Second transplant recipients (STR) constituted the relative group of interest. Recipients older than 18 years at the date of transplantation between 1996 and 2010 were selected from the French DIVAT (Données Informatisées et VAlidées en Transplantation - http://www.divat.fr/en) multicentric prospective cohort 19 . Codes were used to assure donor and recipient anonymity and blind assay. The ’Comité National Informatique et Liberté’ approved the study (N° CNIL 891735) and written informed consent was obtained from the participants. Only recipients with a maintenance therapy with calcineurin inhibitors, mammalian target of rapamycin inhibitors or belatacept, in addition to mycophenolic acid and steroids, were included. Simultaneous transplantations were excluded. Recipients with at least one missing data for all the variables taken into account in the expected hazard were excluded. The same criteria were applied to the reference group composed of first transplant recipients (FTR). The principal outcome was the time between transplantation and graft failure, which was the first event between return to dialysis and patient death with a functioning graft.

Let the individuals be indexed by j (j = 1,...,n _e ) in the reference group and by i (i = 1,...,n _r ) in the relative group. n _e  and n _r  represent the sample sizes of the respective groups. We note h ^o (t _i |z _i ) the observed instantaneous hazard function at time t _i  for the ith individual, where z _i is the vector of explicative variables. The observed hazard of the ith subject of the relative group can be decomposed in the multiplication of two hazards 18 20

h o ( t i | z i ) = h e ( t i | z i e ) h r ( t i | z i r )

where h e ( t i | z i e ) is the expected hazard for an individual in the reference group with similar characteristics to the ith individual. z i e is a subset of z _i  and represents these common characteristics. h r ( t i | z i r ) is the relative hazard with z i r being a subset of z _i .

Parameters of the expected hazard can be estimated assuming a semi-parametric and proportional hazards (PH) model 21 . For the jth individual (j = 1,...,n _e )

h e ( t j | z j e ) = h 0 e ( t j ) exp ( β e z j e )

where h 0 e ( t j ) is an unknown expected baseline hazard function and β ^e  is the vector of regression coefficients associated with z j e . The estimations β ̂ e are obtained by maximizing the partial log-likelihood among the n _e  patients of the reference group

log P ℒ e ( β e ) = ∑ j = 1 n e δ j β e z j e - log ∑ k : t k ≥ t j exp ( β e z k e )

where δ _j  equals 1 if the failure was observed for the jth subject and 0 otherwise. The estimation of the variance-covariance matrix, V ̂ ( β ̂ e ) , is obtained via the corresponding information matrix.

Parameters of the relative hazard can also be estimated using a semi-parametric PH model. For the ith individual (i = 1,...,n _r ), the instantaneous hazard is defined as

h r ( t i | z i ) = h 0 r ( t i ) exp ( β r z i r )

where h 0 r ( t i ) is an unknown relative baseline hazard function and β ^r  is the vector of regression coefficients associated with z i r . By adapting the partial likelihood function (3) and assuming the previous estimations of expected parameters as constants, the regression coefficients β ^r  are estimated by maximizing

log P ℒ r ( β r ) = ∑ i = 1 n r δ i β ̂ e z i e + β r z i r - log ∑ k : t k ≥ t i exp ( β ̂ e z k e )

Of note, for explicative variables not taken into account in the expected hazard in the model (1), exp(β ^r ) represents the observed hazard ratios (HR) in the reference group, just as exp(β ^r ) estimated from the PH model (4) among the relative group. In contrast, for explicative variables taken into account in the expected hazard model in the model (1), exp(β ^r ) represents the weighting factors between the expected HR, i.e. exp ( β ̂ e ) , and the observed HR in the relative group, i.e. exp ( β ̂ e ) × exp ( β r ) . In other words, for explicative variables involved in both models (2) and (4), β ^r  = 0 means that the variable has the same effect in both groups. If β ^r  > 0, the hazard ratio increases in the relative group compared to the reference group. If β ^r  < 0, the hazard ratio decreases.

Nevertheless, in contrast to the traditional relative survival models based on life tables, the expected hazards cannot be reasonably assumed as constants since the corresponding parameters were estimated from the reference sample. To take into account the variability associated with the expected model (2) in the estimation of the relative model (4), we used Monte-Carlo simulations associated with bootstrap resampling 22 . At each of the B iterations ( b = 1 , … , B ), this procedure can be divided into the following steps

• Generation of a vector of parameters β ̂ b e ∗ using the multivariate normal distribution N β ̂ e , V ̂ ( β ̂ e ) obtained from the maximisation of the partial likelihood (3). This first step takes into account the variance of the expected hazard.

• Generation of a bootstrap sample from the relative sample comprising n _r  subjects. This second step takes into account the variance due to sample-to-sample fluctuation.

