Estimating and modelling relative survival

Transcription

1 Estimating and modelling relative survival Paul W. Dickman Department of Medical Epidemiology and Biostatistics Karolinska Institutet, Stockholm, Sweden Regstat 2009 Workshop on Statistical Methods for Cancer Patient Survival 1 Sept 2009 Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept 2009 Welcome to Regstat! Welcome to what we hope will be the first in a series of annual workshops organised by MEB in statistical methods for registry-based epidemiology. This year s theme is statistical methods for population-based cancer patient survival. Today s workshop has been organised in collaboration with FMS (Föreningen för Medicinsk Statistik). The organisers gratefully acknowledge financial support from Cancerfonden (The Swedish Cancer Society). In my presentation I will attempt to simultaneously introduce the field as well as the program for today s workshop. Reprints available at Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept 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). Cause-specific mortality provides an estimate of net mortality. Excess mortality provides an estimate of net mortality. excess = total expected mortality mortality mortality Excess mortality is generally preferred when using data collected by population-based cancer registries [1]. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

2 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 Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Relative survival example Table 1: Number of cases (N) and 5-year observed (p), expected (p ), and relative (r) survival for males diagnosed with localised skin melanoma in Finland during Age N p p r Note that 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. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Applying relative survival to diseases other than cancer In order to interpret excess mortality as mortality due to the disease of interest we need to accurately estimate expected mortality (the mortality that would have been observed in the absence of the disease). General population mortality rates may not satisfy this criteria. Excess mortality (compared to the general population) may nevertheless still be of interest. Recent applications in cardiovascular disease[2] and HIV/AIDS[3, 4]. Nelson et al. Relative survival: what can cardiovascular disease learn from cancer? Eur Heart J. 2008;29: Bhaskaran et al. Changes in the risk of death after HIV seroconversion compared with mortality in the general population. JAMA 2008;300:51-9. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

3 More on the choice of measure for cancer patient survival The fact that excess mortality is generally preferred when using data collected by population-based cancer registries does not necessarily mean it is the uniformly best measure for all such studies (Kathy and others will talk more about this). Cause-specific mortality is the measure of choice in clinical trials assessing cancer patient survival. Should excess mortality be given greater consideration for trials? Additional (unpublished) analyses of data from a randomised trial of the effect of screening for prostate cancer using PSA and DRA showed a greater benefits of screening on excess mortality than cause-specific mortality. It is plausible that contact with health care professionals through screening benefits health (lowers mortality) via mechanisms other than those directly related to prostate cancer. Such benefits are captured with excess mortality but not cause-specific mortality. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Statistical cure The life table is a useful tool for describing the survival experience of the patients over a long follow-up period. In particular, an interval-specific relative survival ratio equal to one indicates that, during the specified interval, mortality in the patient group was equivalent to that of the general population. The attainment and maintenance of an interval-specific RSR of one indicates that there is no excess mortality due to cancer and the patients are assumed to be statistically cured. An individual is considered to be medically cured if he or she no longer displays symptoms of the disease. Statistical cure applies at a group, rather than individual, level. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept r Cancer of the stomach Annual follow-up interval Figure 1: Plots of the annual (interval-specific) relative survival ratios (r) for males and females diagnosed with cancer of the stomach in Finland and followed up to the end of Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

4 Plots of the interval-specific RSR are also useful for assessing the quality of follow-up. If the interval-specific RSR levels out at a value greater than 1, this generally indicates that some deaths have been missed in the follow-up process. An interval-specific relative survival ratio of unity is generally not achieved for smoking-related cancers, such as cancer of the lung and kidney. Compared to the general population, these patients are subject to excess mortality due to the cancer in addition to excess mortality due to other conditions caused by smoking, such as cardiovascular disease. We ll return to these concepts later when we discuss cure models. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Interpreting relative survival estimates The cumulative relative survival ratio can be interpreted as the proportion of patients alive after i years of follow-up in the hypothetical situation where the cancer in question is the only possible cause of death. 1-RSR can be interpreted as the proportion of patients who will die of cancer within i years of follow-up in the hypothetical situation where the cancer in question is the only possible cause of death. We do not live in this hypothetical world. Estimates of the proportion of patients who will die of cancer in the presence of competing risks can also be made. Cronin and Feuer (2000) [5] showed how crude and net mortality could be estimated based on relative survival for grouped data (implemented in the Stata command strs) and Lambert et al [6] showed how how individual-level estimates could be obtained. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Estimating relative survival using a period approach In 1996 Hermann Brenner suggested estimating cancer patient survival using a period, rather than cohort, approach [7]. Time at risk is left truncated at the start of the period window and right censored at the end. This suggestion was initially met with scepticism although studies based on historical data [8] have shown that period analysis provides very good predictions of the prognosis of newly diagnosed patients; and highlights temporal trends in patient survival sooner than cohort methods. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

