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Table 1 Results of the simulation study 1 (scenario I) using a weighted generalised linear model

From: Assessing the effect of a partly unobserved, exogenous, binary time-dependent covariate on survival probabilities using generalised pseudo-values

    Truth Waiting times wGLMk wGLM ad-hocl
Patientsm Donor \( w \) Survival S log(−log(S))d \( {f}_{01}(w) \) \( \overline{q}(w) \) e \( {\overline{\gamma}}_1 \) f \( {\widehat{f}}_{01}(w) \) g Biash SEesti Coveragej Biash SEesti Coveragej
\( n=1000 \) No 0.333a 0.10 −0.003 0.125 95.5% −0.003 0.125 95.5%
  Yes 0.620b −0.74 −0.002 0.078 92.2% −0.002 0.098 94.0%
   0.5 0.733c −1.17 0.33 0.46 0.72 0.33 −0.003 0.164 94.7% −0.002 0.172 94.0%
   1 0.681c −0.96 0.33 0.39 0.86 0.33 −0.010 0.134 94.6% −0.011 0.161 95.1%
   3 0.451c −0.23 0.33 0.15 2.26 0.33 0.000 0.106 86.4% 0.004 0.191 93.7%
\( n=400 \) n No 0.333a 0.10 −0.004 0.210 95.6% −0.003 0.210 95.6%
  Yes 0.620b −0.74 −0.007 0.124 92.8% −0.007 0.156 92.0%
   0.5 0.733c −1.17 0.33 0.46 0.72 0.33 −0.024 0.263 95.4% −0.019 0.277 95.4%
   1 0.681c −0.96 0.33 0.39 0.86 0.33 −0.019 0.224 93.8% −0.026 0.269 94.9%
   3 0.451c −0.23 0.33 0.15 2.26 0.33 −0.001 0.170 85.6% −0.017 0.309 93.4%
  1. Weighted generalised linear model (wGLM) with log-log link; 0–6 years uniform censoring was used
  2. aTrue survival \( {S}_0(5) \) in patients without a donor available
  3. bTrue survival \( {S}_1(5) \) in patients with a donor available
  4. cTrue survival \( {S}_1\left(5\left|w\right.\right) \) in patients with a donor available at waiting time w
  5. dlog-log transformation of true survival probabilities \( \mathrm{S} \)
  6. eMean observed proportion of patients with a 0 → 1 transition at \( w \) = 0.5, 1 and 3
  7. fMean estimated weight for \( w \) = 0.5, 1 and 3
  8. gMean estimated probabilities for \( {f}_{01}(w) \) at \( w \) = 0.5, 1 and 3
  9. hMean difference between estimated and true \( \log \left(-\log \left(\mathrm{S}\right)\right) \) values
  10. iMean of standard errors of \( \log \left(-\log \left(\mathrm{S}\right)\right) \) estimates
  11. jCoverage of the 95% confidence intervals for \( \log \left(-\log \left(\mathrm{S}\right)\right) \)
  12. kThe weighted generalised linear model (wGLM) uses \( {\widehat{V}}_{i,1}\left({t}^{\ast}\right) \) according to eq. (6)
  13. lThe weighted generalised linear model (wGLM) uses the ad-hoc correction suggested to estimate \( {\widehat{V}}_{i,1}\left({t}^{\ast}\right) \) (with one repetition per observation per simulation run)
  14. m\( n \) represents the size of the entire sample where 25% of the patients have no donor; 25% of the patients have a donor available at \( w \) =0.5, 25% at \( w \) =1, and 25% at \( w \) =3, respectively
  15. nTwo of 1000 simulation runs were excluded due to non-convergence during parameter estimation