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Choosing the net survival method for cancer survival estimation
European Journal of Cancer (2013) xxx, xxx– xxx
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Choosing the net survival method for cancer survival estimation Karri Seppa¨ a,b, Timo Hakulinen a, Arun Pokhrel a,⇑ a b
Finnish Cancer Registry, Institute for Statistical and Epidemiological Cancer Research, Pieni Roobertinkatu 9, FI-00130 Helsinki, Finland Department of Mathematical Sciences, University of Oulu, Oulu, Finland
KEYWORDS Epidemiologic methods Models Neoplasms Prognosis Relative survival Net survival
Abstract Background: A new net survival method has been introduced by Pohar Perme et al. (2012 [4]) and recommended to substitute the relative survival methods in current use for evaluating population-based cancer survival. Methods: The new method is based on the use of continuous follow-up time, and is unbiased only under non-informative censoring of the observed survival. However, the populationbased cancer survival is often evaluated based on annually or monthly tabulated follow-up intervals. An empirical investigation based on data from the Finnish Cancer Registry was made into the practical importance of the censoring and the level of data tabulation. A systematic comparison was made against the earlier recommended Ederer II method of relative survival using the two currently available computer programs (Pohar Perme (2013) [10] and Dickman et al. (2013) [11]). Results: With exact or monthly tabulated data, the Pohar-Perme and the Ederer II methods give, on average, results that are at five years of follow-up less than 0.5% units and at 10 and 14 years 1–2% units apart from each other. The Pohar-Perme net survival estimator is prone to random variation and may result in biased estimates when exact follow-up times are not available or follow-up is incomplete. With annually tabulated follow-up times, estimates can deviate substantially from those based on more accurate observations, if the actuarial approach is not used. Conclusion: At 5 years, both the methods perform well. In longer follow-up, the Pohar-Perme estimates should be interpreted with caution using error margins. The actuarial approach should be preferred, if data are annually tabulated. Ó 2013 Elsevier Ltd. All rights reserved.
⇑ Corresponding author: Tel.: +358 9 135 33 274; fax: +358 9 135 5378.
E-mail address: arun.pokhrel@cancer.fi (A. Pokhrel). 0959-8049/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejca.2013.09.019
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1. Introduction The population-based cancer registries have used relative survival to give estimates of patients’ net survival, i.e. as far as the patients’ cancer is concerned when eliminating the effects of the other causes of death [1,2]. In this way, no information on causes of death has been needed as the mortality from the other causes (often called expected mortality) has been estimated from life tables of the underlying general population. Recently, a recommendation of using the Ederer II relative survival method was made based on both theoretical and empirical arguments [3]. This recommendation has been also followed, e.g. by the pan-European EUROCARE-5 study (European cancer registry based study on survival and care of cancer patients). Even more recently, a new method to estimate net survival has been proposed by Pohar Perme et al. [4] as a substitute of the relative survival approach. This method is not based on a direct comparison of an observed survival proportion of the patients against an expected survival proportion in the comparable general population group as the relative survival methods. It still uses the general population mortality as an estimate of mortality due to the other causes, so that no information on the actual causes of death is needed. This method, unlike the relative survival methods, has been shown to provide an unbiased estimator of the true net survival, if there is no informative censoring of the observed survival (e.g. censoring that would vary by patients’ age [5]) and continuous time is used in survival calculations. The international CONCORD-2 (Global surveillance of cancer survival) study will use the Pohar-Perme net survival method. Also the relative survival methods, including the Ederer II method, aim to estimate net survival. The Ederer II estimator calculates the cumulative product of the interval-specific relative survival ratios, which are based on unweighted observations of patients alive at the beginning of the corresponding intervals. Therefore, patients who have a high probability of dying due to other causes than cancer get too small weights in estimation of net survival, as a patient’s contribution to net survival is omitted in subsequent intervals after dying. Because net survival depends almost always on the same demographic variables as the expected hazard due to other causes than cancer, the estimator of the Ederer II method becomes biased. In the classical relative survival methods, stratified analyses and their summarisations, e.g. by (age-)standardisation, have been conducted to reduce this bias. In the method of Pohar Perme et al., a patient’s contribution to net survival is weighted on the basis of the patient’s expected survival, i.e. the probability of being alive for a healthy person in the national or other population (comparable with respect to demographic variables e.g. sex, age and calendar year). The method may be viewed also as a generalisation of the gold
standard used in an earlier study [3] into a situation where each patient makes her own group defined by sex, age and year of diagnosis. The choice of weights for each group can also be viewed natural, as in a true gold standard, depending on the cancer-related excess hazard of death only. The present study investigates systematically, using data from the population-based Finnish Cancer Registry and the two publicly available computer programs, how crucial these two assumptions (no informative censoring of the observed survival and use of continuous time) are, particularly the latter one, when a change of method from the traditional relative to the new net survival is done. It is important to know, for national and international population-based cancer survival analyses, how much results obtained by the two methods differ and under which conditions the new method can be recommended in practice. 2. Patients and methods Patients diagnosed in Finland in 1981–1995 and followed-up until the end of 2010 were included in the analysis with stratification by the most common 26 sites. Table 1 shows the list of the sites and the numbers of Table 1 The 26 cancer sites included in the analyses and the numbers of patients diagnosed in Finland in 1981–1995 by site and sex. Cancer site
International Total number of Classification of patients Diseases (ICD)-10 code Males Females
Oesophagus Stomach Colon Rectum, rectosigma, anus Liver Gall bladder, bile ducts Pancreas Larynx Lung, trachea Skin, melanoma Skin, non-melanoma Soft tissues Breast Cervix uteri Corpus uteri Ovary Prostate Testis Kidney Bladder, ureter, urethra Central nervous system Thyroid Hodgkin lymphoma Non-Hodgkin lymphoma Multiple myeloma Leukaemia
