Estimating the loss of lifetime function using flexible parametric relative survival models
Abstract
Background
Within cancer care, dynamic evaluations of the loss in expectation of life provides useful information to patients as well as physicians. The loss of lifetime function yields the conditional loss in expectation of life given survival up to a specific time point. Due to the inevitable censoring in timetoevent data, loss of lifetime estimation requires extrapolation of both the patient and general population survival function. In this context, the accuracy of different extrapolation approaches has not previously been evaluated.
Methods
The loss of lifetime function was computed by decomposing the allcause survival function using the relative and general population survival function. To allow extrapolation, the relative survival function was fitted using existing parametric relative survival models. In addition, we introduced a novel mixture cure model suitable for extrapolation. The accuracy of the estimated loss of lifetime function using various extrapolation approaches was assessed in a simulation study and by data from the Danish Cancer Registry where complete followup was available. In addition, we illustrated the proposed methodology by analyzing recent data from the Danish Lymphoma Registry.
Results
No uniformly superior extrapolation method was found, but flexible parametric mixture cure models and flexible parametric relative survival models seemed to be suitable in various scenarios.
Conclusion
Using extrapolation to estimate the loss of lifetime function requires careful consideration of the relative survival function outside the available followup period. We propose extensive sensitivity analyses when estimating the loss of lifetime function.
Keywords
Loss of lifetime Relative survival Extrapolation Cancer survivalAbbreviations
Background
Dynamic survival prediction is important in cancer care, where prognostic assessments are given numerous times during diagnosis, treatment, and posttreatment followup. A popular measure for characterizing the severity of a disease is the expected amount of lifetime lost due to the disease as compared to the general population. This measure is known as the loss in expectation of life and may be computed as the difference between the area under the general population and patient survival curves [1]. The loss in expectation of life has previously been used to characterize the disease burden within colon cancer and acute myeloid leukemia [2, 3]. The loss of lifetime function generalizes this measure by dynamically evaluating the loss in expectation of life, yielding the conditional number of years lost due to cancer given survival up to specific time points.
Due to the occurrence of censoring, computing the loss of lifetime function typically requires extrapolation of both the patient and general population survival function. Generally, extrapolation of survival functions estimated from censored timetoevent data is challenging since there is no way to evaluate the extrapolation accuracy and even a wellfitted model may extrapolate poorly. Nonetheless, in order to provide estimates of the longterm effects of a given treatment, extrapolated survival probabilities are often required in the analysis of data from clinical trials [4].
An extensive literature exists on techniques for extrapolating survival functions. Jackson et al. reviewed methods for incorporating external data, such as register data or national life tables, to extrapolate survival functions [5]. Such approaches require a quantification of how the survival in the present patient population and the external data differ and assumptions about how this will continue beyond the followup. In particular, extrapolation through the relative survival function has been proposed for both grouped and individuallevel data, which has demonstrated improved accuracy in comparison to models for the allcause survival function [1, 6]. Andersson et al. examined the accuracy of the loss in expectation of life estimates calculated by three types of relative survival models [1]. However, none of these assessments were conducted for the entire loss of lifetime function.
In the following article, we compute the loss of lifetime function using previously introduced extrapolation approaches. In addition, a new flexible parametric relative survival model based on mixture cure models and splinebased proportional hazards models is introduced [7, 8]. We expand the study of Andersson et al. [1] by evaluating the accuracy of the entire loss of lifetime function based on various extrapolation approaches in a simulation study and in data from the Danish Cancer Registry where complete followup was available. In addition, as a clinically motivated example, the loss of lifetime function is computed for three lymphoma types using recent data from the Danish Lymphoma Registry.
Methods
Relative survival
where h^{∗}(tz) is the general population hazard function and λ(tz) is termed the excess hazard function or excess mortality. Both h^{∗}(tz) and S^{∗}(tz) are usually computed from publicly available life tables matched on variables such as age, sex, and calendar year. The most popular way to include covariate effects is the proportional excess hazard model with a parametric specification of the baseline excess hazard [9, 10].
Parametric cure models
where π(z) is the, potentially covariate dependent, cure proportion and S_{u}(tz) is the relative survival function of the uncured patients. The cure proportion can be modelled through a link function, e.g., with a logistic, identity, or loglog link function [7]. The function S_{u}(tz) can conveniently be modelled by regular parametric survival models, such as a Weibull model, a lognormal model, or more flexible alternatives such as a WeibullWeibull mixture model [12]. The model is estimated by maximum likelihood where the only external information needed is the general population hazard at the observed event times (see Lambert et al. [7] for the likelihood function).
where the function \(\widetilde {S}(t\boldsymbol {z})\) is a proper survival function which does not have an intuitive interpretation like S_{u}(tz). By rewriting the nonmixture cure model, it can be formulated as a mixture cure model, with \(\left (\pi (\boldsymbol {z})^{1\widetilde {S}(t\boldsymbol {z})}  \pi (\boldsymbol {z})\right)/(1  \pi (\boldsymbol {z}))\) as the relative survival function of the uncured patients [7]. Thus, estimation of the nonmixture cure model can be carried out similarly to that of mixture cure models.
Flexible parametric cure models
for j=2,...,K_{i}−1, where \(\lambda _{ij} = \frac {k_{iK_{i}}  k_{ij}}{k_{iK_{i}}  k_{i1}}\) and x_{+}= max(x,0). Generally, the number and placement of the knots in the different spline functions do not need to be the same.
Similarly to the more simple cure models presented in Lambert et al. [7], π(z) can be modelled by various link functions and the relative survival cannot fall below π(z), thus ensuring statistical cure. The model is fitted by maximum likelihood using the likelihood of the mixture cure model. This cure model enables flexible modelling of the relative survival without the strong assumption of cure after the last knot while providing the more intuitive interpretation of a mixture cure model. Additionally, in this model, the modelling of the cure proportion becomes more clearly separated from the modelling of S_{u}(t).
The loss of lifetime function
which is the difference in expected residual lifetime after time point t between the general population and the patients.
Extrapolation of both S^{∗}(·z) and S(·z) is required to compute (9) since the survival distributions typically cannot be fully estimated due to censoring. Similarly to Andersson et al. [1], the extrapolation of the expected survival, S^{∗}(·), can be accomplished by using the method of Ederer et al. [14] (Ederer I) and by making assumptions about the future population mortality rates. The latter can be carried out by using mortality rates from the last available time point or, if available, by using predicted future mortality rates.

