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Guidance on individualized treatment rule estimation in high dimensions

  • Philippe Boileau EMAIL logo , Ning Leng und Sandrine Dudoit
Veröffentlicht/Copyright: 22. Mai 2025
The International Journal of Biostatistics
Aus der Zeitschrift The International Journal of Biostatistics Band 21 Heft 1

Abstract

Individualized treatment rules, cornerstones of precision medicine, inform patient treatment decisions with the goal of optimizing patient outcomes. These rules are generally unknown functions of patients’ pre-treatment covariates, meaning they must be estimated from clinical or observational study data. Myriad methods have been developed to learn these rules, and these procedures are demonstrably successful in traditional asymptotic settings with moderate number of covariates. The finite-sample performance of these methods in high-dimensional covariate settings, which are increasingly the norm in modern clinical trials, has not been well characterized, however. We perform a comprehensive comparison of state-of-the-art individualized treatment rule estimators, assessing performance on the basis of the estimators’ rule quality, interpretability, and computational efficiency. Sixteen data-generating processes with continuous outcomes and binary treatment assignments are considered, reflecting a diversity of randomized and observational studies. We summarize our findings and provide succinct advice to practitioners needing to estimate individualized treatment rules in high dimensions. Owing to individualized treatment rule estimators’ poor interpretability, we propose a novel pre-treatment covariate filtering procedure based on recent work for uncovering treatment effect modifiers. We show that it improves estimators’ rule quality and interpretability. All code is made publicly available, facilitating modifications and extensions to our simulation study.

Keywords: clinical trials; heterogeneous treatment effects; observational studies; precision medicine
Corresponding author: Philippe Boileau, Department of Epidemiology, Biostatistics and Occupational Health, Department of Medicine, McGill University, Montreal, Canada, E-mail: 

Acknowledgments

Philippe Boileau gratefully acknowledges the support of the Fonds de recherche du Québec – Nature et technologies and the Natural Sciences and Engineering Research Council of Canada.

  1. Research ethics: Not applicable.

  2. Informed consent: Not applicable.

  3. Author contributions: The authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  4. Use of Large Language Models, AI and Machine Learning Tools: Not used to write this manuscript.

  5. Conflict of interest: The authors state no conflict of interest.

  6. Research funding: None declared.

  7. Data availability: Not applicable.

  8. Software availability: All code and software used to generate the results of this manuscript are made publicly available.

Appendix
Table A1:

Simulation results: RCT with sparse linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.99 1.00 0.95 0.95 1.00 0.95
Empirical FDR (%) 54.88 51.82 53.33 5.41 4.60 4.84
Empirical TPR (%) 100.00 100.00 100.00 91.60 99.90 100.00
Empirical TNR (%) 97.28 97.56 97.42 99.88 99.89 99.89
Mean fit time (s) 0.51 0.66 1.08 67.43 75.54 87.46
Plug-in XGBoost Rule quality 0.34 0.54 0.63 0.67 0.79 0.80
Empirical FDR (%) NA NA NA 5.34 4.69 4.77
Empirical TPR (%) NA NA NA 91.50 99.90 100.00
Empirical TNR (%) NA NA NA 99.88 99.89 99.89
Mean fit time (s) 2.18 5.21 14.19 66.17 74.09 86.73
Modified covariates LASSO Rule quality 0.47 0.69 0.84 0.84 0.95 0.92
Empirical FDR (%) 57.72 64.50 71.96 5.42 4.69 4.77
Empirical TPR (%) 40.50 74.40 98.40 91.60 99.90 100.00
Empirical TNR (%) 97.20 95.72 93.91 99.88 99.89 99.89
Mean fit time (s) 0.44 0.86 0.93 65.80 73.78 85.99
Modified covariates XGBoost Rule quality 0.31 0.44 0.58 0.66 0.79 0.82
Empirical FDR (%) NA NA NA 5.35 4.76 4.77
Empirical TPR (%) NA NA NA 91.50 99.90 100.00
Empirical TNR (%) NA NA NA 99.88 99.89 99.89
Mean fit time (s) 10.26 16.26 32.96 74.98 85.32 101.38
Augmented modified covariates LASSO Rule quality 0.98 0.99 0.95 0.94 1.00 0.95
Empirical FDR (%) 77.63 75.85 73.33 5.18 4.69 4.84
Empirical TPR (%) 100.00 100.00 100.00 91.40 99.90 100.00
Empirical TNR (%) 91.72 92.65 93.67 99.89 99.89 99.89
Mean fit time (s) 1.17 1.67 2.81 67.30 75.87 88.60
Augmented modified covariates XGBoost Rule quality 0.83 0.93 0.91 0.88 0.97 0.93
Empirical FDR (%) NA NA NA 5.42 4.60 4.77
Empirical TPR (%) NA NA NA 91.60 99.90 100.00
Empirical TNR (%) NA NA NA 99.88 99.89 99.89
Mean fit time (s) 86.46 114.07 154.01 102.34 122.81 138.76
AIPW-based LASSO Rule quality 0.98 0.99 0.95 0.94 1.00 0.95
Empirical FDR (%) 68.27 66.45 63.50 5.41 4.69 4.69
Empirical TPR (%) 100.00 100.00 100.00 91.50 99.90 100.00
Empirical TNR (%) 94.91 95.28 95.93 99.88 99.89 99.89
Mean fit time (s) 718.99 803.38 977.49 77.62 88.79 106.51
AIPW-based Super Learner Rule quality 0.98 0.99 0.95 0.94 1.00 0.95
Empirical FDR (%) NA NA NA 5.28 4.69 4.75
Empirical TPR (%) NA NA NA 91.50 99.90 100.00
Empirical TNR (%) NA NA NA 99.88 99.89 99.89
Mean fit time (s) 749.06 857.28 1,092.48 88.86 99.19 123.61
Causal Random Forests Rule quality 0.28 0.35 0.35 0.73 0.85 0.83
Empirical FDR (%) NA NA NA 5.53 4.69 4.77
Empirical TPR (%) NA NA NA 91.50 99.90 100.00
Empirical TNR (%) NA NA NA 99.88 99.89 99.89
Mean fit time (s) 11.64 31.43 79.28 72.61 87.78 111.26
Figure A1: Rule quality: the relative rule quality computed over the 100 test set replicates versus sample size, for each combination of DGP. Relative rule quality is defined as the mean ITR outcome divided by the optimal ITR defined in Proposition 1, which is approximated using a Monte Carlo procedure for each DGP. The dotted line corresponds to the idealized relative rule quality.
Figure A1:

Rule quality: the relative rule quality computed over the 100 test set replicates versus sample size, for each combination of DGP. Relative rule quality is defined as the mean ITR outcome divided by the optimal ITR defined in Proposition 1, which is approximated using a Monte Carlo procedure for each DGP. The dotted line corresponds to the idealized relative rule quality.

Figure A2: Accurate interpretability, FDR: empirical FDR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired nominal Type I error rate of 5 %.
Figure A2:

Accurate interpretability, FDR: empirical FDR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired nominal Type I error rate of 5 %.

Figure A3: Accurate interpretability, TPR: empirical TPR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired TPR of 100 %.
Figure A3:

Accurate interpretability, TPR: empirical TPR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired TPR of 100 %.

Figure A4: Accurate interpretability, TNR: empirical TNR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired TNR of 100 %.
Figure A4:

Accurate interpretability, TNR: empirical TNR computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to desired TNR of 100 %.

Figure A5: Computational efficiency: fit time computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to the minimum mean fit time observed in each DGP.
Figure A5:

Computational efficiency: fit time computed over the 100 learning set replicates versus sample size for each combination of DGP. The dotted line corresponds to the minimum mean fit time observed in each DGP.