C15 C16 C18 C19–20 C22 C23–24 C25 C32 C33–34 C43 C44 C48–49 C50 C53 C54 C56 C61 C62 C64–65 C67–68 C70–72 C73 C81 C82–85, C96 C90 C91–95
1545 8071 5905 5006 1555 1020 4266 1672 25,992 3331 3538 901 – – – – 21,359 893 4626 7235 3747 783 1007 4274 1625 3600
1516 7297 8449 4991 1340 2762 5166 – 5260 3577 4236 970 35,399 2420 7777 6043 – – 3867 2389 5102 3128 775 4620 2035 3299
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diagnosed patients by site and sex. Cancer sites with less than 500 patients were not included. The effect of censoring was studied by using the ends of 1995 and 1999 as the alternative closing dates of follow-up. The overall non-standardised net survival estimates were obtained by the Ederer II relative survival method [6] and by the method proposed by Pohar Perme et al. [4]. The results of applying the methods were compared at 5, 10 and 14 years of follow-up by using exact follow-up times as well as by applying annual and monthly follow-up intervals as a basis of grouping the data. In addition to the point estimates, also the precision of the point estimates was evaluated by investigating the lengths of confidence intervals. In an empirical comparison, a true ‘gold standard’ is not available, as even an unbiased method with no censoring is prone to give estimates with random error. Nevertheless, due to unbiasedness and the recent recommendation [7], the results of the Pohar-Perme method with exact and uncensored follow-up times (i.e. followed-up until the end of 2010) were selected as the gold standard against which the other approaches were compared. The other approaches included the use of monthly or annually grouped observations, also subject to empirical patterns of censoring due to earlier common closing dates (1995 or 1999) and the use of the Ederer II method instead of the Pohar-Perme method. Results were calculated by site, but the estimates of site-specific gold standards are very unstable. Overall survival combining patients of all sites is more stable but a less reasonable measure in practice [8]. Therefore, we focused on results averaged over the various sites with equal weights. The average gives a summary measure that treats the estimation for each site equally important and has a smaller random error than the site-specific results. Age-standardised results were produced by using internal age-standardisation [3,9] based on five age groups: 0–44, 45–54, 55–64, 65–74 and 75+ years. The calculations for the monthly and annually grouped observations were conducted by using the both available computer programs: the original program in R by Pohar Perme [10] (version 2.0-4) and another program in STATA by Dickman et al. [11] (version 1.3.8). The most accurate follow-up time was called the exact follow-up time, although it was based on the exact date at exit (day of death, emigration or the 31st December 2010) and an approximated date of diagnosis, as the exact date of diagnosis is not available. The date of diagnosis was set to be the 15th day of a month of diagnosis, or, if the month of diagnosis and exit were the same, the day in the middle between the 1st day and the day of exit. As the R program had been designed for exact observations, in its grouped data application, following the traditional life table practice, all the deaths and censoring events were placed in mid-points of the respective
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follow-up intervals. We slightly modified the variance of the actuarial estimator of the Pohar-Perme method in STATA program to obtain better approximation for the weighted person-time at risk. Implementations of these different approaches in R and STATA are presented in the Supplementary Web Appendix.
3. Results Colon cancer in males is shown as an example on the comparisons (Fig. 1). The results depend quite a lot on the choice of the method, level of grouping of the data and on patterns of censoring. With annually grouped follow-up times, the Pohar-Perme and the Ederer II methods tend to give much higher values in R, particularly when the data are censored (closing year 1995). The actuarial approach in STATA provides estimates that are much closer to those based on exact follow-up times. Age-standardisation does not remove these differences although it brings the Ederer II and the Pohar-Perme estimates closer to each other when the data are censored. In incomplete follow-up with exact observations, the Pohar-Perme method tends to underestimate longterm net survival, whereas the estimates of Ederer II method are closer to the gold standard. The site-specific results of males are summarised in Supplementary Figs. 1–3. Averaged over the sites, the gold standards of net survival for males were 49.9%, 42.6% and 38.9% for the 5-, 10- and 14-year follow-up, respectively. Annually grouped data in R caused a marked overestimation, particularly when the data were censored and the follow-up was long (Table 2). At 10 and 14 years, the average overestimations in the most heavily censored situation were 3.4% and 6.0% units, respectively, in the Pohar-Perme estimates and 3.3% and 5.3% units, respectively, in the Ederer II estimates. Even with no censoring, the average overestimations were 1.5% and 2.5% units in the PoharPerme estimates and 1.8% and 2.8% units in the Ederer II estimates. The analyses based on the actuarial approach in STATA virtually removed the large differences to the gold standard observed in R with annually grouped data. The use of monthly grouped data mostly reproduced the same average differences as the exact data. With censored observations, the Pohar-Perme method using exact follow-up times tended to underestimate net survival at 10 and 14 years. With the heaviest censored data (closing year 1995), the underestimation was 2.0% units at 14 years (Table 2). The Ederer II method based on exact follow-up times did not give the same negative differences to the gold standard as the Pohar-Perme method, particularly when the follow-up was long and the data heavily censored. But when there was no censoring, this approach
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Closing year 1995 Age-standardised 0.70 0.60
0.65
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.55
0.60
0.35
0.35
0.40
0.45
0.45
0.50
0.50
0.55
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.40
Cumulative net survival
0.65
0.70
Non-standardised
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Years from diagnosis
Years from diagnosis
Closing year 2010
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.55
0.60
0.65
0.70
Age-standardised
0.40
0.45
0.45
0.50
0.50
0.55
0.60
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.35
0.35
0.40
Cumulative net survival
0.65
0.70
Non-standardised
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Years from diagnosis
Years from diagnosis
Fig. 1. Cumulative net survival curves of male colon-cancer patients diagnosed in Finland 1981–1995 and followed-up until the end of 1995 and until the end of 2010 by using the Pohar-Perme (solid lines) and the Ederer II method (dashed lines), two different levels of grouping the data (annually grouped (Anl) and exact follow-up times (Exact)) and the programs in R and STATA. In R, all the events have been placed in the midpoints of the follow-up intervals. Both non-standardised and internally age-standardised curves are shown.
