About me. An overview and some recent advances in statistical methods for population-based cancer survival analysis. Research interests

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1 An overview and some recent advances in statistical methods for population-based cancer survival analysis Paul W Dickman Paul C Lambert,2 Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden 2 Department of Health Sciences, University of Leicester, UK 9 April 24 About me Born in Sydney Australia; studied mathematics and statistics in Newcastle (Australia). Worked in health services research; dabbled in industrial process control and quality improvement. Arrived in Sweden November 993 for a month visit to cancer epidemiology unit at Radiumhemmet. Stayed in Sweden for most of my PhD. Short Postdoc periods at Finnish Cancer Registry and Karolinska Institutet (clinical epidemiology). Joined Hans-Olov Adami s new department (now MEB) in March 999, attracted by the strong research environment and possibilities for register-based epidemiology. 9 April 24 2 Research interests Primary research interests are in the development and application of methods for population-based cancer survival analysis, particularly the estimation and modeling of relative survival. Recent interest has been in presenting information on patient survival in a manner relevant for patients and clinicians. General interest in statistical aspects of the design, analysis, and reporting of epidemiological studies along with studies of disease aetiology, with particular focus on cancer epidemiology and perinatal/reproductive epidemiology. Collaborate closely with Paul Lambert (Biostatistician at University of Leicester) and Magnus Björkholm (Haematologist). Overview of today s talk (to what I understand is a heterogeneous audience) Measures used in cancer control; why study patient survival. Intro to relative survival (excess mortality) and why it is the measure of choice for population-based cancer survival analysis. Flexible parametric models (Royston-Parmar models). The concept of statistical cure; cure models. Estimating crude and net probabilities of death. Partitioning excess mortality; treatment-related CVD mortality.. 9 April April 24 4 r year, per y rate per or mortality ncidence o In Incidence Mortality Lung cancer incidence, mortality and survival (age-standardised) England, , by sex Year of death / Men Women Women Men al (%) ve surviva ear relativ Five-ye Lung cancer incidence in Sweden 9 April 24 9 April 24 6 How might we measure the prognosis of cancer patients? Need to consider competing risks [] Total mortality (among the patients). Our interest is typically in net mortality (mortality associated with a diagnosis of cancer). Cause-specific mortality provides an estimate of net mortality (under certain assumptions). When estimating cause-specific mortality only those deaths which can be attributed to the cancer in question are considered to be events. cause-specific mortality = number of deaths due to cancer person-time at risk The survival times of patients who die of causes other than cancer are censored. Figure The competing risks multi-state model. 9 April April 24 8

2 Many synonyms for the same concept Net probability of death due to cancer Crude probability of death due to cancer = = Probability of death in a hypothetical world where the cancer under study is the only possible cause of death ProbabilityProstate of deathcancer in the real world where you may die of other causes before the cancer kills you Net (left) and crude (right) probabilities of death in men with localized prostate cancer aged 7+ at diagnosis (Cronin and Feuer [2]) Net probability also known as the marginal probability. Crude probability also known as the cause-specific cumulative incidence function (Geskus) or the cumulative Helene incidence Hartvedtfunction. Grytli a, Morten Wang Fagerland b, Sophie D. Fosså c,d, Kristin Austlid Taskén a,d, * 9 April a April 24 Department of Tumor Biology, Institute of Cancer Research, Oslo University Hospital, Oslo, Norway; b Unit of Biostatistics and Epidemiology, Oslo University Hospital, Oslo, Norway; c Department of Oncology, Oslo University Hospital, Oslo, Norway; d Institute of Clinical Medicine, University of Oslo, Oslo, Norway Should we use crude or net survival? available at journal homepage: EUROPEAN UROLOGY 6 (24) Association Between Use of b-blockers and Prostate Cancer Specific : A Cohort Study of 36 Prostate Cancer Patients with High-Risk or Metastatic Disease Article info Should we use crude or net survival? Abstract Comparing patient survival between countries. 2 Studying temporal trends in patient survival. 3 Communicating prognosis to patients. Article history: Accepted January 6, 23 Published online ahead of print on January 4, 23 Keywords: ASA b-adrenergic receptor antagonist b-blocker Epidemiology High-risk prostate cancer Metastasis Norway Prostate cancer Prostate cancer specific mortality Statin 9 April 24 Cause-specific survival can estimate net survival (assuming conditional independence) Background: We recently reported reduced prostate cancer (PCa) specific mortality for b-blocker users among patients receiving androgen-deprivation therapy in a health survey cohort including 6 PCa patients. Information on clinical characteristics was limited. Objective: To assess the association between b-blockers and PCa-specific mortality in a cohort of 36 prostate cancer patients with high-risk or metastatic disease, and to address potential confounding from the use of statins or