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
$\theta \left(t\right)=E\left[R\left(t{Z}_2\right)\right]$	$\theta \left(t\right)=E\left[R\left(t|Z_2\right)\right]$
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$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\right||Z_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{R}\left(t\right|Z_2=z_{2i})$
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$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{w}_i\widehat{R}\left(t\right|,Z_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{w}_i\widehat{R}\left(t\right|Z_2=z_{2i})$
Corrections in section: Marginal all-cause survival
Equation 3
$\theta \left(t\right)=E\left[S\left(tZ\right)\right]=E\left[S^{*}\left(t{Z}_1\right)R\left(t|Z_2\right)\right]$	$\theta \left(t\right)=E\left[S\left(t|Z\right)\right]=E\left[S^{*}\left(t|Z_1\right)R\left(t|Z_2\right)\right]$
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$\widehat{\theta }\left(t\right)=\frac{1}{N} \sum _{i=1}^N{S}^{*}\left(t|Z_1=z_{1i}\right)\widehat{R}\left(t\right|{|Z}_2=z_{2i})$	$\widehat{\theta }\left(t\right)=\frac{1}{N}\sum _{i=1}^N{S}^{*}\left(t|Z_1=z_{1i}\right)\widehat{R}\left(t\right|Z_2=z_{2i})$
Corrections in section: Marginal crude probabilities of death
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$F_c\left(tZ\right)$	$F_c\left(t|Z\right)$
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$F_o\left(tZ\right)$	$F_o\left(t|Z\right)$
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${\theta }_c\left(t\right)=E\left[F_c\left(tZ\right)\right]=E\left[{\int }_0^t{S}^{*}\left(u\left|\right|Z_1\right)R\left(u\left|\right|Z_2\right)\lambda \left(u\left|\right|Z_2\right)du\right]$	${\theta }_c\left(t\right)=E\left[F_c\left(t|Z\right)\right]=E\left[{\int }_0^t{S}^{*}\left(u|Z_1\right)R\left(u|Z_2\right)\lambda \left(u|Z_2\right)du\right]$
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${\widehat{\theta }}_c\left(t\right)=\frac{1}{N}{\sum }_{i=1}^N\widehat{F_c}\left(tZ=z_i\right) =\frac{1}{N}{\sum }_{i=1}^N{\int }_0^t{S}^{*}\left(u\right||Z_1=z_{1i}) \widehat{R}\left(u||{Z_2=z}_{2i}\right)\widehat{\lambda }(u\left|{Z_2=z}_{2i}\right)du$	${\widehat{\theta }}_c\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_c}\left(t|Z=z_i\right) =\frac{1}{N}\sum _{i=1}^N{\int }_0^t{S}^{*}\left(u|Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)\widehat{\lambda }\left(u\right|{Z_2=z}_{2i})du$
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${\theta }_o\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_o}\left(t{z}_i\right) =\frac{1}{N}{\sum }_{i=1}^N{\int }_0^t{S}^{*}\left(u||Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)h^{*}\left(u\right|Z_1{=z}_{1i})du$	${\widehat{\theta }}_o\left(t\right)=\frac{1}{N}\sum _{i=1}^N\widehat{F_o}\left(t|Z=z_i\right) =\frac{1}{N}\sum _{i=1}^N{\int }_0^t{S}^{*}\left(u|Z_1=z_{1i}\right)\widehat{R}\left(u|{Z_2=z}_{2i}\right)h^{*}\left(u\right|{Z_1=z}_{1i})du$
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
$E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t\left|\right|X=0,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$	$E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t|X=0,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$
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$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=0, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=0, {Z_2^{X=1}=z}_{2i}\right)$	$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=0, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=0, {Z_2^{X=1}=z}_{2i}\right)$
Corrections in section: Eliminating cancer-related differences
Equation 7
$E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t\left|\right|X=1,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$	$E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]-E\left[S^{*}\left(t|X=1,Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]$
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$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=0, {Z_2^{X=1}=z}_{2i}\right)$	$\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=1, {Z_2^{X=1}=z}_{2i}\right)-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=0, {Z_2^{X=1}=z}_{2i}\right)$
Corrections in section: Avoidable deaths
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${{\mathrm{D}}_1\left(t\left|\right|\mathrm{X}=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t\left|\right|X=1, Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]\right]$	${{\mathrm{D}}_1\left(t|X=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t|X=1, Z_1^{X=1}\right)R\left(t|X=1,Z_2^{X=1}\right)\right]\right]$
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${{\mathrm{D}}_{R_0}\left(t\left|\right|\mathrm{X}=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t\left|\right|X=1, Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]\right]$	${{\mathrm{D}}_{R_0}\left(t|X=1\right)=\mathrm{N}}^{\mathrm{*}}\left[1-E\left[S^{*}\left(t|X=1, Z_1^{X=1}\right)R\left(t|X=0,Z_2^{X=1}\right)\right]\right]$
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
$N^{*}\left[1-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t\left|\right|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t\left|\right|X=x, {Z_2^{X=1}=z}_{2i}\right)\right]$	$N^{*}\left[1-\frac{1}{N^{X=1}}\sum _{i=1}^{N^{X=1}}{S}^{*}\left(t|X=1, {Z_1^{X=1}=z}_{1i}\right)\widehat{R}\left(t|X=x, {Z_2^{X=1}=z}_{2i}\right)\right]$