overestimated the gold standard on average with 1.0% and 1.8% units at 10 and 14 years of follow-up, respectively. Lengths of confidence intervals (CIs) of the gold standard of net survival for males were, on average, 4.1%, 5.7% and 8.1% units at 5, 10 and 14 years, respectively. With censored data, the Pohar-Perme method tended to give longer confidence intervals than the gold standard
(Table 2). The Ederer II method did so only with the heaviest censored data, whereas otherwise the confidence intervals were shorter than those of the gold standard. Within data of the same closing year, the average lengths of the confidence intervals of the Pohar-Perme method were 5–8%, 28–39% and 66–75% longer at 5, 10 and 14 years, respectively, than those of the Ederer II method.
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Table 2 Differences (in % units, average of 20 sitesa) to the point estimate and the length of confidence interval of gold standard of net survival (Pohar Perme, exact follow-up times for closing year 2010) at 5, 10 and 14 years by method, program and level of data grouping, for male cancer patients diagnosed in Finland in 1981–1995 and followed-up until the end of three different closing years. Follow-up time (years)
Closing year
Ederer II
Pohar Perme
Annualb R
Monthly STATA
Difference to the point estimate of gold standard 5 1995 1.93 1999 0.95 2010 0.74
R
(49.88%, 42.60% and 0.20 0.15 0.26 0.45 0.23 0.40
STATA
Exact
Annual
R
R
Monthly STATA
38.91% at 5, 10 and 14 years, respectively) 0.00 0.02 1.82 0.36 0.42 0.39 0.76 0.03 0.40 0.37 0.51 0.02
R
Exact STATA
R
0.14 0.11 0.05
0.32 0.06 0.02
0.32 0.04 0
10
1995 1999 2010
3.27 2.49 1.77
0.37 0.42 0.60
0.35 0.97 1.11
0.10 0.86 1.07
0.08 0.82 1.04
3.41 2.36 1.45
0.95 0.35 0.15
0.51 0.06 0.13
0.85 0.27 0.01
0.86 0.29 0
14
1995 1999 2010
5.33 4.14 2.80
0.24 0.92 1.21
0.68 1.75 1.94
0.27 1.55 1.89
0.19 1.49 1.84
6.02 4.35 2.48
1.77 0.35 0.15
1.24 0.01 0.22
1.91 0.36 0.03
2.02 0.42 0
respectively) 0.45 0.49 0.01 0.03 0.03 0.01
0.48 0.02 0
Difference to the length of confidence 5 1995 1999 2010
interval of gold standard (4.11%, 0.16 0.17 0.15 0.54 0.24 0.27 0.54 0.26 0.28
5.66% and 8.09% units 0.18 0.18 0.23 0.24 0.25 0.25
at 5, 10 and 14 years, 0.13 0.42 0.29 0.03 0.29 0.05
10
1995 1999 2010
0.07 1.10 1.47
0.06 0.93 1.34
0.10 0.90 1.31
0.13 0.88 1.29
0.12 0.89 1.30
2.13 0.63 0.12
2.13 0.67 0.12
2.26 0.80 0.01
2.36 0.85 0.02
2.27 0.82 0
14
1995 1999 2010
0.54 2.14 3.41
0.90 2.07 3.37
1.11 1.97 3.31
0.54 2.64 4.02
1.18 1.96 3.30
7.00 2.53 0.12
6.83 1.95 0.28
7.49 2.38 0.01
7.96 2.46 0.07
7.60 2.32 0
a
Cancers of the liver and gallbladder not included as no results were estimable for them at 14-year follow-up with closing date at the end of 1995. Levels of data grouping: annually grouped, monthly grouped and exact follow-up times. In R, all the events (deaths and censorings) were placed in mid-points of the respective follow-up intervals. b
The results obtained for females were quite comparable with those obtained for males (available from the first author on request).