acetylsalicylic acid (ASA). Design, setting, and participants: Clinical information from all men reported to the Cancer Registry of Norway with a PCa diagnosis between 24 and 29 (n = 24 7) was coupled with information on filled prescriptions between 24 and 2 from the Norwegian Prescription Database. Exclusion criteria were low- or intermediate-risk disease; planned radiotherapy or radical prostatectomy; initiation of b-blocker, ASA, or statin use after diagnosis where applicable; missing information on baseline Gleason score, prostate-specific antigen level, T stage or performance status; and missing follow-up. Outcome measurements and statistical analysis: Cox proportional hazards modelling and competing risk regression modelling were used to analyse the effects of b-blocker use on all-cause and PCa-specific mortality, respectively. Differences between b-blocker users and nonusers regarding baseline clinical characteristics were assessed by the Wilcoxon-Mann-Whitney U test, Pearson chi-square test, and Student t test. Results and limitations: Median follow-up was 39 mo. b-blocker use was associated with reduced PCa mortality (adjusted subhazard ratio:.79; 9% confidence interval [CI], 8.9; p value:.). The observed reduction in PCa mortality was independent of the use of statins or ASA. We observed no association with all-cause mortality 9 April 24 2 (adjusted hazard ratio:.92; 9% CI, 3 ). The main limitations of the study were the observational study design and short follow-up. Conclusions: b-blocker use was associated with reduced PCa-specific mortality in patients with high-risk or metastatic disease at the time of diagnosis. Our findings need validation from further observational studies. # 23 European Association of Urology. Published by Elsevier B.V. All rights reserved. All-cause mortality for males with colon cancer and Finnish population Using cause-specific methods requires that reliably coded information on cause of death is available. Even when cause of death information is available to the cancer registry via death certificates, it is often vague and difficult to determine whether or not cancer is the primary cause of death. How do we classify, for example, deaths due to treatment complications? Consider a patient treated with radiation therapy and chemotherapy who dies of cardiovascular disease. Do we classify this death as due entirely to cancer or due entirely to other causes? * Corresponding 8 author. Oslo University General Hospital, populationinstitute of Cancer Research, Department of Tumor Biology, P.O. Box 493 Nydalen, Colon NO-424 cancer Oslo, patients Norway. Tel ; Fax: address: 7 k.a.tasken@medisin.uio.no (K.A. Taskén). Mortality Rate (per, person years) /$ see back matter # 23 European Association of Urology. Published by Elsevier B.V. All rights reserved Age 9 April April 24 4 Relative survival aims to estimate net survival (still need conditional independence) Cervical cancer in New Zealand Life table estimates of patient survival We estimate excess mortality: the difference between observed (all-cause) and expected mortality. excess = observed expected mortality mortality mortality Relative survival is the survival analog of excess mortality the relative survival ratio is defined as the observed survival in the patient group divided by the expected survival of a comparable group from the general population. relative survival ratio = observed survival proportion expected survival proportion Women diagnosed with follow-up to the end of 22 Interval- Interval- Effective specific specific Cumulative Cumulative Cumulative number observed relative observed expected relative I N D W at risk survival survival survival survival survival April 24 9 April 24 6

3 Relative survival example (skin melanoma) Table : Number of cases (N) and -year observed (p), expected (p ), and relative (r) survival for males diagnosed with localised skin melanoma in Finland during Age N p p r Relative survival controls for the fact that expected mortality depends on demographic characteristics (age, sex, etc.). In addition, relative survival may, and usually does, depend on such factors. 9 April 24 7 Which approach for estimating net survival? A common question in our teaching. Unless there is reason to prefer a cause-specific approach, we recommend one of the following: Ederer II (99 [?]) Pohar Perme (22 [3]) Model-based The choice of method depends on the research question and practical considerations. We were recently critical [4] of a paper [] that advocated the Pohar Perme approach; our critisism was of that particular paper and not the Pohar Perme approach per se, of which we are great admirers. Breakthrough paper in 22; Biometrics 68, 3 2 An unbiased estimator for net survival [3] March 22 On Estimation in Relative DOI:./j-424.x Maja Pohar Perme,, Janez Stare, and Jacques Estève 2 Department of Biostatistics and Medical Informatics, University of Ljubljana, Vrazov trg 2, SI- Ljubljana, Slovenia 2 Université Claude Bernard, Hospices Civils de Lyon, Service de Biostatistique, 62, Avenue Lacassagne Lyon Cedex 3, France maja.pohar@mf.uni-lj.si Summary. Estimation of relative survival has become the first and the most basic step when reporting cancer survival statistics. Standard estimators are in routine use by all cancer registries. However, it has been recently noted that these estimators do not provide information on cancer mortality that is independent of the national general population mortality. Thus they are not suitable for comparison between countries. Furthermore, the commonly