0.007257 0.129100 -0.122417
#> norm103 norm105 norm107 norm149 norm162 norm165 norm182 norm183
#> 0.004531 -0.013386 -0.005617 -0.037282 0.009964 0.039251 0.024600 -0.010102
#> norm184 norm188 norm195 norm197 c
#> 0.006549 -0.003960 -0.007028 0.027865 0.032417
#> Cross-validation time: 0.1114 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET of pye
#> -> algorithm: pye_KS_proximal_gradient_method ; fold = 3 ;
#> lambda: 0.1 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2715 ; youden_index: 0.377 ; sensitivity: 0.7879 ; specificity: 0.7576 ; geometric_mean: 0.7726 ; fdr: 0.2353 ; mcc: 0.5457 ; auc: 0.877 ; corrclass: 0.7727
#> TP: 26 ; TN: 25 ; FP: 8 ; FN: 7 ; betas:
#> norm2 norm5 norm7 norm8 norm10 norm12 norm15
#> -4.290e-02 -2.879e-04 1.389e-03 9.275e-04 -5.310e-02 -8.563e-02 9.420e-04
#> norm17 norm19 norm20 norm25 norm29 norm32 norm33
#> -1.871e-03 2.089e-04 3.063e-04 4.820e-02 7.561e-03 -1.685e-02 -3.377e-02
#> norm34 norm36 norm37 norm38 norm39 norm45 norm46
#> 1.156e-03 3.197e-02 6.458e-02 1.787e-02 -5.395e-02 1.632e-03 4.784e-04
#> norm47 norm49 norm50 norm51 norm52 norm55 norm56
#> 5.086e-04 -2.856e-02 -3.736e-02 1.261e-01 -3.731e-01 -3.093e-02 -2.121e-03
#> norm60 norm61 norm64 norm71 norm72 norm75 norm79
#> 3.431e-02 9.420e-03 -2.163e-03 1.105e-02 -8.339e-04 -2.544e-03 -7.339e-02
#> norm81 norm83 norm84 norm92 norm98 norm99 norm100
#> -8.987e-02 -1.354e-03 5.617e-04 8.175e-04 1.015e-01 -4.660e-03 5.484e-03
#> norm101 norm102 norm103 norm104 norm105 norm107 norm109
#> 2.363e-01 -2.057e-01 5.936e-02 -2.986e-03 -3.003e-02 -3.129e-02 4.541e-02
#> norm111 norm113 norm118 norm119 norm120 norm121 norm122
#> 5.318e-02 1.871e-02 -8.288e-03 -6.275e-04 2.150e-04 8.578e-04 -2.501e-02
#> norm123 norm124 norm126 norm128 norm129 norm131 norm132
#> 2.125e-02 -1.282e-02 -2.332e-03 -1.685e-03 -2.183e-02 2.090e-03 1.085e-03
#> norm135 norm136 norm139 norm141 norm147 norm149 norm152
#> -4.937e-03 8.464e-04 -5.587e-03 -2.260e-03 5.751e-02 -8.837e-02 -3.511e-02
#> norm153 norm157 norm159 norm162 norm165 norm166 norm167
#> -1.181e-03 -8.656e-03 -1.536e-03 2.372e-02 5.915e-02 -4.926e-04 -1.652e-02
#> norm171 norm173 norm175 norm177 norm180 norm181 norm182
#> 1.446e-04 9.811e-04 -4.880e-03 2.186e-03 -3.292e-04 5.592e-03 6.995e-02
#> norm183 norm184 norm185 norm187 norm188 norm190 norm191
#> -3.580e-03 7.231e-02 3.415e-04 -1.765e-03 -3.118e-02 3.307e-02 1.816e-02
#> norm192 norm194 norm195 norm197 norm200 c
#> 6.796e-06 1.111e-03 -1.070e-02 4.203e-02 4.269e-02 1.490e-02
#> Cross-validation time: 0.1143 ; Number of iterations: 10
#>
#>
#>
#> Starting the CV of tau using as lambda: 0.6082 , that is the best value of lambda as per: ccr
#> ----------------------------------------------------------------
#> | starting with the 1 -th fold for the CV of tau |
#> ----------------------------------------------------------------
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ;
#> total iters: 6 ; Backtraking iters: 638 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3781 ; youden_index: 0.7603 ; sensitivity: 0.9552 ; specificity: 0.9394 ; geometric_mean: 0.9473 ; fdr: 0.05882 ; mcc: 0.8948 ; auc: 0.9878 ; corrclass: 0.9474 ;
#> TP: 64 ; TN: 62 ; FP: 4 ; FN: 3 ; betas:
#> norm2 norm12 norm16 norm18 norm23 norm35 norm39
#> -0.0834877 -0.0024775 0.0014795 -0.0001693 -0.0024886 -0.0055629 -0.0139270
#> norm45 norm48 norm51
#> 0.0060359 -0.0003150 0.0702604
#> [ reached 'max' / getOption("max.print") -- omitted 23 entries ]
#> Estimation time: 0.3188 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ; total iters: 9 ; Backtraking iters: 151 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7618 ; youden_index: 0.7903 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9394 ; geometric_mean: 0.962 ; fdr: 0.05714 ; mcc: 0.9257 ; auc: 0.9878 ; aauc: 0 ; corrclass: 0.9624 ;
#> TP: 66 ; TN: 62 ; FP: 4 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.03618
#> Estimation time: 0.0161 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ; total iters: 10 ; Backtraking iters: 144 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7709 ; youden_index: 0.7923 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9394 ; geometric_mean: 0.962 ; fdr: 0.05714 ; mcc: 0.9257 ; auc: 0.9878 ; aauc: 0 ; corrclass: 0.9624 ;
#> TP: 66 ; TN: 62 ; FP: 4 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.04226
