An overview and some recent advances in statistical methods for populationbased cancer survival analysis


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1 An overview and some recent advances in statistical methods for populationbased cancer survival analysis Paul W Dickman 1 Paul C Lambert 1,2 1 Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Stockholm, Sweden 2 Department of Health Sciences, University of Leicester, UK 9 April 214
2 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 1993 for a 1 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 HansOlov Adami s new department (now MEB) in March 1999, attracted by the strong research environment and possibilities for registerbased epidemiology. Paul Dickman PopulationBased Cancer Survival 9 April Research interests Primary research interests are in the development and application of methods for populationbased 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). Paul Dickman PopulationBased Cancer Survival 9 April 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 populationbased cancer survival analysis. Flexible parametric models (RoystonParmar models). The concept of statistical cure; cure models. Estimating crude and net probabilities of death. Partitioning excess mortality; treatmentrelated CVD mortality. Loss in expectation of life. Paul Dickman PopulationBased Cancer Survival 9 April 214 4
3 r year 1, per y rate per 1 In ncidence or mortality Incidence Mortality Survival Lung cancer incidence, mortality and survival (agestandardised) England, , by sex Men Women Women Men Fiveye ear relativ ve surviva al (%) Year of death / Year of diagnosis Paul Dickman PopulationBased Cancer Survival 9 April Lung cancer incidence in Sweden Paul Dickman PopulationBased Cancer Survival 9 April How might we measure the prognosis of cancer patients? Total mortality (among the patients). Our interest is typically in net mortality (mortality associated with a diagnosis of cancer). Causespecific mortality provides an estimate of net mortality (under certain assumptions). When estimating causespecific mortality only those deaths which can be attributed to the cancer in question are considered to be events. causespecific mortality = number of deaths due to cancer persontime at risk The survival times of patients who die of causes other than cancer are censored. Paul Dickman PopulationBased Cancer Survival 9 April 214 7
4 Need to consider competing risks [1] Figure 1 The competing risks multistate model. Paul Dickman PopulationBased Cancer Survival 9 April Many synonyms for the same concept Net 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 Crude probability of death due to cancer = Probability of death 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 causespecific cumulative incidence function (Geskus) or the cumulative incidence function. Paul Dickman PopulationBased Cancer Survival 9 April Net (left) and crude (right) probabilities of death in men with localized prostate cancer aged 7+ at diagnosis (Cronin and Feuer [2]) Paul Dickman PopulationBased Cancer Survival 9 April 214 1
5 lable at nal homepage: EUROPEAN UROLOGY 65 (214) Should we use crude or net survival? tate Cancer 1 Comparing patient survival between countries. 2 Studying temporal trends in patient survival. 3 Communicating prognosis to patients. ociation Between Use of bblockers and Prostate cer Specific Survival: A Cohort Study of 3561 Prostate cer Patients with HighRisk or Metastatic Disease ne Hartvedt Grytli a, Morten Wang Fagerland b, Sophie D. Fosså c,d, tin Austlid Taskén a,d, * Paul Dickman PopulationBased Cancer Survival 9 April rtment of Tumor Biology, Institute of Cancer Research, Oslo University Hospital, Oslo, Norway; b Unit of Biostatistics and Epidemiology, Oslo University al, Oslo, Norway; c Department of Oncology, Oslo University Hospital, Oslo, Norway; d Institute of Clinical Medicine, University of Oslo, Oslo, Norway le info Should we use crude or net survival? Abstract le history: pted January 6, 213 shed online ahead of on January 14, 213 ords: renergic receptor onist cker miology risk prostate cancer stasis ay ate cancer ate cancer specific ality Background: We recently reported reduced prostate cancer (PCa) specific mortality for bblocker users among patients