4. Discussion The recommendation [7] to use the Pohar-Perme method [4] is based on the fact that, unlike the traditional relative survival methods, it gives unbiased estimates. This, however, holds true provided that the follow-up times are recorded accurately and used as such and when there is no informative censoring of the observed survival. The former condition cannot always be met in practical applications due to, e.g. non-availability or confidentiality of the data whereas the latter condition can be guaranteed only with a complete follow-up. Fortunately, with the Finnish Cancer Registry’s data, both of these two conditions can be met, and thus it is possible to study the importance of these conditions when they are not met in practice. The site-specific gold standards were prone to random variation. Therefore, it was more difficult to assess the magnitude of bias by site. Averaging the net survival estimates over the sites retains the unbiasedness of the
gold standard and gives case-mix (site) adjusted comparisons, in which each site has the same weight. As the computer program [10] in R requires accurate follow-up times it was necessary to decide how to produce data grouped into follow-up intervals by year or month of follow-up. The old actuarial choice was to place all the events in the middle of the interval. With annually grouped data, this approach proved to be clearly unacceptable. There were many ties between observed and censored survival times, and patients whose survival times were censored at the mid-point of the interval were assumed to remain at risk of dying at the mid-point according to the practice suggested originally by Breslow [12] for handling tied survival observations in the Cox proportional hazards analyses. Exact dates of deaths and diagnosis may not be available or accessible (e.g. due to confidentiality and data protection regulations). On the other hand, the closing date of the study is known allowing more accurate follow-up times for censored patients. In an alternative analysis, ties were removed by using exact follow-up times for patients alive at the end of follow-up. The estimates of this alternative were very close to those based on the exact data in the heaviest censored situation. However, the corresponding estimates were deviating,
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when there was no censoring. It seemed that the R program worked better, when event times were more heterogeneous and not centred at the mid-points of follow-up intervals. A feasible solution in R might be to draw event times of each interval from a uniform distribution. In a routine use of the net survival method, however, these kinds of tricks are not really applicable. Censoring of the observed survival was informative, because patients were diagnosed over a long calendar period during which the distributions of covariates that affect survival (e.g. age at diagnosis) have changed. This emerges, e.g. in ageing populations, when the mean age of diagnosed patients increases over the period of diagnosis, and therefore, older patients have on average shorter times from their diagnosis to the end of the study. The impact of this type of informative censoring on various net survival methods has been studied using simulated accurate follow-up data under various scenarios [13]. Recently, Rebolj Kodre and Pohar Perme proposed a method of inverse probability weighting that allows this type of informative censoring [14]. The method requires estimating probabilities of censoring times and was not used in our study, as it is not available in the current computer programs. Moreover, the Ederer II method was not considered in that paper. A second reason for informative censoring was that the prognosis of the patients changed over the period of diagnosis: patients whose follow-up times were censored earlier had often better prognosis than earlier diagnosed patients who remained under follow-up. This type of informative censoring cannot be corrected for without extrapolation of survival beyond the closing date, and therefore, it is not a problem of the methods, as any corrections are subject to pure guessing [14]. The different sources of bias can be controlled for in simulation-based studies. In empirical data, the second type of informative censoring cannot be eliminated without eliminating the first type of informative censoring, too. In this study, results are based on real data with real progressive censoring owing to early closing dates of follow-up. These real patterns of censoring may well be different in different countries, but a good net survival method should be resistant against biases any pattern of informative censoring might cause. As opposed to simulated data, however, the targeted gold standard under complete follow-up and accurate follow-up times is still a random quantity with a standard error. When the data are censored, the Ederer II method could be preferred as it gives results closer to the gold standard. This may, however, be a characteristic due to the particular censoring pattern in the Finnish data, as the positive bias in the Ederer II method was compensated by the negative bias due to informative censoring of the observed survival. In the setting of cause-specific survival, deaths due to other causes than cancer are considered as censoring
events. The Kaplan–Meier estimator is biased under informative censoring but can be corrected for by following the idea of inverse probability weighting [15] that was adapted to the framework of relative survival by Pohar Perme et al. [4]. Of course, informative censoring caused by changes in patients’ prognosis cannot be corrected for in cause-specific survival, either. In cause-specific survival, cause of death is not always correct, whereas, in the framework of relative survival, the expected survival estimated from the mortality rates of national population may not always be relevant for the patients [8]. This is the main reason for differences between results of the two approaches which both aim to estimate net survival. In the Pohar-Perme method the few observations in the old age groups get large weights, because the competing risks of death do not leave for older ages sufficiently sizable materials on which to base reliable estimation [16]. This can be seen in the standard errors and the confidence intervals based on them. The Ederer II method gives estimates that have distinctively narrower confidence intervals than those derived by the Pohar-Perme method. It is likely that in this respect the gold standard is far from a true gold standard. The age-standardisation is not a solution for removing biases related to the level of grouping and informative censoring or inaccuracies related to interval estimation. Statistical modelling [17–19] is capable of finding the essence also in net survival analyses and should be developed into a standard for routine use on a large scale. Otherwise, it is crucial to report in which way the net survival results have been obtained. Net survival is especially useful for evaluating differences in cancer survival between population groups and over time, when the expected mortality differs across the groups we wish to compare. Thus, in future studies, it would be important to assess whether the choice of the approach could actually affect results of comparisons between population groups. A clear recommendation to use the net survival method by Pohar Perme et al. is conditional on the completeness of follow-up and the time point of follow-up at which the net survival is wished to be estimated. Both methods perform well in the estimation of net survival until 5 years. In complete follow-up, the Pohar-Perme estimator can be preferred in terms of bias but point estimates of the long-term net survival should be interpreted with due caution, because the estimator becomes prone to random variation. In incomplete follow-up, the estimator of the Pohar-Perme method may be biased if censoring of the observed survival is informative, even if the recently developed weighting method [14] was used. Irrespective of the method, the actuarial approach in STATA should be utilised, if data are grouped into annual follow-up intervals. Following this recommendation, the results
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given by different approaches differ on average by 1–2% units depending on the context. Conflict of interest statement None declared. Acknowledgement This work was supported by a grant from the Finnish Cancer Foundation. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/ 10.1016/j.ejca.2013.09.019. References [1] Ries LAG, Melbert D, Krapcho M, et al., editors. SEER cancer statistics review, 1975–2004. National Cancer Institute: Bethesda, MD; 2007. [2] Coleman MP, Quaresma M, Berrino F, et al. Cancer survival in five continents: a worldwide population-based study (CONCORD). Lancet Oncol 2008;9:730–56. [3] Hakulinen T, Seppa¨ K, Lambert PC. Choosing the relative survival method for cancer survival estimation. Eur J Cancer 2011;47:2202–10. [4] Pohar Perme M, Stare J, Este`ve J. On estimation in relative survival. Biometrics 2012;68:113–20. [5] Hakulinen T. Cancer survival corrected for heterogeneity in patient withdrawal. Biometrics 1982;38:933–42. [6] Ederer F, Heise H. Instructions to IBM 650 programmers in processing survival computations. Methodological note no. 10. Bethesda, MD: End Results Evaluation Section, National Cancer Institute; 1959.
