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. Author manuscript; available in PMC: 2022 Dec 1.
Published in final edited form as: Health Serv Outcomes Res Methodol. 2022 Jun 6;22(4):435–453. doi: 10.1007/s10742-022-00275-x

Table 3.

Results for top 10 algorithms in modeling four spine-related RVUs

Algorithma MSEb Relative MSE R2 REc MAE
Spine-related RVUs (5% zero)
 Two-stage Super Learner + Thresholding 6253.45 1.0000 0.0713 1.0000 19.6995
 Two-stage Super Learner 6261.24 1.0012 0.0701 0.9838 19.7632
 One-stage Super Learner 6298.74 1.0072 0.0646 0.9057 19.8006
 Discrete two-stage Learner + Thresholding 6316.67 1.0101 0.0619 0.8683 19.8365
 Discrete two-stage Learner 6323.14 1.0111 0.0609 0.8548 19.8581
 RFd 6327.06 1.0118 0.0604 0.8474 19.8749
 S1e: RF + S2f: RF 6330.65 1.0123 0.0599 0.8401 19.8695
 I(P(Y>0)>=0.5) * (S1: RF + S2: RF) 6331.10 1.0124 0.0598 0.8386 19.8656
 I(P(Y>0)>=0.75) * (S1: RF + S2: RF) 6331.62 1.0125 0.0597 0.8379 19.8767
 I(P(Y>0)>=0.75) * (S1: GLMg + S2: RF) 6331.98 1.0126 0.0596 0.8364 19.8851
Spine-related imaging RVUs (55% zero)
 Two-stage Super Learner 54.9361 1.0000 0.0979 1.0000 4.2369
 Two-stage Super Learner + Indicator 55.2834 1.0063 0.0922 0.9417 4.2539
 Discrete two-stage Learner 55.4711 1.0097 0.0891 0.9102 4.2934
 Discrete two-stage Learner + Indicator 55.6390 1.0128 0.0863 0.8820 4.3064
 S1: RF + S2: RF 55.7874 1.0155 0.0842 0.8606 4.3505
 S1: RF + S2: Lasso 55.8547 1.0167 0.0828 0.8459 4.3264
 I(P(Y>0)>=0.2) * (S1: RF + S2: RF) 55.8598 1.0168 0.0827 0.8450 4.3232
 S1: GLM + S2: Lasso 55.8673 1.0169 0.0826 0.8437 4.3036
 One-stage Super Learner 55.8768 1.0171 0.0824 0.8421 4.2730
 S1: GLM + S2: RF 55.8850 1.0173 0.0823 0.8408 4.3299
Spine-related physical therapy RVUs (85% zero)h
 Two-stage Super Learner 2.4698 1.0000 0.2846 1.0000 0.5388
 Two-stage Super Learner + Thresholding 2.4741 1.0017 0.2834 0.9957 0.5391
 Discrete two-stage Learner 2.5111 1.0167 0.2734 0.9605 0.5446
 Discrete two-stage Learner + Thresholding 2.5163 1.0188 0.2719 0.9554 0.5452
 S1: Lassoi + S2: Log-OLS smearingj 2.5229 1.0215 0.2692 0.9460 0.5445
 I(P(Y>0)>=0.02) * (S1: Lasso + S2: Log-OLS smearing) 2.5242 1.0220 0.2689 0.9447 0.5450
 S1: Lasso + S2: GLM-Gamma-Identity 2.5288 1.0239 0.2674 0.9396 0.5473
 I(P(Y>0)>=0.02) * (S1: Lasso + GLM-Gamma-Identityk) 2.5294 1.0241 0.2673 0.9393 0.5490
 S1: Lasso + S2: Adaptive GLM 2.5304 1.0245 0.2671 0.9387 0.5529
 I(P(Y>0)>=0.02) * (S1: Lasso + S2: Adaptive GLM) 2.5307 1.0247 0.2670 0.9381 0.5483
 One-stage Super Learner 2.5337 1.0259 0.2661 0.9350 0.5552
Spine-related injection RVUs (91% zero)h
 Two-stage Super Learner + Thresholding 16.8832 1.0000 0.1323 1.0000 1.2578
 Two-stage Super Learner 16.9414 1.0034 0.1293 0.9774 1.2632
 Discrete two-stage Learner + Thresholding 17.1150 1.0137 0.1204 0.9100 1.2753
 Discrete two-stage Learner 17.1581 1.0163 0.1182 0.8932 1.2781
 I(P(Y>0)>=0.01) * (S1: Lasso + S2: Lasso) 17.2029 1.0189 0.1159 0.8758 1.2853
 S1: Lasso + S2: Lasso 17.2074 1.0192 0.1157 0.8741 1.2855
 I(P(Y>0)>=0.05) * (S1: Lasso + S2: Lasso) 17.2116 1.0195 0.1154 0.8724 1.2939
 I(P(Y>0)>=0.01) * (S1: GLM + S2: GLM-Gamma-Identity) 17.2284 1.0204 0.1146 0.8659 1.2647
 S1: GLM + S2: GLM-Gamma-Identity 17.2304 1.0206 0.1145 0.8651 1.2673
 I(P(Y>0)>=0.01) * (S1: GLM + S2: Log-OLS smearing) 17.2361 1.0209 0.1142 0.8629 1.2627
 One-stage Super Learner 17.2599 1.0223 0.1130 0.8537 1.3134
a

Algorithms are presented in ascending order based on MSE.

b

Lower MSE and relative MSE indicate better performance.

c

Greater R2 and RE indicate better performance.

d

RF refers to Random Forest.

e

S1 refers to stage-1 algorithms.

f

S2 refers to stage-2 algorithms.

g

GLM in S1 refers to logistic regression.

h

For spine-related physical therapy RVUs and spine-related injection RVUs, the one-stage super learner is not among the top 10 algorithms, we still list its results for comparison.

i

Lasso in S1 refers to logistic Lasso regression.

j

Log-OLS smearing refers to logarithmic OLS with smearing retransformation.

k

GLM-Gamma-Identity refers to GLM with Gamma distribution and Identity link function.