receiving androgendeprivation therapy in a health survey cohort including 655 PCa patients. Information on clinical characteristics was limited. Objective: To assess the association between bblockers and PCaspecific mortality in a cohort of 3561 prostate cancer patients with highrisk 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 = ) was coupled with information on filled prescriptions between 24 and 211 from the Norwegian Prescription Database. Exclusion criteria were low or intermediaterisk disease; planned radiotherapy or radical prostatectomy; initiation of bblocker, ASA, or statin use after diagnosis where applicable; missing information on baseline Gleason score, prostatespecific antigen level, T stage or performance status; and missing followup. Outcome measurements and statistical analysis: Cox proportional hazards modelling and competing risk regression modelling were used to analyse the effects of bblocker use on allcause and PCaspecific mortality, respectively. Differences between bblocker users and nonusers regarding baseline clinical characteristics were assessed by the WilcoxonMannWhitney U test, Pearson chisquare test, and Student t test. Results and limitations: Median followup was 39 mo. bblocker use was associated with reduced PCa mortality (adjusted subhazard ratio:.79; 95% confidence interval [CI],.68.91; p value:.1). The observed reduction in PCa mortality was independent of the use of statins or ASA. We observed no association with allcause mortality Paul Dickman PopulationBased Cancer Survival 9 April (adjusted hazard ratio:.92; 95% CI, ). The main limitations of the study were the observational study design and short followup. Conclusions: bblocker use was associated with reduced PCaspecific mortality in patients with highrisk or metastatic disease at the time of diagnosis. Our findings need validation from further observational studies. # 213 European Association of Urology. Published by Elsevier B.V. All rights reserved. Causespecific survival can estimate net survival (assuming conditional independence) * Corresponding author. Oslo University Hospital, Institute of Cancer Research, Department of Tumor Biology, P.O. Box 4953 Nydalen, NO424 Oslo, Norway. Tel ; Fax: Using causespecific methods requires that reliably coded address: (K.A. Taskén). 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. 2838/$ see back matter # 213 European Association of Urology. Published by Elsevier B.V. All rights reserved. /dx.doi.org/1.116/j.eururo 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? Paul Dickman PopulationBased Cancer Survival 9 April
6 Allcause mortality for males with colon cancer and Finnish population Mortality Rate (per 1, person years) General population Colon cancer patients Age Paul Dickman PopulationBased Cancer Survival 9 April Relative survival aims to estimate net survival (still need conditional independence) We estimate excess mortality: the difference between observed (allcause) 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 Paul Dickman PopulationBased Cancer Survival 9 April Cervical cancer in New Zealand Life table estimates of patient survival Women diagnosed with followup 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 Paul Dickman PopulationBased Cancer Survival 9 April
7 Relative survival example (skin melanoma) Table 1: Number of cases (N) and 5year 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. Paul Dickman PopulationBased Cancer Survival 9 April Breakthrough paper in 212; Biometrics 68, An unbiased estimator for net survival [3] March 212 DOI: /j x On Estimation in Relative Survival Maja Pohar Perme, 1, Janez Stare, 1 and Jacques Estève 2 1 Department of Biostatistics and Medical Informatics, University of Ljubljana, Vrazov trg 2, SI1 Ljubljana, Slovenia 2 Université Claude Bernard, Hospices Civils de Lyon, Service de Biostatistique, 162, Avenue Lacassagne Lyon Cedex 3, France 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; Survival analysis. 