used interpretation of the relative survival curve is vague and misleading. The present article attempts to remedy these basic problems. The population quantities of the traditional estimators are carefully described and their interpretation discussed. We then propose a new estimator of net survival probability that enables the desired comparability between countries. The new estimator requires no modeling and is accompanied with a straightforward variance estimate. The methods are described on real as well as simulated data. Key words: Age standardization; Cancer registry data; Competing risks; Net survival; Relative survival; analysis.. Introduction observed hazard is larger than the population hazard. probability of cancer patients Population-Based has been used Cancer for many Under 9 April this 24 assumption, we use the term excess hazard for the hazard due to the disease and we have the 8 years as one of the main tools for evaluation of therapeutic advances. With improved treatments and prognosis, studies relation often now have long follow-up times and it is common to have observed hazard = population hazard+ excess hazard. a substantial proportion of deaths from causes other than the cancer under study. In the usual situation, the cause of death () is unavailable or unreliable. Hence the field of relative survival A survival function derived from the excess hazard has developed in which observed deaths are compared with alone is termed the net survival. those expected from general population life tables. We distinguish two goals: In this article, we shall focus on three estimators in widespread use in relative survival. We will refer to them () To We compare arethenot observed suggesting survival (SO) tothesurvival that theasederer Ederer I (Ederer, II approach Axtell, and Cutler, is superior 96), Hakulinen to of a disease-free group having the same demographic (Hakulinen, 982), and Ederer II (Ederer et al., 96, p. ) characteristics the Pohar as theperme study group. approach. If the expected general population survival is SP, the comparison is made Although all three were proposed as variants for estimat- and define them in Section 3 below. using However, the ratio we argue that it is not ing as the denominator inferiorof as the relative somesurvival others ratio, they (e.g., have SO (t) also been interpreted as estimators of net survival. We will Roche et al SR (t) []) = would SP (t). have us show believe. that this is not generally correct. When the excess and the population hazard are not affected by any common This has been named the relative survival ratio. covariates, all methods to be discussed in this article estimate the same population quantity. In practice, however, ex- We do not agree with Roche et al [] that In estimating net (2) To estimate survival in the hypothetical situation where the disease under study would be the only possible cess hazard almost always depends on demographic variables survival, cancer registries should abandon all classical methods cause of death. (e.g., age) and one of the aims of this article is to find the This estimation is made possible by decomposing the population quantities that the estimators are estimating in and adopt the new Pohar-Perme estimator because great observed hazard into the hazard due to the disease and such situations (Sections 2 and 3). that due to other causes. This decomposition can be Although the concept of net survival may seem too hypothetical to be of interest by itself, it becomes crucial errors may occur.... carried out if the time to death due to the disease and the time to death due to other causes are conditionally when trying to compare cancer burden between countries, because it is independent Ederer of the IIgeneral estimates population mortality. of independent Internally given astandardised, known set of covariates. or Weage-specific, then assume that the hazard due to other causes is given Up to now, most authors who have produced large sets of by -year the population andmortality -year and, therefore, net survival that the survival are biased, statistics (Sant although et al., 29, and the references biastherein) is C 2, The International Biometric Society 3 We are not suggesting Ederer II is superior generally so small that it makes no practical difference. In short, don t panic if you have used, or are using, Ederer II. 9 April April 24 2 Modelling excess mortality Review Period analysis for up-to-date cancer survival data: theory, empirical evaluation, computational realisation and applications H. Brenner a, *, O. Gefeller b, T. Hakulinen c,d a Department of Epidemiology, German Centre for Research on Ageing, Bergheimer Str. 2, D-69 Heidelberg, Germany b Department of Medical Informatics, Biometry and Epidemiology, University of Erlangen-Nuremberg, Waldstrasse 6, D-94 Erlangen, Germany Abstract European Journal of Cancer 4 (24) c Finnish Cancer Registry, Liisankatu 2 B, FIN-7 Helsinki, Finland d Department of Public Health, University of Helsinki, FIN-7 Helsinki, Finland Received 6 August 23; received in revised form 3 September 23; accepted 2 October 23 Long-term survival rates are the most commonly used outcome measures for patients with cancer. However, traditional longterm survival statistics, which are derived by cohort-based types of analysis, essentially reflect the survival expectations of patients diagnosed many years ago. They are therefore often severely outdated at the time they become available. A couple of years ago, a new method of survival analysis, denoted period analysis, has