the Nelson et al. [10] relative survival (NRS) model, which is linear on the log cumulative excess hazard scale after the last knot,

the Andersson et al. [13] relative survival (ARS) model, which is constant on the log cumulative excess hazard scale after the last knot and thereby incorporates statistical cure, and

the flexible mixture cure (FMC) model in (8), which incorporates statistical cure, but is not restricted to a constant log cumulative excess hazard after the last knot.
Due to their flexibility, the three models typically behave similarly within the first part of the followup, but may produce different survival trajectories beyond the available followup. In cure models, the relative survival cannot fall below π, and thus these models have a parameter to control the asymptote of the relative survival. Therefore, in cases where statistical cure occurs, cure models may improve extrapolation as compared to noncure models. In cases where statistical cure does not occur, cure models may provide too optimistic extrapolations and hence may not be appropriate. However, in such cases, the introduced FMC model is expected to estimate π close to zero such that the fit is mainly based on the flexible survival function, S_{u}(t). In the ARS model, letting π=0, substantially affects the survival function since this forces R(k_{K})=0. Therefore, we consider the FMC model a hybrid between the NRS and ARS models.
Implementation
is subtracted from the loglikelihood, where t_{j} and z_{j} are the observed followup time and covariate vector, respectively, of patient j. Initially, κ is 1, but doubles until no negative values of h_{u} are obtained. Orthogonalization of the base functions of the restricted cubic splines has previously been recommended due to the potential correlation between the base functions [16]. We employed a QRdecomposition approach to carry out the orthogonalization.
Choosing the number and location of the knots is a key issue in splinebased models. Similarly to Royston and Parmar, the knots of the FMC model were selected according to the quantiles of the uncensored event times [8]. In a simulation study, Rutherford et al. [16] concluded that complex hazard shapes can adequately be captured by the splinebased model of Royston and Parmar [8] provided that a sufficient number of knots are selected. In particular, the survival model was rather insensitive to the number of knots and it was argued that the results should also be valid in relative survival and cure models [16].
All analyses were performed in the statistical programming language R. For the purpose of this article, the NRS and ARS models were fitted using the package rstpm2 [17]. Functions for estimating the presented FMC model and computing the loss of lifetime function were assembled in the Rpackage cuRe (see https://github.com/LasseHjort/cuRe). The package also enables estimation of the expected residual lifetime, restricted expected residual lifetime, and restricted loss of lifetime using any of the models considered here. The integrals of the loss of lifetime function were computed numerically by GaussLegendre quadrature, while the pointwise variance of the loss of lifetime function was estimated using the delta method and numerical differentiation.
Results
Simulation study
Simulation design

Draw a general population survival time T_{E} from S^{∗}.