1. Introduction observed hazard is larger than the population hazard. Survival probability Paul Dickman of cancer patients PopulationBased has been used Cancer for many Survival Under 9 April this 214 assumption, we use the term excess hazard for the hazard due to the disease and we have the 18 years as one of the main tools for evaluation of therapeutic advances. With improved treatments and prognosis, studies relation often now have long followup 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 (1) is unavailable or unreliable. Hence the field of relative survival has developed in which observed deaths are compared with A survival function derived from the excess hazard 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 (1) To compare the observed survival (SO) tothesurvival as Ederer I (Ederer, Axtell, and Cutler, 1961), Hakulinen of Aa diseasefree common groupquestion having the same indemographic our teaching. (Hakulinen, 1982), and Ederer II (Ederer et al., 1961, p. 11) characteristics as the study group. If the expected general population survival SP, the comparison is made and define them in Section 3 below. using Unless the ratiothere is reason to prefer aalthough causespecific all three were proposed approach, as variants for we estimat the denominator of the relative survival ratio, they have recommend oneso of (t) the following: also been interpreted as estimators of net survival. We will SR (t) = SP (t). show that this is not generally correct. When the excess Ederer II (1959 [?]) 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 (2) To estimate Pohar survival in Perme the hypothetical (212 situation [3]) where the disease under study would be the only possible cess hazard almost always depends on demographic variables cause of death. Modelbased (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 observed hazard into the hazard due to the disease and such situations (Sections 2 and 3). The choice of method depends on the research question and 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 carried out if the time to death due to the disease and practical considerations. the time to death due to other causes are conditionally when trying to compare cancer burden between countries, because it is independent of the general population mortality. independent given a known set of covariates. We then assume We that were the hazard recently due to other critical causes is [4] givenof Up a to paper now, most [5] authors that whoadvocated have produced large the sets of by the population mortality and, therefore, that the survival statistics (Sant et al., 29, and references therein) Pohar Perme approach; our critisism was of that particular paper C 211, The International Biometric Society 113 Which approach for estimating net survival? and not the Pohar Perme approach per se, of which we are great admirers. Paul Dickman PopulationBased Cancer Survival 9 April
8 We are not suggesting Ederer II is superior We are not suggesting that the Ederer II approach is superior to the Pohar Perme approach. However, we argue that it is not as inferior as some others (e.g., Roche et al [5]) would have us believe. We do not agree with Roche et al [5] that In estimating net survival, cancer registries should abandon all classical methods and adopt the new PoharPerme estimator because great errors may occur.... Internally standardised, or agespecific, Ederer II estimates of 5year and 1year net survival are biased, although the bias is generally so small that it makes no practical difference. In short, don t panic if you have used, or are using, Ederer II. Paul Dickman PopulationBased Cancer Survival 9 April European Journal of Cancer 4 (24) Review Period analysis for uptodate 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 Heidelberg, Germany b Department of Medical Informatics, Biometry and Epidemiology, University of ErlangenNuremberg, Waldstrasse 6, D9154 Erlangen, Germany c Finnish Cancer Registry, Liisankatu 21 B, FIN17 Helsinki, Finland d Department of Public Health, University of Helsinki, FIN17 Helsinki, Finland Received 6 August 23; received in revised form 3 September 23; accepted 21 October 23 Abstract Longterm survival rates are the most commonly used outcome measures for patients with cancer. However, traditional longterm survival statistics, which are derived by cohortbased 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 uptodate estimates of longterm survival rates. Paul WeDickman give a comprehensive PopulationBased review of the Cancer new methodology, Survival its 