• From this bootstrap sample, the model (4) is estimated by maximizing (5) in which the simulated parameters β ̂ b e ∗ are used instead of β ̂ e . β ̂ b r is the resulting estimation of the relative regression coefficients.

Means, standard deviations and 95% confidence intervals can be calculated from the B estimations of β ̂ b e ∗ .

For the ith individual of the overall sample (i = 1,...,n), i.e. the reference and the relative group together (n = n _e  + n _r ), let z _i  be the vector of covariates that are applied to the model for either first or second transplant recipients. Let k be the indicator of the strata with k = e for the reference group and k = r for the relative group. The stratified Cox model is given by

h k ( t i | z i ) = h k , 0 ( t i ) exp ( β z i ) ; for k = e , r.

with h _k,0(t _i ) the baseline hazard function in the strata k. The vector z _i  can be decomposed into four different subvectors: (i) the vector z i e of explicative variables that enter in the reference hazard only (their values equal 0 if k = r); (ii) the vector z i r of explicative variables that enter in the relative hazard only (their values equal 0 if k = e); (iii) the vector z i c of explicative variables associated to both groups; and (iv) the vector z i s a subvector of explicative variables included in z i c but with separate effects. The model (6) can be developed as follows

h k ( t i | z i ) = h k , 0 ( t i ) exp ( β e z i e + β c z i c + β s z i s δ ir + β r z i r ) ; for k = e , r.

with β = (β ^e ,β ^c ,β ^s ,β ^r ) the vector of regression coefficients and δ _{i r}  equals 1 if the subject i belongs to the strata k = r and 0 otherwise. Then, for the reference group, we obtain

h e ( t i | z i ) = h e , 0 ( t i ) exp ( β e z i e + β c z i c )

and for the relative group, we obtain

h r ( t i | z i ) = h r , 0 ( t i ) exp ( β c z i c + β s z i s + β r z i r )

Therefore, e x p(β ^e ) represents the HR associated with specific variables for the reference group. And e x p(β ^r ) represents the HR associated with specific variables for the relative group. For variables only included in z ^c  (and not in z ^s ), e x p(β ^c ) represents the common HR in both groups associated with z ^c . For variables included in both z ^c  and z ^s , the HR associated with z ^c  equals e x p(β ^c ) in the reference group and e x p(β ^c +β ^s ) in the relative group. Thus, e x p(β ^s ) represents the weighting factors between the expected HR, i.e. e x p(β ^c ), and the observed HR in the relative group, i.e. e x p(β ^c ) × e x p(β ^s ).

In models (2), (4) and (7), hazard proportionality was checked for each explicative variable by plotting log-minus-log survival curves obtained by the Kaplan and Meier estimator 23 and by testing the scaled Schoenfeld residuals 24 separately in the reference and relative samples. If the observed hazard ratios are constant regardless of time in both groups, the ratio between both observed hazard ratios, i.e. the weighting factor e x p(β ^r ), will also be constant.

All statistical analyses were performed using R version 2.15.1 25 . The proposed multiplicative-regression model for relative survival was implemented in an R package MRsurv available at http://www.divat.fr/en/softwares/mrsurv. The adapted stratified Cox model was implemented by using the R package survival (function coxph, option strata).

641 STR potentially made up the relative group of interest, but 75 STR (11.7%) with missing data for explicative variables of the expected hazard were excluded. Finally, 566 STR were included in the group of interest. The mean follow-up was 3.1 years with a maximum of 13.1 years. During the observation period, 72 returns to dialysis and 34 deaths were observed. We identified 2462 FTR who met the inclusion criteria. We excluded 256 FTR (10.3%) with one missing data for at least one of the variables taken into account in the expected hazard model. Finally, 2206 FTR made up the reference group. The mean follow-up was 3.4 years with a maximum of 13.7 years. During the observation period, 191 returns to dialysis and 109 deaths were observed.

The demographic and baseline characteristics at the time of transplantation are presented in Table 1. Regardless of the group, the majority of patients received a transplant from a deceased donor and the recipient gender was comparable between groups. However, STR were younger and their transplants were provided by younger donors. Recurrent nephropathy, past history of cardiac disease, hepatitis and malignancy were more frequent among STR, but STR had less diabetes and were less likely to be obese at the time of transplantation. Compared to FTR, STR received better HLA-matched transplants, but their cold ischemia time was longer and they were more immunized against HLA class I and class II antigens (historical Panel Reactive Antibodies) than FTR. They were also more frequently exposed to induction therapy with a lymphocyte-depleting agent.

p-values were obtained by using the Chi-square statistic.

