Department of Biostatistics EA 4275, Clinical Research and Subjective Measures in Health Sciences, University of Nantes, 1 rue Gaston Veil, Nantes 44035, France
Transplantation, Urology and Nephrology Institute (ITUN), Nantes Hospital and University, Inserm U1064, 30 bd Jean Monnet, Nantes 44093, France
Abstract
Background
Whereas the prognosis of second kidney transplant recipients (STR) compared to the first ones has been frequently analyzed, no study has addressed the issue of comparing the risk factor effects on graft failure between both groups.
Methods
Here, we propose two alternative strategies to study the heterogeneity of risk factors between two groups of patients:
Results
We demonstrate, for the first time in renal transplantation, that:
Conclusion
While the traditional Cox model did not provide original results based on the renal transplantation literature, the proposed relative and stratified models revealed new findings that are useful for clinicians. These methodologies may be of interest in other medical fields when the principal objective is the comparison of risk factors between two populations.
Background
In patients facing a first allograft loss, repeat kidney transplantation provides a better chance for both longterm survival and quality of life than a return to dialysis
For this purpose, traditional survival models can be used by merging STR and FTR. Nevertheless, one can notice two important limitations of these traditional approaches. Firstly, the comparisons of risk factors between both groups would imply testing all the interactions with the graft rank. Secondly, STRspecific explicative variables (survival time of the first transplant, transplantectomy or time in dialysis before retransplantation) cannot be included despite the knowledge that their use would improve risk evaluation
To overcome these limits, in this paper we propose two alternative strategies. The first one is an adaptation of a multiplicativeregression model for relative survival (MRS). This type of relative approach is often used to study the net survival of patients with cancer, i.e. the survival of patients if the only cause of death is related to the disease
Moreover, we propose a second method by adapting a stratified Cox model (SCM) specifying the graft rank as strata and assuming a vector of explicative variables decomposed into subvectors of variables that enter either in the reference hazard only, or in the relative hazard only, or in both groups but with common or separate effects.
Methods
Study population
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 
The multiplicativeregression model for relative survival (MRS)
Let the individuals be indexed by
where
Parameters of the expected hazard can be estimated assuming a semiparametric and proportional hazards (PH) model
where
where
Parameters of the relative hazard can also be estimated using a semiparametric PH model. For the
where
Of note, for explicative variables not taken into account in the expected hazard in the model (1), exp(
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 MonteCarlo simulations associated with bootstrap resampling
• Generation of a vector of parameters
• Generation of a bootstrap sample from the relative sample comprising
• From this bootstrap sample, the model (4) is estimated by maximizing (5) in which the simulated parameters
Means, standard deviations and 95% confidence intervals can be calculated from the
The stratified Cox model (SCM)
For the
with
with
and for the relative group, we obtain
Therefore,
Evaluation of the proportional hazards assumption
In models (2), (4) and (7), hazard proportionality was checked for each explicative variable by plotting logminuslog survival curves obtained by the Kaplan and Meier estimator
Software
All statistical analyses were performed using R version 2.15.1
Results
Description of the cohort
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 followup 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 followup 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
All (N = 2772)
FTR (N = 2206)
STR (N = 566)
pvalues were obtained by using the Chisquare statistic.
Demographic characteristics
Number (%)
Number (%)
Number (%)
pvalue
Transplantation period < 2005
594
(21.4)
457
(20.7)
137
(24.2)
0.0806
Recipient ≥ 55 years of age
1175
(42.4)
994
(45.1)
181
(32.0)
<0.0001
Male recipient
1705
(61.5)
1362
(61.7)
343
(60.6)
0.6536
Recurrent causal nephropathy
906
(32.7)
666
(30.2)
240
(42.4)
<0.0001
History of diabetes
306
(11.0)
269
(12.2)
37
(6.5)
0.0002
History of hypertension
2263
(81.6)
1804
(81.8)
459
(81.1)
0.7545
History of vascular disease
352
(12.7)
272
(12.3)
80
(14.1)
0.2804
History of cardiac disease
903
(32.6)
686
(31.1)
217
(38.3)
0.0012
History of dyslipemia
799
(28.8)
661
(30.0)
138
(24.4)
0.0104
History of malignancy
228
(8.2)
147
(6.7)
81
(14.3)
<0.0001
History of hepatitis B or C
168
(6.1)
96
(4.4)
72
(12.7)
<0.0001
Recipient BMI ≥ 30 kg.m ^{2}
263
(9.5)
235
(10.7)
28
(4.9)
<0.0001
Positive anticlass I PRA
706
(25.5)
355
(16.1)
351
(62.0)
<0.0001
Positive anticlass II PRA
733
(26.4)
319
(14.5)
414
(73.1)
<0.0001
Donor ≥ 55 years of age
1172
(42.3)
973
(44.1)
199
(35.2)
0.0002
Deceased donor
2470
(89.1)
1940
(87.9)
530
(93.6)
0.0002
Donor serum creatinine ≥ 133
342
(12.5)
279
(12.8)
63
(11.4)
0.3807
Positive donor EBV serology
2613
(94.3)
2087
(94.6)
526
(92.9)
0.1540
HLAABDR incompatibilities > 4
365
(13.2)
326
(14.8)
39
(6.9)
<0.0001
Cold ischemia time ≥ 24h
754
(27.2)
552
(25.0)
202
(35.7)
<0.0001
Lymphocytedepleting induction
1223
(44.1)
793
(35.9)
430
(76.0)
<0.0001
First graft survival < 1 year