9 statistical April 214background, empirical evaluation, computational realisation and applications. We conclude that period analysis is a powerful tool to provide more uptodate cancer sur 21 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. Modelling Keywords: Neoplasms; Prognosis; excess Registries; Survival mortality 1. Introduction Relative Survival Model Longterm 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. [1 4]). Furthermore, cancer survival statistics are increasingly accessible through the Internet to clinicians and cancer patients, and their knowledge has Observed a strong impact on both clinicians management of the disease as well as patients coping strategies. However, traditional longterm survival rates, which have been derived by cohortbased types of analysis [5 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: (H. Brenner) /$  see front matter # 23 Elsevier Ltd. All rights reserved. doi:1.116/j.ejca h(t) = h (t) + λ(t) survival analysis, denoted period analysis, has been introduced to derive more uptodate estimates of longterm survival rates [8,9]. Meanwhile, this methodology has undergone extensive empirical evaluation [1 14], which showed that the method provides much more uptodate estimates of longterm 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 [15]. The method is now applied to derive more uptodate longterm survival rates in an increasing Mortality number of Rate countries [16 24]. These analyses suggest that longterm survival rates achieved by the end of the 2th century are much higher than previously suggested by traditional cohortbased analysis. For example, a recent period analysis of cancer patient survival in the United States [22] indicated that 2year relative survival rates for all cancers combined are now approximately 51% rather than 4% as suggested by traditional cohortbased analysis (see Fig. 1). Even larger differences are seen for many common forms of cancer, such as breast cancer (65% versus 52%) or ovarian cancer (5% versus 35%). Mortality Rate = Expected Mortality Rate + Excess 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]). Paul Dickman PopulationBased Cancer Survival 9 April
9 Flexible Parametric Survival Models [7, 1, 11] 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 211) [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). Paul Dickman PopulationBased Cancer Survival 9 April This paper has been cited over 27, times [12] Paul Dickman PopulationBased Cancer Survival 9 April The Cox model[12] 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. Paul Dickman PopulationBased Cancer Survival 9 April
10 Quote from Sir David Cox (Reid 1994 [13]) Reid What do you think of the cottage industry that s grown up around [the Cox model]? 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 nonparametric 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. Paul Dickman PopulationBased Cancer Survival 9 April Example: survival of patients diagnosed with colon carcinoma in Finland Patients diagnosed with colon carcinoma in Finland Potential followup to end of 1995; censored after 1 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? Paul Dickman PopulationBased Cancer Survival 9 April Smoothed empirical hazards (cancerspecific mortality rates) sts graph, by(distant) hazard kernel(epan2) Empirical hazard Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April
11 Smoothed empirical hazards on log scale sts graph, by(distant) hazard kernel(epan2) yscale(log) Empirical hazard Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April Fit a Cox model to estimate the mortality rate ratio. stcox distant No. of subjects = 1328 Number of obs = 1328 No. of failures = 7122 Time at risk = LR chi2(1) = Log likelihood = Prob > chi2 = _t Haz. Ratio Std. Err. z P> z [95% C.I.] distant Paul Dickman PopulationBased Cancer Survival 9 April Fitted hazards from Cox model with Efron method for ties stcox distant, efron Hazard Hazard ratio: 6.64 Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April
12 Hazard Fitted hazards from parametric survival model (exponential) Not distant Distant Hazard ratio: 1.4 Hazard Ratios Cox: 6.64 Exponential: Paul Dickman PopulationBased Cancer Survival 9 April Fitted hazards from parametric survival model (Weibull) Hazard Hazard ratio: 7.41 Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April Fitted hazards from parametric survival model (Weibull) Not distant Distant Hazard Paul Dickman PopulationBased Cancer Survival 9 April