Among FTR meeting the inclusion criteria, some patients were also part of the STR group as they had received two transplants during the observation period. These 37 patients, who were included in both cohorts, represented 2% and 7% of the FTR and STR groups respectively. Given the large number of explicative variables, it seemed reasonable to assume conditional independence of the two transplantations of a given patient. In order to validate this assumption, we performed a frailty Cox model 26 based on the 37 individuals who were included in both groups. The frailty term was assumed to be Gamma distributed. The variance of the random variable was estimated at 5.10 ^-9 (p = 0.9948). Therefore, no intra-individual dependency was demonstrated. In order to validate the robustness of the results, we also performed both models after exclusion of the 37 STR also included in FTR. These results are presented in Additional files 1 and 2.

As previously illustrated in Table 1, it is well-established that FTR and STR are not intrinsically comparable. Thus, for the analysis of risk factors in the FTR population, adjustments were made (i) for all of the possible pre- or per-transplant immunological and non-immunological confounding factors according to experts and (ii) for all the baseline parameters differentially distributed between FTR and STR. All together, the expected hazard of graft failure was estimated according to recipient age and gender, causal nephropathy, comorbidities (including history of diabetes, hypertension, cardiac or vascular disease, dyslipemia, hepatitis B or C, and malignancy), obesity, pre-transplant immunization (panel reactive antibody, PRA) against class I and class II antigens), donor age, deceased or living donor status, Epstein-Barr Virus (EBV) serology, period of transplantation, level of HLA-A-B-DR mismatches, induction therapy and cold ischemia time. This modelling is explained in detail in the paper by Trébern-Launay et al. 8 . The final multivariate model in the reference group of FTR is presented in the first three columns of Table 2.

PRA, panel reactive antibody; EBV, Epstein-Barr virus; HLA, human leukocyte antigen.

A first selection of variables was performed (p < 0.20) followed by a step-by-step descending procedure (Wald test with p < 0.05). In line with the requirements of additive-regression models, adjustments were forced for recipient gender and age and transplantation period. All the variables were categorized in order to avoid any log-linearity assumption and to obtain interpretable results.

The final relative model is presented in the last three columns of Table 2. Expected HR previously estimated in FTR are presented in the first columns to enable a direct comparison between FTR (Cox model) and STR (relative model). Donor gender and waiting time before retransplantation were not taken into account in the expected hazards for FTR. Donor gender was not a significant risk factor for FTR and waiting time is by definition a specific factor for STR. More precisely, we estimated a 1.5-fold increase in risk of graft failure for STR with grafts from males compared to STR with grafts from females (p = 0.0320). Moreover, STR who waited more than 3 years in dialysis before retransplantation had a 1.9-fold increased risk compared to STR with a shorter waiting time (p < 0.0001).

In contrast, the effect of recipient age and donor age seemed significantly different between FTR and STR (p < 0.05). More precisely, if we assumed a similar effect of recipient age between both groups, the expected HR associated with recipient age ≥ 55 years would be 1.39 in the STR group, regarding the HR observed in the FTR group. In fact, the relative model showed that this HR was 1.6-fold higher for STR compared to FTR (CI95% = [1.01-2.72], p = 0.0480). Similarly, the effect of donor age ≥ 55 years was nearly two fold lower for STR than for FTR (CI95% = [0.33-0.99], p = 0.0440), while it was identified as a significant risk factor for FTR (HR = 1.34, p = 0.0313). Of note, the relationship between the recipient gender and the risk of graft failure was not found to be significantly different between FTR and STR (p = 0.0720).

As an alternative, we performed the SCM based on the same variables as those used in the previous MRS. Donor gender and waiting time before retransplantation were included in variables applied only in the relative part, i.e. z ^r . The other four variables (transplantation period, recipient gender and age, and donor age) were included in z ^c  and z ^s  to evaluate the difference in their effect between both groups. The results are presented in Table 3.

PRA, panel reactive antibody; EBV, Epstein-Barr virus; HLA, human leukocyte antigen.

Estimations and corresponding 95% confidence intervals were very similar to those obtained in the MRS. Indeed, as in the previous model, we estimated a 2-fold increase in risk of graft failure for STR who waited more than 3 years in dialysis before retransplantation compared to STR who waited less than 3 years (p = 0.0019). The relationship between the donor gender and the risk of graft failure among STR was similar to that obtained in the MRS but was not found to be significant (HR = 1.51, p = 0.0674).

Transplantation period, recipient gender, recipient age and donor age were included in variables applied in both models. Results were also concordant with the MRS. For the four explicative variables, estimations and corresponding 95% confidence intervals were similar to those obtained in the MRS. However, conversely to the MRS, recipient age and donor age were not found to be significantly differently associated with the risk of graft failure between the two groups.