Draw a relative survival time, T_{R} from R.

Draw a censoring time T_{C} from C.

The observed followup time is given by T= min(T_{R},T_{E},T_{C}) and the event indicator is δ=1[min(T_{R},T_{E})≤T_{C}].
To mimic typical register data, the censoring times were simulated from a uniform distribution, C, between 0 and 15 years. Using S^{∗} and R, the true loss of lifetime function was obtained by inserting into (9). All scenarios were simulated 500 times with a sample size of 1000.
Specification of models used to estimate the loss of lifetime function
Model  Model  Nr. knots  Knot locations 

A  NRS  6  0%, 20%, 40%, 60%, 80%, and 100% quantiles of the uncensored event times. 
B  ARS  7  0%, 20%, 40%, 60%, 80%, and 100% quantiles of the uncensored event times with an additional knot placed at 10 years. 
C  ARS  7  0%, 20%, 40%, 60%, and 80% quantiles of the uncensored event times. The last knot is placed at 80 years and an additional knot is placed at 10 years. 
D  FMC  5  0%, 25%, 50%, 75%, and 100% quantiles of the uncensored event times. 
E  FMC  5  First uncensored event time, 0.5, 1, 2, and 5 years. 
For each model, the loss of lifetime function was computed and the bias was measured by \(D(t) = \widehat {L}(t)  L(t)\). The integral, \(\int _{0}^{15}D(u)du\), was used to measure the bias of the loss of lifetime estimate during the entire followup period.
Simulation results
Generally, the FMC models, D and E, showed good performance both in scenarios with statistical cure occurring within and beyond the available followup. In scenarios where statistical cure did not occur, the performance of the FMC models was comparable to model A, but the biases were more dispersed for later time point, especially in scenario 4 and 6. At ten years, the biases of model E were slightly less dispersed compared to model D.
The integrated loss of lifetime bias in the Weibull scenario, computed by integrating D(t) from 0 to 15 years
Age  Scenario  pi  Model A  Model B  Model C  Model D  Model E 

50  1  0.40  8.6(0.939.2)  2.4(0.510.7)  4.1(0.919.5)  2.4(0.218.5)  2.6(0.328.6) 
2  0.40  28.9(5.566.9)  11.7(6.023.1)  12.7(3.735.3)  11.1(0.668.0)  9.3(0.752.1)  
3  0.75  8.9(0.742.7)  8.9(4.315.6)  4.8(0.916.4)  7.5(0.531.5)  7.2(0.223.9)  
4  0.00  6.5(0.334.5)  144.8(123.4171.2)  37.2(8.982.1)  24.9(0.3111.8)  18.1(0.2104.6)  
5  0.00  11.0(0.243.3)  25.4(14.835.4)  13.0(3.231.0)  12.8(0.433.8)  9.6(0.330.9)  
6  0.00  18.1(0.565.4)  106.6(71.6127.7)  49.2(7.992.4)  36.2(0.6103.7)  23.9(0.289.4)  
60  1  0.40  6.0(1.319.4)  1.9(0.48.0)  3.1(0.613.5)  2.1(0.214.7)  2.4(0.217.6) 
2  0.40  14.9(2.645.4)  6.4(3.414.9)  7.7(2.026.2)  7.7(0.640.2)  6.6(0.242.5)  
3  0.75  7.2(0.439.6)  4.2(1.910.0)  4.0(0.422.7)  5.4(0.328.2)  4.7(0.319.6)  
4  0.00  5.7(0.324.1)  79.5(64.693.2)  21.0(6.144.4)  14.2(0.362.4)  10.2(0.149.8)  
5  0.00  7.5(0.333.4)  10.7(5.118.1)  5.6(1.518.6)  7.2(0.526.0)  5.0(0.217.9)  
6  0.00  10.9(0.637.3)  48.2(36.461.2)  18.5(4.142.5)  16.8(1.250.9)  11.2(0.345.3)  
70  1  0.40  3.6(0.912.4)  1.5(0.24.8)  2.2(0.47.3)  1.7(0.18.8)  2.0(0.112.9) 
2  0.40  6.2(1.220.8)  3.4(1.58.8)  4.0(0.914.2)  4.7(0.319.4)  4.3(0.319.2)  
3  0.75  4.9(0.320.6)  2.4(0.87.0)  3.3(0.312.9)  3.6(0.219.0)  2.8(0.211.2)  
4  0.00  4.3(0.316.1)  34.9(26.944.4)  9.6(3.523.3)  7.3(0.231.3)  5.7(0.127.5)  
5  0.00  5.3(0.425.2)  3.9(1.78.9)  3.5(0.614.8)  4.3(0.218.5)  2.9(0.110.5)  
6  0.00  6.0(0.321.7)  16.5(9.923.8)  6.2(1.816.6)  6.9(0.223.7)  5.1(0.217.4) 
The integrated loss of lifetime bias in the generalized gamma scenario, computed by integrating D(t) from 0 to 15 years
Age  Scenario  pi  Model A  Model B  Model C  Model D  Model E 