13 Fitted cumulative hazards from Weibull model Not distant Distant Cumulative Hazard Paul Dickman PopulationBased Cancer Survival 9 April Demography and epidemiology: Practical use of the Lexis diagram in the computer age. or: Who needs the Coxmodel anyway? Annual meeting of Finnish Statistical Society May 25 Revised December 25. Bendix Carstensen Steno Diabetes Center, Gentofte, Denmark & Department of Biostatistics, University of Copenhagen Paul Dickman PopulationBased Cancer Survival 9 April Hazard Fitted hazards from Poisson model (yearly intervals) Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Poisson (annual): 6.89 Not distant Distant The contents of this paper was presented at the meeting of the Finnish Statistical Society in May 25 in Oulu. TheHazard slides presented ratio: 6.89 can be found on my homepage as Paul Dickman PopulationBased Cancer Survival 9 April
14 Hazard Fitted hazards from Poisson model (3months) Hazard ratio: 6.65 Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Poisson (annual): 6.89 Poisson (quarter): 6.65 Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April Hazard Fitted hazards from Poisson model (months) Hazard ratio: 6.64 Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Poisson (annual): 6.89 Poisson (quarter): 6.65 Poisson (months): 6.64 Not distant Distant Paul Dickman PopulationBased Cancer Survival 9 April Hazard Hazard ratio: 6.64 Poisson (rcs 5df for ln(time)) Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Poisson (annual): 6.89 Poisson (quarter): 6.65 Poisson (months): 6.64 Not distant Distant Poisson (spline): Paul Dickman PopulationBased Cancer Survival 9 April 214 4
15 Hazard Fitted hazards from flexible parametric model (5df) Hazard ratio: 6.63 Hazard Ratios Cox: 6.64 Exponential: 1.4 Weibull: 7.41 Poisson (annual): 6.89 Poisson (quarter): 6.65 Poisson (months): 6.64 Not distant Distant Poisson (spline): 6.65 Flexible parametric: Paul Dickman PopulationBased Cancer Survival 9 April Flexible Parametric Models: Basic Idea 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) This is a linear function of ln(t) Introducing covariates gives ln [H(t x i )] = ln(λ) + γ ln(t) + x i β Rather than assuming linearity with ln(t) flexible parametric models use restricted cubic splines for ln(t). Paul Dickman PopulationBased Cancer Survival 9 April Fitted cumulative hazards from Weibull model Not distant Distant Cumulative Hazard Paul Dickman PopulationBased Cancer Survival 9 April
16 Flexible Parametric Models: Incorporating Splines We thus model on the log cumulative hazard scale. ln[h(t x i )] = ln [H (t)] + x i 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 x i )] = η i = s (ln(t) γ, k ) + x i β For example, with 4 knots we can write ln [H(t x i )] = η i = γ + γ 1 z 1i + γ 2 z 2i + γ 3 z }{{ 3i } log baseline cumulative hazard + x i β }{{} log hazard ratios Paul Dickman PopulationBased Cancer Survival 9 April We are fitting a linear predictor on the log cumulative hazard scale. Survival and Hazard Functions We can transform to the survival scale S(t x i ) = exp( exp(η i )) The hazard function is a bit more complex. h(t x i ) = ds (ln(t) γ, k ) dt exp(η i ) This involves the derivatives of the restricted cubic splines functions, although these are relatively easy to calculate. Paul Dickman PopulationBased Cancer Survival 9 April Paul Dickman PopulationBased Cancer Survival 9 April
17 Sensitivity to choice of knots; Simulation study by Rutherford et al. [14] 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. Paul Dickman PopulationBased Cancer Survival 9 April
18 Simulation Study (Rutherford et al.) [14] Generate data assuming a mixture Weibull distribution. 2.5 Scenario Scenario Hazard rate Hazard rate Time Since Diagnosis (Years) Time Since Diagnosis (Years) 2.5 Scenario Scenario Hazard rate Hazard rate Time Since Diagnosis (Years) Time Since Diagnosis (Years) Fit models using restricted cubic splines. Paul Dickman PopulationBased Cancer Survival 9 April Scenario 3 comparison of Log Hazard Ratios .4 Cox Model Flexible Parametric Model Paul Dickman PopulationBased Cancer Survival 9 April Choice of knots: Scenario 3 8 knots (7 df) 1. Survival Function 1.6 Hazard Function S(t) h(t) Time since diagnosis (years) Paul Dickman PopulationBased Cancer Survival 9 April