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Choosing the net survival method for cancer survival estimation Karri Seppa¨ a,b, Timo Hakulinen a, Arun Pokhrel a,⇑ a b
Finnish Cancer Registry, Institute for Statistical and Epidemiological Cancer Research, Pieni Roobertinkatu 9, FI-00130 Helsinki, Finland Department of Mathematical Sciences, University of Oulu, Oulu, Finland
KEYWORDS Epidemiologic methods Models Neoplasms Prognosis Relative survival Net survival
Abstract Background: A new net survival method has been introduced by Pohar Perme et al. (2012 [4]) and recommended to substitute the relative survival methods in current use for evaluating population-based cancer survival. Methods: The new method is based on the use of continuous follow-up time, and is unbiased only under non-informative censoring of the observed survival. However, the populationbased cancer survival is often evaluated based on annually or monthly tabulated follow-up intervals. An empirical investigation based on data from the Finnish Cancer Registry was made into the practical importance of the censoring and the level of data tabulation. A systematic comparison was made against the earlier recommended Ederer II method of relative survival using the two currently available computer programs (Pohar Perme (2013) [10] and Dickman et al. (2013) [11]). Results: With exact or monthly tabulated data, the Pohar-Perme and the Ederer II methods give, on average, results that are at five years of follow-up less than 0.5% units and at 10 and 14 years 1–2% units apart from each other. The Pohar-Perme net survival estimator is prone to random variation and may result in biased estimates when exact follow-up times are not available or follow-up is incomplete. With annually tabulated follow-up times, estimates can deviate substantially from those based on more accurate observations, if the actuarial approach is not used. Conclusion: At 5 years, both the methods perform well. In longer follow-up, the Pohar-Perme estimates should be interpreted with caution using error margins. The actuarial approach should be preferred, if data are annually tabulated. Ó 2013 Elsevier Ltd. All rights reserved.
⇑ Corresponding author: Tel.: +358 9 135 33 274; fax: +358 9 135 5378.
E-mail address: arun.pokhrel@cancer.fi (A. Pokhrel). 0959-8049/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ejca.2013.09.019
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1. Introduction The population-based cancer registries have used relative survival to give estimates of patients’ net survival, i.e. as far as the patients’ cancer is concerned when eliminating the effects of the other causes of death [1,2]. In this way, no information on causes of death has been needed as the mortality from the other causes (often called expected mortality) has been estimated from life tables of the underlying general population. Recently, a recommendation of using the Ederer II relative survival method was made based on both theoretical and empirical arguments [3]. This recommendation has been also followed, e.g. by the pan-European EUROCARE-5 study (European cancer registry based study on survival and care of cancer patients). Even more recently, a new method to estimate net survival has been proposed by Pohar Perme et al. [4] as a substitute of the relative survival approach. This method is not based on a direct comparison of an observed survival proportion of the patients against an expected survival proportion in the comparable general population group as the relative survival methods. It still uses the general population mortality as an estimate of mortality due to the other causes, so that no information on the actual causes of death is needed. This method, unlike the relative survival methods, has been shown to provide an unbiased estimator of the true net survival, if there is no informative censoring of the observed survival (e.g. censoring that would vary by patients’ age [5]) and continuous time is used in survival calculations. The international CONCORD-2 (Global surveillance of cancer survival) study will use the Pohar-Perme net survival method. Also the relative survival methods, including the Ederer II method, aim to estimate net survival. The Ederer II estimator calculates the cumulative product of the interval-specific relative survival ratios, which are based on unweighted observations of patients alive at the beginning of the corresponding intervals. Therefore, patients who have a high probability of dying due to other causes than cancer get too small weights in estimation of net survival, as a patient’s contribution to net survival is omitted in subsequent intervals after dying. Because net survival depends almost always on the same demographic variables as the expected hazard due to other causes than cancer, the estimator of the Ederer II method becomes biased. In the classical relative survival methods, stratified analyses and their summarisations, e.g. by (age-)standardisation, have been conducted to reduce this bias. In the method of Pohar Perme et al., a patient’s contribution to net survival is weighted on the basis of the patient’s expected survival, i.e. the probability of being alive for a healthy person in the national or other population (comparable with respect to demographic variables e.g. sex, age and calendar year). The method may be viewed also as a generalisation of the gold
standard used in an earlier study [3] into a situation where each patient makes her own group defined by sex, age and year of diagnosis. The choice of weights for each group can also be viewed natural, as in a true gold standard, depending on the cancer-related excess hazard of death only. The present study investigates systematically, using data from the population-based Finnish Cancer Registry and the two publicly available computer programs, how crucial these two assumptions (no informative censoring of the observed survival and use of continuous time) are, particularly the latter one, when a change of method from the traditional relative to the new net survival is done. It is important to know, for national and international population-based cancer survival analyses, how much results obtained by the two methods differ and under which conditions the new method can be recommended in practice. 2. Patients and methods Patients diagnosed in Finland in 1981–1995 and followed-up until the end of 2010 were included in the analysis with stratification by the most common 26 sites. Table 1 shows the list of the sites and the numbers of Table 1 The 26 cancer sites included in the analyses and the numbers of patients diagnosed in Finland in 1981–1995 by site and sex. Cancer site
International Total number of Classification of patients Diseases (ICD)-10 code Males Females
Oesophagus Stomach Colon Rectum, rectosigma, anus Liver Gall bladder, bile ducts Pancreas Larynx Lung, trachea Skin, melanoma Skin, non-melanoma Soft tissues Breast Cervix uteri Corpus uteri Ovary Prostate Testis Kidney Bladder, ureter, urethra Central nervous system Thyroid Hodgkin lymphoma Non-Hodgkin lymphoma Multiple myeloma Leukaemia
C15 C16 C18 C19–20 C22 C23–24 C25 C32 C33–34 C43 C44 C48–49 C50 C53 C54 C56 C61 C62 C64–65 C67–68 C70–72 C73 C81 C82–85, C96 C90 C91–95
