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

Pötschger, Ulrike; Heinzl, Harald; Valsecchi, Maria Grazia; Mittlböck, Martina

doi:10.1186/s12874-017-0430-5

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			wGLM^k			wGLM ad-hoc^l
Patients^m	Donor	\( w \)	Survival S	log(−log(S))^d	\( {f}_{01}(w) \)	\( \overline{q}(w) \) ^e	\( {\overline{\gamma}}_1 \) ^f	\( {\widehat{f}}_{01}(w) \) ^g	Bias^h	SE_estⁱ	Coverage^j	Bias^h	SE_estⁱ	Coverage^j
\( n=1000 \)	No	–	0.333^a	0.10	–	–	–	–	−0.003	0.125	95.5%	−0.003	0.125	95.5%
	Yes	–	0.620^b	−0.74	–	–	–	–	−0.002	0.078	92.2%	−0.002	0.098	94.0%
		0.5	0.733^c	−1.17	0.33	0.46	0.72	0.33	−0.003	0.164	94.7%	−0.002	0.172	94.0%
		1	0.681^c	−0.96	0.33	0.39	0.86	0.33	−0.010	0.134	94.6%	−0.011	0.161	95.1%
		3	0.451^c	−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	–	0.333^a	0.10	–	–	–	–	−0.004	0.210	95.6%	−0.003	0.210	95.6%
	Yes	–	0.620^b	−0.74	–	–	–	–	−0.007	0.124	92.8%	−0.007	0.156	92.0%
		0.5	0.733^c	−1.17	0.33	0.46	0.72	0.33	−0.024	0.263	95.4%	−0.019	0.277	95.4%
		1	0.681^c	−0.96	0.33	0.39	0.86	0.33	−0.019	0.224	93.8%	−0.026	0.269	94.9%
		3	0.451^c	−0.23	0.33	0.15	2.26	0.33	−0.001	0.170	85.6%	−0.017	0.309	93.4%

Weighted generalised linear model (wGLM) with log-log link; 0–6 years uniform censoring was used
^aTrue survival \( {S}_0(5) \) in patients without a donor available
^bTrue survival \( {S}_1(5) \) in patients with a donor available
^cTrue survival \( {S}_1\left(5\left|w\right.\right) \) in patients with a donor available at waiting time w
^dlog-log transformation of true survival probabilities \( \mathrm{S} \)
^eMean observed proportion of patients with a 0 → 1 transition at \( w \) = 0.5, 1 and 3
^fMean estimated weight for \( w \) = 0.5, 1 and 3
^gMean estimated probabilities for \( {f}_{01}(w) \) at \( w \) = 0.5, 1 and 3
^hMean difference between estimated and true \( \log \left(-\log \left(\mathrm{S}\right)\right) \) values
ⁱMean of standard errors of \( \log \left(-\log \left(\mathrm{S}\right)\right) \) estimates
^jCoverage of the 95% confidence intervals for \( \log \left(-\log \left(\mathrm{S}\right)\right) \)
^kThe weighted generalised linear model (wGLM) uses \( {\widehat{V}}_{i,1}\left({t}^{\ast}\right) \) according to eq. (6)
^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)
^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
ⁿTwo of 1000 simulation runs were excluded due to non-convergence during parameter estimation

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