19 Model Selection 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. Paul Dickman PopulationBased Cancer Survival 9 April Implementation in Stata [1] stpm2 available from SSC. ssc install stpm2 All cause survival. stpm2 eng, scale(hazard) df(5) Relative survival. stpm2 eng, scale(hazard) df(5) hazard(rate) Timedependent effects. stpm2 eng, scale(hazard) df(5) hazard(rate) tvc(eng) dftvc(3) Cure model. stpm2 eng, scale(hazard) df(5) hazard(rate) tvc(eng) dftvc(3) cure Paul Dickman PopulationBased Cancer Survival 9 April Example using attained age as the timescale Study from Sweden [6] comparing incidence of hip fracture of, 17,731 men diagnosed with prostate cancer treated with bilateral orchiectomy. 43,23 men diagnosed with prostate cancer not treated with bilateral orchiectomy. 362,354 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 timescale. Actually, two timescales, but will initially ignore time from diagnosis. Paul Dickman PopulationBased Cancer Survival 9 April
20 Estimates from a proportional hazards model stset using age as the timescale. stset dateexit,fail(frac = 1) enter(datecancer) origin(datebirth) /// id(id) scale(365.25) exit(time datebirth + 1*365.25) Cox Model. stcox noorc orc Incidence rate ratio (no orchiectomy) = 1.37 (1.28 to 1.46) Incidence rate ratio (orchiectomy) = 2.9 (1.93 to 2.27) Flexible Parametric Model. stpm2 noorc orc, df(5) scale(hazard) Incidence rate ratio (no orchiectomy) = 1.37 (1.28 to 1.46) Incidence rate ratio (orchiectomy) = 2.9 (1.93 to 2.27) Paul Dickman PopulationBased Cancer Survival 9 April Proportional Hazards Incidence Rate (per 1 py's) Control No Orchiectomy Orchiectomy Age Paul Dickman PopulationBased Cancer Survival 9 April Non Proportional Hazards Incidence Rate (per 1 py's) Control No Orchiectomy Orchiectomy Age Paul Dickman PopulationBased Cancer Survival 9 April
21 Incidence Rate Ratio Orchiectomy vs Control Incidence Rate Ratio Age Paul Dickman PopulationBased Cancer Survival 9 April Incidence Rate Difference Difference in Incidence Rates (per 1 person years) Orchiectomy vs Control Age Paul Dickman PopulationBased Cancer Survival 9 April stpm2 postestimation commands Fit model (nonproportional hazards) stpm2 noorc orc, df(5) scale(h) tvc(noorc orc) dftvc(3) Fitted hazards predict h1, h zeros predict h2, h at(noorc 1 orc ) predict h3, h at(noorc orc 1) Hazard ratios predict hr2, hrnum(noorc 1) ci predict hr3, hrnum(orc 1) ci Hazard differences predict hdiff2, hdiff1(noorc 1) ci predict hdiff3, hdiff1(orc 1) ci Paul Dickman PopulationBased Cancer Survival 9 April
22 Net Survival (revisited) Relative Survival 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 causespecific survival. We also assume that we have modelled covariates appropriately. Paul Dickman PopulationBased Cancer Survival 9 April Crude and Net Probabilities Net 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 Crude Probability of Death Due to Cancer = 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 causespecific cumulative incidence function (Geskus) or the cumulative incidence function. Paul Dickman PopulationBased Cancer Survival 9 April Brief Mathematical Details [15] h(t) = h (t) + λ(t)  allcause mortality rate h (t)  expected mortality rate λ(t)  excess mortality rate S (t)  Expected Survival R(t)  Relative Survival ( t ) Net Prob of Death = 1 R(t) = 1 exp λ(t) Crude Prob of Death (cancer) = Crude Prob of Death (other causes) = t t S (t)r(t)λ(t) S (t)r(t)h (t) Paul Dickman PopulationBased Cancer Survival 9 April
23 Probabilities of death due to prostate cancer [16] Net probability of death (1 relative survival) Dead (PC) Alive Age 75 at diagnosis Crude probability of death Dead (PC) Dead (other causes) alive Age 75 at diagnosis Net probability of death (1 relative survival) Dead (PC) Alive Age 6 at diagnosis Crude probability of death Dead (PC) Dead (other causes) Alive Age 6 at diagnosis The inserted numbers represent the 1 year probabilities of death Paul Dickman PopulationBased Cancer Survival 9 April What is cure? Medical cure occurs when all signs of cancer have been removed in a patient; this is an individuallevel 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 populationlevel definition of cure. When the excess mortality reaches (and stays) at zero, the relative survival curve is seen to reach a plateau. Paul Dickman PopulationBased Cancer Survival 9 April Plateau for relative survival 1..8 Relative Survival Years from Diagnosis Paul Dickman PopulationBased Cancer Survival 9 April