50  1  0.40  6.8(0.630.1)  2.4(0.411.3)  4.0(0.521.0)  2.6(0.329.8)  3.1(0.330.5) 
2  0.40  23.1(5.048.1)  10.2(5.718.3)  7.1(1.822.8)  8.5(0.639.3)  10.3(0.743.8)  
3  0.75  17.4(1.574.8)  10.2(4.219.8)  7.6(1.031.0)  11.2(0.863.1)  10.0(0.356.2)  
4  0.00  6.7(0.232.6)  146.5(121.6166.8)  39.2(10.475.0)  22.8(0.2123.1)  15.4(0.295.0)  
5  0.00  14.8(0.688.3)  35.9(22.745.6)  18.0(3.239.4)  18.4(0.677.5)  13.7(0.340.3)  
6  0.00  16.1(0.672.9)  130.5(108.5153.0)  56.1(9.899.7)  36.4(0.5117.8)  21.3(0.8100.1)  
60  1  0.40  4.7(0.916.5)  1.9(0.27.5)  3.0(0.312.1)  2.2(0.214.3)  2.8(0.121.5) 
2  0.40  12.0(2.735.4)  5.5(3.110.9)  4.6(1.019.4)  5.7(0.432.1)  7.2(0.535.0)  
3  0.75  11.1(1.053.0)  5.2(2.211.4)  6.1(0.731.7)  7.4(0.837.6)  7.5(0.431.0)  
4  0.00  5.8(0.222.2)  80.3(65.196.0)  21.8(7.342.4)  13.9(0.366.4)  9.5(0.249.4)  
5  0.00  10.2(0.755.2)  14.8(6.922.4)  7.4(1.629.4)  9.6(0.333.6)  6.8(0.222.0)  
6  0.00  9.8(0.337.7)  62.2(44.777.0)  24.1(5.951.5)  17.1(0.764.5)  11.4(0.654.9)  
70  1  0.40  3.1(0.611.0)  1.4(0.24.8)  2.2(0.27.4)  1.8(0.28.8)  2.3(0.112.7) 
2  0.40  5.5(1.016.7)  2.6(1.46.0)  2.9(0.411.1)  3.2(0.213.6)  4.2(0.314.6)  
3  0.75  6.4(0.430.7)  2.8(1.08.8)  4.1(0.320.0)  4.6(0.424.6)  4.5(0.417.6)  
4  0.00  4.2(0.314.9)  35.4(26.743.6)  10.2(3.923.9)  7.4(0.330.9)  5.4(0.224.0)  
5  0.00  6.1(0.226.8)  4.9(2.310.5)  3.9(0.617.2)  4.8(0.426.0)  3.6(0.113.1)  
6  0.00  6.1(0.423.2)  22.4(14.530.4)  8.0(2.422.7)  7.2(0.729.7)  5.2(0.425.1) 
Analysis of Danish cancer registry data
Data description
To investigate the performance of the models in Table 1 in cancer survival data, we analyzed data from the Danish Cancer Registry [20] on patients with colon cancer (n=4558), breast cancer (n=21,731), bladder cancer (n=11,738) and malignant melanoma (n=2404). To achieve (almost) complete followup, we included patients diagnosed in the period 1960–1975, who were older than 50 years at diagnosis. The diseases were chosen based on the same considerations as in Andersson et al. [1], i.e., colon cancer typically displays statistical cure, bladder cancer a constant excess hazard, melanoma a rather high survival rate, and breast cancer is seen in both young and old patients. Patients were followed until the end of 2016, where alive patients were censored and followup was measured from diagnosis until death or censoring. For the purpose of investigating the extrapolation performance, we restricted the followup to 16 years by censoring patients alive in January 1976 and divided patients into age groups; 50–59, 60–69, 70–79, 80+. The true loss of lifetime was calculated by inserting the KaplanMeier estimate into (9), and the bias was computed by D(t). For both the true and estimated loss of lifetime, the upper limit of the integrals in (9) was set to 40 years at which time the true survival was close to zero.
Results
Overall, no model was consistently superior to the others, but in scenarios of statistical cure, there was a slight advantage of using cure models. However, in scenarios without statistical cure, models B and C were substantially biased.
Analysis of Danish lymphoma registry data
Data description
To illustrate a potential clinical application of the proposed extrapolation techniques, we analyzed patient data from the Danish Lymphoma Registry, which covers 94.9% of all lymphoma cases in Denmark [21]. We included adult patients (≥18 years of age) diagnosed with diffuse large Bcell lymphoma (DLBCL, n=6639), follicular lymphoma (FL, n=3204), or mantle cell lymphoma (ML, n=980) in the period from 2000 to 2016. The followup period was terminated in June 2017 and the followup time was measured from time of diagnostic biopsy to death or censoring.
Populationbased loss of lifetime
Median age, 5year relative survival (RS), and loss of lifetime estimates at time zero in Danish diffuse large Bcell lymphoma (DLBCL), follicular lymphoma (FL), and mantle cell lymphoma (ML) patients
Model  DLBCL  FL  ML  