1545 8071 5905 5006 1555 1020 4266 1672 25,992 3331 3538 901 – – – – 21,359 893 4626 7235 3747 783 1007 4274 1625 3600
1516 7297 8449 4991 1340 2762 5166 – 5260 3577 4236 970 35,399 2420 7777 6043 – – 3867 2389 5102 3128 775 4620 2035 3299
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diagnosed patients by site and sex. Cancer sites with less than 500 patients were not included. The effect of censoring was studied by using the ends of 1995 and 1999 as the alternative closing dates of follow-up. The overall non-standardised net survival estimates were obtained by the Ederer II relative survival method [6] and by the method proposed by Pohar Perme et al. [4]. The results of applying the methods were compared at 5, 10 and 14 years of follow-up by using exact follow-up times as well as by applying annual and monthly follow-up intervals as a basis of grouping the data. In addition to the point estimates, also the precision of the point estimates was evaluated by investigating the lengths of confidence intervals. In an empirical comparison, a true ‘gold standard’ is not available, as even an unbiased method with no censoring is prone to give estimates with random error. Nevertheless, due to unbiasedness and the recent recommendation [7], the results of the Pohar-Perme method with exact and uncensored follow-up times (i.e. followed-up until the end of 2010) were selected as the gold standard against which the other approaches were compared. The other approaches included the use of monthly or annually grouped observations, also subject to empirical patterns of censoring due to earlier common closing dates (1995 or 1999) and the use of the Ederer II method instead of the Pohar-Perme method. Results were calculated by site, but the estimates of site-specific gold standards are very unstable. Overall survival combining patients of all sites is more stable but a less reasonable measure in practice [8]. Therefore, we focused on results averaged over the various sites with equal weights. The average gives a summary measure that treats the estimation for each site equally important and has a smaller random error than the site-specific results. Age-standardised results were produced by using internal age-standardisation [3,9] based on five age groups: 0–44, 45–54, 55–64, 65–74 and 75+ years. The calculations for the monthly and annually grouped observations were conducted by using the both available computer programs: the original program in R by Pohar Perme [10] (version 2.0-4) and another program in STATA by Dickman et al. [11] (version 1.3.8). The most accurate follow-up time was called the exact follow-up time, although it was based on the exact date at exit (day of death, emigration or the 31st December 2010) and an approximated date of diagnosis, as the exact date of diagnosis is not available. The date of diagnosis was set to be the 15th day of a month of diagnosis, or, if the month of diagnosis and exit were the same, the day in the middle between the 1st day and the day of exit. As the R program had been designed for exact observations, in its grouped data application, following the traditional life table practice, all the deaths and censoring events were placed in mid-points of the respective
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follow-up intervals. We slightly modified the variance of the actuarial estimator of the Pohar-Perme method in STATA program to obtain better approximation for the weighted person-time at risk. Implementations of these different approaches in R and STATA are presented in the Supplementary Web Appendix.
3. Results Colon cancer in males is shown as an example on the comparisons (Fig. 1). The results depend quite a lot on the choice of the method, level of grouping of the data and on patterns of censoring. With annually grouped follow-up times, the Pohar-Perme and the Ederer II methods tend to give much higher values in R, particularly when the data are censored (closing year 1995). The actuarial approach in STATA provides estimates that are much closer to those based on exact follow-up times. Age-standardisation does not remove these differences although it brings the Ederer II and the Pohar-Perme estimates closer to each other when the data are censored. In incomplete follow-up with exact observations, the Pohar-Perme method tends to underestimate longterm net survival, whereas the estimates of Ederer II method are closer to the gold standard. The site-specific results of males are summarised in Supplementary Figs. 1–3. Averaged over the sites, the gold standards of net survival for males were 49.9%, 42.6% and 38.9% for the 5-, 10- and 14-year follow-up, respectively. Annually grouped data in R caused a marked overestimation, particularly when the data were censored and the follow-up was long (Table 2). At 10 and 14 years, the average overestimations in the most heavily censored situation were 3.4% and 6.0% units, respectively, in the Pohar-Perme estimates and 3.3% and 5.3% units, respectively, in the Ederer II estimates. Even with no censoring, the average overestimations were 1.5% and 2.5% units in the PoharPerme estimates and 1.8% and 2.8% units in the Ederer II estimates. The analyses based on the actuarial approach in STATA virtually removed the large differences to the gold standard observed in R with annually grouped data. The use of monthly grouped data mostly reproduced the same average differences as the exact data. With censored observations, the Pohar-Perme method using exact follow-up times tended to underestimate net survival at 10 and 14 years. With the heaviest censored data (closing year 1995), the underestimation was 2.0% units at 14 years (Table 2). The Ederer II method based on exact follow-up times did not give the same negative differences to the gold standard as the Pohar-Perme method, particularly when the follow-up was long and the data heavily censored. But when there was no censoring, this approach
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Closing year 1995 Age-standardised 0.70 0.60
0.65
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.55
0.60
0.35
0.35
0.40
0.45
0.45
0.50
0.50
0.55
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.40
Cumulative net survival
0.65
0.70
Non-standardised
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Years from diagnosis
Years from diagnosis
Closing year 2010
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.55
0.60
0.65
0.70
Age-standardised
0.40
0.45
0.45
0.50
0.50
0.55
0.60
Anl, R (EdII) Anl, STATA (EdII) Exact, R (EdII) Anl, R (PP) Anl, STATA (PP) Exact, R (PP)
0.35
0.35
0.40
Cumulative net survival
0.65
0.70
Non-standardised
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Years from diagnosis
Years from diagnosis
Fig. 1. Cumulative net survival curves of male colon-cancer patients diagnosed in Finland 1981–1995 and followed-up until the end of 1995 and until the end of 2010 by using the Pohar-Perme (solid lines) and the Ederer II method (dashed lines), two different levels of grouping the data (annually grouped (Anl) and exact follow-up times (Exact)) and the programs in R and STATA. In R, all the events have been placed in the midpoints of the follow-up intervals. Both non-standardised and internally age-standardised curves are shown.