24 Mixture cure model Mixture cure model S(t) = S (t)(π + (1 π)s u (t)); λ(t) = h (t) + (1 π)fu(t) π+(1 π)s u(t) S (t) is the expected survival. π is the proportion cured (the cure fraction). (1 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. [17], Verdecchia et al. [18] and Lambert et al.[19] for details. Paul Dickman PopulationBased Cancer Survival 9 April Cure models: Interpreting changes over time Survival of Uncured (c) Cure Fraction Adapted from Verdeccia (1998) (a) (d) (b) (a) General Improvement (b) Selective Improvment (c) Improved palliative care or lead time (d) Inclusion of subjects with no excess risk Paul Dickman PopulationBased Cancer Survival 9 April Time trends for cancer of the colon age <5 [2] Cure Fraction and Median Survival of Uncured Age Group: <5 1. Cure Fraction Median Survival Cure Fraction Median Survival Year of Diagnosis. Paul Dickman PopulationBased Cancer Survival 9 April 214 7
25 Andersson 21 [21]: trends for AML Aged 19 4 at diagnosis Aged 41 6 at diagnosis Cure proportion Median survival time Cure proportion Median survival time Year of diagnosis Year of diagnosis Aged 61 7 at diagnosis Aged 71 8 at diagnosis Cure proportion Median survival time Cure proportion Median survival time Year of diagnosis Year of diagnosis Cure proportion Median survival time 95% CI Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models 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 211, 11:96 RESEARCH ARTICLE Open Access Estimating and modelling cure in populationbased cancer studies within the framework of flexible parametric survival models Therese ML Andersson 1*, Paul W Dickman 1, Sandra Eloranta 1 and Paul C Lambert 1,2 Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models 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 Paul Dickman PopulationBased Cancer Survival 9 April
26 Flexible parametric cure models Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models 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 1 z K 1 )) which can be written as where π = exp( exp(γ )). R(t) = π exp(γ 2z γ K 1 z K 1 ), Paul Dickman PopulationBased Cancer Survival 9 April Comparison with a nonmixture model The flexible parametric cure model is a special case of a nonmixture cure model with and π = exp( exp(γ )) F Z (t) = exp(γ 2 z γ K 1 z K 1 ). This is a proportional hazards model as long as no timedependent effects are modelled. Paul Dickman PopulationBased Cancer Survival 9 April
27 Flexible parametric cure models Incorporating covariates gives, and π = exp( exp(γ + xβ)) F Z (t) = exp(γ 2 z γ K 1 z K 1 + D s (ln(t); γ i ) x i ) the constant parameters are used to model the cure proportion i=1 the timedependent 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 Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models Flexible parametric cure models can be fitted in Stata by adding a cure option to the stpm2 command stpm2 cage2cage4, scale(h) df(5) bhazard(rate) /// tvc(cage2cage4) dftvc(3) cure 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 Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models Log likelihood = Number of obs = xb Coef. Std. Err. z P> z [95% Conf. Interval] cage cage cage _rcs _rcs _rcs _rcs _rcs5 (omitted) _rcs_cage _rcs_cage _rcs_cage23 (omitted) _rcs_cage _rcs_cage _rcs_cage33 (omitted) _rcs_cage _rcs_cage _rcs_cage43 (omitted) _cons Paul Dickman PopulationBased Cancer Survival 9 April
28 Flexible parametric cure models 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(5) uncured. predict survunc, survival uncured. predict surv, survival Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models Paul Dickman PopulationBased Cancer Survival 9 April