Median age (range)  68(18101)  63(1897)  70(2899)  
5year RS (95% CI)  NRS  0.66(0.650.68)  0.9(0.880.91)  0.61(0.570.65) 
ARS  0.66(0.650.67)  0.9(0.880.91)  0.61(0.570.65)  
FMC  0.66(0.640.67)  0.9(0.880.91)  0.61(0.580.65)  
Loss of lifetime (95% CI)  NRS  7.43(7.067.80)  4.58(3.735.42)  7.66(6.868.46) 
ARS  6.70(6.426.98)  3.57(3.134.02)  6.92(6.267.59)  
FMC  7.21(6.867.55)  3.97(3.244.70)  7.74(6.958.53) 
Clearly, the three models, despite being similar in the beginning of the followup, produce rather different conditional loss of lifetime estimates. At time zero, the maximal difference between the models is seen to be around 1 year for FL, for which the assumption of statistical cure is typically not reasonable. The model differences increased as time progressed, with the largest difference seen in ML patients. For DLBCL patients, the presented FMC model yielded a compromise between the NRS and ARS models which was seen by an intermediate loss of lifetime function. However, for the FL and ML patients where cure cannot usually be assumed, this model resembled the NRS model and even provided slightly higher loss of lifetime estimates.
Age dependent loss of lifetime
was chosen. Since none of the diseases showed a clear statistical cure trajectory, we did not consider the ARS model here. The number and location of the knots for the baseline spline function, s_{0}(x), remained unchanged from “Populationbased loss of lifetime” section. For s_{a}(a), 4 knots placed at the 0%, 33%, 66%, and 100% quantiles of the patient age distribution were selected and the intercept was removed since this is already modelled by the baseline splines and β_{0}. For s_{1}(x), the number of knots was chosen to be 3 and 2 for the NRS model and the FMC model, respectively, yielding the same total number of parameters.
For DLBCL, the two models seemed to be in agreement across patient age. However, the agreement between the two models for 60–70 year old FL patients was poor, likely due to the different model assumptions. For ML, the model differences were larger for younger patients, likely due to the additional extrapolation needed to compute the loss of lifetime for these patients.
Discussion
In (8) we introduced a novel model, which incorporates statistical cure by combining regular mixture cure models with splinebased survival models. This model was compared to the NRS model, which has a linear effect in the spline function after the last knot and the ARS model, which is constant after the last knot and thereby incorporates statistical cure. The simulations demonstrated a consistently good performance of the NRS model and the FMC model. The analysis of data from the Danish Cancer Registry did not show consistently satisfactory performance of any model, but in general assuming statistical cure at the end of the followup can lead to substantial biases in cases where this assumption is violated, while yielding good estimates when cure is reached. The NRS model performed slightly better than the FMC model in scenarios where statistical cure did not occur. This is likely due to the lack of identifiability often seen in cure models in cases where cure is not reached within the observed followup period [22], which ultimately may produce inaccurate extrapolations.
The present article expanded on the study of Andersson et al. [1] by evaluating the accuracy of the entire loss of lifetime function using three extrapolation approaches. While the loss of lifetime estimates at time zero in Fig. 4 seemed to be in agreement with the results reported by Andersson et al., where only 10 years of followup were used, the biases were not constant over time.
The general population survival probabilities for young patients are high and precise extrapolation of the relative survival is required to avoid a biased loss of lifetime function for these patients. Confirming this, we observed a higher bias among young patients which should be kept in mind when reporting loss of lifetime results. With longer followup and higher age, the bias will decrease and in future studies it would be of interest to estimate for a fixed age distribution, the amount of followup needed to provide sufficiently unbiased loss of lifetime estimates.
