Summarizing and communicating on survival data according to the audience: a tutorial on different measures illustrated with population-based cancer registry data

Figures & data

Figure 1 Graphical representation of the different measures using simulated data: the overall survival probability (dashed black curve), the 10-year RMST (lower shaded area), the NLYL at 10 years according to each cause (NLYL_cancer – upper shaded area and NLYL_other – middle shaded area, which sum up to give the RMTL), and the curves of the CPD due to cancer (CPD_cancer) and due to other causes (CPD_other), using a (reverse) stacked display format.

Note: Simulated data were used for this graphical representation; therefore, the values do not match the estimated values from the manuscript (which were based on real data).
Abbreviations: CPD, crude probability of death; NLYL, number of life years lost; RMST, restricted mean survival time; RMTL, restricted mean time lost.

Table 1 Equation, interpretation and general comments of the statistical measures detailed in this work for summarizing survival data, where λ defines the overall mortality hazard, and λ_N(u) the net mortality hazard due to the disease under study. The cumulative net hazard is estimated using the Pohar-Perme estimator.²³ Refer the “Theoretical framework” section for the definition of λ_N(u), λ_C(u) and λ_P(u).

Setting	Equation	Interpretation	Purpose and general comments
Measure
Overall survival setting
Survival probability	$S(t)=\exp (-{\displaystyle \underset{0}{\overset{t}{\int }}\lambda (u)du})$	Probability of being alive after time t	Clinical purpose; defined on the probability scale providing the overall prognosis of patients
CS probability	$CS(t|s)=\frac{S(s+t)}{S(s)}=\exp (-{\displaystyle \underset{s}{\overset{s+t}{\int }}\lambda (u)du})$	Probability of being alive further t years, given alive after s years	Clinical purpose; defined on the probability scale, providing an updated prognosis for patients alive at s years; relevant quantitative information for survivors; number of patients at risks should be checked for long-term estimates
τ-RMST (years)	$RMST(\tau )={\displaystyle \underset{0}{\overset{\tau }{\int }}S(u)du}$	Mean survival time over the time window [0,τ] years	Clinical purpose; defined on the timescale, corresponding to the life expectancy over a τ-year window; easy to interpret and to communicate; depends on the choice of τ, thus recommend to always display the survival curves for a better appreciation Assume that i) the other-cause mortality hazard is well approximated by the general population mortality hazard, and that ii) the disease under study represents a small part of the general population mortality
Relative survival settinga
NS probability	$S_N(t)=\exp (-{\displaystyle \underset{0}{\overset{t}{\int }}{\lambda }_N(u)du})$	Probability of being alive after time t, if the disease under study is the only possible cause of death	Population health purpose; defined on the probability scale, providing a measure of the prognosis of the disease under study; hypothetical situation
CNS probability (%)	$CNS(t|s)=\frac{S_N(s+t)}{S_N(s)}=\exp (-{\displaystyle \underset{s}{\overset{s+t}{\int }}{\lambda }_N(u)du})$	Probability of being alive further t years, given alive after s years, if the disease under study is the only possible cause of death	Population health purpose; defined on the probability scale, providing an updated measure of the prognosis of the disease under study for patients alive at s years if they could only die from the disease under study; hypothetical situation
τ-RMNST (years)	$RMNST(\tau )={\displaystyle \underset{0}{\overset{\tau }{\int }}{S}_N(u)du}$	Mean net survival time over the time window [0, τ] years, if the disease under study is the only possible cause of death	Population health purpose; defined on the timescale; hypothetical situation; depends on the choice of τ, thus recommend to always display the net survival curves for a better appreciation
Crude probability of death i) due to the disease under study or ii) due to other causes	$F_C(t)={\displaystyle \underset{0}{\overset{t}{\int }}S(u){\lambda }_C(u)du}$ $F_P(t)={\displaystyle \underset{0}{\overset{t}{\int }}S(u){\lambda }_P(u)du}$	Probability of death i) due to the disease under study at time t or ii) due to the other cause at time t	Clinical and population health purposes; defined on the probability scale; split the actual prognosis of patients according to each cause (disease under study or other causes); recommend to always display both probabilities, as well as the two hazards, λC(u) and λP(u)
NLYL i) due to the disease under study or ii) due to other causes (years)	$NLY{L}_C(\tau )={\displaystyle \underset{0}{\overset{\tau }{\int }}{F}_C(t)dt}$ $NLY{L}_P(\tau )={\displaystyle \underset{0}{\overset{\tau }{\int }}{F}_P(t)dt}$	NLYL i) due to the disease under study or ii) due to the other causes	Clinical and population health (health economist) purposes; defined on the timescale; easy to interpret and communicate

Notes:

a Relative survival setting corresponds to the particular competing risk setting in which information of the cause of death is not reliably known. This situation is common, for example, when analyzing population-based cancer registry data.

