Erratum to: Marginal measures and causal effects using the relative survival framework

First published online: 18 January 2020, Int J Epidemiol 2020;49:619–28. doi: https://doi.org/10.1093/ije/dyz268

The originally published version of this article contained some errors in the setting of some equations (see below). These have now been corrected. OUP apologises for this error.

Published version (wrong version)	How it should be (RIGHT version)
Corrections in section: Marginal relative survival
Page 3
$R\left(t{Z}_2\right)$	$R\left(t|Z_2\right)$
Equation 2
$\theta \left(t\right)=E\left[R\left(t{Z}_2\right)\right]$	$\theta \left(t\right)=E\left[R\left(t|Z_2\right)\right]$
Page 4
$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\right||Z_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\right|Z_2=z_{2i})$
Page 4
$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{w}_i\widehat{R}\left(t\right|,Z_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{w}_i\widehat{R}\left(t\right|Z_2=z_{2i})$
Corrections in section: Marginal all-cause survival
Equation 3
$\theta \left(t\right)=E\left[S\left(tZ\right)\right]=E\left[S^{*}\left(t{Z}_1\right)R\left(t|Z_2\right)\right]$	$\theta \left(t\right)=E\left[S\left(t|Z\right)\right]=E\left[S^{*}\left(t|Z_1\right)R\left(t|Z_2\right)\right]$
Page 4
$\widehat{\theta }\left(t\right)=\frac{1}{N} \sum _{i=1}^N{S}^{*}\left(t|Z_1=z_{1i}\right)\widehat{R}\left(t\right|{|Z}_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t|Z_1=z_{1i}\right)\widehat{R}\left(t\right|Z_2=z_{2i})$
Corrections in section: Marginal crude probabilities of death
Page 4
$F_c\left(tZ\right)$	$F_c\left(t|Z\right)$
Page 4
$F_o\left(tZ\right)$	$F_o\left(t|Z\right)$
Page 4
${\theta }_c\left(t\right)=E\left[F_c\left(tZ\right)\right]=E\left[{\int }_0^t{S}^{*}\left(u\left|\right|Z_1\right)R\left(u\left|\right|Z_2\right)\lambda \left(u\left|\right|Z_2\right)du\right]$	${\theta }_c\left(t\right)=E\left[F_c\left(t|Z\right)\right]=E\left[{\int }_0^t{S}^{*}\left(u|Z_1\right)R\left(u|Z_2\right)\lambda \left(u|Z_2\right)du\right]$
Page 4
${\widehat{\theta }}_c\left(t\right)=\frac{1}{N}{\sum }_{i=1}^N\widehat{F_c}\left(tZ=z_i\right) =\frac{1}{N}{\sum }_{i=1}^N{\int }_0^t{S}^{*}\left(u\right||Z_1=z_{1i}) \widehat{R}\left(u||{Z_2=z}_{2i}\right)\widehat{\lambda }(u\left|{Z_2=z}_{2i}\right)du$	${\widehat{\theta }}_c\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_c}\left(t|Z=z_i\right) =\frac{1}{N}\sum _{i=1}^N{\int }_0^t{S}^{*}\left(u|Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)\widehat{\lambda }\left(u\right|{Z_2=z}_{2i})du$
Page 4
${\theta }_o\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_o}\left(t{z}_i\right) =\frac{1}{N}{\sum }_{i=1}^N{\int }_0^t{S}^{*}\left(u||Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)h^{*}\left(u\right|Z_1{=z}_{1i})du$	${\widehat{\theta }}_o\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_o}\left(t|Z=z_i\right) =\frac{1}{N}\sum _{i=1}^N{\int }_0^t{S}^{*}\left(u|Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)h^{*}\left(u\right|{Z_1=z}_{1i})du$
Corrections in section: Relative survival differences
Page 5
$\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\left|\right|X=1, {Z_2=z}_{2i}\right)-\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\left|\right|X=0, {Z_2=z}_{2i}\right)$	$\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t|X=1, {Z_2=z}_{2i}\right)-\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t|X=0, {Z_2=z}_{2i}\right)$
Corrections in section: All-cause survival differences
Equation 5
$E\left[S^{*}\left(t\left|\right|X=1, Z_1\right)R\left(t|X=1,Z_2\right)\right]-E\left[S^{*}\left(t\left|\right|X=0, Z_1\right)R\left(t|X=0,Z_2\right)\right]$	$E\left[S^{*}\left(t|X=1, Z_1\right)R\left(t|X=1,Z_2\right)\right]-E\left[S^{*}\left(t|X=0, Z_1\right)R\left(t|X=0,Z_2\right)\right]$
Page 5
$\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t\left|\right|X=1, Z_1=z_{1i}\right)\widehat{R}\left(t\left|\right|X=1, {Z_2=z}_{2i}\right)-\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t\left|\right|X=0, Z_1=z_{1i}\right)\widehat{R}\left(t\left|\right|X=0, {Z_2=z}_{2i}\right)$	$\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t|X=1, Z_1=z_{1i}\right)\widehat{R}\left(t|X=1, {Z_2=z}_{2i}\right)-\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t|X=0, Z_1=z_{1i}\right)\widehat{R}\left(t|X=0, {Z_2=z}_{2i}\right)$
Corrections in section: Forming contrasts within subsets of the population
Equation 6
$E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t\left|\right|X=0,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$	$E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t|X=0,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$
Page 6
$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=0, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=0, {Z_2^{X=1}=z}_{2i}\right)$	$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=0, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=0, {Z_2^{X=1}=z}_{2i}\right)$
Corrections in section: Eliminating cancer-related differences
Equation 7
$E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$	$E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$
Page 6
$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=0, {Z_2^{X=1}=z}_{2i}\right)$	$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=0, {Z_2^{X=1}=z}_{2i}\right)$
Corrections in section: Avoidable deaths
Page 7
${{\mathrm{D}}_1\left(t\left|\right|\mathrm{X}=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t\left|\right|X=1, Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]\right]$	${{\mathrm{D}}_1\left(t|X=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t|X=1, Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]\right]$
Page 7
${{\mathrm{D}}_{R_0}\left(t\left|\right|\mathrm{X}=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t\left|\right|X=1, Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]\right]$	${{\mathrm{D}}_{R_0}\left(t|X=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t|X=1, Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]\right]$
Equation 8
${{\mathrm{AD}}_{R_0}=\mathrm{D}}_1\left(t\left|\right|\mathrm{X}=1\right)-{\mathrm{D}}_{R_0}\left(t|\mathrm{X}=1\right)$	${{\mathrm{AD}}_{R_0}=\mathrm{D}}_1\left(t|X=1\right)-{\mathrm{D}}_{R_0}\left(t|X=1\right)$
Equation 9
$N^{*}\left[1-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=x, {Z_2^{X=1}=z}_{2i}\right)\right]$	$N^{*}\left[1-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=x, {Z_2^{X=1}=z}_{2i}\right)\right]$