29 Flexible parametric cure models Paul Dickman PopulationBased Cancer Survival 9 April Flexible parametric cure models Paul Dickman PopulationBased Cancer Survival 9 April Partitioning Excess Mortality (Eloranta [22, 23]) 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 nondcs 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. Paul Dickman PopulationBased Cancer Survival 9 April
30 Excess mortality for males with Hodgkin lymphoma 25 Age at Diagnosis: Age at Diagnosis: Excess Mortality (per 1, p yrs) Excess Mortality (per 1, p yrs) Years Since Diagnosis Years Since Diagnosis Excess DCS Mortality Remaining Excess Mortality 95% C.I Paul Dickman PopulationBased Cancer Survival 9 April Temporal trends in 2year probability of death 2 year Probability of Death Age at Diagnosis: Year of Diagnosis Predicted future 2 year Probability of Death Age at Diagnosis: Year of Diagnosis 2 year Probability of Death Age at Diagnosis: Year of Diagnosis Predicted future 2 year Probability of Death Age at Diagnosis: Year of Diagnosis Dead (Excess DCS Mortality) Dead (Other Causes) Dead (Remaining Excess HL Mortality) 95 % CI Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life 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 lifeyears are lost due to the disease? Quantify differences between socioeconomic groups or countries, how many lifeyears are lost in the population due to differences in cancer patient survival between groups? how many lifeyears 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 Paul Dickman PopulationBased Cancer Survival 9 April
31 Loss in expectation of life Survival Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life Survival Mean observed survival 5.3 years Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life Survival Mean expected survival 1.5 years Paul Dickman PopulationBased Cancer Survival 9 April
32 Loss in expectation of life Survival Difference 5.2 years Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life Survival Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life Even though we are now interested in the allcause 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 Paul Dickman PopulationBased Cancer Survival 9 April
33 Loss in expectation of life Approaches for extrapolating relative survival 1 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 allcause survival can then be estimated by multiplying with the (extrapolated) expected survival Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life, colon cancer Survival Colon cancer, Age 6 69, Males Mean survival: 5.3 years 6.6 years 5.1 years 5.4 years Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life, breast cancer Breast cancer Survival Age 5 59 Age 6 69 Survival Age 7 79 Age K M estimates of all cause survival Extrapolating relative survival 95% CI for K M Relative survival, constant EH Paul Dickman PopulationBased Cancer Survival 9 April
34 Loss in expectation of life, melanoma Melanoma Survival Age 5 59 Age 6 69 Survival Age 7 79 Age K M estimates of all cause survival Extrapolating relative survival 95% CI for K M Relative survival, cure Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life, bladder cancer Bladder cancer Survival Age 5 59 Age 6 69 Survival Age 7 79 Age K M estimates of all cause survival Extrapolating relative survival 95% CI for K M Relative survival, constant EH Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life, colon cancer Relative survival 1 Males 1 Females year RSR year RSR Year of diagnosis Year of diagnosis Age 55 Age 65 Age 75 Age 85 Paul Dickman PopulationBased Cancer Survival 9 April 214 1
35 Loss in expectation of life, colon cancer Males, Age 55 Males, Age Years 15 1 Years Year of diagnosis Year of diagnosis Males, Age 75 Males, Age Years 15 1 Years Year of diagnosis Year of diagnosis Life expectancy of general population Life expectancy of cancer patients Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life, colon cancer Loss in expectation of life Males Females Years Years Year of diagnosis Year of diagnosis Age 55 Age 65 Age 75 Age 85 Paul Dickman PopulationBased Cancer Survival 9 April Loss in expectation of life Males Females Years Proportion Years Proportion Age at diagnosis Age at diagnosis Years of expected life lost (left y axis) Proportion of expected life lost (right y axis) Paul Dickman PopulationBased Cancer Survival 9 April
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