overestimated the gold standard on average with 1.0% and 1.8% units at 10 and 14 years of follow-up, respectively. Lengths of confidence intervals (CIs) of the gold standard of net survival for males were, on average, 4.1%, 5.7% and 8.1% units at 5, 10 and 14 years, respectively. With censored data, the Pohar-Perme method tended to give longer confidence intervals than the gold standard
(Table 2). The Ederer II method did so only with the heaviest censored data, whereas otherwise the confidence intervals were shorter than those of the gold standard. Within data of the same closing year, the average lengths of the confidence intervals of the Pohar-Perme method were 5–8%, 28–39% and 66–75% longer at 5, 10 and 14 years, respectively, than those of the Ederer II method.
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Table 2 Differences (in % units, average of 20 sitesa) to the point estimate and the length of confidence interval of gold standard of net survival (Pohar Perme, exact follow-up times for closing year 2010) at 5, 10 and 14 years by method, program and level of data grouping, for male cancer patients diagnosed in Finland in 1981–1995 and followed-up until the end of three different closing years. Follow-up time (years)
Closing year
Ederer II
Pohar Perme
Annualb R
Monthly STATA
Difference to the point estimate of gold standard 5 1995 1.93 1999 0.95 2010 0.74
R
(49.88%, 42.60% and 0.20 0.15 0.26 0.45 0.23 0.40
STATA
Exact
Annual
R
R
Monthly STATA
38.91% at 5, 10 and 14 years, respectively) 0.00 0.02 1.82 0.36 0.42 0.39 0.76 0.03 0.40 0.37 0.51 0.02
R
Exact STATA
R
0.14 0.11 0.05
0.32 0.06 0.02
0.32 0.04 0
10
1995 1999 2010
3.27 2.49 1.77
0.37 0.42 0.60
0.35 0.97 1.11
0.10 0.86 1.07
0.08 0.82 1.04
3.41 2.36 1.45
0.95 0.35 0.15
0.51 0.06 0.13
0.85 0.27 0.01
0.86 0.29 0
14
1995 1999 2010
5.33 4.14 2.80
0.24 0.92 1.21
0.68 1.75 1.94
0.27 1.55 1.89
0.19 1.49 1.84
6.02 4.35 2.48
1.77 0.35 0.15
1.24 0.01 0.22
1.91 0.36 0.03
2.02 0.42 0
respectively) 0.45 0.49 0.01 0.03 0.03 0.01
0.48 0.02 0
Difference to the length of confidence 5 1995 1999 2010
interval of gold standard (4.11%, 0.16 0.17 0.15 0.54 0.24 0.27 0.54 0.26 0.28
5.66% and 8.09% units 0.18 0.18 0.23 0.24 0.25 0.25
at 5, 10 and 14 years, 0.13 0.42 0.29 0.03 0.29 0.05
10
1995 1999 2010
0.07 1.10 1.47
0.06 0.93 1.34
0.10 0.90 1.31
0.13 0.88 1.29
0.12 0.89 1.30
2.13 0.63 0.12
2.13 0.67 0.12
2.26 0.80 0.01
2.36 0.85 0.02
2.27 0.82 0
14
1995 1999 2010
0.54 2.14 3.41
0.90 2.07 3.37
1.11 1.97 3.31
0.54 2.64 4.02
1.18 1.96 3.30
7.00 2.53 0.12
6.83 1.95 0.28
7.49 2.38 0.01
7.96 2.46 0.07
7.60 2.32 0
a
Cancers of the liver and gallbladder not included as no results were estimable for them at 14-year follow-up with closing date at the end of 1995. Levels of data grouping: annually grouped, monthly grouped and exact follow-up times. In R, all the events (deaths and censorings) were placed in mid-points of the respective follow-up intervals. b
The results obtained for females were quite comparable with those obtained for males (available from the first author on request).