been introduced to derive more up-to-date estimates of long-term survival rates. Paul WeDickman give a comprehensive Population-Based review of the Cancer new methodology, its 9 statistical April 24background, empirical evaluation, computational realisation and applications. We conclude that period analysis is a powerful tool to provide more up-to-date cancer sur- 2 vival rates. More widespread use by cancer registries should help to increase the use of cancer survival statistics for patients, clinicians, and public health authorities. # 23 Elsevier Ltd. All rights reserved. Flexible Parametric Models [7,, ] Keywords: Neoplasms; Prognosis; Registries; Relative Model h(t) = h (t) + λ(t) Observed Mortality Rate = Expected Mortality Rate + Excess Mortality Rate Cox model cannot be applied to model a difference in two rates. It is the observed mortality that drives the variance. Can use Poisson regression (Dickman et al. 24) [6]. Even better: flexible parametric models (Royston and Parmar 22 [7], Nelson et al. [8]). 9 April This paper has been cited over 27, times [2]. Introduction Long-term survival rates are the most commonly used outcome measures for patients with cancer. They are widely used to monitor progress in cancer care over time, or to compare quality of cancer care between different populations (e.g. [ 4]). Furthermore, cancer survival statistics are increasingly accessible through the Internet to clinicians and cancer patients, and their knowledge has a strong impact on both clinicians management of the disease as well as patients coping strategies. However, traditional long-term survival rates, which have been derived by cohort-based types of analysis [ 7], have essentially reflected the survival expectations of patients diagnosed many years ago. They have often been severely outdated at the time they became available as they failed to account for ongoing improvements in survival over time. A few years ago, a new method of * Corresponding author. Tel.: ; fax address: brenner@dzfa.uni-heidelberg.de (H. Brenner) /$ - see front matter # 23 Elsevier Ltd. All rights reserved. doi:/j.ejca3..3 survival analysis, denoted period analysis, has been introduced to derive more up-to-date estimates of longterm survival rates [8,9]. Meanwhile, this methodology has undergone extensive empirical evaluation [ 4], which showed that the method provides much more upto-date estimates of long-term survival rates than traditional methods of survival analysis indeed. Furthermore, software has been developed which allows easy implementation of this new analytical tool for both absolute and relative survival rates []. The method is now applied to derive more up-to-date long-term survival rates in an increasing number of countries [6 24]. These analyses suggest that long-term survival rates achieved by the end of the 2th century are much higher than previously suggested by traditional cohort-based analysis. For example, a recent period analysis of cancer patient survival in the United States [22] indicated that 2-year relative survival rates for all cancers combined are now approximately % rather than 4% as suggested by traditional cohort-based analysis (see Fig. ). Even larger differences are seen for many common forms of cancer, such as breast cancer (6% versus 2%) or ovarian cancer (% versus 3%). First introduced by Royston and Parmar (22) [7]. Parametric estimate of the baseline hazard without the usual restrictions on the shape (i.e, flexible). Applicable for standard and relative survival models. Can fit relative survival cure models (Andersson 2) [9]. Once we have a parametric expression for the baseline hazard we derive other quantities of interest (e.g., survival, hazard ratio, hazard differences, expectation of life). 9 April April 24 24

4 The Cox model[2] Quote from Sir David Cox (Reid 994 [3]) Reid What do you think of the cottage industry that s grown up around [the Cox model]? h i (t x i, β) = h (t) exp (x i β) Advantage: The baseline hazard, h (t) is not directly estimated from a Cox model. Disadvantage: The baseline hazard, h (t) is not directly estimated from a Cox model. 9 April 24 2 Cox In the light of further results one knows since, I think I would normally want to tackle the problem parametrically.... I m not keen on non-parametric formulations normally. Reid So if you had a set of censored survival data today, you might rather fit a parametric model, even though there was a feeling among the medical statisticians that that wasn t quite right. Cox That s right, but since then various people have shown that the answers are very insensitive to the parametric formulation of the underlying distribution. And if you want to do things like predict the outcome for a particular patient, it s much more convenient to do that parametrically. 9 April Example: survival of patients diagnosed with colon carcinoma in Finland Patients diagnosed with colon carcinoma in Finland Potential follow-up to end of 99; censored after years. Outcome is death due to colon carcinoma. Interest is in the effect of clinical stage at diagnosis (distant metastases vs no distant metastases). How might we specify a statistical model for these data? Empirical hazard Smoothed empirical hazards (cancer-specific mortality rates) sts graph, by(distant) hazard kernel(epan2) April April Smoothed empirical hazards on log scale sts graph, by(distant) hazard kernel(epan2) yscale(log) Fit a Cox model to estimate the mortality rate ratio Empirical hazard April Fitted hazards from Cox model with Efron method for ties ratio: 64 stcox distant, efron. stcox distant No. of subjects = 328 Number of obs = 328 No. of failures = 722 Time at risk = LR chi2() = 44 Log likelihood = Prob > chi2 = _t Haz. Ratio Std. Err. z P> z [9% C.I.] distant April 24 3 Fitted hazards from parametric survival model (exponential) ratio: Ratios Cox: 64 Exponential: April April 24 32