For some cancer types, the general population survival will likely not reflect the survival of the patients had they remained diseasefree. The life style of patients diagnosed with, e.g., lung or skin cancer is likely different from the general population life style and hence the relative survival will not reflect the diseasespecific (net) survival. However, this does not change the usability of the general population mortality rates to provide extrapolations of the survival function.
In contrast to net survival measures which are interpreted in the setting where the patient can only die from the disease of interest, the loss of lifetime measure provides a crude measure of the cancerrelated mortality. In net measures, such as relative survival, it is often seen that elderly patients have an increased mortality since deaths from other causes are not taken into account. For young patients, even a small excess mortality may have a large impact on the loss of lifetime function as the expected lifetime without cancer is long. Therefore, it is often seen that young patients have a higher loss of lifetime than elderly patients.
An alternative to the unrestricted loss of lifetime, where extrapolation is avoided, can be obtained by replacing the upper limit of the integrals in (9) by a fixed time point τ. In this setting, pseudovalues and flexible parametric survival models have previously been recommended for computing the mean survival time [23] and could also be used for estimating the loss of lifetime function. Using the three models to estimate the restricted loss of lifetime would likely yield fairly similar estimates due to the model similarities in the first part of the followup (Additional file 1: Figure S5). However, interpretation of the restricted loss of lifetime is not straightforward and the measure does not capture the full disease burden.
Conclusion
Since there is no way of assessing the performance of extrapolations applied to data with limited followup, the inconsistencies between the simulation results and the full followup data analysis emphasize the need for sensitivity analyses.
We therefore recommend that extensive sensitivity analyses are performed both with respect to the assumptions of the relative survival model as well as the number and location of the knots of the splines as recommended previously [10, 13].
Notes
Acknowledgements
Not applicable.
Funding
No funding was received for the study.
Availability of data and materials
Data used to generate the findings of the study were obtained from the Danish Clinical Registries (Danish Lymphoma Registry) and the Danish Cancer Registry after approval of our study plan by both registries and the Danish Data Protection Agency. The registries contain patient identifiable information and therefore sharing of these data is not allowed per the terms of the agreement with the registries. However, from the registries, access to the data is granted on a case to case basis after submission and approval of an appropriate study plan and reasonable data request. Data from the Danish Cancer Registry can be applied for at https://sundhedsdatastyrelsen.dk/da/forskerservice, and data from the Danish Lymphoma Registry can be applied for at http://www.rkkp.dk/forskning. The code used for generating the results can be found at https://github.com/LasseHjort/LossOfLifetimeEstimation.
Authors’ contributions
The idea was conceived by LHJ, MB, and TMLA. LHJ performed all analyses and wrote the first draft of the manuscript. MB, TMLA, JLB, and TCEG contributed with essential feedback and suggestions for the methodology and the data analysis. All authors read and approved the final manuscript.
Ethics approval and consent to participate
The study was approved by the Danish Data Protection Agency (2008580028). In Denmark, no informed consent is required in order to use data from the Danish Cancer Registry and the Danish Lymphoma Registry for research purposes.
Consent for publication
Not applicable.
Competing interests
T. M.L. Andersson is involved in an ongoing publicprivate real world evidence collaboration between Karolinska Institutet and Janssen Pharmaceuticals. However, the current project is not related to this collaboration.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary material
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