4. Discussion The recommendation [7] to use the Pohar-Perme method [4] is based on the fact that, unlike the traditional relative survival methods, it gives unbiased estimates. This, however, holds true provided that the follow-up times are recorded accurately and used as such and when there is no informative censoring of the observed survival. The former condition cannot always be met in practical applications due to, e.g. non-availability or confidentiality of the data whereas the latter condition can be guaranteed only with a complete follow-up. Fortunately, with the Finnish Cancer Registry’s data, both of these two conditions can be met, and thus it is possible to study the importance of these conditions when they are not met in practice. The site-specific gold standards were prone to random variation. Therefore, it was more difficult to assess the magnitude of bias by site. Averaging the net survival estimates over the sites retains the unbiasedness of the
gold standard and gives case-mix (site) adjusted comparisons, in which each site has the same weight. As the computer program [10] in R requires accurate follow-up times it was necessary to decide how to produce data grouped into follow-up intervals by year or month of follow-up. The old actuarial choice was to place all the events in the middle of the interval. With annually grouped data, this approach proved to be clearly unacceptable. There were many ties between observed and censored survival times, and patients whose survival times were censored at the mid-point of the interval were assumed to remain at risk of dying at the mid-point according to the practice suggested originally by Breslow [12] for handling tied survival observations in the Cox proportional hazards analyses. Exact dates of deaths and diagnosis may not be available or accessible (e.g. due to confidentiality and data protection regulations). On the other hand, the closing date of the study is known allowing more accurate follow-up times for censored patients. In an alternative analysis, ties were removed by using exact follow-up times for patients alive at the end of follow-up. The estimates of this alternative were very close to those based on the exact data in the heaviest censored situation. However, the corresponding estimates were deviating,
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when there was no censoring. It seemed that the R program worked better, when event times were more heterogeneous and not centred at the mid-points of follow-up intervals. A feasible solution in R might be to draw event times of each interval from a uniform distribution. In a routine use of the net survival method, however, these kinds of tricks are not really applicable. Censoring of the observed survival was informative, because patients were diagnosed over a long calendar period during which the distributions of covariates that affect survival (e.g. age at diagnosis) have changed. This emerges, e.g. in ageing populations, when the mean age of diagnosed patients increases over the period of diagnosis, and therefore, older patients have on average shorter times from their diagnosis to the end of the study. The impact of this type of informative censoring on various net survival methods has been studied using simulated accurate follow-up data under various scenarios [13]. Recently, Rebolj Kodre and Pohar Perme proposed a method of inverse probability weighting that allows this type of informative censoring [14]. The method requires estimating probabilities of censoring times and was not used in our study, as it is not available in the current computer programs. Moreover, the Ederer II method was not considered in that paper. A second reason for informative censoring was that the prognosis of the patients changed over the period of diagnosis: patients whose follow-up times were censored earlier had often better prognosis than earlier diagnosed patients who remained under follow-up. This type of informative censoring cannot be corrected for without extrapolation of survival beyond the closing date, and therefore, it is not a problem of the methods, as any corrections are subject to pure guessing [14]. The different sources of bias can be controlled for in simulation-based studies. In empirical data, the second type of informative censoring cannot be eliminated without eliminating the first type of informative censoring, too. In this study, results are based on real data with real progressive censoring owing to early closing dates of follow-up. These real patterns of censoring may well be different in different countries, but a good net survival method should be resistant against biases any pattern of informative censoring might cause. As opposed to simulated data, however, the targeted gold standard under complete follow-up and accurate follow-up times is still a random quantity with a standard error. When the data are censored, the Ederer II method could be preferred as it gives results closer to the gold standard. This may, however, be a characteristic due to the particular censoring pattern in the Finnish data, as the positive bias in the Ederer II method was compensated by the negative bias due to informative censoring of the observed survival. In the setting of cause-specific survival, deaths due to other causes than cancer are considered as censoring
events. The Kaplan–Meier estimator is biased under informative censoring but can be corrected for by following the idea of inverse probability weighting [15] that was adapted to the framework of relative survival by Pohar Perme et al. [4]. Of course, informative censoring caused by changes in patients’ prognosis cannot be corrected for in cause-specific survival, either. In cause-specific survival, cause of death is not always correct, whereas, in the framework of relative survival, the expected survival estimated from the mortality rates of national population may not always be relevant for the patients [8]. This is the main reason for differences between results of the two approaches which both aim to estimate net survival. In the Pohar-Perme method the few observations in the old age groups get large weights, because the competing risks of death do not leave for older ages sufficiently sizable materials on which to base reliable estimation [16]. This can be seen in the standard errors and the confidence intervals based on them. The Ederer II method gives estimates that have distinctively narrower confidence intervals than those derived by the Pohar-Perme method. It is likely that in this respect the gold standard is far from a true gold standard. The age-standardisation is not a solution for removing biases related to the level of grouping and informative censoring or inaccuracies related to interval estimation. Statistical modelling [17–19] is capable of finding the essence also in net survival analyses and should be developed into a standard for routine use on a large scale. Otherwise, it is crucial to report in which way the net survival results have been obtained. Net survival is especially useful for evaluating differences in cancer survival between population groups and over time, when the expected mortality differs across the groups we wish to compare. Thus, in future studies, it would be important to assess whether the choice of the approach could actually affect results of comparisons between population groups. A clear recommendation to use the net survival method by Pohar Perme et al. is conditional on the completeness of follow-up and the time point of follow-up at which the net survival is wished to be estimated. Both methods perform well in the estimation of net survival until 5 years. In complete follow-up, the Pohar-Perme estimator can be preferred in terms of bias but point estimates of the long-term net survival should be interpreted with due caution, because the estimator becomes prone to random variation. In incomplete follow-up, the estimator of the Pohar-Perme method may be biased if censoring of the observed survival is informative, even if the recently developed weighting method [14] was used. Irrespective of the method, the actuarial approach in STATA should be utilised, if data are grouped into annual follow-up intervals. Following this recommendation, the results
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given by different approaches differ on average by 1–2% units depending on the context. Conflict of interest statement None declared. Acknowledgement This work was supported by a grant from the Finnish Cancer Foundation. Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/ 10.1016/j.ejca.2013.09.019. References [1] Ries LAG, Melbert D, Krapcho M, et al., editors. SEER cancer statistics review, 1975–2004. National Cancer Institute: Bethesda, MD; 2007. [2] Coleman MP, Quaresma M, Berrino F, et al. Cancer survival in five continents: a worldwide population-based study (CONCORD). Lancet Oncol 2008;9:730–56. [3] Hakulinen T, Seppa¨ K, Lambert PC. Choosing the relative survival method for cancer survival estimation. Eur J Cancer 2011;47:2202–10. [4] Pohar Perme M, Stare J, Este`ve J. On estimation in relative survival. Biometrics 2012;68:113–20. [5] Hakulinen T. Cancer survival corrected for heterogeneity in patient withdrawal. Biometrics 1982;38:933–42. [6] Ederer F, Heise H. Instructions to IBM 650 programmers in processing survival computations. Methodological note no. 10. Bethesda, MD: End Results Evaluation Section, National Cancer Institute; 1959.
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