5 Fitted hazards from parametric survival model (Weibull) Fitted hazards from parametric survival model (Weibull) ratio: 7 Ratios Cox: 64 Exponential: Weibull: April April Cumulative Fitted cumulative hazards from Weibull model Demography and epidemiology: Practical use of the Lexis diagram in the computer age. or: Who needs the Cox-model anyway? Annual meeting of Finnish Statistical Society May 2 Revised December Bendix Carstensen Steno Diabetes Center, Gentofte, Denmark & Department of Biostatistics, University of Copenhagen bxc@steno.dk 9 April April Fitted hazards from Poisson model (yearly intervals) ratio: 69 Ratios Cox: 64 Exponential: Weibull: 7 Poisson (annual): Fitted hazards from Poisson model (3-months) Ratios Cox: 64 Exponential: Weibull: 7 Poisson (annual): 69 Poisson (quarter): 6 The contents of this paper was presented at the meeting of the Finnish Statistical Society in May 2 ratio: 6 in Oulu. The slides presented can be found on my homepage as April April Fitted hazards from Poisson model (months) ratio: 64 Ratios Cox: 64 Exponential: Weibull: 7 Poisson (annual): 69 Poisson (quarter): 6 Poisson (months): 64 ratio: 64 Poisson (rcs df for ln(time)) Ratios Cox: 64 Exponential: Weibull: 7 Poisson (annual): 69 Poisson (quarter): 6 Poisson (months): 64 Poisson (spline): April April 24 4

6 Flexible Parametric Models: Basic Idea Fitted hazards from flexible parametric model (df) Ratios Cox: 64 Exponential: Weibull: 7 Poisson (annual): 69 Poisson (quarter): 6 Poisson (months): 64 Consider a Weibull survival curve. S(t) = exp ( λt γ ) If we transform to the log cumulative hazard scale. ln [H(t)] = ln[ ln(s(t))] ln [H(t)] = ln(λ) + γ ln(t) Poisson (spline): 6 Flexible parametric: 63 This is a linear function of ln(t) Introducing covariates gives ratio: 63 ln [H(t xi )] = ln(λ) + γ ln(t) + xi β April 24 Rather than assuming linearity with ln(t) flexible parametric models use restricted cubic splines for ln(t). 4 9 April Flexible Parametric Models: Incorporating Splines Fitted cumulative hazards from Weibull model We thus model on the log cumulative hazard scale. Cumulative ln[h(t xi )] = ln [H (t)] + xi β This is a proportional hazards model. Restricted cubic splines with knots, k, are used to model the log baseline cumulative hazard. ln[h(t xi )] = ηi = s (ln(t) γ, k ) + xi β For example, with 4 knots we can write April ln [H(t xi )] = ηi = γ + γ zi + γ2 z2i + γ3 z3i {z } log baseline cumulative hazard 9 April 24 + xi β {z} log hazard ratios 44 We are fitting a linear predictor on the log cumulative hazard scale. and Functions We can transform to the survival scale S(t xi ) = exp( exp(ηi )) The hazard function is a bit more complex. h(t xi ) = ds (ln(t) γ, k ) exp(ηi ) dt This involves the derivatives of the restricted cubic splines functions, although these are relatively easy to calculate. 9 April April 24 46

7 Sensitivity to choice of knots; Simulation study by Rutherford et al. [4] Simulation Study (Rutherford et al.) [4] Generate data assuming a mixture Weibull distribution. 2. Scenario 2. Scenario 2 Through the use of simulation, we show that provided a sufficient number of knots are used, the approximated hazard functions given by restricted cubic splines fit closely to the true function for a range of complex hazard shapes. The simulation results also highlight the insensitivity of the estimated relative effects (hazard ratios) to the correct specification of the baseline hazard. rate rate Time Since Diagnosis () Scenario Time Since Diagnosis () rate rate Time Since Diagnosis () Scenario Time Since Diagnosis () Fit models using restricted cubic splines. 9 April April 24 Scenario 3 comparison of Log Ratios Choice of knots: Scenario 3 Cox Model Flexible Parametric Model S(t) Function knots (7 df) h(t) Time since diagnosis (years) Function April 24 9 April 24 2 Model Selection Implementation in Stata [] stpm2 available from SSC. ssc install stpm2 Estimated hazard and survival functions fairly insensitive to knot location. AIC and BIC can be used as rough guides to choose models. Not crucial (within reason) to inference based on the model. We often present a sensitivity analysis to show this. Could treat number of knots and their locations as unknowns. However, it is an area where more work is still required. All cause survival. stpm2 eng, scale(hazard) df() Relative survival. stpm2 eng, scale(hazard) df() hazard(rate) Time-dependent effects. stpm2 eng, scale(hazard) df() hazard(rate) tvc(eng) dftvc(3) Cure model. stpm2 eng, scale(hazard) df() hazard(rate) tvc(eng) dftvc(3) cure 9 April April 24 4 Example using attained age as the time-scale Study from Sweden [6] comparing incidence of hip fracture of, 7,73 men diagnosed with prostate cancer treated with bilateral orchiectomy. 43,23 men diagnosed with prostate cancer not treated with bilateral orchiectomy. 362,34 men randomly selected from the general population. Study entry is 6 months post diagnosis. Outcome is femoral neck (hip) fracture. Risk of fracture varies by age. Attained age is used as the primary time-scale. Actually, two timescales, but will initially ignore time from diagnosis. Estimates from a proportional hazards model stset using age as the time-scale. stset dateexit,fail(frac = ) enter(datecancer) origin(datebirth) /// id(id) scale(36) exit(time datebirth + *36) Cox Model. stcox noorc orc Incidence rate ratio (no orchiectomy) =.37 (8 to 6) Incidence rate ratio (orchiectomy) = 2.9 (.93 to 27) Flexible Parametric Model. stpm2 noorc orc, df() scale(hazard) Incidence rate ratio (no orchiectomy) =.37 (8 to 6) Incidence rate ratio (orchiectomy) = 2.9 (.93 to 27) 9 April 24 9 April 24 6