#> Estimation time: 0.01553 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ; total iters: 10 ; Backtraking iters: 144 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7832 ; youden_index: 0.825 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9697 ; geometric_mean: 0.9774 ; fdr: 0.02941 ; mcc: 0.955 ; auc: 0.9878 ; aauc: 0 ; corrclass: 0.9774 ;
#> TP: 66 ; TN: 64 ; FP: 2 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11 norm_cov19
#> -0.04714 0.03486 -0.03110
#> Estimation time: 0.01583 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 1 ; total iters: 10 ; Backtraking iters: 162 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7963 ; youden_index: 0.8268 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9545 ; geometric_mean: 0.9697 ; fdr: 0.04348 ; mcc: 0.9403 ; auc: 0.9878 ; aauc: 0 ; corrclass: 0.9699 ;
#> TP: 66 ; TN: 63 ; FP: 3 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11 norm_cov19
#> -0.05096 0.03771 -0.03564
#> Estimation time: 0.01593 mins
#>
#>
#>
#> Final results on TEST SET of covYI for the FOLD: 1 :
#> With lambda: 0.6082 , penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
#> pye_KS: 0.219 ; youden_index: 0.4349 ; sensitivity: 0.8529 ; specificity: 0.8182 ; geometric_mean: 0.8354 ; fdr: 0.1714 ; mcc: 0.6717 ; auc: 0.9349 ; corrclass: 0.8358 ;
#> TP: 29 ; TN: 27 ; FP: 6 ; FN: 5 ; betas:
#> norm2 norm12 norm16 norm18 norm23 norm35 norm39
#> -8.349e-02 -2.477e-03 1.480e-03 -1.693e-04 -2.489e-03 -5.563e-03 -1.393e-02
#> norm45 norm48 norm51 norm52 norm60 norm66 norm78
#> 6.036e-03 -3.150e-04 7.026e-02 -1.272e-01 8.754e-03 -6.666e-04 -4.795e-03
#> norm98 norm99 norm101 norm102 norm103 norm107 norm118
#> 5.411e-03 -1.746e-02 9.307e-02 -9.918e-02 1.797e-04 -5.460e-03 -2.026e-02
#> norm122 norm131 norm139 norm141 norm150 norm155 norm161
#> 7.857e-03 9.453e-03 -2.590e-02 -5.859e-04 -1.075e-03 -5.640e-04 4.895e-03
#> norm170 norm177 norm193 norm197 c
#> 2.545e-04 5.668e-03 3.508e-03 1.465e-14 1.074e-02
#> Cross-validation time: 0.3188 ; Number of iterations: 6
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.6147 ; youden_index: 0.7059 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.7879 ; geometric_mean: 0.8476 ; fdr: 0.1842 ; mcc: 0.706 ; auc: 0.9349 ; aauc: 0 ; corrclass: 0.8507 ;
#> TP: 31 ; TN: 26 ; FP: 7 ; FN: 3 ; gammas:
#> norm_cov6
#> -0.03618
#> Cross-validation time: 0.0161 ; Number of iterations: 9
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.6269 ; youden_index: 0.7059 ; aYI: 0 ; sensitivity: 0.9706 ; specificity: 0.7879 ; geometric_mean: 0.8745 ; fdr: 0.175 ; mcc: 0.7731 ; auc: 0.9349 ; aauc: 0 ; corrclass: 0.8806 ;
#> TP: 33 ; TN: 26 ; FP: 7 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.04226
#> Cross-validation time: 0.01553 ; Number of iterations: 10
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.6264 ; youden_index: 0.7059 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.8788 ; geometric_mean: 0.8951 ; fdr: 0.1143 ; mcc: 0.7913 ; auc: 0.9349 ; aauc: 0 ; corrclass: 0.8955 ;
#> TP: 31 ; TN: 29 ; FP: 4 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11 norm_cov19
#> -0.04714 0.03486 -0.03110
#> Cross-validation time: 0.01583 ; Number of iterations: 10
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.6386 ; youden_index: 0.7059 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.8788 ; geometric_mean: 0.8951 ; fdr: 0.1143 ; mcc: 0.7913 ; auc: 0.9349 ; aauc: 0 ; corrclass: 0.8955 ;
#> TP: 31 ; TN: 29 ; FP: 4 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11 norm_cov19
#> -0.05096 0.03771 -0.03564
#> Cross-validation time: 0.01593 ; Number of iterations: 10
#>
#>
#>
#> ----------------------------------------------------------------
#> | starting with the 2 -th fold for the CV of tau |
#> ----------------------------------------------------------------
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ;
#> total iters: 7 ; Backtraking iters: 1124 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3894 ; youden_index: 0.7422 ; sensitivity: 0.9403 ; specificity: 0.9242 ; geometric_mean: 0.9322 ; fdr: 0.07353 ; mcc: 0.8647 ; auc: 0.9916 ; corrclass: 0.9323 ;
#> TP: 63 ; TN: 61 ; FP: 5 ; FN: 4 ; betas:
#> norm2 norm9 norm14 norm26 norm33 norm35 norm39
#> -0.0221927 0.0094674 -0.0043182 0.0079288 -0.0002049 -0.0029455 -0.0003129
#> norm50 norm51 norm52
#> -0.0010401 0.0553048 -0.1180324
#> [ reached 'max' / getOption("max.print") -- omitted 21 entries ]