5 Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) Subject 2 (0, 4) Subject 3 Subject Year (0, 6) (0, 3) Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) Subject 2 (0, 4) Subject 3 Subject 4 Period of Interest Year (0, 6) (0, 3) Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Subject 1 Diagnosis Death or Censoring Start and Stop at Risk Times Standard Period (0, 2) (0, 2) Subject 2 (0, 4) (2, 4) Subject 3 Subject 4 Period of Interest Year (0, 6) (0, 3) (5, 6) (, ) Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

6 Age-standardisation of relative survival The problem is more complex than age-standardisation of, for example, incidence rates since the age-distribution of the patients changes during follow-up. Which weights do we use and how does one interpret the resulting estimates? See the papers by Pokhrel et al and Brenner et al. [9, 10, 11, 12, 13]. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Modelling excess mortality (relative survival) Thehazardattimesincediagnosist for persons diagnosed with cancer is modelled as the sum of the known baseline hazard, λ (t), and the excess hazard due to a diagnosis of cancer, ν(t) [14, 15, 16, 17, 18]. λ(t) =λ (t)+ν(t) It is common to assume that the excess hazards are piecewise constant and proportional (although there are better approaches). Such models can be estimated in the framework of generalised linear models using standard statistical software (e.g., SAS, Stata, R) [14]. Non-proportional excess hazards are common but can be incorporated by introducing follow-up time by covariate interaction terms. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept Overview of approaches to modelling excess mortality Poisson regression (piecewise exponential) [14]. Poisson regression with fine splitting and modelling the baseline excess hazard using splines or fractional polynomials [19, 20, 21, 22]. Flexible parametric models on the log cumulative hazard scale [23]. Analogue to the Cox model where no assumptions are made about the baseline excess hazard [24, 25]. Cure models [26, 27, 28, 29, 30]. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

7 Modelling excess mortality using Poisson regression The model can be written as ln(μ j d j)=ln(y j )+xβ, (1) where μ j = E(d j ), d j the expected number of deaths, and y j person-time. This implies a generalised linear model with outcome d j, Poisson error structure, link ln(μ j d j ), and offset ln(y j). Such models have previously been described by Breslow and Day (1987) [31, pp ] and Berry (1983) [17]. The usual regression diagnostics (residuals, influence statistics) and method for assessing model fit for generalised linear models can be utilised. Hakulinen and Tenkanen [32] and Estève et al. [15] describe alternative approaches to fitting similar/identical models. Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept References [1] Dickman PW, Adami HO. Interpreting trends in cancer patient survival. J Intern Med 2006;260: [2] Nelson CP, Lambert PC, Squire IB, Jones DR. Relative survival: what can cardiovascular disease learn from cancer? Eur Heart J 2008;29: [3] Bhaskaran K, Hamouda O, Sannes M, Boufassa F, Johnson AM, Lambert PC, et al.. Changes in the risk of death after HIV seroconversion compared with mortality in the general population. JAMA 2008;300: [4] Harrison KM, Ling Q, Song R, Hall HI. County-level socioeconomic status and survival after HIV diagnosis, United States. Ann Epidemiol 2008;18: [5] Cronin K, Feuer E. Cumulative cause-specific mortality for cancer patients in the presence of other causes: a crude analogue of relative survival. Stat Med 2000;19: [6] 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 2009;(in press). [7] Brenner H, Gefeller O. An alternative approach to monitoring cancer patient survival. Cancer 1996;78: [8] Brenner H, Gefeller O, Hakulinen T. Period analysis for up-to-date cancer survival data: Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept theory, empirical evaluation, computational realisation and applications. European Journal of Cancer 2004;40: [9] Pokhrel A, Hakulinen T. Age-standardisation of relative survival ratios of cancer patients in a comparison between countries, genders and time periods. Eur J Cancer 2009;45: [10] Pokhrel A, Hakulinen T. How to interpret the relative survival ratios of cancer patients. European Journal of Cancer 2008;00: [11] Brenner H, Arndt V, Gefeller O, Hakulinen T. An alternative approach to age adjustment of cancer survival rates. Eur J Cancer 2004;40: [12] Brenner H, Hakulinen T. Age adjustment of cancer survival rates: methods, point estimates and standard errors. Br J Cancer 2005;93: [13] Brenner H, Hakulinen T. On crude and age-adjusted relative survival rates. J Clin Epidemiol 2003;56: [14] Dickman PW, Sloggett A, Hills M, Hakulinen T. Regression models for relative survival. Stat Med 2004;23: [15] Estève J, Benhamou E, Croasdale M, Raymond L. Relative survival and the estimation of net survival: Elements for further discussion. Statistics in Medicine 1990;9: [16] Hakulinen T, Tenkanen L. Regression analysis of relative survival rates. Applied Statistics 1987;36: [17] Berry G. The analysis of mortality by the subject-years method. Biometrics 1983; 39: Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept

8 [18] Pocock S, Gore S, Kerr G. Long term survival analysis: the curability of breast cancer. Stat Med 1982;1: [19] Bolard P, Quantin C, Abrahamowicz M, Esteve J, Giorgi R, Chadha-Boreham H, et al.. Assessing time-by-covariate interactions in relative survival models using restrictive cubic spline functions. J Cancer Epidemiol Prev 2002;7: [20] Giorgi R, Abrahamowicz M, Quantin C, Bolard P, Esteve J, Gouvernet J, Faivre J. A relative survival regression model using B-spline functions to model non-proportional hazards. Stat Med 2003;22: [21] Remontet L, Bossard N, Belot A, Estève J, French network of cancer registries FRANCIM. An overall strategy based on regression models to estimate relative survival and model the effects of prognostic factors in cancer survival studies. Stat Med 2007;26: [22] Lambert PC, Smith LK, Jones DR, Botha JL. Additive and multiplicative covariate regression models for relative survival incorporating fractional polynomials for time-dependent effects. Stat Med 2005;24: [23] Nelson CP, Lambert PC, Squire IB, Jones DR. Flexible parametric models for relative survival, with application in coronary heart disease. Stat Med 2007;26: [24] Sasieni PD. Proportional excess hazards. Biometrika 1996;83: [25] Perme MP, Henderson R, Stare J. An approach to estimation in relative survival regression. Biostatistics 2009;10: Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept [26] Lambert PC, Thompson JR, Weston CL, Dickman PW. Estimating and modeling the cure fraction in population-based cancer survival analysis. Biostatistics 2007;8: [27] Lambert PC. Modeling of the cure fraction in survival studies. The Stata Journal 2007; 7: [28] De Angelis R, Capocaccia R, Hakulinen T, Söderman B, Verdecchia A. Mixture models for cancer survival analysis: Application to population-based data with covariates. Statistics in Medicine 1999;18: [29] Gamel JW, Weller EA, Wesley MN, Feuer EJ. Parametric cure models of relative and cause-specific survival for grouped survival times. Computer Methods and Programs in Biomedicine 2000;61: [30] Sposto R. Cure model analysis in cancer: an application to data from the Children s Cancer Group. Stat Med 2002;21: [31] Breslow NE, Day NE. Statistical Methods in Cancer Research: Volume II - The Design and Analysis of Cohort Studies. IARC Scientific Publications No. 82. Lyon: IARC, [32] Hakulinen T, Tenkanen L. Regression analysis of relative survival rates. Applied Statistics 1987;36: Regstat 2009, Workshop on Statistical Methods for Cancer Patient Survival, 1 Sept