Testing the Functional Form of Continuous
Predictors and their Interactions in the Outcomes
of Graft Survival in Kidney Transplantations in
the elderly Dutch Population
----> Moving towards a Prediction Model
Priya S. Gopalrai
Dept. of Nephrology &
Medical Informatics
ACADEMIC MEDICAL CENTER – UNIVERSITY OF AMSTERDAM (AMC-UVA)
MASTER - MEDICAL INFORMATICS (MI)
SCIENTIFIC RESEARCH PROJECT (SRP)
MAY 14, 2018
Academic Medical Center - University of Amsterdam (AMC-UVA)
MASTER - MEDICAL INFORMATICS (MI)
Scientific Research Project (SRP)
Testing the Functional Form of Continuous
Predictors and their Interactions in the
Outcomes of Graft Survival in Kidney
Transplantations in the elderly Dutch
Population
Moving towards a Prediction Model
P
REFACE
A great opportunity, a great honor and above all an intensive learning period it was during the entire course of the Master Medical Informatics at the Academic Medical Center and the University of Amsterdam. To accomplish the master’s degree in medical informatics I am submitting this thesis on “Testing the functional forms of continuous predictors and their interactions with outcomes in graft survival in kidney transplantation in the elderly Dutch population”, for my Scientific Research Project(SRP). Although, there are many possibilities for one holding a master’s degree in Medical Informatics, it is important to find a direction as a medical informatician to use and develop the strengths and skills in one or even more particular fields of this phenomenal and broad specialism. As I always felt the connection with the analytical medical field, this study was a great opportunity to express my affinity with data science and research in clinical fields and I hope the latter will be recognizable in this thesis.
The study has both statistical- mathematical and clinical characteristics. I compared the statistical outcomes with clinical practice to find out whether there are similarities, differences and most of all (possible) directions to new developments. Each part is analyzed as objective as possible and the thesis is written and organized in such way to try to build up and maintain the interest of the reader.
Before heading the reader towards the content of the study, I sincerely, want to thank each person that contributed not only to this SRP, but also during the entire Master course. Special thanks go to my parents and husband, for their support, the coordinator of the SRP, Prof. Dr. Ameen Abu Hanna, the tutor, Prof. Dr. Kitty Jager, the daily mentor of the study, Drs. H. Peter-Sengers and the mentor Prof. Dr. F.J. Bemelman for believing in me and their incredible guidance through the SRP-process.
R
EADING
D
IRECTIONS
Abbreviations and definitions can be found in a specific list after the keywords of the study. The first chapter describes and contains some preliminary information for reader’s knowledge and the background on the subject, before moving to chapter 2 for problem description, goals definition and related work. The third chapter includes all the used methods in the analysis process. Results are reported in chapter 4 and 5. For each result part (chapter 4 & 5) a summary was added in the end. Chapter 6 contains the discussion on the results and the conclusion with references.
In the result section visualization is done using tables and graphs. However, due to a large amount of analyses and results, it was not possible to show all the results in these sections and moreover to preserve the clarity of the thesis. Based on personal preference one could possibly prefer to see most of the relevant results. This was solved by adding supplementary appendices for chapter 4 and 5. Each Supplementary appendix has coded headings linked to the headings of the result section, where the referral is done. For a good understanding of the matter, it is recommended that the reader has basic statitistical and mathematical knowledge.
TABLE OF CONTENTS
PREFACE --- READING DIRECTIONS--- TABLE OF CONTENTS --- 1 KEYWORDS --- 1 LIST OF ABBREVIATIONS --- 2 ABSTRACT --- 3 ABSTRACT (DUTCH) --- 4 1 BACKGROUND --- 51.1 RENAL REPLACEMENT THERAPY IN THE NETHERLANDS --- 6
1.2 KIDNEY TRANSPLANTATION IN ELDERLY PATIENTS --- 6
1.3 REFERENCES --- 7
2 PROBLEM DEFINITION: THE FUNCTIONAL FORMS AND ASSOCIATIONS OF CONTINUOUS PREDICTORS IN KIDNEY TRANSPLANTATION - TOWARDS A PREDICTION MODEL --- 8
2.1 RELATED WORK --- 9
2.2 REFERENCES --- 10
3 MATERIALS AND METHODS --- 14
3.1 DATABASE --- 14
3.2 COVARIATES OF INTEREST --- 14
3.3 MISSING VALUES --- 15
3.4 STATISTICAL ANALYSIS --- 15
3.5 GRAFT SURVIVAL –THE SURVIVAL ANALYSIS –COX REGRESSION --- 15
3.6 COX ASSUMPTIONS --- 16
3.7 TRANSFORMATIONS --- 18
3.8 MOVING TOWARDS THE PREDICTION MODEL --- 20
3.9 MODEL VALIDATION --- 21
3.10 EXTENDED ANALYSIS (DICHOTOMIZATION)--- 22
3.11 STUDY CONDUCT, STUDY DESIGN SUMMARY --- 22
3.12 REFERENCES --- 24
4 PRIMARY OUTCOME-DEATH CENSORED GRAFT SURVIVAL --- 25
4.1 COVARIATES OF INTEREST --- 25
4.2 BASELINE CHARACTERISTICS --- 26
4.3 MISSING VALUES AND IMPUTATIONS --- 28
4.4 THE SURVIVAL CURVE-KAPLAN MEIER --- 29
4.5 COX REGRESSION AND RISKS (UNIVARIATE- REGRESSION ANALYSIS) --- 30
4.6 COX ASSUMPTIONS --- 30
4.7 TRANSFORMATIONS --- 37
4.8 TOWARDS THE PREDICTION MODEL --- 42
4.9 MODEL VALIDATION AND CALIBRATION --- 44
4.11 SUMMARY --- 47
5 SECONDARY OUTCOME -PATIENT SURVIVAL --- 48
5.1 SURVIVAL CURVE -KAPLAN MEIER --- 48
5.2 COX REGRESSION AND COX ASSUMPTIONS --- 48
5.3 TRANSFORMATIONS --- 50
5.4 TOWARDS THE PREDICTION MODEL --- 50
5.5 MODEL VALIDATION --- 51 5.6 SUMMARY --- 52 6 DISCUSSION--- 53 6.1 CONCLUSION --- 56 6.2 REFERENCES --- 57 ACKNOWLEDGEMENTS --- 58
SUPPLEMENTARY APPENDIX A- CHAPTER 4 --- 59
1
K
EYWORDS
Kidney Transplantation
Death Censored Graft Survival (DCGS) Patient Survival Survival Analysis Functional Forms Proportional Hazards Cox Regression Continuous predictors Linearity
Akaike Information Criterion (AIC) Discrimination
2
L
IST OF
A
BBREVIATIONS
AIC: Akaike Information Criterion ANOVA: Analysis of the Variance AUC: Area under CurveCIT: Cold Ischemic Time DGF: Delayed Graft Function ECD: Extended Criteria Donor ESRD: End Stage Renal Disease DBD: Donation after Brain Death DCD: Donation after Cardiac Death GS: Graft Survival
DCGS: Death Censored Graft Survival MFP: Multivariable Fractional Polynomials NS: Natural Spline
PS: Patient Survival / Recipient Survival PSpline: Penalized Spline
RCS: Resticted Cubic Spline WIT1: First Warm Ischemic Time WIT2: Second Warm Ischemic Time BMI: Body Mass Index
Donor, Transplantation and Recipient factors
Donorage: Age of the donorDonorCRP: The last measured C-reactive proteins of the donor Donor SCr: Last creatinine level of the donor
1st WIT: First Warmischemia time
2nd WIT: Second Warischemia time
Donor cause of death-CVA: cerebrovascular accident as cause of donor’s death Recipientage: Age of the Recipient
3
A
BSTRACT
IntroductionIn this study we investigated the functional form of continuous predictors and their interaction with graft survival and mortality after kidney transplantation for elderly recipients (65+), as early research showed that these were not linearly associated with the outcomes in graft survival. With the organ shortage, there is need to set reliable cut-off criteria for selection of donor kidney organs while expanding the donor pool safely to transplant elderly patients with comorbidities.
Objective
Performing comprehensive analysis of continuous predictors to test their functional forms, moving towards a prediction model for death-censored graft survival (DCGF) as primary - and patientsurvival (PS) as secondary outcome, after testing the cox assumptions and taking actions to meet these assumptions and finally setting possible cut off criteria for the selection of kidney transplants in elderly patients. Methods
Using the NOTR database from 2005-2017, the Kaplan Meier Curve and Cox regression were used to perform survival analysis for the total dataset, the donation after brain death group (DBD) and the donation after cardiac death (DCD). To meet and deal with the Cox assumptions, proportional hazards were tested for each selected predictor, while linearity was checked for the continuous predictors. Several quadratic, power, spline and fractional polynomials transformations were performed to deal with the non-linearity of continuous predictors. Transformed predictors were used to build a prognostic prediction model. An internal validation, with discrimination and calibration was performed to validate the prediction model(s).
Results
For DCGS, continuous non-linear predictors were age of the donor, Body Mass Index (BMI) of the donor, last serum creatinine of the donor, final C-reactive proteins (CRP) CIT, the age of the recipient, BMI of the recipient and Panel Reactive Proteins (PRA). BMI of the donor, last Serum Creatinine of the donor, C-Reactive proteins, WIT2, CIT, and BMI of the recipient were not lineary associated with Patient Survival (PS). WIT1 was only considered in the DCD group and was non-linear for both DCGS and PS. Transformation criteria were met by using the Natural Spline for each non-linear continuous predictor. Multivariate analysis delivered global proportionalities of > 0.05. After transformations lower (Akaike Information Criterion (AIC) values were observed. Several donor, transplantation and recipient factors were significant in univariate and multivariate analysis (p< 0.05) for both DCGS and patient survival. The overall graft survival (GS) rates after year 1, year 3 and year 5 were repectively 90 %, 87% and 82 %. PS rates after year 1, 3 and 5 were 89%, 77% and 60 %. For DCGS, the discrimination, area under the curve (AUC) value of the final model for the total set was 0.73 and 0.79 for both DBD and DCD. AUC values for the final model for PS for the total dataset, for DBD and DCD were 0.66, 0.73 and 0.72. Calibration showed the best slopes at 2 years cumulative survival time for PS and DCGS. Cutoff Criteria could be derived from the analogous and dynamic nomograms.
Conclusion
Our chosen transformations did not only correct the hazard ratios of the continuous predictors, but also increased the model quality and model performance by respectively decreasing the AIC values and increasing the discrimation values. However, from the calibration plots we could derive that the model accuracy decreases with increased survival years.
4
A
BSTRACT
(D
UTCH
)
IntroductieIn deze studie wordt de interactie van continue predictiefactoren met transplantaat overleving en mortaliteit onderzocht voor oudere ontvangers (65+), aangezien eerdere studies hebben uitgewezen dat deze variabelen niet lineair geassocieerd zijn met transplantaat overleving.
Doel
Het uitvoeren van (uitgebreide) analyses om de functionele vorm van de continue variabelen te testen en na het testen van specifieke cox assumpties, en het voldoen aan deze, werken naar een predictiemodel voor dood gecencureerde transplaat overleving (DCGF) als primaire uitkomst en ontvangers mortaliteit (PS) als secundaire uitkomst.
Methode
The NOTR-database (2005-2017) werd gebruikt voor deze studie. Kaplan Meier en Cox regressieanalyses zijn uitgevoerd voor de totale set met alle variabelen, the DCD en DBD-groep. Cox assumpties voor proportional hazards werden uitgetest voor elke geselecteerde variabele, terwijl de lineariteit werd uitgetest voor alle continue variabelen. BMI van de donor, laatste serum creatinine, CRP, CIT en ontvanger’s BMI waren niet linear geassocieerd met mortaliteit. Leeftijd van de donor, BMI, laatste serum creatinine, CRP, WIT1, CIT en ontvanger’s BMI waren niet linear geassocieerd met mortaliteit. WIT1 was alleen geincludeerd in de DCD-groep. Verschillende transformaties waaronder kwadratische, macht-, spline en fractionele polynomials tranformaties werden toegepast om de niet lineaire predictoren te transformeren. Deze getransformeerde variabelen werden als covariaten meegenomen in het predictiemodel. Een interne validatie op basis van discriminatie en calibratie werd uitgevoerd op het finale model.
Resultaten
De lineariteitstest liet zien dat de predictiefactoren o.a. leeftijd van de donor, gewicht/massa ratio/index (BMI) van de donor, CIT leeftijd van de ontvanger en panel reactieve proteinen (PRA) niet lineair waren voor DCGS. BMI van de donor, laatste serum creatinine, CRP, WIT1, CIT en ontvanger’s BMI waren niet linear geassocieerd met mortaliteit. WIT 1 werd alleen geincludeerd in de DCD-groep en bleek zowel voor DCGS als PS niet linear te zijn. De Natural spline tranformatie voldeed het best aan de gestelde selectiecriteria voor de uitgevoerde transformaties. In multivariate analyses werden de proportionaliteiten niet overschreden (P>0.05). Na tranformaties werden lagere AIC-waarden verkregen. Verschillende donor-, transplantatie en ontvangersfactoren waren significant voor DCGS en PS (P<0.05). De transplantaat overleving was 90 %, 87% en 82 % na jaar 1, jaar 3 en jaar 5. Patient overleving na jaar 1,3 en 5 waren 89%, 77 % en 60 %. AUC van het finale model was 0.73. Voor DBD en DCD was deze waarde 0.79. AUC-waarden voor de predictie van mortaliteit voor de totale set en voor DBD en DCD waren respectievelijk 0.66, 0.73 en 0.72. Calibratie toonde de beste predicties op een cumulatieve overlevingstijd van 2 jaar voor PS en DCGS. Cutoff criteria kon afgeleid worden van de analoge en dynamische nomogrammen.
Conclusie
De gekozen transformaties hebben niet alleen hazard ratio’s van de factoren veranderd (verhoogd), maar resulteerden ook in een verhoogde model kwaliteit en predictie vermogen door respectievelijk the AIC-waarden te verlagen en de predictie-AIC-waarden te verhogen. Desondanks, blijkt uit de calibratie plots dat met toenemende overlevingsjaren het predictie vermogen achteruitgaat.
5
1 B
ACKGROUND
Organ transplantation is one of the treatment possibilities for end stage organ failure. The procurement of organs for transplantation involves the removal of organs from the bodies of deceased persons and/or
living donors. This removal must follow legal requirements, including the definition of death and consent. The kidney is one of the most highly differentiated organs in the body. After embryologic development,
nearly 30 different cell types form a multitude of filtering capillaries and segmented nephrons enveloped by a dynamic interstitium. This cellular diversity modulates a variety of complex physiologic processes, endocrine functions, the regulation of blood pressure and intraglomerular hemodynamics, solute and water transport, acid-base balance, and removal of drug metabolites are all accomplished by intricate mechanisms of renal response [1,2].
Chronic kidney disease (CKD) is due to conditions that damage the kidneys and decrease their ability for example for creatinine clearance, resulting in high serum creatinine levels, proteinuria (proteins in the urine) and decrease in electrolyte regulations. Kidney disease can cause high blood pressure, anemia (low blood count), weak bones, poor nutritional health, and nerve damage. It also increases the risk of having cardiovascular disease [3]. CKD may be caused by diabetes, high blood pressure or other disorders. Early detection and treatment can often keep chronic kidney disease from getting worse. When it progresses, kidney failure can be the result, which requires kidney replacement therapy, like dialysis (haemo- or peritoneal dialysis) or kidney transplantation [3].
Although end stage renal disease (ESRD) patients can be treated with other renal replacement therapies, kidney transplantation is frequently carried out and is generally accepted as the best treatment for both quality of life and cost effectiveness [4].
Kidney donation by well selected living donors in good health carries negligible risks. This can only be ensured through rigorous selection procedures, careful surgical nephrectomy and follow up of the donor to ensure the optimal management of untoward consequences [4].
Unfortunately, in most countries the supply of kidneys for transplantation is insufficient to meet the demand. Therefore, hemodialysis is the replacement therapy for patients with renal failure [5].
Kidney transplantation in most instances involves a donor who is genetically and therefore antigenitically distinct from the recipient, the consequence of which is an immune response termed rejection. The major histocompatibility complex (MHC) generates a uniquely diverse set of antigens, the responses to which the mature immune system cross-react on established memory. Cellular events that contribute to rejection have led to the development of a range of biological agents that target specific aspects of the immune response, with a view to optimizing the risk-benefit of immunosuppression [6]. Before transplantation, patients should first be assessed for their suitability. There are a few absolute contraindications, and the decision must be based on an overall assessment of the patient’s current fitness and ability to withstand surgery, immunosuppression, and subjective assessment of compliance and estimated life expectancy. Availability of kidneys may also influence selection policy. The best first option is a pre-emptive transplant from a living related donor due to the much longer survival of living donor transplants, the lower morbidity and the reduced doses of immunosuppression. If living donor transplantation is not possible, the next best option is to put the patient for a pre-emptive deceased donor transplant. Deceased donor renal donation remains the predominant source of transplantable kidneys. They are divided into subgroups of donation after brain death(DBD)1 and donation after circulatory death
(DCD)2. While total proportions of standard criteria donors and extended criteria donors have remained
stable over the last decade, the contribution of the DCD donors has become increasingly important.
1 DBD: Heart Beating Donor 2 DCD: Non Heartbeating Donor
6 However, the total Ischemia time should be kept as low as possible [4]. There are 3 ischemia times in kidney transplantation. The first warm ischemic time (WIT1), the second warm ischemic time (WIT2)3 and
the cold ischemic time (CIT). 4
1.1 RENAL REPLACEMENT THERAPY IN THE NETHERLANDS
Ever more Dutch citizens depend on renal replacement therapy which consist of dialysis and kidney transplantation. In the end of 2016, the prevalence was 17000 patients from which 60 % had a functioning transplant. Also, in 2016 the increase in the number of patients could be due to the increase of the existing population which already had a transplantation in 2016 or before (prevalence). Although, there is an increase in the age of the patient population, the inflow of new patients (incidence) is stable for both dialysis and transplantation for the last few years. In 2016, 1656 patients started their dialysis treatment while 991 patients received a kidney transplantation [7].
The number of elderly patients, older than 75 years has increased in the past decennium with 33% for dialysis and transplantation. However, the survival with the dialysis treatments is significantly improving for all age categories [7].
There is an increase of transplantable patients on the waiting list. This can be partly explained by the lower number of deceased kidney transplantations and partly by the increase of elderly patients being put on the waiting list. The waiting time for patients with blood group O is longer in comparison to patients with other blood groups. Although patients are screened and placed on the waiting list of Eurotransplant in the pre- and dialysis phase, it is remarkable that the majority of these patients do not receive a transplant due to the fact that they still not meet the kidney transplant criteria of Eurotransplant and the shortage of kidney transplants [7].
Comparison to other European countries shows that the prevalence and incidence of renal replacement therapy are average in the Netherlands compared with the European average. Same numbers show that there are less deceased donor transplantations in the Netherlands, while being in the top-3 for living-donor kidney transplantations [7].
1.2 KIDNEY TRANSPLANTATION IN ELDERLY PATIENTS
As Kidney transplantation is considered as the optimal treatment for patients with end-stage renal disease (ESRD) [8-10], a successful kidney transplantation offers patients the greatest potential for increased longevity and enhanced quality of life [11,12]. There is a growing discrepancy between availability of and the need for donor kidneys. The shortage of available organs for kidney transplantation has led to several strategies to increase the donor pool [13]. One strategy is the use of extended criteria donors (ECD). ECDs are defined as donors more than 60 years (or 50 years with clinical characteristics), eligible for transplantation but expected to have diminished posttransplant function [14-18]. However, this definition is not used in the Netherlands. Most of transplantations are ECDs and for about 50% of donors are in the Eurotransplant Senior Program (ESP) with donor age categories of >65 years [7].
The morbidity and mortality of the dialysis population is significant and the prognosis for elderly patients in dialysis is especially poor [19-22]. Increased time on dialysis has been shown to be a significant risk
3 Warm ischemia time: In surgery, the time a tissue, organ, or body part remains at body temperature after its
blood supply has been reduced or cut off but before it is cooled or reconnected to a blood supply.
4 Cold ischemia time: In surgery, the time between the chilling of a tissue, organ, or body part after its
blood supply has been reduced or cut off and the time it is warmed by having its blood supply restored. This can occur while the organ is still in the body or after it is removed from the body if the organ is to be used for transplantation.
7 factor for poor posttransplant outcome in elderly kidney transplant recipients [23]. Elderly patients may spend the rest of their lives waiting for a deceased-donor kidney. As a result, transplant physicians are often confronted with the dilemma of accepting an ECD kidney for transplantation to avoid extensive waiting time in dialysis [24]. In previous years, the median waiting period for a kidney from a deceased donor was more than three years and 74 patients died while waiting for a kidney transplantation in the Netherlands in 2015. To address the organ shortage, accepting older deceased donors with more co-morbidities has become common practice. Stretching these criteria facilitates the risk of delayed graft function (DGF), prolonged hospitalization, rejection, and graft failure. However, much uncertainty still exists about the correct form of the relationships between continuous donor factors and graft failure [23].
1.3 REFERENCES
1. HARRISON’s principles of INTERNAL MEDICINE 18th Edition Volume 2 CHAPTER 277 Cellular
and Molecular Biology of the Kidney
2. Richard J. Johnson, John Feehally, Jürgen Floege COMPREHENSIVE CLINICAL NEPHROLOGY CHAPTER 2 ,1,3
3. https://www.kidney.org/atoz/content/about-chronic-kidney-disease 4. http://www.who.int/transplantation/organ/en/
5. HARRISON’s principles of INTERNAL MEDICINE 18th Edition Volume 2 CHAPTER 277 Cellular
and Molecular Biology of the Kidney
6. Peter J. Morris, Stuart J. Knechtle Kidney Transplantation Principles and Practice- Seventh Edition
7. Nierfunctie vervangende behandeling in Nederland- jaarboek 2016
8. JohnsonDW, Herzig K, Purdie D, Brown AM, Rigby RJ, Nicol DL, etal. A comparison of the effects of dialysis and renal transplantation on the survival of older uremic patients. Transplantation 2000; 69: 794.
9. OniscuG C, Brown H, Forsythe JL. How great is the survival advantage of transplantation over dialysis in elderly patients? Nephrol Dial Transplant 2004; 19: 945.
10. Fabrizii V, Winkelmayer WC, Klauser R, Kletzmayr J, Sa¨emann MD, Steininger R, etal. Patient and graft survival in older kidney transplant recipients: does age matter? J Am Soc Nephrol 2004; 15(4): 1052.
11. Wolfe RA, McCullough KP, Schaubel D.E, Kalbfleisch JD, Murray S, Stegall MD, Leichtman AB. Calculating life years from transplant (LYFT): methods for kidney and kidney-pancreas candidates. Am J Transplant 2008; 8(4 Pt 2): 997.
12. Liem YS, Bosch JL, Arends LR, Heijenbrok-Kal MH, Hunink MG- Quality of life assessed with the Medical Outcomes Study Short Form 36-Item Health Survey of patients on renal replacement therapy: a systematic review and meta-analysis. Value Health 2007; 10(5): 390.
13. Rosen gard BR, FengS, Alfrey EJ, Zaroff JG, Emond JC, Henry M.L., etal. Report of the Crystal City meeting to maximize the use of organs recovered from the cadaver donor. Am J Transplant 2002; 2: 701.
14. Port FK, Bragg-Gresham JL, Metzger RA, Dykstra DM, Gillespie BW, Young EW, et al. Donor characteristics associated with reduced graft survival: an approach to expanding the pool of kidney donors. Transplantation 2002; 74: 1281.
15. Baskin-Bey ES, Nyberg SL. Matching graft to recipient by predicted survival: can this be an acceptable strategy to improve utilization of deceased donor kidneys? Transplant Rev 2008; 22: 167.
8 16. Meier-Kriesche HU, Cibrik DM, Ojo AO, Hanson JA, Magee JC, Rudich SM, Leichtman AB, etal. Interaction between donor and recipient age in determining the risk of chronic renal allograft failure. J Am Geriatr Soc 2002; 50: 14.
17. Rosendale JD, Chabalewski FL, McBride MA, Garrity ER, Rosengard BR, Delmonico FL, etal. Increased transplanted organs from the use of a standardized donor management protocol. Am J Transplant 2002; 2: 761.
18. Gill J, Bunnapradist S, Danovitch GM, Gjertson D, Gill JS, Cecka M. Outcomes of kidney transplantation from older living donors to older recipients. Am J Kidney Dis 2008; 52: 541. 19. Munshi SK, Vijayakumar N, Taub NA, Bhullar H, Lo TC, Warwick G. Outcome of renal
replacement therapy in the very elderly. Nephrol Dial Transplant 2001; 16: 128.
20. Mandigers CM, de Jong W, van den Wall Bake AW, Gerlag PG. Renal replacement therapy in the elderly. Neth J Med 1996; 49: 135.
21. Macrae J, Friedman AL, Friedman EA, Eggers P. Live and deceased donor kidney transplantation in patients aged 75 years and older in the United States. Int Urol Nephrol 2005; 37: 641.
22. ERA-EDTA Registry Annual Report 2005 available on http://www.era-edta-reg.org/files/annualreports/pdf/AnnRep2005.pdf
23. Heldal K, Hartmann A, Leivestad T, Svendsen MV, Foss A, Lien BH, Midtvedt K. Clinical outcomes in elderly kidney transplant recipients are related to acute rejection episodes rather than pre-transplant co morbidity. Transplantation in Press. 2009 Apr 15;87(7):1045-51
24. Snoeijs MG1, Schaubel D.E, Hené R, Hoitsma AJ, Idu MM, Ijzermans JN, Ploeg RJ, Ringers J,
Christiaans MH, Buurman WA, van Heurn LW. Kidneys from donors after cardiac death provide survival benefit. J Am Soc Nephrol. 2010 Jun; 21(6): 1015–1021
2 P
ROBLEM DEFINITION
:
THE FUNCTIONAL FORMS AND ASSOCIATIONS OF
CONTINUOUS PREDICTORS IN KIDNEY TRANSPLANTATION
-
TOWARDS A
PREDICTION MODEL
Organ shortage has resulted in increased use of expanded criteria (deceased) donors for transplantation, in particularly older donors. Important continuous factors are age, serum creatinine, donor height and weight cold ischemic time (CIT), 1st warm ischemic time (WIT1), and anastomosis time. There is evidence that the donor age, donor weight, and CIT are not linearly associated with graft failure [1,2]. If a nonlinear situation is studied with a linear model, serious errors may emerge and is often a poor approximation of the truth [1,2].
If the correct functional form5 of the associations of deceased donor factors with transplant outcomes
would be known and incorporated in a prediction model, transplant professionals could improve the selection of the donors. This may help to increase the donor pool safely. The functional forms of the associations of these factors and their interactions have not yet been investigated in the Dutch population, and results may put a new focus on selecting elderly DCD donors for transplantation. From there we can
5 Functional form: The correct relationship of a continous variabele with the outcome variabele.
9 base the decision on donor criteria on evidence instead of gut-feeling of clinicians, which is often current practice. Especially for the acceptance of elderly (65+) kidneys from circulatory-death donors, which are associated with higher mortality risk in the elderly (65+) recipient [3].
The aim of this Scientific Research Project is to develop a prognostic prediction model for graft survival, taking into account all the factors and their correct functional form in their association with outcome. Furthermore, working towards setting reliable cut-off criteria for selecting elderly kidneys from deceased donors. Specific Research questions and a subquestions were defined to contribute to the results:
• What is the functional form of continuous factors (e.g. age, serum creatinine, donor height and weight, cold ischemic time (CIT), 1st warm ischemic time (WIT), and anastomosis time) in their association with outcomes in graft survival?
• What is the predictive performance of an appropriately developed model for graft survival in kidney transplantation in the Dutch population with the emphasis on elderly kidneys from deceased donors?
The subquestion
• What are reliable cut-off criteria for selecting elderly kidneys from deceased donors?
Prediction research is a distinct field of epidemiologic research, which should be clearly separated from aetiological research. Both prediction and aetiology make use of multivariable modelling, but the underlying research aim and interpretation of results are very different. Aetiology aims at uncovering the causal effect of a specific risk factor on an outcome, adjusting for confounding factors that are selected based on pre-existing knowledge of causal relations. In contrast, prediction aims at accurately predicting the risk of an outcome using multiple predictors collectively, where the final prediction model is usually based on statistically significant, but not necessarily causal, associations in the data at hand [4,5].
2.1 RELATED WORK
Heading up towards the methods we looked up for similar studies and their approaches.
When considering the patient population, studies for elderly recipients have been performed to find out whether donor age is an important predictor to stretch the donor pool. The type of donor, donation after circulatory death (DCD) or donation after brain death (DBD) plays an important role in selection, post-transplant management based on the (patient) outcomes. Wong et al. showed that there was a significant interaction between total ischemic time, donor age and graft loss (P=0.01). The study stated that there is an interaction between donor age, the pathway of donor death and total ischemic time on graft outcomes, such that the duration of the ischemic time has the greatest impact on the composite endpoint of graft survival in recipients with older donations after circulatory death [6]. Other studies were more focused on combinations of (different clinical) outcomes. Pretagostini et al. found that donorage ≥60 years (p< 0.0001) is associated with delayed graft function. Delayed graft function (DGF), a state in which the transplant has delayed function due to ischemia time and the recipient need dialysis within 7 days after transplantation, represents one of the most common complications after kidney transplantation [7]. Sozen et al. contradicted the latter by stating that in their study with donors ≥ 55 years old there was only a 3.6 % of DGF displayed in their study group. However, graft outcomes were not affected by DGF [8]. Another study investigated outcomes, in which they compared a younger donor group with the older donor group, which correspond to deceased donors with age of 70 years and older. Jozwik et al. reported a 13.4 % higher frequency of DGF in the older group. A 3-year survival rate of 85 % and 80 % compared to the control group of 92.5% and 88.6% was reported. However, they stated that these results were comparable to other studies and acceptable [9]. Ruta et al. showed that the glomerular filtration rate(GFR) was worse in their older group compared to the younger group. Donor age (OR=1.07, P=0.001) and recipient age (OR = 1.047, P=0.022) were one of the 9 variables associated with worse renal transplant function after one-year post transplantation [10]. While Impedovo et al. stated that high donor age is not an exclusion criterion for Kidney transplantation as there were no significant differences in graft survival between their study group (old donors > 60 years) and the control group (donors younger than 60 years). However, elderly
10 patients experienced worse patient survival. The most common causes of patient death were myocardial infarction, other cardiovascular complications and tumors. Si Nga et al. stated that donor age is a predictor for graft outcomes. Study outcomes showed that each additional year in donor age increased the relative risk of DGF by 8% (1.08 (1.0-1.13; p=.01)) [11,12]. DGF is more common in recipients of DCD kidney transplantation compared to DBD cited Singha et al. in their study. Results showed that there were differences in DBD and DCD donor outcomes [13].
The functional forms of continuous predictors can be dealt with using different approaches. While some studies state that the Multivariable Fractional Polynomials model is better, others use the martingale residuals method, followed by parametric and non-parametric transformations to deal with non-linearity [14-16]. Sauerbrei et al. mentioned that after more than a half century of research, the question of the “best” way of selecting the multivariable model is still unresolved [17]. It is generally agreed that subject matter knowledge, when available, should guide model building. However, such knowledge is often limited, and data dependent model building is required. Assuming linear functions they used both Fractional Polynomials and Spline methods for model building and compared these two methods. Given the results of model quality, they chose for the fractional polynomials approach for model building with linear predictors [17]. Another study in cancer survival on model building showed a simple approach as cited in this study by using cubic splines to deal with non-linearity to get the right functional association between the predictors and the outcome [18]. Croxford et al. discussed the methods of the book ‘Clinical prediction models’ by Ewout Steyerberg. The approaches, from dichotomization, restricted cubic splines, fractional polynomials are discussed in detail describing the fractional polynomials to be a more flexible method while dealing with non-linearity [19]. In contrast to Steyerberg’s first shown approach of dichotomization, Royston et al. cited that the latter with use of cut off points may harm the power of a study drastically and that this bias could result in non-direct interaction or association between predictors and their outcomes [20].
Studies on graft failure prediction seem to be less frequent. Kabore etal. performed a systematic review on prognostic models, validation of these models and tools (nomograms) and discussed the methological approaches [21]. From 134 studies, 39 met the inclusion criteria, from which 34 developed and validated a new risk model [22-55], 5 validated an existing model [55-60]. Discrimination performance was reported in 87% of studies, while calibration was reported in 56%. Performance indicators were estimated using both internal and external validation in 13 studies and using external validation only in 6 studies. Although some of the models showed good predictive performance [31], there is still room for improvement in the development of predictive models in kidney transplantation [21].
Overall Remarks on related studies:
• Most of the studies were performed to stretch the donor pool to address the organ shortage (using ECD donors)
• Some of the studies made a distinction between DCD and DBD and used extended donor criteria. • The Cox Regression model is used in most of the studies to perform survival analysis.
• Most of the studies were retrospective studies.
2.2 R
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13. Rajinder P. Singha, Alan C. Farneya, Jeffrey Rogersa, Jack Zuckermana, Amber ReevesDanielb, Erica Hartmannb, Samy Iskandarc, Patricia Adamsb and Robert J. Stratta
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Regression Models in Clinical Studies: Determining Relationships Between Predictors and Response- J Natl Cancer Inst. 1988 Oct 5;80(15):1198-202
12 19. Ruth Croxford, Institute for Clinical Evaluative Sciences, Toronto, Ontario, Canada
Continuous Predictors in Regression Analyses.Paper 288-2017. 20. Patrick Royston; *; †, Douglas G. Altman and Willi Sauerbrei
Dichotomizing continuous predictors in multiple regression: a bad idea. Stat Med: 2006 Jan 15;25(1):127-41
21. Remi Kabore, Maria C. Haller, Jerome Harambat, Georg Heinze and Karen Leffondre: Risk prediction models for graft failure in kidney transplantation: a systematic review Nephrol Dial Transplant (2017) 32: ii68-ii76
22. Akl A, Ismail AM, Ghoneim M. Prediction of graft survival of living-donor kidney transplantation: nomograms or artificial neural networks? Transplantation2008; 86:1401–1406
23. Baskin-Bey ES, Kremers W, Nyberg SL. A recipient risk score for deceased donor renal allocation. AmJ Kidney Dis 2007; 49:284–293
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25. Van Wal raven C, Austin PC, Knoll G. Predicting potentialsurvivalbenefitof renal transplantation in patients with chronic kidney disease. Can Med AssocJ 2010; 182:666–672
26. TiongHY, Goldfarb DA, Kattan MW etal. Nomograms for predicting graft function and survival in living donor kidney transplantation based on the UNOSRegistry.JUrol2009;181:1248–1255 27. Dahle DO, Eide IA, A ˚sberg A etal. Aortic stiffness in a mortalityrisk calculator for kidney transplant
recipients. Transplantation 2015; 99:1730–1737
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29. Einecke G, Reeve J, Sis B et al. A molecular classifier for predicting future graft loss in late kidney transplant biopsies. J Clin Invest 2010; 120: 1862–1872
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32. Fritsche L, Hoerstrup J, Budde K et al. Accurate prediction of kidney allograft outcome based on creatinine course in the first 6 months posttransplant. Transplant Proc 2005; 37:731–733
33. Grams ME, Kucirka LM, Hanrahan CF et al. Candidacy for kidney transplantation of older adults. J Am Geriat Soc 2012; 60:1–7
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14
3 M
ATERIALS AND METHODS
To perform our research, we used the NOTR database (2005-2017), with relevant information of elderly transplant recipients. The software used as tool for data analysis was R. For replication and follow up purposes an R- script with specific Syntax for this study has been produced by the author (Appendix C-4).
3.1 D
ATABASEThe NOTR database for elderly patients was used for this study. This database consisted of 1287 recipients of 65 years or older that received a kidney transplantation from a deceased donor. The database consisted of donor information, transplantation information and recipient information. All data was retrieved pseudonymized from the Dutch Transplant Foundation from all eight Dutch transplant centers. For further information on the variables see Appendix C “Codebook for the database”.
3.1.1 Study Population
All elderly (65+) transplant recipients who received their (deceased) donor kidney in the NOTR database between 2005 and 2017 were included in the analysis.
3.2 C
OVARIATES OF INTERESTWe collected baseline data of all patients and included age at transplantation, gender, transplant era, the number of human leukocyte antigen (HLA) mismatches, panel-reactive antibodies (PRA), dialysis duration, recipient comorbidities such as cardiovascular disease, body mass index and relevant donor factors, such as donor age, donor sex, donor body mass index, terminal serum creatinine, the duration of (total) ischemia6, and the stratification of donor death, brain death (DBD) and donation after cardiac death (DCD).
Each recipient with a recipient number is linked to a donor with a fictive donor number. Cases of both left and right kidney donation (dual) for a specific recipient were considered as one organ. The donor and recipient variables consisted of binary, categorical and continuous variables. Important information for donor cause of death were also reported. This information was useful to be able to make distinctions between Donation after Circulatory Death and Donation after Brain death. Also, for the functioning of the transplant and to take into account donor pathway related outcomes.
In case of the recipient this information was important to relate the recipient’s death to kidney failure or other renal problems. Primary diseases of the recipients were also included to make sure that graft loss could not be related to autoimmune diseases or other specific causes rather than donor related problems. For better insight in older donors we also created subsets DBD and DCD to perform analysis, reported and compared results with the outcomes of the total dataset. Another reason is that WIT1 can only be anlysed in the DCD group.
3.2.1 Measurement of Primary and secondary outcome(s)
Time to event information was used to define the outcomes. The primary outcome of the study was death censored graft survival (DCGS). The secondary outcome was patient survival. Time-to-event data present themselves in different ways which create special problems in analyzing such data. One peculiar feature, often present in time-to-event data, is known as censoring, which, broadly speaking, occurs when some lifetimes or survival periods when an event of interest could not be reached, because 1. Lost to follow-up, 2. Other outcome of interest reached first, 3. End of study date reached. The remainder of the lifetimes
6 Total Ischemia time: Transplant surgery- The time that an organ is outside the body when the heart is not beating or supplied with O2 by the coronary arteries
15 are known exactly. To exclude graft loss due to patient death we used death censored graft failure as our primary outcome.
3.3 M
ISSING VALUESMultiple imputation is a statistical technique for analyzing incomplete data sets, that is, data sets for which some entries are missing. Application of the technique requires three steps: imputation, analysis and pooling. Rubin (1987) has shown that if the method to create imputations is 'proper', then the resulting inferences will be statistically valid.
Missing data that occur in more than one variable presents a special challenge. In this study missing values were handled with multiple imputations by chained equations using the “MICE” package. The R package mice imputes incomplete multivariate data by chained equations [1].
To the uninitiated, multiple imputation is a bewildering technique that differs substantially from conventional statistical approaches. The philosophy behind the MICE methodology is that multiple imputation is best done as a sequence of small steps, each of which may require diagnostic checking. Based on the number of missing values (max 20%), 20 imputations with 50 iterations per imputation were performed. The variable with the highest missing values was the BMI of the donor (20%). One of the 20 imputations was randomly selected to test the cox assumptions.
3.4
STATISTICALA
NALYSISThe baseline characteristics of the study cohort were expressed as counts and percentages (for categorical variables) or mean ± SD (for numeric variables). Both categorical and numeric covariates were analyzed in univariate and multivariate Cox regression. For the continuous donor predictors univariate and multivariate analysis were used and these were plotted to show the functional form of these variables. The Cox assumptions, proportional hazards and linearity, were checked and dealt with according to methods described below.
3.5 G
RAFTS
URVIVAL–
T
HES
URVIVALA
NALYSIS–
C
OXR
EGRESSIONFor survival analysis we used the “Survival “package [3]. Survival analysis is used to analyze data in which the time until the event is of interest. There are several models, non-parametric, semi parametric and parametric7 models for survival time data. Three specific examples of parametric survival time models are
the exponential, the Weibull and the log-logistic regression models [4]. In our study we used the semiparametric Cox proportional hazards regression model. The response variable is the time until that event and is often called a failure time, survival time, or event time.
The hazard ratio (HR) plays a central role in Survival analysis, since it is a convenient summary of the relationship between two survival curves. The crucial assumption of proportional hazards (PHs) is equivalent to saying that the hazard ratio is independent of time. If a non-PH is detected, then the Cox model may be extended in various ways to accommodate it [5].
First Univariate analysis were performed to see the” 1:1” association between the predictor and outcome(s). A Multivariate analysis was performed afterwards to check the differences in associations and to develop the prediction model
16
3.6 C
OXA
SSUMPTIONSThe Cox proportional hazards model makes several assumptions. Thus, it is important to assess whether a fitted Cox regression model adequately describes the data. The three types of diagnostics for the Cox model are testing the proportional hazards assumption, examining influential observations (or outliers), detecting nonlinearity in relationship between the log hazard and the covariates. To check the three model assumptions, we inspected the residuals8 based on different methods. The common residuals for the Cox
model included the Schoenfeld residuals to check the proportional hazards assumption, Martingale residuals to assess nonlinearity and the deviance residual (symmetric transformation of the Martingale residuals), to examine influential observations. In addition to the Survival package, for this part of analysis, the Survminer package is required [6].
3.6.1 Proportionality
Scaled Schoenfeld residuals are based on score residuals and are useful in a visual assessment of the proportional Hazard (PH) assumption. Under PH, the mean of these residuals is zero, and is independent of time. A systematic pattern in the smoothed residuals, when plotted against time suggests a time varying effect of the covariate.
3.6.1.1 Testing the proportional hazard assumption with the Schoenfeld Residuals
The proportional hazards (PH) assumption can be checked using statistical tests and graphical diagnostics based on the earlier mentioned scaled Schoenfeld residuals. In principle, the Schoenfeld residuals are independent of time. A plot that shows a non-random pattern against time is evidence of violation of the PH assumption. The function cox.zph () [in the survival package] provides a convenient solution to test the proportional hazards assumption for each covariate included in a Cox regression model fit. For each covariate, the function cox.zph () correlates the corresponding set of scaled Schoenfeld residuals with time, to test for independence between residuals and time. In a multivariate analysis a global proportionality can be extracted. The global proportionality is the proportionality of the total model. Based on a chisquare test a violation of the PH is indicated by a P value < 0.05.
The proportional hazard assumption is supported by a non-significant relationship between residuals and time and refuted by a significant relationship. A graphical diagnostic using the function ggcoxzph () [in the survminer package] is done, which produces, for each covariate, graphs of the scaled Schoenfeld residuals against the transformed time. A violation of proportional hazards assumption can be resolved by adding covariate*time interaction or stratification. If the global proportionality (the total proportionality in a multivariate analysis), is not violated there is no need to resolve individual proportionality violations. 3.6.1.2 Influential observations
To test influential observations or outliers, either the deviance9 residuals for distributional assumptions or
dfbeta 10 for differences in parameters can be used. The deviance residual is a normalized transform of the
martingale residual. These residuals should be roughly symmetrically distributed around zero with a standard deviation of 1. If the latter is not the case, data could be winsorized by limiting extreme values into the data to reduce the effect of possible spurious outliers. The function ggcoxdiagnostics () [in survminer package] provides a convenient solution for checking influential observations. It’s also possible to check for outliers by visualizing the deviance residuals graphically.
8 Residual: of an observed value is the difference between the observed value and the estimated value of the
quantity of interest
9 Deviance: are used to check distributional assumptions in regression models
17 3.6.1.3 Testing the Normality assumption (distribution of the sample)
When you take the parametric approach to inferential statistics, the values that are assumed to be normally distributed are the means across samples. The assumption of normality, that underlies parametric statistics does not assert that the observations within a given sample are normally distributed, nor does it assert that the values within the population (from which the sample was taken) are normal. The core element of the assumption of normality asserts that the distribution of sample means (across independent samples) follow a normal distribution [7]. As we are dealing with the functional forms of continuous donor predictors, the normal distribution assumption is also taken into consideration for these specific variables. In most of the cases a non-normal distribution can show non-linearity of the regression. In regression analysis, it is more important to check the distribution of the residuals rather than the distribution of the data [8].
There were 6 methods used to check normality: • The histogram of the variable • The histogram of the residuals
• The quantile- quantile (Q-Q) norm and the quantile quantile line • Density plots
• The Shapiro Wilk test.
• The Normal and Non-Normal errors approach
Q-Q plots take the sample data, sort it in ascending order, and then plot them versus quantiles calculated from a theoretical distribution. The number of quantiles is selected to match the size of the sample data. While Normal Q-Q Plots are the ones most often used in practice due to so many statistical methods assuming normality, Q-Q Plots can be created for any distribution [9].
3.6.2 Testing the linearity assumption; Dealing with continuous donor predictors and their functional forms
Often, the assumption is made that continuous covariates have a linear form. However, this assumption should be checked to make sure that the factors have properly described associations with the outcomes. There are several methods to check the linearity assumption. The very simplest way is dichotomization with 4 quartiles and by checking whether the regression coefficients are proportional with created dummy variables. However, there are more convenient methods to check the linearity assumption.
3.6.2.1 The Martingale Residuals
Unscaled martingale residuals give a local estimate of the difference between the observed and predicted number of events. The pattern of the (smoothed) martingale residuals provides information on the functional form of a continuous covariate X in a model. For a proposed function of X, systematic patterns seen in a plot of the smoothed martingale residuals against X indicate lack of fit and may suggest how the chosen function of X may be improved [10]. Plotting the Martingale residuals against continuous covariates is a common approach used to detect nonlinearity or, in other words, to assess the functional form of a covariate. For a given continuous covariate, patterns in the plot may suggest that the variable is not properly fit. Non-linearity is not an issue for categorical variables, so only plots of martingale residuals and partial residuals against a continuous variable are examined. Martingale residuals may present any value in the range (-INF, +1):
For the outcome graft survival:
• Positive values correspond to individuals that “had graft failure too soon” compared to expected survival times.
• Negative values correspond to individual that “had (too) long graft survival”.
18 For the outcome patient survival:
• Positive values correspond to individuals that “died too soon” compared to expected survival times.
• Negative values correspond to individual that “lived too long”.
• Very large or small values are outliers, which are poorly predicted by the model [11].
To assess the functional form of a continuous variable in a Cox proportional hazards model, the function ggcoxfunctional () [in the survminer R package] is used. The function ggcoxfunctional () displays graphs of continuous covariates against martingale residuals of null cox proportional hazards model. This might help to properly choose the functional form of continuous variables in the Cox model. Fitted lines with lows function should be linear to satisfy the Cox proportional hazards model assumptions. The martingale residual plot is resembled by plotting the covariates against their martingale residuals which is the difference between the observed and the expected number of events.
3.6.3 Penalized Spline Smoothing
Penalized spline 11(pspline)smoothing is a very flexible concept. Different basis functions, form of the
penalties, amount and location of knots all provide a wide spectrum of smoothers. The idea is to define a flexible function and control the smoothness through a penalty term, usually on the second derivative [12].
Advantages:
• Relatively low-rank basis and simplified penalties • Completely parametric form
• Number and location of knots not critical • Automatic smoothing selection
• Efficient computational methods
• Well-grounded theoretical framework [13,14]
In our case we used the splines to check linearity assumptions visually and statistically. In case of a p-value <0.05 the linearity was considered as violated. Plots were generated by plotting the log hazard against the predictor. For spline transformations the R Package “Spline” is required [15].
3.7 T
RANSFORMATIONSWhen we consider a continuous predictor as a linear term in a prediction model, we assume that the effect is the same for each unit increase of the predictor. If a non-linear function is expected, various options can readily be considered in regression models. In the next sections we discuss non-linear modelling of continuous predictors with common transformations, polynomials, fractional polynomials, and spline functions [16].
3.7.1 Common transformations
Besides adding polynomial terms as extensions to a model with a linear term, commonly, square and cubic terms are considered. For example, we can examine models with X, X +X2 and X +X2 +X3, where X is a continuous predictor. This results in nested models, and we can statistically test each extension [15].
Common transformations we considered were the inverse (X−1) and square root (X0.5), and logarithmic
19
(log (X), exponential (X)). We may use these terms as replacement of the linear term X, as extension to a model with X as a linear term included or X as added term [17].
3.7.2 Spline transformations
Very flexible transformations are provided by spline functions. Various types of spline functions can be considered, such as natural splines. These can be well fitted with generalized additive models (GAM). The extreme flexibility leads sometimes to wiggly patterns of predictions, which are unlikely to be reproduced in new data. Smoothness can be enforced by parameters in the model fitting process, e.g. penalty terms in the likelihood function. Without such penalty, splines may easily overfit patterns in the data. Restricted cubic spline (RCS) functions have been proposed for a more stable approach for prediction models. RCSs are cubic splines (containingX3 terms) that are restricted to be linear in the tails. These splines are still very
flexible and can take more forms than a parametric transformation with the same df in the model. For example, addingX2 restricts the relationship to be parabolic, while an RCS with 2 df (3 knots) incorporates a wider family of functions. A spline function requires the specification of knots. The spline will bend around these knots. Fortunately, the exact position of the knots is usually not critical to the shape that the spline will take. It is common to specify the location from the distribution of the predictor variable. It may often be reasonable to use linear terms or splines with 3 knots (2 df) [16]. We used the linear spline, restricted cubic spline with manual knots and the Natural spline form. Remarkably, the two latter splines are similar. The only difference is in the packages used for these 2 splines in R.
Advantages of Regression spline over other methods:
• Parametric splines are piecewise polynomials and can be fitted using any existing regression program after the constructed predictors are computed. Spline regression is equally suitable to multiple linear regression, survival models, and logistic models for discrete outcomes.
• Regression coefficients for the spline function are estimated using standard techniques (maximum likelihood or least squares), and statistical inferences can readily be drawn. Formal tests of no overall association, linearity, and additivity can readily be constructed. Confidence limits for the estimated regression function are derived by standard theory.
• The fitted spline function directly estimates the transformation that a predictor should receive to yield linearity in C(Y|X). The fitted spline transformation sometimes suggests a simple transformation (e.g. square root) of a predictor that can be used if one is not concerned about the proper number of degrees of freedom for testing association of the predictor with the response. • The spline function can be used to represent the predictor in the final model. Nonparametric
methods do not yield a prediction equation.
• Splines can be extended to non-additive models. Multidimensional nonparametric estimators often require burdensome computations [17].
The disadvantages of Spline transformations:
• The Risk of overfitting (based on the number of knots)
• The Regression coefficients are less interpretable (x^3 coefficients with knots are only interpretable through a graph).
3.7.3 Fractional Polynomials
Following the previous section for continuous variables, fractional polynomials were also used. In this approach only, the continuous predictors were added. In section 3.7.3.1 we emphasize the usage and the expectations using this approach.
