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Table 1 Simulation results for Model 1 for 1:1, 1:2, or 1:4 matched case-control data including about 100 cases arising from populations of 1000 or 5000 subjects, based on 1000 replications

From: On the proportional hazards model for occupational and environmental case-control analyses

Population size Case: control ratio Intensity patterns (a) Exposure variables β Method (b) Relative bias (%) (c) Relative bias / Cox pop (%) (c) Relative efficiency (d) RMSE × 10-3 (e) ASE/SDE (e) Cov. rate (e)
1 000 1:1 A Intensity 1.39 WC1 2.9 2.4 0.61 158 0.87 89.1
    WC2 - - - - 1.17 97.5
    CLR 5.9 5.5 0.14 327 0.95 96.5
    ULR −2.6 −3.0 0.31 218 0.97 93.1
    Duration 0.05 WC1 3.3 2.0 0.41 14 0.82 88.3
    WC2 - - - - 1.08 97.1
    CLR 6.2 4.6 0.19 20 0.96 95.7
    ULR −5.3 −6.6 0.35 15 1.03 95.1
  1:1 B Intensity 1.39 WC1 2.6 2.7 0.59 158 0.88 89.9
    WC2 - - - - 1.18 98.7
    CLR 3.4 3.4 0.14 315 0.94 94.9
    ULR −3.8 −3.7 0.31 219 0.98 92.0
    Duration 0.05 WC1 2.0 2.4 0.45 14 0.79 88.3
    WC2 - - - - 1.04 96.1
    CLR 1.9 2.2 0.21 21 0.94 94.2
    ULR −8.6 −8.3 0.39 16 0.99 93.4
5 000 1:1 B Intensity 1.39 WC1 7.3 9.3 0.20 254 0.72 76.1
      WC2 - - - - 0.85 85.6
      CLR −0.9 0.7 0.10 325 0.87 89.8
      ULR −3.3 −1.7 0.23 219 0.92 91.8
    Duration 0.05 WC1 1.6 7.0 0.17 28 0.79 89.0
      WC2 - - - - 0.90 93.0
      CLR −15.7 −12.5 0.19 27 0.92 90.8
      ULR −15.4 −11.9 0.32 22 0.94 90.4
  1:2 B Intensity 1.39 WC1 −0.3 1.6 0.25 203 0.78 86.7
      WC2 - - - - 0.93 92.8
      CLR −3.0 −1.2 0.22 218 0.98 93.1
      ULR −3.5 −1.7 0.34 181 0.96 91.8
    Duration 0.05 WC1 −3.4 1.3 0.27 22 0.85 89.2
      WC2 - - - - 1.00 94.5
      CLR −10.0 −6.5 0.33 21 0.95 92.3
      ULR −10.2 −6.5 0.45 18 0.96 93.3
  1:4 B Intensity 1.39 WC1 −5.3 −3.6 0.37 191 0.80 85.0
      WC2 - - - - 0.99 91.5
      CLR −3.9 −2.2 0.36 187 0.93 89.8
      ULR −3.6 −1.8 0.47 164 0.93 91.0
    Duration 0.05 WC1 −10.6 −6.5 0.39 19 0.86 88.9
      WC2 - - - - 1.06 94.6
      CLR −11.1 −7.3 0.49 17 0.95 91.7
      ULR −10.9 −6.9 0.58 16 0.95 92.6
  1. (a) Exposure intensity was either constant over lifetime for 85% of the subjects, highly increasing for 6%, moderately decreasing for 6%, and moderately increasing intensity for 3% (Scenario A); or, was highly increasing for 50% and moderately decreasing for 50% (Scenario B).
  2. (b) WC1, weighted Cox models with robust sandwich variance; WC2, weighted Cox model with superpopulation variance; CLR, conditional logistic regression on age; ULR, unconditional logistic regression adjusted for age as a continuous covariate.
  3. (c) Relative bias as compared to the true effect and as compared to the estimated effect of the Cox model using the full population source data. Each of these two bias was the same for WC1 and WC2 since these models used the same regression parameter estimator β ^ .
  4. (d) Relative efficiency as compared to the Cox model estimated on the full population source. This quantity was the same for WC1 and WC2 since these models used the same regression parameter estimator β ^ .
  5. (e) RMSE, root mean squared error (same for WC1 and WC2 which used the same regression parameter estimator β ^ ); ASE, average of the 1000 standard errors s β ^ ; SDE, empirical standard deviation of the 1000 β ^ estimates; cov. rate, coverage rate of the 95% confidence interval of β ^ .