#> Estimation time: 0.4538 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ; total iters: 10 ; Backtraking iters: 143 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7469 ; youden_index: 0.8038 ; aYI: 0 ; sensitivity: 0.9701 ; specificity: 0.9848 ; geometric_mean: 0.9775 ; fdr: 0.01515 ; mcc: 0.955 ; auc: 0.9916 ; aauc: 0 ; corrclass: 0.9774 ;
#> TP: 65 ; TN: 65 ; FP: 1 ; FN: 2 ; gammas:
#> norm_cov6 norm_cov11
#> -0.03511 0.03673
#> Estimation time: 0.01681 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ; total iters: 9 ; Backtraking iters: 90 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7651 ; youden_index: 0.8075 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9697 ; geometric_mean: 0.9774 ; fdr: 0.02941 ; mcc: 0.955 ; auc: 0.9916 ; aauc: 0 ; corrclass: 0.9774 ;
#> TP: 66 ; TN: 64 ; FP: 2 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11
#> -0.04008 0.04315
#> Estimation time: 0.01461 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ; total iters: 10 ; Backtraking iters: 131 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7783 ; youden_index: 0.809 ; aYI: 0 ; sensitivity: 0.9851 ; specificity: 0.9697 ; geometric_mean: 0.9774 ; fdr: 0.02941 ; mcc: 0.955 ; auc: 0.9916 ; aauc: 0 ; corrclass: 0.9774 ;
#> TP: 66 ; TN: 64 ; FP: 2 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11
#> -0.04342 0.04703
#> Estimation time: 0.01653 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 2 ; total iters: 10 ; Backtraking iters: 93 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7898 ; youden_index: 0.8183 ; aYI: 0 ; sensitivity: 0.9701 ; specificity: 0.9545 ; geometric_mean: 0.9623 ; fdr: 0.04412 ; mcc: 0.9249 ; auc: 0.9916 ; aauc: 0 ; corrclass: 0.9624 ;
#> TP: 65 ; TN: 63 ; FP: 3 ; FN: 2 ; gammas:
#> norm_cov3 norm_cov6 norm_cov11
#> 0.01923 -0.04686 0.04676
#> Estimation time: 0.01523 mins
#>
#>
#>
#> Final results on TEST SET of covYI for the FOLD: 2 :
#> With lambda: 0.6082 , penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
#> pye_KS: 0.1794 ; youden_index: 0.3913 ; sensitivity: 0.9118 ; specificity: 0.8182 ; geometric_mean: 0.8637 ; fdr: 0.1622 ; mcc: 0.7339 ; auc: 0.8913 ; corrclass: 0.8657 ;
#> TP: 31 ; TN: 27 ; FP: 6 ; FN: 3 ; betas:
#> norm2 norm9 norm14 norm26 norm33 norm35 norm39
#> -0.0221927 0.0094674 -0.0043182 0.0079288 -0.0002049 -0.0029455 -0.0003129
#> norm50 norm51 norm52 norm61 norm71 norm72 norm74
#> -0.0010401 0.0553048 -0.1180324 0.0001712 0.0189608 -0.0086096 -0.0011047
#> norm75 norm79 norm82 norm83 norm101 norm102 norm103
#> -0.0003637 -0.0084191 0.0005923 0.0066332 0.1199288 -0.0834833 0.0087865
#> norm107 norm131 norm142 norm152 norm165 norm171 norm188
#> -0.0096822 0.0114343 -0.0483673 -0.0001453 0.0077519 0.0145624 -0.0089935
#> norm192 norm196 c
#> 0.0001891 -0.0001453 0.0070873
#> Cross-validation time: 0.4538 ; Number of iterations: 7
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5389 ; youden_index: 0.7353 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.8788 ; geometric_mean: 0.8951 ; fdr: 0.1143 ; mcc: 0.7913 ; auc: 0.8913 ; aauc: 0 ; corrclass: 0.8955 ;
#> TP: 31 ; TN: 29 ; FP: 4 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11
#> -0.03511 0.03673
#> Cross-validation time: 0.01681 ; Number of iterations: 10
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5589 ; youden_index: 0.7353 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.8788 ; geometric_mean: 0.8951 ; fdr: 0.1143 ; mcc: 0.7913 ; auc: 0.8913 ; aauc: 0 ; corrclass: 0.8955 ;
#> TP: 31 ; TN: 29 ; FP: 4 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11
#> -0.04008 0.04315
#> Cross-validation time: 0.01461 ; Number of iterations: 9
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5737 ; youden_index: 0.7353 ; aYI: 0 ; sensitivity: 0.9118 ; specificity: 0.8788 ; geometric_mean: 0.8951 ; fdr: 0.1143 ; mcc: 0.7913 ; auc: 0.8913 ; aauc: 0 ; corrclass: 0.8955 ;
#> TP: 31 ; TN: 29 ; FP: 4 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11
#> -0.04342 0.04703
#> Cross-validation time: 0.01653 ; Number of iterations: 10
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5761 ; youden_index: 0.7353 ; aYI: 0 ; sensitivity: 0.8824 ; specificity: 0.8485 ; geometric_mean: 0.8653 ; fdr: 0.1429 ; mcc: 0.7315 ; auc: 0.8913 ; aauc: 0 ; corrclass: 0.8657 ;
#> TP: 30 ; TN: 28 ; FP: 5 ; FN: 4 ; gammas:
#> norm_cov3 norm_cov6 norm_cov11
#> 0.01923 -0.04686 0.04676
#> Cross-validation time: 0.01523 ; Number of iterations: 10
#>
#>
#>
#> ----------------------------------------------------------------
#> | starting with the 3 -th fold for the CV of tau |