8 Proportional s Non Proportional s Incidence Rate (per py's) 7 2. Control No Orchiectomy Orchiectomy Incidence Rate (per py's) 7 2. Control No Orchiectomy Orchiectomy Age Age 9 April April 24 8 Incidence Rate Ratio Incidence Rate Difference Incidence Rate Ratio 2 2 Orchiectomy vs Control Age Difference in Incidence Rates (per person years) 3 2 Orchiectomy vs Control Age 9 April April 24 6 stpm2 postestimation commands Fit model (non-proportional hazards) stpm2 noorc orc, df() scale(h) tvc(noorc orc) dftvc(3) Fitted hazards predict h, h zeros predict h2, h at(noorc orc ) predict h3, h at(noorc orc ) ratios predict hr2, hrnum(noorc ) ci predict hr3, hrnum(orc ) ci differences predict hdiff2, hdiff(noorc ) ci predict hdiff3, hdiff(orc ) ci 9 April 24 6 Net (revisited) Relative aims to estimate net survival. This is the probability of not dying of cancer in the hypothetical world where it is impossible to die of other causes. Key Assumptions Independence between mortality due to cancer and mortality due to other causes & an appropriate estimate of expected survival. Same interpretation/assumption for cause-specific survival. We also assume that we have modelled covariates appropriately. 9 April Crude and Net Probabilities Brief Mathematical Details [] Net Probability of Death Due to Cancer Crude Probability of Death Due to Cancer = = Probability of death due to cancer in a hypothetical world, where the cancer under study is the only possible cause of death Probability of death due to cancer in the real world, where you may die of other causes before the cancer kills you Net probability also known as the marginal probability. Crude probability also known as the cause-specific cumulative incidence function (Geskus) or the cumulative incidence function. h(t) = h (t) + λ(t) - all-cause mortality rate h (t) - expected mortality rate λ(t) - excess mortality rate S (t) - Expected R(t) - Relative ( Net Prob of Death = R(t) = exp Crude Prob of Death (cancer) = Crude Prob of Death (other causes) = t t t ) λ(t) S (t)r(t)λ(t) S (t)r(t)h (t) 9 April April 24 64

9 Probabilities of death due to prostate cancer [6] What is cure? Net probability of death ( relative survival) Net probability of death ( relative survival) Dead (PC) Alive Age 7 at diagnosis Dead (PC) Alive Age 6 at diagnosis Crude probability of death Crude probability of death Dead (PC) Dead (other causes) alive Age 7 at diagnosis Dead (PC) Dead (other causes) Alive Age 6 at diagnosis Medical cure occurs when all signs of cancer have been removed in a patient; this is an individual-level definition of cure. It is difficult to prove that a patient is medically cured. Population or statistical cure occurs when mortality among patients with the disease returns to the same level as that expected for the general population. Equivalently the excess mortality rate approaches zero. This is a population-level definition of cure. When the excess mortality reaches (and stays) at zero, the relative survival curve is seen to reach a plateau. The inserted numbers represent the year probabilities of death 9 April April Plateau for relative survival Mixture cure model. Mixture cure model S(t) = S (t)(π + ( π)s u (t)); λ(t) = h (t) + ( π)fu(t) π+( π)s u(t) Relative from Diagnosis S (t) is the expected survival. π is the proportion cured (the cure fraction). ( π) is the proportion uncured (those bound to die ). S u (t) is the net survival for the uncured group. The excess mortality rate has an asymptote at zero. See De Angelis et al. [7], Verdecchia et al. [8] and Lambert et al.[9] for details. 9 April April Cure models: Interpreting changes over time Time trends for cancer of the colon age < [2] Cure Fraction and Median of Uncured Age Group: <. Cure Fraction 2. of Uncured (c) (a) (d) (b) Cure Fraction Adapted from Verdeccia (998) (a) General Improvement (b) Selective Improvment (c) Improved palliative care or lead time (d) Inclusion of subjects with no excess risk Cure Fraction. Median Median Year of Diagnosis 9 April April 24 7 Andersson 2 [2]: trends for AML Cure proportion Cure proportion Aged 9 4 at diagnosis Aged 6 7 at diagnosis Median survival time Median survival time Cure proportion Cure proportion Aged 4 6 at diagnosis Aged 7 8 at diagnosis Cure proportion Median survival time 9% CI 9 April Median survival time Median survival time One limitation of parametric cure models is that a functional form has to be specified Hard to fit survival functions flexible enough to capture high excess hazard within a few months from diagnosis Hard to fit high cure proportion Flexible parametric approach for cure models enable inclusion of these patient groups Andersson et al. BMC Medical Research Methodology 2, :96 RESEARCH ARTICLE Estimating and modelling cure in populationbased cancer studies within the framework of flexible parametric survival models Therese ML Andersson *, Paul W Dickman, Sandra Eloranta and Paul C Lambert,2 Open Access 9 April 24 72