131
(24.1)

Waiting time before regraft ≥ 3 years




272
(49.8)

Transplantectomy of the first graft




220
(38.9)

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
Multivariate Cox model for FTR (N = 2206) and results of the MRS in the STR group after exclusion of the 37 STR also included in FTR, based on 507 STR (22 recipients presenting missing data for the waiting time before retransplantation were excluded).
Click here for file
Results of the stratified Cox model after exclusion of the 37 STR also included in FTR, based on 2713 patients with 2206 FTR and 507 STR (22 STR presenting missing data for the waiting time before retransplantation were excluded).
Click here for file
Analysis of risk factors in the FTR sample
As previously illustrated in Table
Cox model in
MRS in
the FTR group
the STR group
PRA, panel reactive antibody; EBV, EpsteinBarr virus; HLA, human leukocyte antigen.
HR
95% CI
pvalue
HR
95% CI
pvalue
Variables entering in the model for FTR only
Causal nephropathy (recurrent / non recurrent)
1.24
0.961.59
0.0987



History of diabetes (positive / negative)
1.34
0.961.85
0.0819



History of hypertension (positive / negative)
0.77
0.571.05
0.0986



History of cardiac disease (positive / negative)
1.41
1.111.79
0.0051



History of vascular disease (positive / negative)
1.10
0.811.51
0.5351



History of dyslipemia (positive / negative)
1.12
0.871.45
0.3828



History of hepatitis B/C (positive / negative)
0.82
0.451.47
0.4969



History of malignancy (positive / negative)
1.25
0.841.86
0.2698



Body mass index (≥ 30 kg.m2 / < 30 kg.m2)
1.58
1.122.14
0.0084



Anticlass I PRA (positive / negative)
1.45
1.071.97
0.0182



Anticlass II PRA (positive / negative)
1.09
0.781.52
0.6299



Donor status (deceased/living)
2.50
1.414.43
0.0016



Donor EBV serology (positive / negative)
1.65
0.982.78
0.0606



Number of HLAABDR mismatches (> 4 / ≤ 4)
1.30
0.971.76
0.0824



Induction therapy (depleting / non depleting)
0.79
0.601.05
0.1091



Cold ischemia time (≥ 24 h / < 24 h)
1.29
1.011.66
0.0441



Variables entering in both models
Transplantation period (< 2005 / ≥ 2005)
1.33
0.971.82
0.0693
0.97
0.551.74
0.9360
Recipient gender (male / female)
1.17
0.911.51
0.2186
0.61
0.381.05
0.0720
Recipient age (≥ 55 years / < 55 years)
1.39
1.051.83
0.0204
1.65
1.012.72
0.0480
Donor age (≥55 years / <55 years)
1.34
1.031.74
0.0313
0.59
0.330.99
0.0440
Variables entering in the model for STR only
Donor gender (male / female)