#> ----------------------------------------------------------------
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ;
#> total iters: 6 ; Backtraking iters: 619 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.394 ; youden_index: 0.7675 ; sensitivity: 0.9706 ; specificity: 0.9242 ; geometric_mean: 0.9471 ; fdr: 0.07042 ; mcc: 0.8963 ; auc: 0.9828 ; corrclass: 0.9478 ;
#> TP: 66 ; TN: 61 ; FP: 5 ; FN: 2 ; betas:
#> norm2 norm7 norm8 norm10 norm12 norm33 norm36
#> -8.924e-03 1.760e-04 3.634e-03 -1.820e-03 -3.012e-02 -2.632e-03 9.833e-04
#> norm38 norm45 norm51
#> 5.766e-05 2.123e-04 5.859e-02
#> [ reached 'max' / getOption("max.print") -- omitted 22 entries ]
#> Estimation time: 0.3033 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ; total iters: 9 ; Backtraking iters: 414 ; tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7808 ; youden_index: 0.817 ; aYI: 0 ; sensitivity: 0.9853 ; specificity: 0.9545 ; geometric_mean: 0.9698 ; fdr: 0.04286 ; mcc: 0.9407 ; auc: 0.9828 ; aauc: 0 ; corrclass: 0.9701 ;
#> TP: 67 ; TN: 63 ; FP: 3 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.05835
#> Estimation time: 0.02049 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ; total iters: 9 ; Backtraking iters: 414 ; tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7922 ; youden_index: 0.8186 ; aYI: 0 ; sensitivity: 0.9853 ; specificity: 0.9545 ; geometric_mean: 0.9698 ; fdr: 0.04286 ; mcc: 0.9407 ; auc: 0.9828 ; aauc: 0 ; corrclass: 0.9701 ;
#> TP: 67 ; TN: 63 ; FP: 3 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.06452
#> Estimation time: 0.01991 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ; total iters: 10 ; Backtraking iters: 453 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8004 ; youden_index: 0.8193 ; aYI: 0 ; sensitivity: 0.9853 ; specificity: 0.9545 ; geometric_mean: 0.9698 ; fdr: 0.04286 ; mcc: 0.9407 ; auc: 0.9828 ; aauc: 0 ; corrclass: 0.9701 ;
#> TP: 67 ; TN: 63 ; FP: 3 ; FN: 1 ; gammas:
#> norm_cov6
#> -0.06855
#> Estimation time: 0.0214 mins
#>
#>
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> fold = 3 ; total iters: 10 ; Backtraking iters: 447 ; tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.8081 ; youden_index: 0.8283 ; aYI: 0 ; sensitivity: 0.9853 ; specificity: 0.9545 ; geometric_mean: 0.9698 ; fdr: 0.04286 ; mcc: 0.9407 ; auc: 0.9828 ; aauc: 0 ; corrclass: 0.9701 ;
#> TP: 67 ; TN: 63 ; FP: 3 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11
#> -0.06353 0.02332
#> Estimation time: 0.02136 mins
#>
#>
#>
#> Final results on TEST SET of covYI for the FOLD: 3 :
#> With lambda: 0.6082 , penalty: SCAD and weight: 0.5 . Accuracy measures only pye:
#> pye_KS: 0.2052 ; youden_index: 0.4229 ; sensitivity: 0.8485 ; specificity: 0.7576 ; geometric_mean: 0.8017 ; fdr: 0.2222 ; mcc: 0.6086 ; auc: 0.9229 ; corrclass: 0.803 ;
#> TP: 28 ; TN: 25 ; FP: 8 ; FN: 5 ; betas:
#> norm2 norm7 norm8 norm10 norm12 norm33 norm36
#> -8.924e-03 1.760e-04 3.634e-03 -1.820e-03 -3.012e-02 -2.632e-03 9.833e-04
#> norm38 norm45 norm51 norm52 norm64 norm66 norm79
#> 5.766e-05 2.123e-04 5.859e-02 -1.341e-01 -1.386e-03 -1.653e-04 -5.358e-03
#> norm81 norm98 norm101 norm102 norm104 norm107 norm139
#> -3.771e-02 3.610e-02 1.091e-01 -9.620e-02 -2.750e-03 -1.400e-02 -2.828e-03
#> norm141 norm149 norm162 norm165 norm171 norm182 norm183
#> -1.449e-03 -1.951e-03 2.269e-03 1.886e-02 3.859e-03 4.895e-03 -3.751e-03
#> norm188 norm195 norm197 c
#> -2.126e-02 -4.267e-03 4.607e-03 1.371e-02
#> Cross-validation time: 0.3033 ; Number of iterations: 6
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.15 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5645 ; youden_index: 0.697 ; aYI: 0 ; sensitivity: 0.8788 ; specificity: 0.7576 ; geometric_mean: 0.8159 ; fdr: 0.2162 ; mcc: 0.6411 ; auc: 0.9229 ; aauc: 0 ; corrclass: 0.8182 ;
#> TP: 29 ; TN: 25 ; FP: 8 ; FN: 4 ; gammas:
#> norm_cov6
#> -0.05835
#> Cross-validation time: 0.02049 ; Number of iterations: 9
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.104 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5738 ; youden_index: 0.697 ; aYI: 0 ; sensitivity: 0.8788 ; specificity: 0.7576 ; geometric_mean: 0.8159 ; fdr: 0.2162 ; mcc: 0.6411 ; auc: 0.9229 ; aauc: 0 ; corrclass: 0.8182 ;
#> TP: 29 ; TN: 25 ; FP: 8 ; FN: 4 ; gammas:
#> norm_cov6
#> -0.06452
#> Cross-validation time: 0.01991 ; Number of iterations: 9
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5807 ; youden_index: 0.697 ; aYI: 0 ; sensitivity: 0.8788 ; specificity: 0.7576 ; geometric_mean: 0.8159 ; fdr: 0.2162 ; mcc: 0.6411 ; auc: 0.9229 ; aauc: 0 ; corrclass: 0.8182 ;
#> TP: 29 ; TN: 25 ; FP: 8 ; FN: 4 ; gammas:
#> norm_cov6
#> -0.06855
#> Cross-validation time: 0.0214 ; Number of iterations: 10
#>
#>
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> tau: 0.05 ; weight: 0.5 ; penalty: L12 ; covYI_KS_L12 : 0.5986 ; youden_index: 0.697 ; aYI: 0 ; sensitivity: 0.9091 ; specificity: 0.7576 ; geometric_mean: 0.8299 ; fdr: 0.2105 ; mcc: 0.6745 ; auc: 0.9229 ; aauc: 0 ; corrclass: 0.8333 ;
#> TP: 30 ; TN: 25 ; FP: 8 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11
#> -0.06353 0.02332
#> Cross-validation time: 0.02136 ; Number of iterations: 10
#>
#>
#>
#> ----------------------> END OF THE CROSS-VALIDATION OF THE PYE METHOD <-----------------
#> -------------> For the whoole Cross-validation it took: 4.284 minutes <------------
# 6. Print results and access optimal lambda/tau
cat("\nOptimal Lambda (based on CCR):", pye_cv_result$lambda_hat_ccr, "\n")
#>
#> Optimal Lambda (based on CCR): 0.6082
if (c_function_of_covariates == TRUE) {
cat("Optimal Tau (based on CCR):", pye_cv_result$tau_hat_ccr, "\n")
}
#> Optimal Tau (based on CCR): 0.07211
# You can access other results like:
pye_cv_result$auc # AUC values
#> $`lambda=0.6082201996`
#> $`lambda=0.6082201996`$train
#> tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
#> fold=1 0.9878 0.9878 0.9878 0.9878
#> fold=2 0.9916 0.9916 0.9916 0.9916
#> fold=3 0.9828 0.9828 0.9828 0.9828
#>
#> $`lambda=0.6082201996`$test
#> tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
#> fold=1 0.9349 0.9349 0.9349 0.9349
#> fold=2 0.8913 0.8913 0.8913 0.8913
#> fold=3 0.9229 0.9229 0.9229 0.9229
pye_cv_result$n_betas # Number of non-zero betas for each lambda
#> lambda=1.5 lambda=0.6082201996 lambda=0.2466212074 lambda=0.1
#> fold=1 6 32 57 37
#> fold=2 6 30 63 54
#> fold=3 5 31 36 96
pye_cv_result$n_gammas # Number of non-zero gammas for each tau
#> $`lambda=0.6082201996`
#> tau=0.15 tau=0.1040041912 tau=0.07211247852 tau=0.05
#> [1,] 1 1 3 3
#> [2,] 2 2 2 3
#> [3,] 1 1 1 2
# 7. Compute the performance over 50 simulations with the optimal lambda and tau
sim_result <- pye_KS_simulation_study(
n = 5,
df = df,
X = X,
y = y,
C = C,
lambda = pye_cv_result$lambda_hat_ccr,
tau = pye_cv_result$tau_hat_ccr,
trace = 1,
penalty = penalty, # Options: "L12", "L1", "EN", "SCAD", "MCP"
penalty_g = penalty_g, # Options: "L12", "L1", "EN", "SCAD", "MCP"
used_cores = 1,
c_function_of_covariates = c_function_of_covariates,
max_iter = max_iter, # Reduced for a quick example
max_iter_g = 20 # Reduced for a quick example
)
#> ------------------------------------------------------------------
#> | Starting simulation study with 5 simulations |
#> ------------------------------------------------------------------
#> Running simulation in sequential mode (used_cores = 1).
#>
#> --------------------> experiment n 1 of 5 <----------------------
#> lambda = 0.6082 ; weight: 0.5 ; penalty = SCAD
#> tau = 0.07211 ; weight_g: 0.5 ; penalty_g = L12
#> mmAPG converged because the maximum number of iteration has been reached.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 10 ; Backtraking iters: 1948 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3619 ; youden_index: 0.7507 ; sensitivity: 0.9718 ; specificity: 0.8986 ; geometric_mean: 0.9345 ; fdr: 0.09211 ; mcc: 0.8735 ; auc: 0.9839 ; corrclass: 0.9357 ;
#> TP: 69 ; TN: 62 ; FP: 7 ; FN: 2 ; betas:
#> norm1 norm2 norm8 norm10 norm12 norm16 norm23 norm33
#> 0.001209 -0.038129 0.001488 -0.002278 -0.020956 0.010696 -0.001369 -0.003117
#> norm50 norm51
#> -0.007848 0.054271
#> [ reached 'max' / getOption("max.print") -- omitted 25 entries ]
#> Estimation time: 0.528 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 12 ; Backtraking iters: 398 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7758 ; youden_index: 0.8194 ; aYI: 0 ; sensitivity: 0.9859 ; specificity: 0.9275 ; geometric_mean: 0.9563 ; fdr: 0.06667 ; mcc: 0.9157 ; auc: 0.9839 ; aauc: 0 ; corrclass: 0.9571 ;
#> TP: 70 ; TN: 64 ; FP: 5 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11 norm_cov16
#> -0.05455 0.04707 -0.02379
#> Estimation time: 0.0266 mins
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 1 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2618 ; youden_index: 0.4633 ; sensitivity: 0.9667 ; fdr: 0.1212 ; mcc: 0.8375 ; auc: 0.9633 ; corrclass: 0.9167
#> TP: 29 ; TN: 26 ; FP: 4 ; FN: 1 ; betas:
#> norm1 norm2 norm8 norm10 norm12 norm16 norm23
#> 0.0012091 -0.0381289 0.0014884 -0.0022777 -0.0209560 0.0106959 -0.0013687
#> norm33 norm50 norm51 norm52 norm60 norm72 norm79
#> -0.0031171 -0.0078483 0.0542708 -0.1294587 0.0136537 -0.0001288 -0.0009872
#> norm81 norm98 norm101 norm102 norm103 norm107 norm108
#> -0.0229569 0.0117141 0.1163116 -0.0791257 0.0045781 -0.0338378 0.0046103
#> norm118 norm131 norm139 norm141 norm149 norm152 norm162
#> -0.0081268 0.0098305 -0.0041641 -0.0064591 -0.0027599 -0.0123368 0.0017866
#> norm165 norm167 norm171 norm188 norm195 norm197 c
#> 0.0103645 -0.0026192 0.0097248 -0.0041283 -0.0062523 0.0019291 0.0172448
#> Estimation time: 0.528 ; Number of iterations: 10
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.6812 ; youden_index: 0.8333 ; aYI: 0 ; sensitivity: 0.9667 ; specificity: 0.9333 ; geometric_mean: 0.9499 ; fdr: 0.06452 ; mcc: 0.9005 ; auc: 0.9633 ; aauc: 0 ; corrclass: 0.95
#> TP: 29 ; TN: 28 ; FP: 2 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11 norm_cov16
#> -0.05455 0.04707 -0.02379
#> Estimation time: 0.0266 ; Number of iterations: 12
#>
#>
#>
#> --------------------> experiment n 2 of 5 <----------------------
#> lambda = 0.6082 ; weight: 0.5 ; penalty = SCAD
#> tau = 0.07211 ; weight_g: 0.5 ; penalty_g = L12
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 6 ; Backtraking iters: 623 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3673 ; youden_index: 0.7172 ; sensitivity: 0.9859 ; specificity: 0.8696 ; geometric_mean: 0.9259 ; fdr: 0.1139 ; mcc: 0.8626 ; auc: 0.9824 ; corrclass: 0.9286 ;
#> TP: 70 ; TN: 60 ; FP: 9 ; FN: 1 ; betas:
#> norm2 norm8 norm9 norm12 norm33 norm35 norm36
#> -0.0375633 0.0006549 0.0079897 -0.0068112 -0.0156184 -0.0005642 0.0056790
#> norm50 norm51 norm52
#> -0.0032703 0.0776131 -0.1250317
#> [ reached 'max' / getOption("max.print") -- omitted 22 entries ]
#> Estimation time: 0.317 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 12 ; Backtraking iters: 382 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7456 ; youden_index: 0.7772 ; aYI: 0 ; sensitivity: 0.9859 ; specificity: 0.913 ; geometric_mean: 0.9488 ; fdr: 0.07895 ; mcc: 0.9022 ; auc: 0.9824 ; aauc: 0 ; corrclass: 0.95 ;
#> TP: 70 ; TN: 63 ; FP: 6 ; FN: 1 ; gammas:
#> norm_cov6 norm_cov11
#> -0.05164 0.04474
#> Estimation time: 0.02619 mins
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 2 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3101 ; youden_index: 0.4778 ; sensitivity: 0.9667 ; fdr: 0.1714 ; mcc: 0.7775 ; auc: 0.9778 ; corrclass: 0.8833
#> TP: 29 ; TN: 24 ; FP: 6 ; FN: 1 ; betas:
#> norm2 norm8 norm9 norm12 norm33 norm35 norm36
#> -3.756e-02 6.549e-04 7.990e-03 -6.811e-03 -1.562e-02 -5.642e-04 5.679e-03
#> norm50 norm51 norm52 norm60 norm72 norm98 norm101
#> -3.270e-03 7.761e-02 -1.250e-01 3.509e-03 -9.589e-04 8.052e-03 1.023e-01
#> norm102 norm103 norm107 norm109 norm119 norm131 norm132
#> -8.943e-02 6.659e-03 -3.394e-02 1.594e-03 -1.059e-02 6.024e-03 -1.181e-12
#> norm146 norm147 norm149 norm150 norm162 norm165 norm167
#> 1.065e-03 2.288e-03 -7.748e-03 -2.460e-12 2.235e-03 9.052e-03 -1.040e-03
#> norm171 norm182 norm195 c
#> 6.935e-05 2.032e-04 -7.612e-03 2.848e-03
#> Estimation time: 0.317 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7123 ; youden_index: 0.9333 ; aYI: 0 ; sensitivity: 0.9333 ; specificity: 0.9333 ; geometric_mean: 0.9333 ; fdr: 0.06667 ; mcc: 0.8667 ; auc: 0.9778 ; aauc: 0 ; corrclass: 0.9333
#> TP: 28 ; TN: 28 ; FP: 2 ; FN: 2 ; gammas:
#> norm_cov6 norm_cov11
#> -0.05164 0.04474
#> Estimation time: 0.02619 ; Number of iterations: 12
#>
#>
#>
#> --------------------> experiment n 3 of 5 <----------------------
#> lambda = 0.6082 ; weight: 0.5 ; penalty = SCAD
#> tau = 0.07211 ; weight_g: 0.5 ; penalty_g = L12
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 6 ; Backtraking iters: 1054 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3643 ; youden_index: 0.7215 ; sensitivity: 0.9437 ; specificity: 0.942 ; geometric_mean: 0.9428 ; fdr: 0.05634 ; mcc: 0.8857 ; auc: 0.9824 ; corrclass: 0.9429 ;
#> TP: 67 ; TN: 65 ; FP: 4 ; FN: 4 ; betas:
#> norm2 norm9 norm10 norm12 norm18 norm22 norm33
#> -3.858e-02 5.095e-03 -1.742e-02 -1.475e-02 -3.994e-03 -2.842e-12 -6.076e-03
#> norm36 norm39 norm51
#> 9.110e-03 -2.370e-03 7.990e-02
#> [ reached 'max' / getOption("max.print") -- omitted 23 entries ]
#> Estimation time: 0.4479 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 12 ; Backtraking iters: 348 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7515 ; youden_index: 0.7962 ; aYI: 0 ; sensitivity: 0.9577 ; specificity: 0.9565 ; geometric_mean: 0.9571 ; fdr: 0.04225 ; mcc: 0.9143 ; auc: 0.9824 ; aauc: 0 ; corrclass: 0.9571 ;
#> TP: 68 ; TN: 66 ; FP: 3 ; FN: 3 ; gammas:
#> norm_cov6 norm_cov11 norm_cov13
#> -0.04531 0.04576 -0.03765
#> Estimation time: 0.02253 mins
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 3 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2898 ; youden_index: 0.4611 ; sensitivity: 0.8 ; fdr: 0.07692 ; mcc: 0.7399 ; auc: 0.9611 ; corrclass: 0.8667
#> TP: 24 ; TN: 28 ; FP: 2 ; FN: 6 ; betas:
#> norm2 norm9 norm10 norm12 norm18 norm22 norm33
#> -3.858e-02 5.095e-03 -1.742e-02 -1.475e-02 -3.994e-03 -2.842e-12 -6.076e-03
#> norm36 norm39 norm51 norm52 norm64 norm71 norm79
#> 9.110e-03 -2.370e-03 7.990e-02 -1.211e-01 -1.023e-03 5.925e-03 -1.137e-02
#> norm96 norm98 norm99 norm101 norm102 norm107 norm119
#> -5.576e-03 1.358e-02 -7.360e-03 8.360e-02 -8.914e-02 -1.190e-02 -2.745e-03
#> norm131 norm139 norm142 norm147 norm148 norm149 norm152
#> 9.464e-03 -3.841e-03 -9.452e-04 1.846e-02 -5.418e-12 -1.337e-03 -5.526e-03
#> norm171 norm188 norm192 norm193 c
#> 3.057e-03 -1.079e-02 3.225e-03 1.213e-11 2.277e-02
#> Estimation time: 0.4479 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.6575 ; youden_index: 0.8 ; aYI: 0 ; sensitivity: 0.9333 ; specificity: 0.8667 ; geometric_mean: 0.8994 ; fdr: 0.125 ; mcc: 0.8018 ; auc: 0.9611 ; aauc: 0 ; corrclass: 0.9
#> TP: 28 ; TN: 26 ; FP: 4 ; FN: 2 ; gammas:
#> norm_cov6 norm_cov11 norm_cov13
#> -0.04531 0.04576 -0.03765
#> Estimation time: 0.02253 ; Number of iterations: 12
#>
#>
#>
#> --------------------> experiment n 4 of 5 <----------------------
#> lambda = 0.6082 ; weight: 0.5 ; penalty = SCAD
#> tau = 0.07211 ; weight_g: 0.5 ; penalty_g = L12
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 6 ; Backtraking iters: 640 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3896 ; youden_index: 0.7653 ; sensitivity: 0.9577 ; specificity: 0.8986 ; geometric_mean: 0.9277 ; fdr: 0.09333 ; mcc: 0.8584 ; auc: 0.9845 ; corrclass: 0.9286 ;
#> TP: 68 ; TN: 62 ; FP: 7 ; FN: 3 ; betas:
#> norm2 norm5 norm9 norm18 norm35 norm45 norm51
#> -0.0413841 -0.0025221 0.0016441 -0.0043512 -0.0002928 0.0029710 0.0681066
#> norm52 norm55 norm72
#> -0.1347239 -0.0049972 -0.0016372
#> [ reached 'max' / getOption("max.print") -- omitted 19 entries ]
#> Estimation time: 0.3128 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 16 ; Backtraking iters: 410 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7956 ; youden_index: 0.8359 ; aYI: 0 ; sensitivity: 0.9859 ; specificity: 0.971 ; geometric_mean: 0.9784 ; fdr: 0.02778 ; mcc: 0.9572 ; auc: 0.9845 ; aauc: 0 ; corrclass: 0.9786 ;
#> TP: 70 ; TN: 67 ; FP: 2 ; FN: 1 ; gammas:
#> norm_cov1 norm_cov6 norm_cov19
#> 0.02419 -0.05397 -0.02924
#> Estimation time: 0.03054 mins
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 4 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.1869 ; youden_index: 0.4067 ; sensitivity: 0.9 ; fdr: 0.2059 ; mcc: 0.6727 ; auc: 0.9067 ; corrclass: 0.8333
#> TP: 27 ; TN: 23 ; FP: 7 ; FN: 3 ; betas:
#> norm2 norm5 norm9 norm18 norm35 norm45 norm51
#> -0.0413841 -0.0025221 0.0016441 -0.0043512 -0.0002928 0.0029710 0.0681066
#> norm52 norm55 norm72 norm74 norm78 norm79 norm98
#> -0.1347239 -0.0049972 -0.0016372 -0.0001541 -0.0104261 -0.0269780 0.0333792
#> norm101 norm102 norm103 norm107 norm109 norm120 norm131
#> 0.0764677 -0.0821419 0.0020298 -0.0315187 0.0180655 0.0001175 0.0393312
#> norm139 norm150 norm152 norm162 norm167 norm177 norm183
#> -0.0197371 -0.0032835 -0.0026031 0.0051906 -0.0008973 0.0015815 -0.0012876
#> c
#> -0.0014933
#> Estimation time: 0.3128 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.5096 ; youden_index: 0.7 ; aYI: 0 ; sensitivity: 0.9 ; specificity: 0.7667 ; geometric_mean: 0.8307 ; fdr: 0.2059 ; mcc: 0.6727 ; auc: 0.9067 ; aauc: 0 ; corrclass: 0.8333
#> TP: 27 ; TN: 23 ; FP: 7 ; FN: 3 ; gammas:
#> norm_cov1 norm_cov6 norm_cov19
#> 0.02419 -0.05397 -0.02924
#> Estimation time: 0.03054 ; Number of iterations: 16
#>
#>
#>
#> --------------------> experiment n 5 of 5 <----------------------
#> lambda = 0.6082 ; weight: 0.5 ; penalty = SCAD
#> tau = 0.07211 ; weight_g: 0.5 ; penalty_g = L12
#> mmAPG converged because alpha is below the threshold min_alpha.
