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Table 3 Results from simulations including a covariate Z, weakly correlated to X. Naive and regression calibrated effect estimates in linear regression with error-prone exposure X and a covariate Z weakly correlated to X (ρ XZ = 0.2), analysing (A) dummy variables, comparing 5th vs. 1st quintile, (B) quintile numbers, and (C) median values within quintile groups. Results from analysis with continuous exposure is included for comparison. We have X, Z and Y ~ N (0,1) and the error U ~ N (0, ), where is chosen such that the reliability ratio λ is either 0.8, 0.5 or 0.2. The results are obtained via simulation, where the correlations ρ XY = ρ ZY are set to either 0.7 or 0.2. The true effects of X and Z are indicated for the various models. All individuals are measured twice. Standard errors for the corrected cases are bootstrapped.

From: Regression analysis with categorized regression calibrated exposure: some interesting findings

ρ XY = ρ ZY model true coef (X) λ naive coef (X) (SE) RC coef (X) (SE) true coef (Z) naive coef (Z) (SE) RC coef (Z) (SE)
    0.8 0.52 (0.001) 0.58 (0.002)   0.60 (0.002) 0.58 (0.001)
  cont 0.58 0.5 0.38 (0.001) 0.58 (0.003) 058 0.62 (0.002) 0.58 (0.002)
    0.2 0.19 (0.001) 0.59 (0.009)   0.66 (0.002) 0.58 (0.003)
    0.8 1.53 (0.010) 1.53 (0.006)   0.61 (0.003) 0.60 (0.002)
  A 1.63 0.5 1.32 (0.007) 1.34 (0.010) 0.60 0.63 (0.003) 0.60 (0.002)
    0.2 0.92 (0.010) 0.95 (0.008)   0.67 (0.003) 0.60 (0.003)
0.7         
    0.8 0.36 (0.002) 0.36 (0.001)   0.61 (0.003) 0.60 (0.002)
  B 0.39 0.5 0.31 (0.002) 0.32 (0.001) 0.60 0.63 (0.003) 0.60 (0.002)
    0.2 0.22 (0.002) 0.23 (0.002)   0.67 (0.003) 0.60 (0.003)
    0.8 0.56 (0.003) 0.63 (0.002)   0.61 (0.003) 0.60 (0.002)
  C 0.63 0.5 0.42 (0.003) 0.63 (0.003) 0.60 0.63 (0.003) 0.60 (0.002)
    0.2 0.21 (0.002) 0.63 (0.009)   0.67 (0.003) 0.60 (0.003)
    0.8 0.15 (0.003) 0.17 (0.003)   0.17 (0.003) 0.17 (0.003)
  cont 0.17 0.5 0.11 (0.003) 0.16 (0.004) 0.17 0.18 (0.003) 0.17 (0.003)
    0.2 0.05 (0.002) 0.17 (0.006)   0.19 (0.003) 0.17 (0.003)
    0.8 0.44 (0.010) 0.45 (0.010)   0.18 (0.003) 0.17 (0.003)
  A 0.47 0.5 0.38 (0.010) 0.38 (0.009) 0.17 0.18 (0.003) 0.17 (0.003)
    0.2 0.27 (0.010) 0.29 (0.010)   0.19 (0.003) 0.17 (0.003)
0.2         
    0.8 0.11 (0.002) 0.11 (0.002)   0.18 (0.003) 0.17 (0.003)
  B 0.11 0.5 0.09 (0.002) 0.09 (0.002) 0.17 0.18 (0.003) 0.17 (0.003)
    0.2 0.06 (0.002) 0.07 (0.002)   0.19 (0.003) 0.17 (0.003)
    0.8 0.16 (0.003) 0.18 (0.004)   0.18 (0.003) 0.17 (0.003)
  C 0.18 0.5 0.12 (0.003) 0.18 (0.004) 0.17 0.18 (0.003) 0.17 (0.003)
    0.2 0.06 (0.002) 0.18 (0.007)   0.19 (0.003) 0.17 (0.003)