Corrigendum

This article refers to:

Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach

de Oliveira, R. P., Achcar, J. A., Peralta, D., & Mazucheli, J. (2018). Discrete and continuous bivariate lifetime models in presence of cure rate: a comparative study under Bayesian approach, Journal of Applied Statistics.

http://dx.doi.org/10.1080/02664763.2018.1495701

A few errors were identified in Sections 2.1 to 2.4 and Table 1, which have now been corrected.

Table 1. Summary of the proposed models and their correlation coefficients.

Model	Type of model	Parameters	Correlation coefficient (ρ)	Correlation range
GBE	Continuous	${\lambda }_1,{\lambda }_2>0$; $0<{\lambda }_3<1$	$1-{\displaystyle \frac{1}{{\lambda }_3}}\exp \left({\displaystyle \frac{1}{{\lambda }_3}}\right)E_i\left({\displaystyle \frac{1}{{\lambda }_3}}\right)$	$0.404\le \rho \le 1$ $\mathrm{if}{\lambda }_3\to 1$; otherwise $0<\rho \le 1$
BBBE	Continuous	${\lambda }_1,{\lambda }_2,{\lambda }_3>0$	Equation (14)	$0<\rho \le 1$
ABG	Discrete	$0<{\theta }_1,{\theta }_2<1$	$-{\displaystyle \frac{{\theta }_1{\theta }_2}{({\theta }_1+{\theta }_2)\left[\right(1-{\theta }_1\left)\right(1-{\theta }_2){]}^{1/2}}}$	$-1\le \rho <0$
BDBG	Discrete	$0<{\theta }_1,{\theta }_2<1$; $0<{\theta }_3\le 1$	${\displaystyle \frac{(1-{\theta }_3)({\theta }_1{\theta }_2{)}^{1/2}}{1-{\theta }_1{\theta }_2{\theta }_3}}$	$0\le \rho \le 1$

The list of changes and the amended Table 1 is provided below:

In Section 2.1 - Gumbel bivariate exponential

The sentence “where ${\lambda }_1,{\lambda }_2\ge 0$” first may please be read “where ${\lambda }_1,{\lambda }_2>0$”

In Section 2.2 - Block and Basu bivariate exponential

The sentence “and ${\lambda }_1,{\lambda }_2,{\lambda }_3\ge 0$.” first may please be read “and ${\lambda }_1,{\lambda }_2,{\lambda }_3>0$”.

The sentence “otherwise $0\le \rho \le 1$” first may please be read “otherwise $0<\rho \le 1$”.

In Section 2.3 - Arnold bivariate geometric

The sentence “otherwise $-1\le \rho \le 0$” first may please be read “otherwise $-1\le \rho <0$”

In Section 2.4 - Basu-Dhar bivariate geometric

The sentence “the correlation is equals to zero when ${\theta }_1=0$ or ${\theta }_2=0$” first may please be read “the correlation is equals to zero when ${\theta }_3=1$”

In Section 4.1. The diabetic retinopathy data (DRS)

The sentence ”…presence of two covariates, the age at diagnosis of diabetes (in years), denoted by $S_1$; and the type of diabetes (juvenile and adult), denoted by $S_2$.” first may please be read ”presence of two covariates, the laser type (xeon or argon), denoted by $X_1$; and the type of diabetes (juvenile and adult), denoted by $X_2$.”

In Section 4.1.1. A Bayesian analysis in the presence of cure rate modeling

In the submission process, Tables 2,3 and 4 files for BBBE model; and Figures 2 and 3 files for BBBE model were sent incorrectly. The corrected estimates of BBBE model in Table 2 are given by:

Table

Model	Par.	Post. Mean (S.D.)	95% Cred. Interval
	${\lambda }_1$	0.0218 (0.0142)	(0.0001, 0.0450)
	${\lambda }_2$	0.0236 (0.0127)	(0.0001, 0.0414)
	${\lambda }_3$	0.0114 (0.0181)	(0.0001, 0.0529)
BBBE	${\phi }_{11}$	0.3461 (0.0734)	(0.2383, 0.5011)
	${\phi }_{10}$	0.0536 (0.0301)	(0.0088, 0.1243)
	${\phi }_{01}$	0.3263 (0.0760)	(0.1455, 0.4655)
	${\phi }_{00}$	0.2740 (0.0488)	(0.1830, 0.3703)

The corrected estimated correlation coefficient for the BBBE model in Table 3, please, read ”Post. Mean$\left(\rho \right)$: $\rho =0.0045$, Std. Dev.: 0.0023, 95% Cred. Interval: (0.0015,0.0069)”

The corrected estimated proportions of non-susceptible patients for the BBBE model in Table 4, please, read ”$T_1({\phi }_{00}+{\phi }_{01})$ = 60.03%” and ”$T_2({\phi }_{00}+{\phi }_{10})$ = 32.76%”

The corrected Figures 2 and 3 for BBBE model are given by:

Figure 2. Kaplan-Meier estimators versus Bayesian estimated survival functions for the marginal survival functions assuming continuous models for $T_1$ (left panel) and $T_2$ (right panel) in presence of cure rate.

Figure 3. Left panel: PP-plots of the proposed models for $T_1$ (time to blindness for the treated eye). Right panel: PP-plots of the proposed models for $T_2$ (time to blindness for the untreated eye).

In Section 4.1.2. A Bayesian analysis in the presence of covariates and cure rate

The sentence ”$X_1$ denoting the age at diagnosis of diabetes” first may please be read ”$X_1$ denoting the laser type (xeon or argon)”

The sentence ”$S_1$ denoting the laser type” first may please be read ”$X_1$ denoting the laser type”.

In Section 4.2. Tobacco-stained fingers data (TSF)

Table 7 title ”The estimated proportions of non-susceptible patients for the application 1.” first may please be read ”The estimated proportions of non-susceptible patients for the TSF dataset.”

The authors apologize for these errors.