1.53
1.032.48
0.0320
Waiting time before regraft ≥ 3 years



1.92
1.223.00
<0.0001
Relative hazard modelling in the STR group using the MRS
A first selection of variables was performed (p < 0.20) followed by a stepbystep descending procedure (Wald test with p < 0.05). In line with the requirements of additiveregression models, adjustments were forced for recipient gender and age and transplantation period. All the variables were categorized in order to avoid any loglinearity assumption and to obtain interpretable results.
The final relative model is presented in the last three columns of Table
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 1.6fold higher for STR compared to FTR (CI95% = [1.012.72], p = 0.0480). Similarly, the effect of donor age ≥ 55 years was nearly two fold lower for STR than for FTR (CI95% = [0.330.99], 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).
Relative hazard modelling in the STR group using the SCM
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.
FTR strata
STR strata
PRA, panel reactive antibody; EBV, EpsteinBarr virus; HLA, human leukocyte antigen.
HR
95% CI
pvalue
HR
95% CI
pvalue
Variables entering in z ^{ e } only
Causal nephropathy (recurrent / non recurrent)
1.12
0.901.39
0.3031



History of diabetes (positive / negative)
1.28
0.951.72
0.1001



History of hypertension (positive / negative)
0.88
0.681.14
0.3270



History of cardiac disease (positive / negative)
1.35
1.101.66
0.0042



History of vascular disease (positive / negative)
1.11
0.851.47
0.4433



History of dyslipemia (positive / negative)
1.19
0.951.49
0.1263



History of hepatitis B/C (positive / negative)
0.97
0.651.46
0.9091



History of malignancy (positive / negative)
1.21
0.871.67
0.2646



Body mass index (≥ 30 kg.m2 / < 30 kg.m2)
1.54
1.132.08
0.0057



Anticlass I PRA (positive / negative)
1.39
1.071.82
0.0153



Anticlass II PRA (positive / negative)
0.98
0.731.31
0.8857



Donor status (deceased/living)
2.06
1.273.36
0.0036



Donor EBV serology (positive / negative)
1.65
1.072.54
0.0235



Number of HLAABDR mismatches (> 4 / ≤ 4)
1.33
1.011.75
0.0397



Induction therapy (depleting / non depleting)
0.88
0.701.11
0.2742



Cold ischemia time (≥ 24 h / < 24 h)
1.20
0.971.49
0.0894



Variables entering in
Transplantation period (< 2005 / ≥ 2005)
1.42
1.091.86
0.0099
0.94
0.541.64
0.8295
Recipient gender (male / female)
1.17
0.911.50
0.2200
0.63
0.401.02
0.0581
Recipient age (≥ 55 years / < 55 years)
1.36
1.031.78
0.0274
1.60
0.952.72
0.0785
Donor age (≥55 years / <55 years)
1.36
1.041.77
0.0238
0.60
0.351.05
0.0725
Variables entering in
Donor gender (male / female)