Table A2:

Simulation results: observational study with sparse linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 1.00 1.00 1.00 0.97 1.00 1.00
Empirical FDR (%) 54.39 53.24 54.06 4.58 4.57 5.25
Empirical TPR (%) 100.00 100.00 100.00 93.30 100.00 100.00
Empirical TNR (%) 97.30 97.44 97.38 99.90 99.89 99.88
Mean fit time (s) 0.54 0.69 1.19 1,285.98 1,498.31 2,196.98
Plug-in XGBoost Rule quality 0.39 0.54 0.68 0.66 0.78 0.85
Empirical FDR (%) NA NA NA 4.32 4.33 5.10
Empirical TPR (%) NA NA NA 93.20 100.00 100.00
Empirical TNR (%) NA NA NA 99.90 99.90 99.88
Mean fit time (s) 2.18 4.89 13.88 1,278.96 1,495.74 2,200.38
Modified covariates LASSO Rule quality 0.42 0.57 0.76 0.84 0.93 0.96
Empirical FDR (%) 62.96 68.30 72.76 4.46 4.33 5.10
Empirical TPR (%) 35.80 72.10 94.40 93.50 100.00 100.00
Empirical TNR (%) 96.17 95.13 93.01 99.90 99.90 99.88
Mean fit time (s) 11.43 15.38 43.12 1,284.50 1,501.51 2,213.89
Modified covariates XGBoost Rule quality 0.37 0.42 0.53 0.62 0.73 0.80
Empirical FDR (%) NA NA NA 4.48 4.33 5.02
Empirical TPR (%) NA NA NA 93.30 100.00 100.00
Empirical TNR (%) NA NA NA 99.90 99.90 99.88
Mean fit time (s) 23.32 34.35 76.25 1,299.77 1,517.25 2,235.83
Augmented modified covariates LASSO Rule quality 0.99 1.00 0.98 0.97 1.00 1.00
Empirical FDR (%) 78.89 76.33 75.00 4.56 4.31 5.25
Empirical TPR (%) 100.00 100.00 97.20 93.40 100.00 100.00
Empirical TNR (%) 90.81 92.56 92.18 99.90 99.90 99.88
Mean fit time (s) 12.05 16.35 45.18 1,286.57 1,503.82 2,212.27
Augmented modified covariates XGBoost Rule quality 0.84 0.94 0.96 0.91 0.97 0.98
Empirical FDR (%) NA NA NA 4.33 4.16 5.02
Empirical TPR (%) NA NA NA 93.50 100.00 100.00
Empirical TNR (%) NA NA NA 99.90 99.90 99.88
Mean fit time (s) 82.00 120.08 200.47 1,316.34 1,552.06 2,283.46
AIPW-based LASSO Rule quality 0.99 1.00 1.00 0.97 1.00 1.00
Empirical FDR (%) 69.89 65.64 66.02 4.30 4.25 5.15
Empirical TPR (%) 100.00 100.00 100.00 93.40 100.00 100.00
Empirical TNR (%) 94.54 95.44 95.51 99.90 99.90 99.88
Mean fit time (s) 747.79 856.14 1,072.22 1,301.05 1,522.12 2,233.43
AIPW-based Super Learner Rule quality 0.99 1.00 1.00 0.96 1.00 1.00
Empirical FDR (%) NA NA NA 4.47 4.50 5.09
Empirical TPR (%) NA NA NA 93.30 100.00 100.00
Empirical TNR (%) NA NA NA 99.90 99.89 99.88
Mean fit time (s) 774.92 909.20 1,187.28 1,310.44 1,534.90 2,244.74
Causal Random Forests Rule quality 0.33 0.32 0.31 0.59 0.74 0.81
Empirical FDR (%) NA NA NA 4.50 4.22 5.32
Empirical TPR (%) NA NA NA 93.10 100.00 100.00
Empirical TNR (%) NA NA NA 99.90 99.90 99.87
Mean fit time (s) 15.72 36.77 94.67 1,290.28 1,514.75 2,236.92
Table A3:

Simulation results: RCT with sparse linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 1.04 1.01 0.98 1.02 1.00 0.98
Empirical FDR (%) 67.91 67.18 62.19 4.60 4.59 5.73
Empirical TPR (%) 78.80 80.00 80.00 67.00 73.90 84.30
Empirical TNR (%) 96.34 96.43 97.20 99.92 99.91 99.88
Mean fit time (s) 0.49 0.61 0.93 65.01 72.45 84.75
Plug-in XGBoost Rule quality 0.77 0.83 0.85 0.92 0.92 0.91
Empirical FDR (%) NA NA NA 4.60 4.59 5.73
Empirical TPR (%) NA NA NA 67.00 73.90 84.30
Empirical TNR (%) NA NA NA 99.92 99.91 99.88
Mean fit time (s) 2.29 5.30 13.97 64.05 70.93 82.80
Modified covariates LASSO Rule quality 0.77 0.88 0.90 0.95 0.97 0.96
Empirical FDR (%) 53.05 58.37 68.66 4.73 4.64 5.73
Empirical TPR (%) 21.90 38.40 54.00 67.00 73.90 84.30
Empirical TNR (%) 98.50 97.85 96.38 99.92 99.91 99.88
Mean fit time (s) 0.34 0.85 6.98 63.89 70.72 82.19
Modified covariates XGBoost Rule quality 0.58 0.72 0.81 0.77 0.84 0.88
Empirical FDR (%) NA NA NA 4.74 4.68 5.73
Empirical TPR (%) NA NA NA 67.00 73.90 84.30
Empirical TNR (%) NA NA NA 99.92 99.91 99.88
Mean fit time (s) 4.94 5.93 10.65 67.24 73.92 85.77
Augmented modified covariates LASSO Rule quality 1.04 1.01 0.98 1.03 1.01 0.98
Empirical FDR (%) 77.21 78.71 77.73 4.51 4.59 5.73
Empirical TPR (%) 74.90 79.70 80.00 67.00 73.90 84.30
Empirical TNR (%) 93.83 93.20 93.41 99.92 99.91 99.88
Mean fit time (s) 1.09 1.50 3.95 64.85 72.83 84.60
Augmented modified covariates XGBoost Rule quality 0.97 0.97 0.96 1.00 0.98 0.97
Empirical FDR (%) NA NA NA 4.76 4.59 5.73
Empirical TPR (%) NA NA NA 67.00 73.90 84.30
Empirical TNR (%) NA NA NA 99.92 99.91 99.88
Mean fit time (s) 19.69 27.49 40.98 72.08 79.84 92.87
AIPW-based LASSO Rule quality 1.04 1.01 0.98 1.03 1.01 0.98
Empirical FDR (%) 72.64 70.42 67.60 4.87 4.59 5.73
Empirical TPR (%) 75.70 79.70 80.00 67.00 73.90 84.30
Empirical TNR (%) 95.07 95.62 96.07 99.92 99.91 99.88
Mean fit time (s) 698.37 781.89 946.50 73.84 84.93 105.01
AIPW-based Super Learner Rule quality 1.04 1.01 0.98 1.03 1.01 0.98
Empirical FDR (%) NA NA NA 4.62 4.59 5.73
Empirical TPR (%) NA NA NA 67.00 73.90 84.30
Empirical TNR (%) NA NA NA 99.92 99.91 99.88
Mean fit time (s) 727.38 838.27 1,097.19 80.72 94.42 121.68
Causal Random Forests Rule quality 0.47 0.57 0.76 0.94 0.92 0.91
Empirical FDR (%) NA NA NA 4.64 4.59 5.73
Empirical TPR (%) NA NA NA 67.00 73.90 84.30
Empirical TNR (%) NA NA NA 99.92 99.91 99.88
Mean fit time (s) 15.01 37.68 83.47 69.82 82.87 106.72
Table A4:

Simulation results: observational study with sparse linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 1.02 1.00 1.00 1.00 1.00 0.99
Empirical FDR (%) 66.68 67.32 59.96 4.09 4.59 4.42
Empirical TPR (%) 79.20 80.00 80.00 67.00 75.90 84.90
Empirical TNR (%) 96.62 96.44 97.38 99.93 99.92 99.91
Mean fit time (s) 0.52 0.64 1.17 1,268.36 1,480.99 2,614.30
Plug-in XGBoost Rule quality 0.76 0.81 0.87 0.89 0.92 0.93
Empirical FDR (%) NA NA NA 4.22 4.53 4.48
Empirical TPR (%) NA NA NA 66.90 76.00 85.20
Empirical TNR (%) NA NA NA 99.93 99.92 99.91
Mean fit time (s) 2.06 5.05 14.30 1,266.69 1,478.83 2,625.31
Modified covariates LASSO Rule quality 0.64 0.76 0.85 0.91 0.94 0.95
Empirical FDR (%) 62.69 59.85 71.13 3.72 4.65 4.35
Empirical TPR (%) 22.70 36.40 55.60 66.90 75.90 84.80
Empirical TNR (%) 97.59 97.54 96.00 99.94 99.91 99.91
Mean fit time (s) 11.43 16.29 91.10 1,271.48 1,485.92 2,640.70
Modified covariates XGBoost Rule quality 0.53 0.62 0.71 0.71 0.75 0.81
Empirical FDR (%) NA NA NA 3.97 4.88 4.41
Empirical TPR (%) NA NA NA 66.80 76.20 84.60
Empirical TNR (%) NA NA NA 99.94 99.91 99.91
Mean fit time (s) 25.78 35.77 114.26 1,279.80 1,505.50 2,652.49
Augmented modified covariates LASSO Rule quality 1.01 1.00 1.00 1.01 1.00 1.00
Empirical FDR (%) 79.01 79.40 76.66 4.09 4.46 4.53
Empirical TPR (%) 75.20 80.00 80.10 66.80 76.00 84.90
Empirical TNR (%) 92.26 92.54 93.71 99.93 99.92 99.91
Mean fit time (s) 12.01 17.02 87.90 1,268.95 1,490.85 2,639.48
Augmented modified covariates XGBoost Rule quality 0.95 0.97 0.97 0.98 0.98 0.98
Empirical FDR (%) NA NA NA 4.22 4.72 4.33
Empirical TPR (%) NA NA NA 66.80 75.70 84.90
Empirical TNR (%) NA NA NA 99.93 99.91 99.91
Mean fit time (s) 60.73 87.24 181.10 1,294.77 1,527.59 2,690.81
AIPW-based LASSO Rule quality 1.02 1.00 1.00 1.01 1.00 1.00
Empirical FDR (%) 75.42 72.76 67.76 3.97 4.58 4.59
Empirical TPR (%) 78.00 79.80 80.00 67.00 75.90 84.70
Empirical TNR (%) 94.48 95.02 95.88 99.94 99.91 99.90
Mean fit time (s) 735.33 848.68 1,220.28 1,280.04 1,505.40 2,635.82
AIPW-based Super Learner Rule quality 1.02 1.00 1.00 1.01 1.00 1.00
Empirical FDR (%) NA NA NA 3.99 4.74 4.34
Empirical TPR (%) NA NA NA 66.80 75.70 84.80
Empirical TNR (%) NA NA NA 99.94 99.91 99.91
Mean fit time (s) 762.44 898.22 1,355.78 1,287.77 1,513.87 2,628.08
Causal Random Forests Rule quality 0.36 0.35 0.38 0.92 0.92 0.92
Empirical FDR (%) NA NA NA 3.72 4.77 4.48
Empirical TPR (%) NA NA NA 67.00 76.00 84.90
Empirical TNR (%) NA NA NA 99.94 99.91 99.91
Mean fit time (s) 17.57 40.15 94.69 1,271.76 1,497.79 2,647.69
Table A5:

Simulation results: RCT with sparse non-linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.72 0.84 0.94 0.50 0.82 0.96
Empirical FDR (%) 61.95 60.18 58.60 12.97 6.63 5.82
Empirical TPR (%) 98.00 100.00 100.00 43.50 92.00 100.00
Empirical TNR (%) 96.43 96.50 96.70 99.82 99.85 99.86
Mean fit time (s) 0.52 0.70 1.15 65.26 72.65 84.65
Plug-in XGBoost Rule quality 0.06 0.07 0.31 0.21 0.41 0.63
Empirical FDR (%) NA NA NA 13.12 6.58 6.04
Empirical TPR (%) NA NA NA 44.10 91.90 100.00
Empirical TNR (%) NA NA NA 99.82 99.85 99.85
Mean fit time (s) 2.21 4.59 12.83 64.29 71.34 83.58
Modified covariates LASSO Rule quality 0.03 0.03 0.20 0.29 0.48 0.69
Empirical FDR (%) 60.20 57.12 62.25 12.75 6.60 6.12
Empirical TPR (%) 7.30 13.70 34.90 44.40 92.10 100.00
Empirical TNR (%) 98.77 98.77 97.54 99.82 99.85 99.85
Mean fit time (s) 0.69 1.81 1.45 64.16 71.11 82.74
Modified covariates XGBoost Rule quality 0.01 −0.02 0.07 0.15 0.23 0.38
Empirical FDR (%) NA NA NA 13.44 6.61 5.97
Empirical TPR (%) NA NA NA 44.00 92.30 100.00
Empirical TNR (%) NA NA NA 99.81 99.85 99.86
Mean fit time (s) 3.59 3.90 6.71 66.92 73.79 85.85
Augmented modified covariates LASSO Rule quality 0.56 0.79 0.92 0.40 0.79 0.95
Empirical FDR (%) 71.30 75.42 74.15 13.58 6.72 5.71
Empirical TPR (%) 84.80 99.90 100.00 42.60 92.00 100.00
Empirical TNR (%) 94.52 92.69 93.35 99.82 99.85 99.86
Mean fit time (s) 1.21 1.99 3.24 65.05 73.01 84.89
Augmented modified covariates XGBoost Rule quality 0.28 0.51 0.70 0.23 0.63 0.84
Empirical FDR (%) NA NA NA 13.77 6.73 5.92
Empirical TPR (%) NA NA NA 44.50 92.00 100.00
Empirical TNR (%) NA NA NA 99.81 99.85 99.86
Mean fit time (s) 8.06 15.47 26.13 67.35 75.95 89.19
AIPW-based LASSO Rule quality 0.59 0.79 0.91 0.38 0.79 0.95
Empirical FDR (%) 69.33 65.56 64.59 13.57 6.39 5.96
Empirical TPR (%) 93.50 99.90 100.00 43.50 91.90 100.00
Empirical TNR (%) 94.91 95.46 95.61 99.82 99.86 99.86
Mean fit time (s) 700.71 784.39 945.12 73.57 86.15 105.23
AIPW-based Super Learner Rule quality 0.59 0.80 0.91 0.37 0.78 0.95
Empirical FDR (%) NA NA NA 13.58 6.52 5.89
Empirical TPR (%) NA NA NA 43.90 92.00 100.00
Empirical TNR (%) NA NA NA 99.82 99.85 99.86
Mean fit time (s) 732.22 846.29 1,072.01 80.19 95.97 123.03
Causal Random Forests Rule quality 0.01 −0.01 0.03 0.28 0.52 0.68
Empirical FDR (%) NA NA NA 12.77 6.53 6.00
Empirical TPR (%) NA NA NA 43.60 92.00 100.00
Empirical TNR (%) NA NA NA 99.82 99.85 99.86
Mean fit time (s) 12.68 31.26 71.44 69.46 84.65 109.05
Table A6:

Simulation results: observational study with sparse non-linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.74 0.89 0.92 0.54 0.88 0.94
Empirical FDR (%) 60.20 61.57 59.33 13.61 8.02 6.84
Empirical TPR (%) 98.00 100.00 100.00 50.40 93.20 100.00
Empirical TNR (%) 96.62 96.32 96.42 99.79 99.81 99.84
Mean fit time (s) 0.56 0.76 1.36 1,288.48 1,498.63 2,214.52
Plug-in XGBoost Rule quality 0.09 0.17 0.30 0.24 0.47 0.59
Empirical FDR (%) NA NA NA 13.34 7.87 7.07
Empirical TPR (%) NA NA NA 50.80 93.50 100.00
Empirical TNR (%) NA NA NA 99.80 99.81 99.83
Mean fit time (s) 2.05 5.14 13.65 1,287.49 1,498.99 2,222.88
Modified covariates LASSO Rule quality 0.05 0.03 0.01 0.21 0.35 0.52
Empirical FDR (%) 69.66 63.56 50.09 13.71 7.96 7.51
Empirical TPR (%) 6.60 14.40 30.00 49.80 93.80 100.00
Empirical TNR (%) 97.88 97.51 97.31 99.79 99.81 99.82
Mean fit time (s) 11.71 16.45 44.60 1,291.44 1,504.89 2,222.43
Modified covariates XGBoost Rule quality 0.01 0.02 −0.02 0.11 0.16 0.16
Empirical FDR (%) NA NA NA 13.32 7.67 7.04
Empirical TPR (%) NA NA NA 51.10 93.60 100.00
Empirical TNR (%) NA NA NA 99.79 99.82 99.83
Mean fit time (s) 21.17 27.25 59.50 1,301.96 1,515.93 2,244.15
Augmented modified covariates LASSO Rule quality 0.51 0.80 0.88 0.35 0.85 0.93
Empirical FDR (%) 73.34 77.93 74.15 13.59 8.24 7.12
Empirical TPR (%) 83.10 99.60 99.80 50.60 93.50 100.00
Empirical TNR (%) 92.94 91.27 92.84 99.80 99.80 99.82
Mean fit time (s) 12.29 16.82 45.95 1,293.00 1,508.69 2,229.23
Augmented modified covariates XGBoost Rule quality 0.26 0.55 0.68 0.20 0.69 0.81
Empirical FDR (%) NA NA NA 12.97 8.04 7.04
Empirical TPR (%) NA NA NA 49.00 93.40 100.00
Empirical TNR (%) NA NA NA 99.80 99.81 99.83
Mean fit time (s) 33.07 57.02 107.41 1,303.81 1,527.92 2,252.42
AIPW-based LASSO Rule quality 0.65 0.85 0.90 0.36 0.86 0.93
Empirical FDR (%) 71.60 69.97 66.02 12.92 8.60 7.17
Empirical TPR (%) 96.10 100.00 100.00 49.20 94.30 100.00
Empirical TNR (%) 94.14 94.55 95.26 99.80 99.80 99.82
Mean fit time (s) 750.51 859.28 1,071.85 1,303.96 1,527.14 2,242.93
AIPW-based Super Learner Rule quality 0.65 0.84 0.90 0.34 0.86 0.93
Empirical FDR (%) NA NA NA 13.69 8.07 7.22
Empirical TPR (%) NA NA NA 49.60 93.70 100.00
Empirical TNR (%) NA NA NA 99.79 99.81 99.82
Mean fit time (s) 779.20 916.65 1,201.11 1,313.50 1,537.09 2,262.57
Causal Random Forests Rule quality 0.01 −0.01 −0.04 0.18 0.38 0.49
Empirical FDR (%) NA NA NA 13.42 8.06 7.24
Empirical TPR (%) NA NA NA 49.20 93.20 100.00
Empirical TNR (%) NA NA NA 99.79 99.81 99.83
Mean fit time (s) 14.93 33.12 82.13 1,293.04 1,518.81 2,252.98
Table A7:

Simulation results: RCT with sparse non-linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.86 0.84 0.93 0.86 0.84 0.94
Empirical FDR (%) 75.47 74.26 74.03 10.65 7.68 8.24
Empirical TPR (%) 47.50 63.00 74.80 48.80 67.20 74.70
Empirical TNR (%) 96.77 95.93 95.32 99.86 99.86 99.84
Mean fit time (s) 0.51 0.67 1.03 64.74 72.23 85.61
Plug-in XGBoost Rule quality 0.29 0.37 0.60 0.67 0.68 0.78
Empirical FDR (%) NA NA NA 11.23 7.02 8.41
Empirical TPR (%) NA NA NA 48.50 66.90 74.70
Empirical TNR (%) NA NA NA 99.86 99.87 99.84
Mean fit time (s) 1.98 4.80 13.95 63.80 70.77 82.57
Modified covariates LASSO Rule quality 0.28 0.29 0.56 0.57 0.61 0.80
Empirical FDR (%) 55.51 56.47 58.35 10.83 7.25 8.46
Empirical TPR (%) 8.10 13.50 24.10 48.20 67.00 74.60
Empirical TNR (%) 98.83 98.82 98.31 99.85 99.87 99.84
Mean fit time (s) 0.76 2.21 9.11 63.66 70.63 81.69
Modified covariates XGBoost Rule quality 0.20 0.11 0.31 0.36 0.32 0.55
Empirical FDR (%) NA NA NA 10.11 7.74 8.35
Empirical TPR (%) NA NA NA 48.70 67.10 74.60
Empirical TNR (%) NA NA NA 99.86 99.86 99.84
Mean fit time (s) 3.56 4.22 7.44 66.41 73.30 85.21
Augmented modified covariates LASSO Rule quality 0.84 0.84 0.94 0.83 0.88 0.97
Empirical FDR (%) 66.22 71.88 72.49 10.69 7.35 8.44
Empirical TPR (%) 42.40 61.60 69.80 48.30 66.90 74.80
Empirical TNR (%) 96.75 95.48 95.37 99.86 99.87 99.84
Mean fit time (s) 1.20 1.89 7.98 64.70 72.54 84.36
Augmented modified covariates XGBoost Rule quality 0.76 0.72 0.83 0.75 0.79 0.88
Empirical FDR (%) NA NA NA 10.55 7.60 8.53
Empirical TPR (%) NA NA NA 48.80 67.10 74.70
Empirical TNR (%) NA NA NA 99.85 99.86 99.84
Mean fit time (s) 7.64 10.37 17.87 67.10 74.71 87.00
AIPW-based LASSO Rule quality 0.85 0.84 0.94 0.83 0.89 0.97
Empirical FDR (%) 70.39 65.97 65.27 10.09 7.55 8.43
Empirical TPR (%) 52.50 63.40 70.30 48.10 67.30 74.60
Empirical TNR (%) 96.57 96.75 96.63 99.86 99.86 99.84
Mean fit time (s) 695.05 778.12 945.00 72.96 84.03 102.92
AIPW-based Super Learner Rule quality 0.85 0.83 0.92 0.83 0.86 0.94
Empirical FDR (%) NA NA NA 9.66 7.39 8.31
Empirical TPR (%) NA NA NA 48.10 66.90 74.70
Empirical TNR (%) NA NA NA 99.87 99.87 99.84
Mean fit time (s) 725.98 839.18 1,168.98 79.53 93.16 119.29
Causal Random Forests Rule quality 0.15 0.10 0.42 0.77 0.74 0.81
Empirical FDR (%) NA NA NA 10.51 7.82 8.65
Empirical TPR (%) NA NA NA 48.60 67.00 74.60
Empirical TNR (%) NA NA NA 99.86 99.86 99.84
Mean fit time (s) 13.52 34.10 80.81 69.20 82.56 105.42
Table A8:

Simulation results: observational study with sparse non-linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.80 0.78 0.87 0.79 0.79 0.88
Empirical FDR (%) 74.10 73.25 71.86 11.16 7.73 7.34
Empirical TPR (%) 48.60 63.20 74.40 50.70 68.60 75.10
Empirical TNR (%) 96.88 96.11 95.79 99.83 99.86 99.86
Mean fit time (s) 0.54 0.72 1.31 1,285.60 1,501.50 2,708.36
Plug-in XGBoost Rule quality 0.32 0.41 0.56 0.61 0.63 0.73
Empirical FDR (%) NA NA NA 10.83 8.01 7.11
Empirical TPR (%) NA NA NA 51.50 68.10 74.80
Empirical TNR (%) NA NA NA 99.84 99.85 99.87
Mean fit time (s) 2.19 4.66 13.72 1,284.61 1,502.65 2,716.66
Modified covariates LASSO Rule quality 0.13 0.08 0.18 0.39 0.40 0.52
Empirical FDR (%) 60.29 63.69 55.52 12.06 8.37 7.10
Empirical TPR (%) 9.10 15.80 22.80 51.00 68.30 75.30
Empirical TNR (%) 98.14 97.44 98.27 99.82 99.85 99.87
Mean fit time (s) 11.92 17.43 95.63 1,286.64 1,507.44 2,709.83
Modified covariates XGBoost Rule quality 0.12 0.02 0.14 0.24 0.18 0.29
Empirical FDR (%) NA NA NA 11.25 8.11 7.21
Empirical TPR (%) NA NA NA 51.00 68.70 74.90
Empirical TNR (%) NA NA NA 99.84 99.85 99.86
Mean fit time (s) 22.86 30.89 109.89 1,295.73 1,518.52 2,725.34
Augmented modified covariates LASSO Rule quality 0.72 0.78 0.88 0.77 0.84 0.92
Empirical FDR (%) 74.77 73.13 75.10 11.36 8.36 6.87
Empirical TPR (%) 46.40 60.40 71.00 51.20 68.40 75.00
Empirical TNR (%) 94.23 95.01 94.74 99.83 99.84 99.87
Mean fit time (s) 12.30 17.69 94.37 1,288.82 1,512.12 2,706.51
Augmented modified covariates XGBoost Rule quality 0.68 0.68 0.78 0.69 0.74 0.82
Empirical FDR (%) NA NA NA 12.15 8.26 7.25
Empirical TPR (%) NA NA NA 50.80 68.30 75.00
Empirical TNR (%) NA NA NA 99.82 99.85 99.86
Mean fit time (s) 33.46 44.63 131.79 1,304.42 1,524.95 2,726.26
AIPW-based LASSO Rule quality 0.79 0.79 0.89 0.77 0.84 0.92
Empirical FDR (%) 72.21 69.47 68.88 11.18 8.77 7.08
Empirical TPR (%) 53.70 64.50 70.90 51.50 68.50 75.10
Empirical TNR (%) 95.99 96.25 96.14 99.83 99.84 99.87
Mean fit time (s) 749.21 865.50 1,247.68 1,303.10 1,525.92 2,690.73
AIPW-based Super Learner Rule quality 0.79 0.79 0.87 0.76 0.83 0.89
Empirical FDR (%) NA NA NA 10.83 8.28 6.97
Empirical TPR (%) NA NA NA 50.60 68.20 75.20
Empirical TNR (%) NA NA NA 99.84 99.85 99.87
Mean fit time (s) 779.23 924.56 1,452.27 1,312.20 1,537.01 2,679.75
Causal Random Forests Rule quality 0.06 −0.06 −0.01 0.69 0.69 0.76
Empirical FDR (%) NA NA NA 11.42 8.37 7.21
Empirical TPR (%) NA NA NA 51.20 68.20 75.30
Empirical TNR (%) NA NA NA 99.83 99.85 99.86
Mean fit time (s) 15.66 37.75 93.58 1,290.05 1,519.02 2,705.30
Table A9:

Simulation results: RCT with non-sparse linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.58 0.92 0.93 0.45 0.69 0.87
Empirical FDR (%) 47.50 43.85 39.97 18.48 7.14 3.58
Empirical TPR (%) 49.82 99.68 100.00 4.68 33.04 82.70
Empirical TNR (%) 94.72 91.23 92.45 99.80 99.72 99.65
Mean fit time (s) 0.55 0.76 1.15 64.45 71.90 83.55
Plug-in XGBoost Rule quality 0.21 0.33 0.41 0.38 0.45 0.49
Empirical FDR (%) NA NA NA 16.82 7.28 3.61
Empirical TPR (%) NA NA NA 4.66 33.08 82.74
Empirical TNR (%) NA NA NA 99.81 99.72 99.64
Mean fit time (s) 2.05 4.60 13.69 63.43 70.61 83.51
Modified covariates LASSO Rule quality 0.43 0.48 0.48 0.42 0.59 0.71
Empirical FDR (%) 52.64 39.92 51.37 19.90 7.25 3.64
Empirical TPR (%) 4.72 7.60 21.74 4.88 32.96 82.80
Empirical TNR (%) 98.20 98.25 95.94 99.78 99.72 99.64
Mean fit time (s) 0.65 2.10 1.21 63.30 70.41 81.51
Modified covariates XGBoost Rule quality 0.44 0.48 0.43 0.44 0.52 0.56
Empirical FDR (%) NA NA NA 17.79 7.12 3.60
Empirical TPR (%) NA NA NA 4.84 33.04 82.68
Empirical TNR (%) NA NA NA 99.79 99.72 99.64
Mean fit time (s) 3.00 4.19 6.49 65.58 72.94 85.37
Augmented modified covariates LASSO Rule quality 0.54 0.84 0.92 0.43 0.62 0.84
Empirical FDR (%) 52.67 64.58 66.10 17.92 7.11 3.60
Empirical TPR (%) 27.84 93.98 100.00 4.74 33.16 82.68
Empirical TNR (%) 94.28 80.28 77.87 99.79 99.72 99.64
Mean fit time (s) 1.34 2.14 3.20 64.45 72.22 83.43
Augmented modified covariates XGBoost Rule quality 0.47 0.65 0.77 0.45 0.54 0.75
Empirical FDR (%) NA NA NA 17.43 7.05 3.55
Empirical TPR (%) NA NA NA 4.74 33.02 82.74
Empirical TNR (%) NA NA NA 99.78 99.73 99.65
Mean fit time (s) 5.47 29.67 98.29 65.78 73.61 93.18
AIPW-based LASSO Rule quality 0.59 0.90 0.92 0.44 0.62 0.84
Empirical FDR (%) 57.47 55.96 52.43 18.89 7.21 3.55
Empirical TPR (%) 52.16 99.12 100.00 5.04 33.18 82.72
Empirical TNR (%) 91.27 85.78 87.49 99.76 99.72 99.65
Mean fit time (s) 690.79 771.17 926.92 71.76 89.88 129.04
AIPW-based Super Learner Rule quality 0.59 0.90 0.92 0.42 0.62 0.84
Empirical FDR (%) NA NA NA 16.44 7.03 3.64
Empirical TPR (%) NA NA NA 5.08 33.12 82.80
Empirical TNR (%) NA NA NA 99.79 99.72 99.64
Mean fit time (s) 723.86 835.28 1,050.34 77.39 102.05 161.40
Causal Random Forests Rule quality 0.44 0.49 0.45 0.45 0.54 0.49
Empirical FDR (%) NA NA NA 18.12 7.18 3.56
Empirical TPR (%) NA NA NA 4.84 33.16 82.76
Empirical TNR (%) NA NA NA 99.80 99.72 99.65
Mean fit time (s) 13.88 30.26 67.22 68.61 86.01 123.85
Table A10:

Simulation results: observational study with non-sparse linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.64 0.93 0.97 0.49 0.70 0.91
Empirical FDR (%) 46.18 43.92 40.70 20.28 5.84 4.31
Empirical TPR (%) 50.94 99.42 100.00 4.26 35.76 83.50
Empirical TNR (%) 95.02 91.23 92.24 99.81 99.73 99.57
Mean fit time (s) 0.56 0.78 1.32 1,244.32 1,450.47 2,129.83
Plug-in XGBoost Rule quality 0.31 0.37 0.44 0.45 0.49 0.53
Empirical FDR (%) NA NA NA 21.00 5.69 4.22
Empirical TPR (%) NA NA NA 4.54 35.60 83.44
Empirical TNR (%) NA NA NA 99.78 99.73 99.58
Mean fit time (s) 2.16 5.05 13.83 1,244.99 1,446.94 2,138.34
Modified covariates LASSO Rule quality 0.49 0.48 0.51 0.50 0.58 0.73
Empirical FDR (%) 53.91 53.01 53.24 22.05 6.22 4.35
Empirical TPR (%) 4.16 7.94 24.36 4.46 35.36 83.44
Empirical TNR (%) 97.54 97.56 95.19 99.79 99.71 99.57
Mean fit time (s) 11.32 15.79 42.52 1,247.94 1,447.56 2,139.97
Modified covariates XGBoost Rule quality 0.49 0.48 0.49 0.50 0.54 0.58
Empirical FDR (%) NA NA NA 20.34 5.69 4.45
Empirical TPR (%) NA NA NA 4.36 35.30 83.48
Empirical TNR (%) NA NA NA 99.81 99.74 99.56
Mean fit time (s) 21.67 28.57 62.41 1,256.05 1,462.82 2,162.50
Augmented modified covariates LASSO Rule quality 0.56 0.84 0.95 0.51 0.62 0.88
Empirical FDR (%) 57.28 65.98 67.46 22.56 6.00 4.41
Empirical TPR (%) 26.32 93.62 100.00 4.48 35.52 83.14
Empirical TNR (%) 94.55 78.76 75.88 99.79 99.72 99.56
Mean fit time (s) 11.95 16.38 44.77 1,250.36 1,449.69 2,140.68
Augmented modified covariates XGBoost Rule quality 0.52 0.64 0.81 0.50 0.57 0.80
Empirical FDR (%) NA NA NA 21.05 5.91 4.36
Empirical TPR (%) NA NA NA 4.78 35.60 83.48
Empirical TNR (%) NA NA NA 99.78 99.73 99.57
Mean fit time (s) 25.82 89.09 236.97 1,258.51 1,466.26 2,194.13
AIPW-based LASSO Rule quality 0.64 0.91 0.96 0.51 0.63 0.88
Empirical FDR (%) 58.46 58.06 53.37 21.07 5.82 4.46
Empirical TPR (%) 56.00 98.96 100.00 4.46 35.28 83.48
Empirical TNR (%) 90.68 84.54 86.95 99.79 99.73 99.56
Mean fit time (s) 719.46 817.62 1,027.26 1,257.46 1,470.86 2,199.02
AIPW-based Super Learner Rule quality 0.63 0.91 0.96 0.49 0.62 0.88
Empirical FDR (%) NA NA NA 22.45 5.81 4.35
Empirical TPR (%) NA NA NA 4.56 35.10 83.10
Empirical TNR (%) NA NA NA 99.78 99.74 99.57
Mean fit time (s) 748.58 875.43 1,154.68 1,259.41 1,483.30 2,220.65
Causal Random Forests Rule quality 0.51 0.49 0.49 0.51 0.49 0.49
Empirical FDR (%) NA NA NA 22.46 5.65 4.44
Empirical TPR (%) NA NA NA 4.50 35.56 83.40
Empirical TNR (%) NA NA NA 99.78 99.74 99.56
Mean fit time (s) 15.10 33.61 79.68 1,244.90 1,467.12 2,185.54
Table A11:

Simulation results: RCT with non-sparse linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.93 0.95 0.95 0.87 0.91 0.93
Empirical FDR (%) 47.78 46.78 46.36 7.62 3.96 3.12
Empirical TPR (%) 42.16 61.14 73.58 26.80 51.36 67.10
Empirical TNR (%) 95.57 93.80 92.82 99.75 99.75 99.75
Mean fit time (s) 0.53 0.69 1.09 64.13 71.02 84.58
Plug-in XGBoost Rule quality 0.32 0.45 0.57 0.60 0.64 0.69
Empirical FDR (%) NA NA NA 7.31 3.92 3.12
Empirical TPR (%) NA NA NA 26.92 51.32 67.10
Empirical TNR (%) NA NA NA 99.75 99.75 99.75
Mean fit time (s) 2.10 4.91 14.47 63.25 70.53 83.11
Modified covariates LASSO Rule quality 0.55 0.62 0.73 0.73 0.78 0.84
Empirical FDR (%) 47.74 50.53 51.99 7.55 3.89 3.13
Empirical TPR (%) 5.72 13.80 25.38 26.94 51.26 67.02
Empirical TNR (%) 98.39 97.29 95.92 99.74 99.76 99.75
Mean fit time (s) 0.65 1.81 8.31 63.07 70.43 81.39
Modified covariates XGBoost Rule quality 0.54 0.52 0.58 0.65 0.68 0.73
Empirical FDR (%) NA NA NA 7.60 4.07 3.07
Empirical TPR (%) NA NA NA 26.84 51.44 67.08
Empirical TNR (%) NA NA NA 99.74 99.74 99.76
Mean fit time (s) 3.00 4.61 9.14 65.59 72.82 84.71
Augmented modified covariates LASSO Rule quality 0.88 0.93 0.94 0.80 0.89 0.94
Empirical FDR (%) 60.10 63.63 66.83 7.73 4.03 3.12
Empirical TPR (%) 35.88 57.70 69.82 26.88 51.34 67.06
Empirical TNR (%) 93.01 88.26 83.55 99.74 99.75 99.75
Mean fit time (s) 1.27 2.00 6.20 64.33 71.79 82.71
Augmented modified covariates XGBoost Rule quality 0.75 0.84 0.87 0.75 0.83 0.88
Empirical FDR (%) NA NA NA 7.60 3.91 3.12
Empirical TPR (%) NA NA NA 27.46 51.36 67.10
Empirical TNR (%) NA NA NA 99.74 99.75 99.75
Mean fit time (s) 11.03 27.33 57.07 66.36 74.97 90.38
AIPW-based LASSO Rule quality 0.91 0.94 0.94 0.81 0.90 0.94
Empirical FDR (%) 55.98 55.23 56.07 7.78 3.94 3.09
Empirical TPR (%) 44.00 60.26 71.14 26.84 51.22 67.08
Empirical TNR (%) 93.16 91.40 89.51 99.74 99.75 99.76
Mean fit time (s) 690.17 770.09 930.39 75.74 93.21 121.72
AIPW-based Super Learner Rule quality 0.91 0.94 0.94 0.81 0.90 0.93
Empirical FDR (%) NA NA NA 7.73 3.84 3.09
Empirical TPR (%) NA NA NA 27.10 51.44 67.08
Empirical TNR (%) NA NA NA 99.74 99.76 99.76
Mean fit time (s) 723.10 833.32 1,115.03 83.23 107.56 150.30
Causal Random Forests Rule quality 0.55 0.52 0.50 0.72 0.72 0.73
Empirical FDR (%) NA NA NA 7.93 3.85 3.12
Empirical TPR (%) NA NA NA 27.12 51.36 67.08
Empirical TNR (%) NA NA NA 99.73 99.76 99.75
Mean fit time (s) 14.71 32.72 79.30 71.31 89.65 124.53
Table A12:

Simulation results: observational study with non-sparse linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.91 0.92 1.00 0.86 0.89 0.98
Empirical FDR (%) 47.76 47.30 45.99 6.55 4.92 3.61
Empirical TPR (%) 43.06 60.40 73.98 29.50 51.86 67.64
Empirical TNR (%) 95.46 93.80 92.87 99.75 99.68 99.71
Mean fit time (s) 0.55 0.71 1.30 1,245.84 1,454.06 2,593.35
Plug-in XGBoost Rule quality 0.40 0.49 0.64 0.58 0.61 0.74
Empirical FDR (%) NA NA NA 7.07 4.72 3.61
Empirical TPR (%) NA NA NA 29.30 51.70 67.66
Empirical TNR (%) NA NA NA 99.72 99.70 99.71
Mean fit time (s) 2.25 4.48 13.69 1,243.62 1,446.91 2,596.23
Modified covariates LASSO Rule quality 0.55 0.54 0.68 0.69 0.73 0.85
Empirical FDR (%) 60.98 48.19 53.87 6.43 4.89 3.62
Empirical TPR (%) 7.52 12.16 24.78 29.06 52.02 67.70
Empirical TNR (%) 97.39 97.19 95.81 99.76 99.68 99.71
Mean fit time (s) 11.46 17.08 91.95 1,250.49 1,457.11 2,599.80
Modified covariates XGBoost Rule quality 0.53 0.51 0.62 0.62 0.62 0.73
Empirical FDR (%) NA NA NA 6.26 4.90 3.60
Empirical TPR (%) NA NA NA 29.24 51.90 67.52
Empirical TNR (%) NA NA NA 99.76 99.68 99.71
Mean fit time (s) 23.46 31.15 112.34 1,256.65 1,468.72 2,624.43
Augmented modified covariates LASSO Rule quality 0.83 0.90 0.99 0.78 0.87 0.98
Empirical FDR (%) 60.93 64.53 67.19 6.77 4.74 3.61
Empirical TPR (%) 36.20 56.52 69.42 29.84 51.90 67.64
Empirical TNR (%) 92.36 87.72 83.32 99.73 99.70 99.71
Mean fit time (s) 12.07 17.10 89.35 1,247.60 1,457.27 2,605.49
Augmented modified covariates XGBoost Rule quality 0.72 0.81 0.92 0.72 0.81 0.93
Empirical FDR (%) NA NA NA 6.49 4.92 3.56
Empirical TPR (%) NA NA NA 29.18 51.98 67.66
Empirical TNR (%) NA NA NA 99.75 99.68 99.71
Mean fit time (s) 45.71 86.56 212.64 1,258.50 1,478.16 2,654.82
AIPW-based LASSO Rule quality 0.89 0.92 0.99 0.80 0.87 0.98
Empirical FDR (%) 57.39 56.95 56.70 6.54 4.81 3.63
Empirical TPR (%) 46.14 60.86 72.20 29.68 52.04 67.64
Empirical TNR (%) 92.66 90.78 89.08 99.74 99.69 99.70
Mean fit time (s) 719.27 824.58 1,195.33 1,257.30 1,481.06 2,634.83
AIPW-based Super Learner Rule quality 0.90 0.92 0.99 0.80 0.88 0.98
Empirical FDR (%) NA NA NA 7.17 5.03 3.60
Empirical TPR (%) NA NA NA 29.70 51.96 67.70
Empirical TNR (%) NA NA NA 99.73 99.68 99.71
Mean fit time (s) 747.26 881.33 1,357.61 1,266.63 1,488.79 2,636.36
Causal Random Forests Rule quality 0.53 0.49 0.56 0.60 0.58 0.68
Empirical FDR (%) NA NA NA 6.41 4.90 3.62
Empirical TPR (%) NA NA NA 28.96 52.12 67.86
Empirical TNR (%) NA NA NA 99.76 99.68 99.70
Mean fit time (s) 15.71 35.59 90.24 1,250.24 1,470.02 2,642.60
Table A13:

Simulation results: RCT with non-sparse non-linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.24 0.56 0.88 0.06 0.27 0.62
Empirical FDR (%) 49.56 45.71 44.93 23.52 14.31 7.42
Empirical TPR (%) 32.00 76.30 99.42 2.48 11.18 47.74
Empirical TNR (%) 96.16 92.64 90.86 99.79 99.76 99.56
Mean fit time (s) 0.55 0.78 1.36 65.97 74.40 86.94
Plug-in XGBoost Rule quality −0.01 0.05 0.10 0.02 0.11 0.20
Empirical FDR (%) NA NA NA 19.73 13.22 7.43
Empirical TPR (%) NA NA NA 2.28 11.20 47.88
Empirical TNR (%) NA NA NA 99.84 99.78 99.55
Mean fit time (s) 1.98 4.86 13.89 64.78 72.69 85.90
Modified covariates LASSO Rule quality 0.01 0.06 0.11 0.03 0.17 0.36
Empirical FDR (%) 55.94 55.01 46.40 17.73 13.05 7.43
Empirical TPR (%) 2.40 2.70 4.94 2.40 10.92 47.60
Empirical TNR (%) 98.68 99.00 98.69 99.86 99.78 99.56
Mean fit time (s) 0.70 2.34 2.62 64.78 72.51 85.18
Modified covariates XGBoost Rule quality −0.02 0.06 0.10 0.00 0.09 0.25
Empirical FDR (%) NA NA NA 21.94 12.43 7.37
Empirical TPR (%) NA NA NA 2.46 11.38 47.90
Empirical TNR (%) NA NA NA 99.83 99.78 99.56
Mean fit time (s) 10.02 11.86 17.79 72.03 81.80 96.96
Augmented modified covariates LASSO Rule quality 0.08 0.42 0.80 0.03 0.18 0.49
Empirical FDR (%) 52.84 56.62 64.47 22.28 13.53 7.35
Empirical TPR (%) 11.92 52.34 96.80 2.50 11.26 47.52
Empirical TNR (%) 97.35 90.48 79.80 99.82 99.78 99.56
Mean fit time (s) 1.49 2.40 3.66 65.77 75.59 88.95
Augmented modified covariates XGBoost Rule quality 0.07 0.23 0.51 0.00 0.11 0.36
Empirical FDR (%) NA NA NA 22.09 13.06 7.41
Empirical TPR (%) NA NA NA 2.50 11.32 47.62
Empirical TNR (%) NA NA NA 99.81 99.77 99.56
Mean fit time (s) 15.59 23.18 75.12 72.71 81.47 100.49
AIPW-based LASSO Rule quality 0.19 0.49 0.81 0.04 0.18 0.50
Empirical FDR (%) 54.44 54.00 53.99 21.95 14.37 7.36
Empirical TPR (%) 27.70 69.78 98.24 2.50 10.98 48.02
Empirical TNR (%) 94.94 90.29 86.88 99.80 99.77 99.56
Mean fit time (s) 709.70 796.43 970.79 75.40 88.13 117.51
AIPW-based Super Learner Rule quality 0.22 0.49 0.81 0.03 0.18 0.53
Empirical FDR (%) NA NA NA 20.33 12.28 7.33
Empirical TPR (%) NA NA NA 2.44 11.20 47.84
Empirical TNR (%) NA NA NA 99.83 99.79 99.56
Mean fit time (s) 742.66 856.73 1,098.70 81.30 98.50 143.29
Causal Random Forests Rule quality 0.00 0.05 0.10 0.03 0.14 0.35
Empirical FDR (%) NA NA NA 19.93 13.72 7.59
Empirical TPR (%) NA NA NA 2.42 11.06 48.04
Empirical TNR (%) NA NA NA 99.83 99.78 99.54
Mean fit time (s) 13.87 29.14 68.09 72.10 86.70 127.29
Table A14:

Simulation results: observational study with non-sparse non-linear outcome model and identity covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.28 0.51 0.74 0.10 0.27 0.53
Empirical FDR (%) 47.71 44.51 45.48 27.09 17.74 8.42
Empirical TPR (%) 32.66 75.22 99.26 3.12 17.14 52.74
Empirical TNR (%) 96.41 92.99 90.64 99.77 99.52 99.42
Mean fit time (s) 0.56 0.79 1.37 1,264.10 1,464.83 2,152.27
Plug-in XGBoost Rule quality 0.02 0.01 −0.01 0.05 0.08 0.06
Empirical FDR (%) NA NA NA 26.45 18.62 8.26
Empirical TPR (%) NA NA NA 3.14 17.08 52.16
Empirical TNR (%) NA NA NA 99.76 99.52 99.44
Mean fit time (s) 2.26 4.65 14.03 1,262.08 1,458.83 2,154.81
Modified covariates LASSO Rule quality 0.02 0.00 −0.04 0.05 0.09 0.11
Empirical FDR (%) 59.57 59.08 46.13 27.80 18.13 8.39
Empirical TPR (%) 2.22 2.96 6.40 2.96 16.68 52.76
Empirical TNR (%) 98.30 98.36 97.51 99.77 99.51 99.42
Mean fit time (s) 11.50 16.41 43.46 1,264.86 1,465.85 2,166.97
Modified covariates XGBoost Rule quality 0.01 −0.01 −0.05 0.02 0.04 0.01
Empirical FDR (%) NA NA NA 30.73 17.68 8.60
Empirical TPR (%) NA NA NA 2.96 16.86 53.26
Empirical TNR (%) NA NA NA 99.76 99.54 99.41
Mean fit time (s) 20.91 26.58 57.12 1,271.27 1,474.22 2,177.19
Augmented modified covariates LASSO Rule quality 0.14 0.34 0.63 0.09 0.13 0.34
Empirical FDR (%) 59.12 60.10 65.89 28.96 18.57 8.45
Empirical TPR (%) 12.56 49.62 93.10 3.34 16.68 53.16
Empirical TNR (%) 96.57 90.51 79.93 99.73 99.50 99.42
Mean fit time (s) 12.24 16.68 45.01 1,260.81 1,465.51 2,168.90
Augmented modified covariates XGBoost Rule quality 0.07 0.19 0.38 0.05 0.06 0.17
Empirical FDR (%) NA NA NA 29.46 17.87 8.31
Empirical TPR (%) NA NA NA 3.04 17.04 52.44
Empirical TNR (%) NA NA NA 99.75 99.54 99.43
Mean fit time (s) 26.00 39.51 121.43 1,269.92 1,473.57 2,183.60
AIPW-based LASSO Rule quality 0.23 0.46 0.69 0.06 0.13 0.36
Empirical FDR (%) 57.64 55.89 54.84 29.34 17.82 8.33
Empirical TPR (%) 32.78 74.06 98.22 2.90 16.62 52.88
Empirical TNR (%) 93.88 89.08 86.49 99.78 99.53 99.42
Mean fit time (s) 729.03 829.63 1,045.16 1,268.48 1,483.21 2,209.84
AIPW-based Super Learner Rule quality 0.25 0.46 0.69 0.06 0.12 0.34
Empirical FDR (%) NA NA NA 31.00 19.25 8.55
Empirical TPR (%) NA NA NA 3.34 16.96 53.18
Empirical TNR (%) NA NA NA 99.72 99.48 99.41
Mean fit time (s) 757.21 884.50 1,166.49 1,274.90 1,494.74 2,225.23
Causal Random Forests Rule quality 0.01 −0.02 −0.06 0.02 0.02 −0.03
Empirical FDR (%) NA NA NA 28.34 18.25 8.53
Empirical TPR (%) NA NA NA 3.10 16.24 53.90
Empirical TNR (%) NA NA NA 99.74 99.53 99.40
Mean fit time (s) 14.96 33.27 78.61 1,259.24 1,473.02 2,204.43
Table A15:

Simulation results: RCT with non-sparse non-linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.68 0.73 0.83 0.57 0.70 0.80
Empirical FDR (%) 50.24 47.74 49.22 11.53 6.96 5.59
Empirical TPR (%) 30.84 44.68 57.02 13.72 36.56 57.18
Empirical TNR (%) 96.33 95.26 93.64 99.76 99.67 99.61
Mean fit time (s) 0.52 0.70 1.06 64.46 72.09 84.93
Plug-in XGBoost Rule quality 0.16 0.00 0.12 0.32 0.27 0.38
Empirical FDR (%) NA NA NA 11.61 6.67 5.63
Empirical TPR (%) NA NA NA 13.62 36.66 57.14
Empirical TNR (%) NA NA NA 99.74 99.69 99.60
Mean fit time (s) 2.07 4.19 13.06 63.38 70.78 81.83
Modified covariates LASSO Rule quality 0.17 0.09 0.23 0.39 0.36 0.53
Empirical FDR (%) 55.78 57.79 48.19 11.11 6.35 5.63
Empirical TPR (%) 2.36 4.86 11.20 13.56 36.50 57.04
Empirical TNR (%) 98.86 98.47 97.72 99.77 99.70 99.60
Mean fit time (s) 0.86 2.82 15.14 63.26 70.11 80.43
Modified covariates XGBoost Rule quality 0.15 0.04 0.11 0.30 0.25 0.36
Empirical FDR (%) NA NA NA 11.26 6.67 5.58
Empirical TPR (%) NA NA NA 13.58 36.56 57.26
Empirical TNR (%) NA NA NA 99.76 99.70 99.60
Mean fit time (s) 3.36 4.35 7.10 65.68 73.23 84.44
Augmented modified covariates LASSO Rule quality 0.55 0.69 0.80 0.44 0.59 0.78
Empirical FDR (%) 52.56 58.05 62.54 10.81 6.69 5.58
Empirical TPR (%) 20.72 39.60 53.12 14.00 36.28 57.18
Empirical TNR (%) 96.06 93.01 89.37 99.75 99.69 99.61
Mean fit time (s) 1.30 2.33 8.29 64.37 72.30 83.50
Augmented modified covariates XGBoost Rule quality 0.40 0.50 0.64 0.34 0.48 0.66
Empirical FDR (%) NA NA NA 11.62 6.79 5.41
Empirical TPR (%) NA NA NA 13.92 36.20 57.22
Empirical TNR (%) NA NA NA 99.75 99.69 99.62
Mean fit time (s) 7.43 12.51 25.32 66.30 74.08 86.95
AIPW-based LASSO Rule quality 0.61 0.70 0.81 0.45 0.60 0.78
Empirical FDR (%) 53.98 50.87 53.03 11.24 6.65 5.48
Empirical TPR (%) 28.78 43.62 55.22 13.74 36.54 57.20
Empirical TNR (%) 95.42 94.46 92.57 99.76 99.69 99.62
Mean fit time (s) 691.70 773.56 936.19 73.93 90.07 118.82
AIPW-based Super Learner Rule quality 0.61 0.69 0.80 0.43 0.60 0.78
Empirical FDR (%) NA NA NA 11.83 6.69 5.45
Empirical TPR (%) NA NA NA 13.68 36.64 57.24
Empirical TNR (%) NA NA NA 99.75 99.69 99.62
Mean fit time (s) 722.40 838.81 1,164.73 80.88 101.80 143.47
Causal Random Forests Rule quality 0.12 0.04 0.11 0.37 0.44 0.58
Empirical FDR (%) NA NA NA 11.75 6.62 5.36
Empirical TPR (%) NA NA NA 13.32 36.32 57.30
Empirical TNR (%) NA NA NA 99.75 99.70 99.62
Mean fit time (s) 13.13 31.45 71.31 69.51 88.07 123.80
Table A16:

Simulation results: observational study with non-sparse non-linear outcome model and block covariance matrix.

CATE estimator Metric Unfiltered TEM-VIP-based filtering
n = 250 n = 500 n = 1,000 n = 250 n = 500 n = 1,000
Plug-in LASSO Rule quality 0.63 0.78 0.84 0.56 0.74 0.81
Empirical FDR (%) 51.46 48.81 47.81 13.67 8.29 7.00
Empirical TPR (%) 30.84 44.66 58.12 18.72 39.18 58.44
Empirical TNR (%) 96.15 95.04 93.82 99.61 99.58 99.48
Mean fit time (s) 0.56 0.73 1.34 1,266.36 1,470.17 2,634.34
Plug-in XGBoost Rule quality 0.09 0.06 0.16 0.23 0.29 0.40
Empirical FDR (%) NA NA NA 14.72 8.31 6.49
Empirical TPR (%) NA NA NA 18.40 39.24 58.18
Empirical TNR (%) NA NA NA 99.59 99.58 99.52
Mean fit time (s) 2.13 4.82 13.82 1,265.64 1,470.82 2,646.84
Modified covariates LASSO Rule quality 0.08 0.08 0.06 0.24 0.32 0.33
Empirical FDR (%) 60.38 54.08 50.15 14.05 8.72 6.75
Empirical TPR (%) 3.18 4.68 10.54 18.60 39.40 58.56
Empirical TNR (%) 98.37 98.07 97.54 99.60 99.55 99.50
Mean fit time (s) 11.82 18.33 99.27 1,271.48 1,481.43 2,650.73
Modified covariates XGBoost Rule quality 0.08 0.09 0.05 0.16 0.21 0.19
Empirical FDR (%) NA NA NA 13.70 8.50 6.37
Empirical TPR (%) NA NA NA 18.66 39.68 58.04
Empirical TNR (%) NA NA NA 99.62 99.56 99.53
Mean fit time (s) 21.97 29.19 104.87 1,278.55 1,489.49 2,667.57
Augmented modified covariates LASSO Rule quality 0.43 0.69 0.81 0.37 0.64 0.78
Empirical FDR (%) 59.66 61.26 62.93 14.35 8.68 6.75
Empirical TPR (%) 19.78 39.04 54.18 18.06 39.90 58.14
Empirical TNR (%) 95.35 91.90 89.12 99.62 99.55 99.50
Mean fit time (s) 12.23 17.80 92.71 1,268.55 1,481.57 2,658.96
Augmented modified covariates XGBoost Rule quality 0.32 0.54 0.64 0.28 0.51 0.66
Empirical FDR (%) NA NA NA 14.47 7.97 6.77
Empirical TPR (%) NA NA NA 18.64 39.42 58.32
Empirical TNR (%) NA NA NA 99.59 99.59 99.50
Mean fit time (s) 30.63 49.91 149.00 1,276.58 1,498.14 2,680.37
AIPW-based LASSO Rule quality 0.60 0.74 0.82 0.39 0.65 0.78
Empirical FDR (%) 60.04 55.65 53.91 13.37 8.29 6.59
Empirical TPR (%) 32.82 45.30 56.80 18.72 39.82 58.24
Empirical TNR (%) 94.02 93.23 92.27 99.62 99.57 99.51
Mean fit time (s) 731.78 843.97 1,211.56 1,276.14 1,502.32 2,674.74
AIPW-based Super Learner Rule quality 0.59 0.74 0.82 0.38 0.66 0.78
Empirical FDR (%) NA NA NA 14.32 8.37 6.56
Empirical TPR (%) NA NA NA 18.92 39.50 58.26
Empirical TNR (%) NA NA NA 99.60 99.57 99.52
Mean fit time (s) 759.28 901.83 1,430.13 1,285.08 1,514.52 2,663.28
Causal Random Forests Rule quality 0.05 0.05 −0.02 0.22 0.36 0.40
Empirical FDR (%) NA NA NA 13.37 8.62 6.74
Empirical TPR (%) NA NA NA 18.24 39.76 58.58
Empirical TNR (%) NA NA NA 99.61 99.56 99.50
Mean fit time (s) 15.22 34.23 84.13 1,268.99 1,494.08 2,682.33

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Received: 2024-01-14
Accepted: 2025-04-07
Published Online: 2025-05-22

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Prognostic adjustment with efficient estimators to unbiasedly leverage historical data in randomized trials
  4. Homogeneity test and sample size of response rates for AC 1 in a stratified evaluation design
  5. A review of survival stacking: a method to cast survival regression analysis as a classification problem
  6. DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data
  7. A hybrid hazard-based model using two-piece distributions
  8. Regression analysis of clustered current status data with informative cluster size under a transformed survival model
  9. Bayesian covariance regression in functional data analysis with applications to functional brain imaging
  10. Risk estimation and boundary detection in Bayesian disease mapping
  11. An improved estimator of the logarithmic odds ratio for small sample sizes using a Bayesian approach
  12. Short Communication
  13. A multivariate Bayesian learning approach for improved detection of doping in athletes using urinary steroid profiles
  14. Research Articles
  15. Guidance on individualized treatment rule estimation in high dimensions
  16. Weighted Euclidean balancing for a matrix exposure in estimating causal effect
  17. Penalized regression splines in Mixture Density Networks
https://doi.org/10.1515/ijb-2024-0005
Schlagwörter für diesen Artikel
clinical trials; heterogeneous treatment effects; observational studies; precision medicine

Artikel in diesem Heft

  1. Frontmatter
  2. Research Articles
  3. Prognostic adjustment with efficient estimators to unbiasedly leverage historical data in randomized trials
  4. Homogeneity test and sample size of response rates for AC 1 in a stratified evaluation design
  5. A review of survival stacking: a method to cast survival regression analysis as a classification problem
  6. DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data
  7. A hybrid hazard-based model using two-piece distributions
  8. Regression analysis of clustered current status data with informative cluster size under a transformed survival model
  9. Bayesian covariance regression in functional data analysis with applications to functional brain imaging
  10. Risk estimation and boundary detection in Bayesian disease mapping
  11. An improved estimator of the logarithmic odds ratio for small sample sizes using a Bayesian approach
  12. Short Communication
  13. A multivariate Bayesian learning approach for improved detection of doping in athletes using urinary steroid profiles
  14. Research Articles
  15. Guidance on individualized treatment rule estimation in high dimensions
  16. Weighted Euclidean balancing for a matrix exposure in estimating causal effect
  17. Penalized regression splines in Mixture Density Networks
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