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 6 ; Backtraking iters: 653 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.3711 ; youden_index: 0.7315 ; sensitivity: 0.9577 ; specificity: 0.8986 ; geometric_mean: 0.9277 ; fdr: 0.09333 ; mcc: 0.8584 ; auc: 0.9722 ; corrclass: 0.9286 ;
#> TP: 68 ; TN: 62 ; FP: 7 ; FN: 3 ; betas:
#> norm2 norm12 norm22 norm38 norm51 norm52 norm71
#> -0.0300373 -0.0121602 -0.0001606 0.0013078 0.0562744 -0.1354793 0.0004489
#> norm78 norm81 norm84
#> -0.0226522 -0.0043947 -0.0057761
#> [ reached 'max' / getOption("max.print") -- omitted 15 entries ]
#> Estimation time: 0.3214 mins
#>
#>
#> mmAPG converged because the convergence error is below the threshold
#> Final estimation:
#> -> algorithm: Accelerated Proximal Gradient for Nonconvex Programming (APG), the monotone version ;
#> total iters: 10 ; Backtraking iters: 224 ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.7505 ; youden_index: 0.7921 ; aYI: 0 ; sensitivity: 1 ; specificity: 0.9275 ; geometric_mean: 0.9631 ; fdr: 0.06579 ; mcc: 0.9309 ; auc: 0.9722 ; aauc: 0 ; corrclass: 0.9643 ;
#> TP: 71 ; TN: 64 ; FP: 5 ; FN: 0 ; gammas:
#> norm_cov6 norm_cov11 norm_cov13
#> -0.04460 0.03878 -0.02852
#> Estimation time: 0.01947 mins
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: pye_KS_proximal_gradient_method ; simulation n. 5 of 5 ; lambda: 0.6082 ; weight: 0.5 ; penalty: SCAD ; pye_KS: 0.2349 ; youden_index: 0.4322 ; sensitivity: 0.9 ; fdr: 0.129 ; mcc: 0.7671 ; auc: 0.9322 ; corrclass: 0.8833
#> TP: 27 ; TN: 26 ; FP: 4 ; FN: 3 ; betas:
#> norm2 norm12 norm22 norm38 norm51 norm52 norm71
#> -0.0300373 -0.0121602 -0.0001606 0.0013078 0.0562744 -0.1354793 0.0004489
#> norm78 norm81 norm84 norm98 norm101 norm102 norm103
#> -0.0226522 -0.0043947 -0.0057761 0.0173802 0.1497855 -0.1158011 0.0080422
#> norm107 norm118 norm139 norm173 norm181 norm182 norm188
#> -0.0003115 -0.0071719 -0.0062219 0.0012471 0.0012946 0.0004397 -0.0049134
#> norm189 norm193 norm197 c
#> 0.0003312 0.0084408 0.0024009 0.0132393
#> Estimation time: 0.3214 ; Number of iterations: 6
#>
#>
#> -> Results on the TEST SET
#> -> algorithm: covYI_KS_proximal_gradient_method ; tau: 0.07211 ; weight: 0.5 ; penalty: L12 ; covYI_KS: 0.6302 ; youden_index: 0.8 ; aYI: 0 ; sensitivity: 0.8667 ; specificity: 0.9 ; geometric_mean: 0.8832 ; fdr: 0.1034 ; mcc: 0.7671 ; auc: 0.9322 ; aauc: 0 ; corrclass: 0.8833
#> TP: 26 ; TN: 27 ; FP: 3 ; FN: 4 ; gammas:
#> norm_cov6 norm_cov11 norm_cov13
#> -0.04460 0.03878 -0.02852
#> Estimation time: 0.01947 ; Number of iterations: 10
#>
#>
#> Total Simulation Time: 2.055 mins mins.
# 8. View summary of simulation results
colMeans(sim_result$corrclass)
#> train test
#> 0.9329 0.8767
colMeans(sim_result$corrclass_covYI)
#> train test
#> 0.9614 0.9000The estimated coefficients \(\hat{\beta}\) (typically element
x1 in the output list) provide the optimal linear
combination of features. Coefficients successfully shrunk to zero by the
penalty are those deemed irrelevant and are effectively excluded from
the final diagnostic model.
The PYE estimation: Salaroli, C. J., & Pardo, M. C. (2023). PYE: A Penalized Youden Index Estimator for selecting and combining biomarkers in high-dimensional data. Chemometrics and Intelligent Laboratory Systems, 236, 104786. https://doi.org/10.1016/j.chemolab.2023.104786
The covYI Estimation: Salaroli, C. J., & Pardo, M. C. (2026). covYI: A Covariate-Adjusted Penalized Youden Index Estimator for Selecting and Combining Covariates in High-Dimensional Data. Submitted for publication, currently under review.