1.51
0.972.36
0.0674
Waiting time before regraft ≥ 3 years



1.99
1.293.07
0.0019
Estimations and corresponding 95% confidence intervals were very similar to those obtained in the MRS. Indeed, as in the previous model, we estimated a 2fold 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.
Discussion
Although the comparison of survival between first and second kidney transplants has been frequently performed, no study has addressed the issue of comparing the risk factors associated with the time to graft failure between both groups. Understanding the factors influencing the longterm evolution of STR compared to FTR would benefit the medical management of graft attribution by identifying patients with the best chances.
The absence of literature focusing on this question may be partially explained by the methodological issues associated with such studies. Indeed, the Cox model is classically used to explore risk factors influencing graft survival and interactions can be included to evaluate risk factor differences between FTR and STR. However, this approach has several limitations. Firstly, it implies testing interactions between the graft rank and each explicative variable, increasing the number of parameters and making interpretations difficult. Secondly and certainly more importantly, only covariates common to both groups can be taken into account. This excludes explicative variables specific for one group. Concerning our application, this constitutes a limitation as several STRspecific explicative variables are known to be associated with second graft prognosis: the first graft transplantectomy
This paper describes two alternative models to overcome these difficulties. Firstly, the adaption of a multiplicativeregression model for relative survival allows a direct comparison of risk factors between two groups of patients without presupposing the role of each variable, i.e. common, different or specific relationships. The corresponding semiparametric models are the Cox model for the expected hazard and the multiplicativeregression for the relative hazard. The main difficulty and limit of these models is the estimation of standard deviations which were obtained by MonteCarlo simulations associated with bootstrap resampling. In this multiplicative modelling, the regression coefficients are straightforward to interpret in terms of their interactions. We propose an R package for a simple way of using the model.
Secondly, we demonstrated that a stratified Cox model specifying the graft rank as strata may be fitted to take into account STRspecific variables as a subvector of variables that enter in the model for STR only. In addition, some variables would enter either in the model for FTR only or in both models (with common or separate effects). The main limit to this approach is that the corresponding structure presupposes knowledge of variables potentially applicable to both models (in contrast with the relative model) unless testing a very large number of models. Indeed, whereas explicative variables entering in a single model (for FTR or STR) would easily be clinically assumed, those applicable to both models and with common or separate effects are not known in advance. Nevertheless, the SCM can be simply estimated by maximising a single partial likelihood function.
As expected, the results were concordant between both approaches. Regression coefficients were similar while standard deviations appeared a little smaller with the MRS approach. The results showed that male donor gender and a long waiting time before retransplantation were two specificSTR risk factors: donor gender was not significantly associated with the risk of graft failure in the FTR population and the waiting time before retransplantation was only related to STR by definition. The interpretations were similar to hazard ratios from a Cox model preformed on the STR group.
Conversely, two explicative variables appeared to be differently associated with the risk of graft failure between STR and FTR. More precisely, we showed for the first time that the adverse effect of recipient age was enhanced for STR as compared to FTR. The main clinical explanation is a cumulative effect of the risk factors for STR, in particular because of the cumulative exposure to immunosuppressive drugs during the first transplantation period. From a clinical point of view, this result may imply that clinicians should pay particular attention to recipient age in second kidney transplantations. Also, for the first time to our knowledge, this study identified an attenuation of the risk factor related to older transplants for STR as compared to FTR. Two explanations are:
Although we illustrated the advantages of both alternative approaches in renal transplantation, this methodology may be useful in number of other clinical and epidemiological applications. For practical use, we propose an R package to compute the MRS. The adaptation of the SCM can be computed by using many statistical software. Of course, the aim of such models is not to replace traditional survival models, but rather to provide a more suitable alternative when the main objective is to compare risk factors between two populations, in particular when populationspecific covariates need to be included.
As always, there are several avenues worth exploring from this work. First, both models can be generalized for timedependent explicative variables by adapting the likelihood functions as proposed by Therneau and Grambsch
Conclusions
MRS and SCM consitute two original approaches to compare risk factors between two populations. The advantage of MRS is to allow a direct modelling strategy but it is not straightforward to estimate the standard deviations. In contrast, SCM allows an overall estimation of parameters and standard deviations but its structure presupposes knowledge of the role of each explicative variable. This study also highlighted novel risk factor differences between first and second kidney transplant recipients. These results could help improve the management of patients waiting for a second graft. They may also encourage the widespread use of this original methodology in other medical fields.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
KTL and YF designed the model. KTL carried out the statistical analysis and wrote the manuscript. YF performed statistical analysis, participated in writing the manuscript and extended the software to enable use of the method. MG and JD participated in research design, performed clinical analysis. MG and JD participated in writing of the manuscript. All authors read and approved the final manuscript.
Acknowledgements
We thank Dr. J.AshtonChess for editing the manuscript. We also wish to thank the DIVAT scientific council and the members of the clinical research assistant team (M. Kessler, M. Ladrière, JP. Soulillou, C. Legendre, H. Kreis, G. Mourad, V. Garrigue, L. Rostaing, N. Kamar, E. Morelon, F. Buron, S. Le Floch, C. Scellier, V. Eschbach, P. Przednowed, K. Zurbonsen, V. Godel, K. Zurbonsen, X. Longy, C. Dagot, F. M’Raiagh) for the collection of the data in the DIVAT cohort and finally Dr. M. Pohar Perme for her helpful advice. This work was partly supported by the RTRS, the Fondation de Coopération Scientifique CENTAURE and Roche Laboratory.
Prepublication history
The prepublication history for this paper can be accessed here: