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Table 2 Simulation results under Scenario 2: single step versus two-step strategy

From: Statistical strategies for constructing health risk models with multiple pollutants and their interactions: possible choices and comparisons

Predictor β Measure (A) One-step regression using all predictors (B) Two-step strategy employing CART at screening step
DSA LASSO PLSR SPCA BMA1 DSA2 LASSO3 PLSR4 SPCA5
X 1 0.50 Estimate (ESE) 0.93 (0.29) 0.08 (0.19) 0.03 (0.01) 0.03 (0.04) 0.32 (0.38) 0.93 (0.29) 0.35 (0.36) 0.11 (0.04) 2.3×10-4 (0.01)
Percent included N/A 28.2% N/A 98.5% 65.2% N/A 68.3% N/A 58.8%
X 2 0.50 Estimate (ESE) 0.75 (0.27) 0.07 (0.22) 0.02 (0.01) 0.02 (0.03) 0.25 (0.32) 0.74 (0.25) 0.33 (0.38) 0.09 (0.04) −2.7×10-4 (0.01)
Percent included N/A 22.6% N/A 94.0% 63.5% N/A 63.8% N/A 58.9%
X 6 0.50 Estimate (ESE) 0.88 (0.29) 0.07 (0.19) 0.03 (0.02) 0.02 (0.03) 0.29 (0.36) 0.88 (0.25) 0.36 (0.36) 0.10 (0.05) −1.2×10-4 (0.01)
Percent included N/A 25.8% N/A 96.2% 63.6% N/A 67.4% N/A 57.9%
X 7 0.50 Estimate (ESE) 0.71 (0.26) 0.04 (0.22) 0.02 (0.01) 0.01 (0.02) 0.24 (0.30) 0.67 (0.26) 0.32 (0.34) 0.08 (0.04) 9.1×10-4 (0.01)
Percent included N/A 18.1% N/A 82.4% 65.6% N/A 64.3% N/A 57.8%
X 1 *X 2 0.20 Estimate (ESE) 0.002 (0.03) 0.17 (0.14) 0.07 (0.04) 0.07 (0.07) 0.24 (0.22) 0.006 (0.06) 0.21 (0.18) 0.27 (0.08) 0.28 (0.13)
Percent included 0.3% 79.2% N/A 96.3% 78.4% 1.1% 84.0% N/A 98.6%
X 1 *X 6 0.20 Estimate (ESE) 0.003 (0.05) 0.20 (0.18) 0.06 (0.03) 0.05 (0.06) 0.21 (0.26) 0.006 (0.08) 0.22 (0.22) 0.23 (0.08) 0.19 (0.11)
Percent included 0.3% 77.3% N/A 99.0% 66.7% 0.9% 78.1% N/A 99.8%
X 6 *X 7 0.20 Estimate (ESE) 0.002 (0.04) 0.17 (0.16) 0.06 (0.03) 0.03 (0.05) 0.25 (0.27) 0.004 (0.05) 0.21 (0.21) 0.19 (0.08) 0.15 (0.11)
Percent included 0.3% 74.4% N/A 94.1% 73.3% 0.5% 76.9% N/A 98.6%
  Average model size 20.1 22.8 210 79.3 6.0 4.2 6.7 10.0 8.2
  1. Average estimated effects, empirical standard errors, percentages of correct identification of non-zero coefficients, and average model size corresponding to four available statistical methods in a cross-sectional study with continuous responses and 20 air pollutants were provide in panel A. Similar results of five statistical methods after an initial CART variable selection using the two-step modeling strategy were summarized in panel B. Sample size for each replicate was N=250. The true model size was 7 without accounting for the intercept, and the possible maximum model size was 210. ESE, empirical standard error of the estimate. Results are based on 1000 replicates.
  2. Estimate of the non-zero predictor is calculated as the mean of the products that estimated regression coefficient of this predictor multiplies the indicator function that this predictor is correctly identified during each replication. The percentage of the non-zero predictor quantifies the proportion of correct identification of this predictor over 1000 replicates in each method. 1In BMA, predictors with their posterior probabilities greater than 10% are regarded as identified. 2In DSA, there is no variable selection for main effects as individual exposures are enforced when their interactions are of interest. Identification of interaction refers to the inclusion of interaction term in the cross-validated best predictive model. 3Predictors with their estimated LASSO regression coefficients not equal to zero are considered identified. 4No variable selection has been applied in PLSR because it uses all predictors. 5In SPCA, predictors are identified if their Wald’s statistics from univariate models are larger than a threshold value.