Abbreviations: CS, conditional survival; CNS, conditional net survival; NLYL, number of life years lost; NS, net survival; RMST, restricted mean survival time; RMNST, restricted mean net survival time.

Table 2 Number of cases (K) and deaths (D) observed before December 31, 2014, in men aged between 15 and 80 years old at diagnosis and diagnosed between 2001 and 2003 in England, by deprivation and age at diagnosis groups (Deprivation 1 corresponding to the less deprived and 5 to the most deprived)

Age at diagnosis		Deprivation group					Total
		1	2	3	4	5
[15;44]	K	122	116	118	123	128	607
	D	57	55	59	57	72	300
[45;54]	K	326	322	294	287	275	1,504
	D	169	181	156	177	163	846
[55;64]	K	1,017	978	898	911	756	4,560
	D	589	583	544	577	531	2,824
[65;74]	K	1,699	1,740	1,680	1,669	1,482	8,270
	D	1,180	1,248	1,200	1,247	1,200	6,075
[75;80]	K	905	1,038	1,052	1,080	837	4,912
	D	766	902	887	944	772	4,271
Total	K	4,069	4,194	4,042	4,070	3,478	19,853
	D	2,761	2,969	2,846	3,002	2,738	14,316

Table 3 Measures estimated in the classical survival setting, in men aged between 15 and 80 years old at diagnosis by deprivation group (Dep), with their 95% CIs: the survival probability at 10 years after diagnosis S(t=10), the conditional probability of surviving further t=5 years given that a patient already survived s = 5 years CS(t=5|s=5), and the restricted mean survival time at 10 years RMST(τ = 10)

	Dep 1	Dep 2	Dep 3	Dep 4	Dep 5
S (t=10) (%)	0.36 (0.34–0.37)	0.33 (0.32–0.35)	0.34 (0.32–0.35)	0.30 (0.28–0.31)	0.25 (0.24–0.27)
CS(t=5|s=5) (%)	0.76 (0.74–0.78)	0.74 (0.72–0.76)	0.75 (0.73–0.77)	0.73 (0.70–0.75)	0.68 (0.66–0.71)
RMST(τ = 10)	5.14 (5.01–5.27)	4.93 (4.80–5.05)	4.92 (4.79–5.05)	4.58 (4.45–4.70)	4.16 (4.03–4.30)

Abbreviations: CS, conditional survival; RMST, restricted mean survival time.

Table 4 Measures estimated in the relative survival setting, in men aged between 15 and 80 years old at diagnosis by deprivation group (Dep), with their 95% CIs: the NS probability at 10 years after diagnosis NS(t = 10), the CNS, CNS(t = 5|s = 5), the RMNST at 10 years RMNST (τ = 10), the crude probability of death at 10 years for cancer F_C(t = 10) and other causes F_P(t = 10), and the number of life years lost due to cancer NLYL_C(τ = 10) and due to other causes NLYL_P(τ = 10) over a 10-year time window

	Dep 1 Estimate (95% CI)	Dep 2 Estimate (95% CI)	Dep 3 Estimate (95% CI)	Dep 4 Estimate (95% CI)	Dep 5 Estimate (95% CI)
NS (t = 10) (%)	0.47 (0.45–0.49)	0.46 (0.44–0.48)	0.50 (0.48–0.52)	0.46 (0.44–0.49)	0.40 (0.38–0.43)
CNS (t = 5|s = 5) (%)	0.89 (0.87–0.92)	0.90 (0.87–0.92)	0.94 (0.91–0.97)	0.93 (0.90–0.96)	0.88 (0.84–0.92)
RMNST (τ = 10)	5.74 (5.58–5.90)	5.61 (5.45–5.76)	5.76 (5.57–5.92)	5.43 (5.29–5.59)	5.02 (4.84–5.19)
Crude probability of death
FC (t = 10)	0.50 (0.49–0.52)	0.51 (0.49–0.53)	0.48 (0.46–0.50)	0.51 (0.49–0.53)	0.56 (0.54–0.58)
FP (t = 10)	0.14 (0.13–0.14)	0.16 (0.15–0.16)	0.19 (0.18–0.19)	0.19 (0.18–0.19)	0.19 (0.19–0.20)
Number of life years lost
NLYLC (τ = 10)	4.14 (3.97–4.28)	4.24 (4.09–4.38)	4.11 (3.95–4.26)	4.4 (4.24–4.55)	4.77 (4.60–4.94)
NLYLP (τ = 10)	0.72 (0.70–0.75)	0.83 (0.80–0.86)	0.97 (0.94–1.01)	1.03 (0.99–1.06)	1.08 (1.03–1.12)

Abbreviations: CNS, conditional net survival; NLYL, number of life years lost; NS, net survival; RMNST, restricted mean net survival time.

PermeMPStareJEstèveJOn estimation in relative survivalBiometrics201268111312021689081

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