10 When cure is reached the excess hazard rate is zero, and the cumulative excess hazard is constant By incorporating an extra constraint on the log cumulative excess hazard after the last knot, so that we force it not only to be linear but also to have zero slope, we are able to estimate the cure proportion This is done by calculating the splines backwards and introduce a constraint on the linear spline parameter in the regression model 9 April April Comparison with a non-mixture model The relative survival function from the flexible parametric survival model, with splines calculated backwards and with restriction on the parameter for the linear spline variable is defined as R(t) = exp( exp(γ + γ 2 z γ K z K )) which can be written as where π = exp( exp(γ )). R(t) = π exp(γ2z2+..+γk zk ), The flexible parametric cure model is a special case of a non-mixture cure model with and π = exp( exp(γ )) F Z (t) = exp(γ 2 z γ K z K ). This is a proportional hazards model as long as no time-dependent effects are modelled. 9 April April Incorporating covariates gives, and π = exp( exp(γ + xβ)) D F Z (t) = exp(γ 2 z γ K z K + s (ln(t); γ i ) x i ) i= can be fitted in Stata by adding a cure option to the stpm2 command stpm2 cage2-cage4, scale(h) df() bhazard(rate) /// tvc(cage2-cage4) dftvc(3) cure the constant parameters are used to model the cure proportion the time-dependent parameters are used to model the distribution function constraint of a zero effect for the linear spline term has to be incorporated for each spline function The spline variables will be calculated backwards, the constraint will be added for the linear spline parameters and the default knot distribution will be slightly different 9 April April Log likelihood = Number of obs = xb Coef. Std. Err. z P> z [9% Conf. Interval] cage cage cage _rcs _rcs _rcs _rcs _rcs (omitted) _rcs_cage _rcs_cage _rcs_cage23 (omitted) _rcs_cage _rcs_cage _rcs_cage33 (omitted) _rcs_cage _rcs_cage _rcs_cage43 (omitted) _cons The cure proportion, the survival function for the uncured and the median survival time for uncured can be estimated using the predict command. predict cure, cure. predict medst, centile() uncured. predict survunc, survival uncured. predict surv, survival 9 April April 24 8

11 9 April April April April Partitioning Excess Mortality (Eloranta [22, 23]) Excess mortality for males with Hodgkin lymphoma We have extended flexible parametric models for relative survival to simultaneously estimate the excess mortality due to diseases of the circulatory system (DCS) and the remaining excess mortality among patients diagnosed with Hodgkin lymphoma. Results are presented both as excess mortality rates and crude probabilities of death. The outcomes (DCS and non-dcs mortality) can be regarded as competing events and the total excess mortality is partitioned using ideas from classical competing risks theory. The model requires population mortality files stratified on cause of death (i.e., DCS and other deaths) to identify those deaths in excess of expected. Excess Mortality (per, p yrs) Age at Diagnosis: Since Diagnosis Excess Mortality (per, p yrs) Age at Diagnosis: 6 2 Since Diagnosis Excess DCS Mortality Remaining Excess Mortality 9% C.I 9 April April Temporal trends in 2-year probability of death 2 year Probability of Death 2 year Probability of Death Age at Diagnosis: Year of Diagnosis Age at Diagnosis: Year of Diagnosis Dead (Excess DCS Mortality) Dead (Other Causes) Predicted future 2 year Probability of Death Predicted future 2 year Probability of Death Age at Diagnosis: Year of Diagnosis Age at Diagnosis: Year of Diagnosis Dead (Remaining Excess HL Mortality) 9 % CI 9 April A useful summary measure of survival is the mean survival, life expectancy The loss in expectation of life is the difference between the mean expected survival (if not diagnosed with cancer) and the mean observed survival (for cancer patients) Quantify disease burden in the society how many life-years are lost due to the disease? Quantify differences between socio-economic groups or countries, how many life-years are lost in the population due to differences in cancer patient survival between groups? how many life-years would be gained if England had the same cancer patient survival as Sweden? Quantify the impact a cancer diagnosis has on a patient s life expectancy 9 April 24 88

12 Mean observed survival.3 years April April 24 9 Mean expected survival. years Difference years April April Even though we are now interested in the all-cause survival we will use a relative survival approach S(t) = S (t) R(t) h(t) = h (t) + λ(t) Easier to extrapolate R(t) than S(t) Has been done for grouped data (life tables) [24], by assuming λ(t) = or λ(t) = c after some point in time 9 April April 24 94, colon cancer Approaches for extrapolating relative survival Assume cure, i.e no excess hazard after a certain point in time, λ(t) = 2 Assume constant excess hazard after a certain point in time, λ(t) = c (exponential) 3 Assume the excess hazard given by the spline parameters from the model without imposing extra constraints, (Weibull) The all-cause survival can then be estimated by multiplying with the (extrapolated) expected survival Colon cancer, Age 6 69, Males Mean survival:.3 years 6 years. years years April April 24 96

13 , breast cancer, melanoma Melanoma Breast cancer Age 7 79 Age 8+ Age 9 Age 6 69 Age 7 79 Age 8+ Age 6 69 Age K M estimates of all cause survival 9% CI for K M K M estimates of all cause survival 9% CI for K M Extrapolating relative survival Relative survival, constant EH Extrapolating relative survival Relative survival, cure 9 April 24 97, bladder cancer April 24 98, colon cancer Bladder cancer Relative survival Males Age 6 69 Females Age 8+ Age 7 79 year RSR year RSR Age K M estimates of all cause survival 9% CI for K M Extrapolating relative survival Relative survival, constant EH 3 99, colon cancer Males, Age Females Life expectancy of general population Life expectancy of cancer patients 9 April Age Age 7 Age 6 Age 8 9 April 24 2 Males Females for males diagnosed with colon cancer, and potential life years gained if gender differences in cancer patient survival could be eliminated Proportion. 2 2 Proportion 2 9 April Age 6 Age Males, Age , colon cancer 96 Males Males, Age Age Age 7 9 April 24 Males, Age Age at diagnosis Age at diagnosis of expected life lost (left y axis) Proportion of expected life lost (right y axis) Age Life years lost Total April 24 3 Life years lost Extra years of life using female rates using female rates April 24 4

14 , conclusion References A relative survival approach is needed for extrapolation of observed survival Extrapolation from a flexible parametric survival model works well Have developed software to enable application of the method (exercise 4) The loss in expectation of life provides a measure of the impact cancer has on society as well as on individual patients life expectancy [] Andersen PK, Abildstrom SZ, Rosthøj S. Competing risks as a multi-state model. Stat Methods Med Res 22;:23 2. [2] Cronin KA, Feuer EJ. Cumulative cause-specific mortality for cancer patients in the presence of other causes: a crude analogue of relative survival. Statistics in Medicine 2;9: [3] Pohar Perme M, Stare J, Estève J. On estimation in relative survival. Biometrics 22; 68:3 2. [4] Dickman PW, Lambert PC, Coviello E, Rutherford MJ. Estimating net survival in population-based cancer studies. Int J Cancer 23;33:9 2. [] Roche L, Danieli C, Belot A, Grosclaude P, Bouvier AM, Velten M, et al.. Cancer net survival on registry data: Use of the new unbiased Pohar-Perme estimator and magnitude of the bias with the classical methods. Int J Cancer 22;32: [6] Dickman PW, Sloggett A, Hills M, Hakulinen T. Regression models for relative survival. Stat Med 24;23: 64. [7] Royston P, Parmar MKB. Flexible parametric proportional-hazards and proportional-odds models for censored survival data, with application to prognostic modelling and estimation of treatment effects. Statistics in Medicine 22;2: April 24 9 April 24 6 References 2 References 3 [8] Nelson CP, Lambert PC, Squire IB, Jones DR. Flexible parametric models for relative survival, with application in coronary heart disease. Stat Med 27;26: [9] Andersson TML, Dickman PW, Eloranta S, Lambert PC. Estimating and modelling cure in population-based cancer studies within the framework of flexible parametric survival models. BMC Med Res Methodol 2;:96. [] Lambert PC, Royston P. Further development of flexible parametric models for survival analysis. The Stata Journal 29;9: [] Royston P, Lambert PC. Flexible parametric survival analysis in Stata: Beyond the Cox model. Stata Press, 2. [2] Cox DR. Regression models and life-tables (with discussion). JRSSB 972;34: [3] Reid N. A conversation with Sir David Cox. Statistical Science 994;9: [4] Rutherford MJ, Hinchliffe SR, Abel GA, Lyratzopoulos G, Lambert PC, Greenberg DC. How much of the deprivation gap in cancer survival can be explained by variation in stage at diagnosis: An example from breast cancer in the east of england. International Journal of Cancer 23;. [] Lambert PC, Dickman PW, Nelson CP, Royston P. Estimating the crude probability of death due to cancer and other causes using relative survival models. Stat Med 2; 29: [6] Eloranta S, Adolfsson J, Lambert PC, Stattin Pär and Akre O, Andersson TML, Dickman PW. How can we make cancer survival statistics more useful for patients and clinicians: An illustration using localized prostate cancer in sweden. Cancer Causes Control 23; 24:. [7] Angelis RD, Capocaccia R, Hakulinen T, Soderman B, Verdecchia A. Mixture models for cancer survival analysis: application to population-based data with covariates. Statistics in Medicine 999;8: [8] Verdecchia A, Angelis RD, Capocaccia R, Sant M, Micheli A, Gatta G, Berrino F. The cure for colon cancer: results from the EUROCARE study. Int J Cancer 998;77: [9] Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating and modeling the cure fraction in population-based cancer survival analysis. Biostatistics 27;8: [2] Lambert PC, Dickman PW, Österlund P, Andersson T, Sankila R, Glimelius B. Temporal trends in the proportion cured for cancer of the colon and rectum: a population-based study using data from the Finnish cancer registry. International Journal of Cancer 27; 2: April April 24 8 References 4 [2] Andersson TML, Lambert PC, Derolf AR, Kristinsson SY, Eloranta S, Landgren O, et al.. Temporal trends in the proportion cured among adults diagnosed with acute myeloid leukaemia in Sweden 973-2, a population-based study. Br J Haematol 2; 48: [22] Eloranta S, Lambert PC, Andersson TML, Czene K, Hall P, Björkholm M, Dickman PW. Partitioning of excess mortality in population-based cancer patient survival studies using flexible parametric survival models. BMC Med Res Methodol 22;2:86. [23] Eloranta S, Lambert PC, Sjöberg J, Andersson TML, Björkholm M, Dickman PW. Temporal trends in mortality from diseases of the circulatory system after treatment for Hodgkin lymphoma: a population-based cohort study in Sweden (973 to 26). Journal of Clinical Oncology 23;3: [24] Hakama M, Hakulinen T. Estimating the expectation of life in cancer survival studies with incomplete follow-up information. Journal of Chronic Diseases 977;3: April 24 9