Survival analysis on kidney failure of kidney transplant patients

“SURVIVAL ANALYSIS ON KIDNEY FAILUREFOR KIDNEY TRANSPLANTPATIENTS”
Summary
The Accelerated Failure Time model presents a way to easily describe and interpret survival regression
data. It approachesthe datadifferentlythanthe widelyusedandwell describedCox proportional hazard
model,byassumingproportional effectof the covariatesonthe log-failure time ratherthanonthe hazard
function.Inthisreport,wepresentsemiparametricmethods(theCox PHmodel)andparametricmethods
(AFT model) for analyzing survival data. We have the data set for 469 patients with kidney transplants
along with the graft survival and failure time. Eight variates have been recognized which might have a
relation with the survival experience.
Introduction
Survival analysisisastatistical methodfordataanalysiswhere the outcome variable of interestisthe
time to the occurrence of an event.Hence,survival analysisisalsoreferredtoas"time to event
analysis",whichisappliedinanumberof appliedfieldssuchasmedicine,publichealth,socialscience,
and engineering.
The Cox proportional hazards(PH) model isnow the mostwidelyusedforthe analysisof survivaldatain
the presence of covariatesorprognosticfactors.Thisisthe mostpopularmodel forsurvival analysis
because of itssimplicity,andnotbeingbasedonanyassumptionsaboutthe survival distribution.The
model assumesthatthe underlyinghazardrate isa functionof the independentcovariates,butno
assumptionsare made aboutthe nature or shape of the hazardfunction.
The acceleratedfailure time (AFT) modelisanotheralternative methodforthe analysisof survival
data. The AFT model assumesacertainparametricdistributionforthe failure timesandthatthe effect
of the covariates onthe failure time ismultiplicative. The appeal of the AFTmodel liesinthe ease of
interpretingthe results,becausethe AFTmodelsthe effectof predictorsandcovariatesdirectlyonthe
survival time insteadof throughthe hazardfunction.
Purpose and Methods
The purpose of this report is to analyze the data using the Cox models and the AFT models. This will be
studied by means of real dataset which is from a randomized data set for 469 patients with kidney
transplants.
We start withthe AFTmodels checkingthe AICandBIC (goodnessof fittests) foreachof the
distributionsincludingall the covariates. We thendiscussbrieflypossible typesof responseand
prognosticvariables.We willselectthe covariate basedonthe backwardselectionprocedure.The final
model will be chosenbasedonthisresult.

Secondpart of the reportconsistsof the Cox’sPH model. We are goingto try to fitthe data basedon
the Cox’sPH model basedondifferenttiesandcompare withthe parametricmodel exponential and
Weibull.Thiswill alsobe followedbyselectionof significantcovariatesandthe final modelwill be
chosen.
SASsoftware package wasusedand proc lifereg,procphregare usedfor estimatingthe resultsforAFT
and Cox PH model.
Conclusion
We applythese methodsto arandomizeddataof 495 kidneytransplants. OurconclusionisthatAge,
Diabetes(DIBT) andALG (animmune drug) are the most significantcovariates,thusbeingthe
interactingvariableswhichispossiblypredictive of the outcome understudy.The majorgoal of this
reportis alsoto supportan argumentforthe considerationof the AFTmodel asan alternative tothe PH
model inthe analysisof some survival databymeansof thisreal dataset.
In conclusion,althoughthe Cox proportional hazardsmodeltendstobe more popularinthe literature,
the AFT model shouldalsobe consideredwhenplanningasurvival analysis.Itshouldgowithoutsaying
that the choice shouldbe drivenbythe desiredoutcome orthe fittothe data,and neverbywhichgives
a significantP value forthe predictorof interest.The choice shouldbe dictatedonlybythe research
hypothesisandbywhichassumptionsof the model are validforthe data beinganalyzed.
Analysis,Procedure along with computation and output with interpretation
Data set
Followingisthe datafor469 patientswithkidneytransplants.The primaryinterestwasgraftsurvival,
and time tograft failure wasrecordedinmonths(whichwassubjecttorightcensoring).Thisstudy
included measurementsof manycovariatesthatmaybe relatedtosurvival experience.Use bothCox's
PH model andthe acceleratedfailure timemodeltoanalyze the dataand write areport.
The 10 covariatesincludedare:
AGE: Age at transplantinyears
SEX: 1=female,0=male
DIALY: Durationof hemodialysispriortotransplantindays
DBT: Diabetes;1=yes,0=no
PTX: Numberof priortransplants
BLOOD: Amountof bloodtransfusion,inbloodunits

MIS: Mismatch score
ALG: Use of ALG, an immune suppressiondrug;1=yes, 0=no
MONTH: Durationtime startingfromtransplant, inmonths
FAIL:status of the newkidney;1=newkidneyfailed,0=functioning
A. AFT Models-
Under AFT modelswe measure the directeffectof the explanatoryvariablesonthe survival
time insteadof hazard,as we do inthe PH model.Thischaracteristicallowsforaneasier
interpretationof the resultsbecause the parametersmeasure the effectof the correspondent
covariate onthe meansurvival time.Currently,the AFTmodel isnotcommonlyusedforthe
analysisof clinical trial data,although itisfairlycommoninthe field of manufacturing.Similarto
the PH model,the AFTmodel describesthe relationshipbetweensurvivalprobabilitiesandaset
of covariates
Procedure-
Usingproc lifereg(SAScode) we performgoodnessof fittestsforeachdistribution(Exponential,
Weibull,Lognormal,GammaandLog-logistic).
Thisprocessis done bytakingall the covariatesintoconsideration.
SASoutput-
Exponential-
Fit Statistics
-2 Log Likelihood 1266.509
AIC (smaller is better) 1284.509
AICC (smaller is better)1284.901
BIC (smaller is better) 1321.864
Weibull-
Fit Statistics

Lognormal-
Fit Statistics
Gamma-
Fit Statistics
Loglogistic-
Fit Statistics
Interpretation-
From the above tablesitisquite evidentthatgammadistributionbeingthe chosenone withthe lowest
AICamongstthe others.
Choice of Covariates-
In thissectionwe will be selectingthe mostsignificantcovariatesamongthe eightcovariatesthatare
giveninthe data set.Covariatesare selectedbyusingthe backwardselectionprocedure.
The backward selectionprocedure isaneliminationprocessinwhichall the covariatesare includedin
the model at the beginningandare removedone by one accordingtoa significancecriterion. The
specificparametersthatdefine the parametricmodelandthe coefficientsof all the covariatesare
estimatedfirst.Thenthe Waldtestisusedtoexamine eachcovariate.
We delete the predictorwiththe highestp-valueandre run the model deletingthe predictorswith
highestp-value until all of themsatisfiesourconstraints.The predictorsleftwill be oursignificant
variable.

To be thoroughwithourselectionprocedureof the covariateswe have done the backwardselection
procedure foreachdistributions.
Age, Diabetes(DBT) andALG(animmune suppressiondrug) are the three significantcovariates.
Followingtablesshowsindetailsthe selectionprocessof the covariatesforeachandeverydistribution.
SASTables-
Exponential distribution-
Analysis of Maximum Likelihood Parameter Estimates
Parameter DF Estimate
Standard
Error95% Confidence LimitsChi-Square Pr > ChiSq
Intercept 1 5.4088 0.3362 4.7500 6.0676 258.90 <.0001
AGE 1 -0.0224 0.0062 -0.0345 -0.0102 13.07 0.0003
SEX 1 0.0393 0.1494 -0.2535 0.3321 0.07 0.7924
DIALY 1 -0.0002 0.0002 -0.0005 0.0001 1.31 0.2516
DBT 1 -0.5263 0.1897 -0.8981 -0.1544 7.70 0.0055
PTX 1 0.0234 0.2159 -0.3998 0.4466 0.01 0.9137
BLOOD 1 -0.0018 0.0049 -0.0113 0.0078 0.13 0.7162
MIS 1 -0.1278 0.0906 -0.3053 0.0497 1.99 0.1582
ALG 1 0.5028 0.1688 0.1720 0.8335 8.87 0.0029
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
We rejectPTX
Standard
Intercept 1 5.4182 0.3252 4.7807 6.0556 277.55 <.0001
AGE 1 -0.0224 0.0061 -0.0345 -0.0104 13.30 0.0003
SEX 1 0.0401 0.1492 -0.2523 0.3325 0.07 0.7880
DIALY 1 -0.0002 0.0002 -0.0005 0.0001 1.35 0.2444
DBT 1 -0.5270 0.1896 -0.8986 -0.1554 7.73 0.0054
BLOOD 1 -0.0016 0.0047 -0.0109 0.0076 0.12 0.7287

Standard
MIS 1 -0.1291 0.0897 -0.3050 0.0467 2.07 0.1500
ALG 1 0.4996 0.1662 0.1738 0.8254 9.03 0.0027
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
We rejectsex
Standard
Intercept 1 5.4301 0.3225 4.7981 6.0621 283.56 <.0001
AGE 1 -0.0223 0.0061 -0.0344 -0.0103 13.24 0.0003
DIALY 1 -0.0002 0.0002 -0.0005 0.0001 1.34 0.2471
DBT 1 -0.5277 0.1896 -0.8992 -0.1561 7.75 0.0054
BLOOD 1 -0.0014 0.0047 -0.0106 0.0077 0.09 0.7596
MIS 1 -0.1310 0.0894 -0.3063 0.0442 2.15 0.1428
ALG 1 0.5021 0.1659 0.1770 0.8272 9.17 0.0025
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
We rejectblood
Standard
Intercept 1 5.4090 0.3147 4.7922 6.0259 295.39 <.0001
AGE 1 -0.0221 0.0061 -0.0340 -0.0102 13.20 0.0003
DIALY 1 -0.0002 0.0002 -0.0005 0.0001 1.80 0.1792
DBT 1 -0.5232 0.1890 -0.8935 -0.1528 7.67 0.0056
MIS 1 -0.1303 0.0893 -0.3052 0.0447 2.13 0.1444
ALG 1 0.4991 0.1655 0.1747 0.8235 9.09 0.0026
Scale 0 1.0000 0.0000 1.0000 1.0000

Standard
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
We rejectDIALY
Standard
Intercept 1 5.3699 0.3150 4.7524 5.9874 290.52 <.0001
AGE 1 -0.0236 0.0060 -0.0353 -0.0119 15.59 <.0001
DBT 1 -0.5050 0.1884 -0.8742 -0.1357 7.19 0.0074
MIS 1 -0.1357 0.0904 -0.3129 0.0416 2.25 0.1336
ALG 1 0.5012 0.1655 0.1768 0.8256 9.17 0.0025
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
We rejectMIS
Standard
Intercept 1 5.1775 0.2850 4.6190 5.7361 330.11 <.0001
AGE 1 -0.0234 0.0060 -0.0351 -0.0117 15.40 <.0001
DBT 1 -0.4895 0.1883 -0.8585 -0.1204 6.76 0.0093
ALG 1 0.4760 0.1648 0.1530 0.7989 8.34 0.0039
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000
Interpretation-The aboveisthe final table showingall the covariatesthatare significantandwill be used
infittingthe model.
AGE, DBT and ALG are chosen.

Weibull distribution-
Standard
Intercept 1 5.7554 0.5405 4.6961 6.8147 113.39 <.0001
AGE 1 -0.0287 0.0099 -0.0482 -0.0093 8.37 0.0038
SEX 1 0.0449 0.2424 -0.4301 0.5200 0.03 0.8529
DIALY 1 -0.0002 0.0003 -0.0007 0.0003 0.50 0.4814
DBT 1 -0.5754 0.3101 -1.1831 0.0323 3.44 0.0635
PTX 1 0.1449 0.3570 -0.5548 0.8447 0.16 0.6848
BLOOD 1 -0.0073 0.0081 -0.0232 0.0085 0.82 0.3663
MIS 1 -0.1537 0.1495 -0.4467 0.1393 1.06 0.3038
ALG 1 0.9736 0.2852 0.4146 1.5327 11.65 0.0006
Scale 1 1.6326 0.1013 1.4457 1.8436
Weibull Shape 1 0.6125 0.0380 0.5424 0.6917
We rejectsex
Standard
Intercept 1 5.7695 0.5356 4.7198 6.8193 116.05 <.0001
AGE 1 -0.0287 0.0099 -0.0481 -0.0092 8.36 0.0038
DIALY 1 -0.0002 0.0003 -0.0007 0.0003 0.48 0.4873
DBT 1 -0.5760 0.3101 -1.1838 0.0317 3.45 0.0632
PTX 1 0.1474 0.3566 -0.5515 0.8463 0.17 0.6793
BLOOD 1 -0.0071 0.0080 -0.0228 0.0086 0.79 0.3750
MIS 1 -0.1551 0.1493 -0.4477 0.1375 1.08 0.2988
ALG 1 0.9761 0.2849 0.4177 1.5345 11.74 0.0006
Scale 1 1.6327 0.1013 1.4458 1.8438

Standard
Weibull Shape 1 0.6125 0.0380 0.5424 0.6917
We rejectPTX-
Standard
Intercept 1 5.8259 0.5200 4.8068 6.8450 125.54 <.0001
AGE 1 -0.0291 0.0099 -0.0485 -0.0097 8.63 0.0033
DIALY 1 -0.0002 0.0003 -0.0007 0.0003 0.54 0.4626
DBT 1 -0.5814 0.3096 -1.1883 0.0255 3.53 0.0604
BLOOD 1 -0.0061 0.0077 -0.0213 0.0090 0.63 0.4265
MIS 1 -0.1644 0.1476 -0.4537 0.1249 1.24 0.2653
ALG 1 0.9553 0.2801 0.4064 1.5042 11.64 0.0006
Scale 1 1.6315 0.1011 1.4448 1.8422
Weibull Shape 1 0.6129 0.0380 0.5428 0.6921
We rejectDIALY-
Standard
Intercept 1 5.8205 0.5225 4.7964 6.8446 124.09 <.0001
AGE 1 -0.0307 0.0097 -0.0496 -0.0118 10.12 0.0015
DBT 1 -0.5696 0.3097 -1.1765 0.0373 3.38 0.0658
BLOOD 1 -0.0078 0.0074 -0.0223 0.0067 1.11 0.2918
MIS 1 -0.1692 0.1488 -0.4608 0.1224 1.29 0.2554
ALG 1 0.9634 0.2805 0.4137 1.5132 11.80 0.0006
Scale 1 1.6346 0.1013 1.4477 1.8457
Weibull Shape 1 0.6118 0.0379 0.5418 0.6908
We rejectBlood-

Standard
Intercept 1 5.6919 0.5056 4.7010 6.6828 126.75 <.0001
AGE 1 -0.0303 0.0096 -0.0491 -0.0115 9.93 0.0016
DBT 1 -0.5447 0.3081 -1.1485 0.0592 3.13 0.0771
MIS 1 -0.1651 0.1486 -0.4563 0.1261 1.24 0.2664
ALG 1 0.9531 0.2796 0.4050 1.5011 11.62 0.0007
Scale 1 1.6321 0.1011 1.4455 1.8428
Weibull Shape 1 0.6127 0.0380 0.5426 0.6918
We rejectMIS-
Standard
Intercept 1 5.4626 0.4582 4.5647 6.3606 142.16 <.0001
AGE 1 -0.0303 0.0096 -0.0492 -0.0114 9.91 0.0016
DBT 1 -0.5236 0.3086 -1.1285 0.0813 2.88 0.0898
ALG 1 0.9318 0.2798 0.3835 1.4801 11.09 0.0009
Scale 1 1.6373 0.1015 1.4500 1.8488
Weibull Shape 1 0.6107 0.0379 0.5409 0.6896
Interpretation-The aboveisthe final table showingall the covariatesthatare significantand will be used
infittingthe model.
AGE, DBT and ALG are chosen.
Log Normal Distribution-

ParameterDF Estimate
Standard
Intercept 1 4.7366 0.5776 3.6046 5.8686 67.26 <.0001
AGE 1 -0.0269 0.0109 -0.0482 -0.0057 6.15 0.0131
SEX 1 0.0886 0.2639 -0.4288 0.6059 0.11 0.7372
DIALY 1 -0.0001 0.0003 -0.0006 0.0005 0.06 0.8096
DBT 1 -0.6132 0.3323 -1.2645 0.0381 3.41 0.0650
PTX 1 0.0198 0.3720 -0.7092 0.7489 0.00 0.9574
BLOOD 1 -0.0064 0.0098 -0.0256 0.0127 0.44 0.5088
MIS 1 -0.1226 0.1617 -0.4396 0.1943 0.58 0.4482
ALG 1 1.2503 0.3281 0.6073 1.8933 14.52 0.0001
We rejectPTX
Standard
Intercept 1 4.7419 0.5689 3.6268 5.8570 69.47 <.0001
AGE 1 -0.0270 0.0108 -0.0482 -0.0057 6.19 0.0128
SEX 1 0.0883 0.2638 -0.4289 0.6054 0.11 0.7380
DIALY 1 -0.0001 0.0003 -0.0006 0.0005 0.06 0.8117
DBT 1 -0.6139 0.3320 -1.2646 0.0368 3.42 0.0644
BLOOD 1 -0.0063 0.0093 -0.0246 0.0120 0.46 0.4995
MIS 1 -0.1239 0.1598 -0.4372 0.1893 0.60 0.4380
ALG 1 1.2484 0.3260 0.6094 1.8874 14.66 0.0001
Scale 1 2.3609 0.1312 2.1173 2.6327
We rejectDialy
Standard
Intercept 1 4.7419 0.5691 3.6265 5.8574 69.42 <.0001
AGE 1 -0.0276 0.0105 -0.0482 -0.0071 6.94 0.0084
SEX 1 0.0834 0.2631 -0.4322 0.5990 0.10 0.7513
DBT 1 -0.6098 0.3315 -1.2596 0.0400 3.38 0.0659
BLOOD 1 -0.0067 0.0092 -0.0247 0.0113 0.54 0.4644
MIS 1 -0.1251 0.1598 -0.4383 0.1881 0.61 0.4338
ALG 1 1.2490 0.3260 0.6099 1.8880 14.67 0.0001
Scale 1 2.3611 0.1312 2.1174 2.6328
We rejectsex

Standard
Intercept 1 4.7819 0.5555 3.6931 5.8706 74.10 <.0001
AGE 1 -0.0279 0.0105 -0.0484 -0.0073 7.08 0.0078
DBT 1 -0.6148 0.3312 -1.2639 0.0343 3.45 0.0634
BLOOD 1 -0.0063 0.0091 -0.0241 0.0115 0.48 0.4882
MIS 1 -0.1256 0.1598 -0.4388 0.1876 0.62 0.4319
ALG 1 1.2508 0.3261 0.6116 1.8899 14.71 0.0001
Scale 1 2.3617 0.1313 2.1179 2.6335
We rejectblood
Standard
Intercept 1 4.6899 0.5380 3.6355 5.7443 76.00 <.0001
AGE 1 -0.0279 0.0105 -0.0483 -0.0074 7.10 0.0077
DBT 1 -0.5998 0.3300 -1.2466 0.0470 3.30 0.0691
MIS 1 -0.1216 0.1596 -0.4343 0.1912 0.58 0.4461
ALG 1 1.2481 0.3257 0.6096 1.8865 14.68 0.0001
Scale 1 2.3597 0.1311 2.1163 2.6311
We rejectMIS
Standard
Intercept 1 4.5301 0.4947 3.5604 5.4998 83.84 <.0001
AGE 1 -0.0284 0.0105 -0.0489 -0.0079 7.38 0.0066
DBT 1 -0.5833 0.3298 -1.2297 0.0632 3.13 0.0770
ALG 1 1.2441 0.3262 0.6048 1.8835 14.55 0.0001
Scale 1 2.3636 0.1313 2.1198 2.6354
infittingthe model.
AGE,DBT and ALG are chosen.

Gamma distribution-
Standard
Intercept 1 3.2686 0.6703 1.9549 4.5824 23.78 <.0001
AGE 1 -0.0186 0.0113 -0.0409 0.0036 2.69 0.1007
SEX 1 0.0996 0.2613 -0.4126 0.6118 0.15 0.7031
DIALY 1 0.0001 0.0003 -0.0005 0.0006 0.05 0.8292
DBT 1 -0.5701 0.3240 -1.2052 0.0650 3.10 0.0785
PTX 1 -0.1666 0.3615 -0.8752 0.5419 0.21 0.6448
BLOOD 1 -0.0008 0.0102 -0.0208 0.0193 0.01 0.9408
MIS 1 -0.0631 0.1533 -0.3636 0.2374 0.17 0.6805
ALG 1 1.1667 0.3428 0.4949 1.8386 11.58 0.0007
Scale 1 2.6270 0.1461 2.3556 2.9296
Shape 1 -1.0848 0.2450 -1.5649 -0.6046
We rejectBlood
Standard
Intercept 1 3.2418 0.6617 1.9449 4.5387 24.00 <.0001
AGE 1 -0.0185 0.0114 -0.0407 0.0038 2.65 0.1037
SEX 1 0.0971 0.2585 -0.4096 0.6038 0.14 0.7072
DIALY 1 0.0001 0.0003 -0.0005 0.0006 0.05 0.8286
DBT 1 -0.5682 0.3235 -1.2022 0.0658 3.09 0.0790
PTX 1 -0.1764 0.3420 -0.8467 0.4939 0.27 0.6060
MIS 1 -0.0625 0.1531 -0.3626 0.2375 0.17 0.6830
ALG 1 1.1628 0.3427 0.4912 1.8344 11.52 0.0007
Scale 1 2.6256 0.1460 2.3544 2.9280
Shape 1 -1.0998 0.2490 -1.5878 -0.6119
We rejectdialy
Standard
Intercept 1 3.2802 0.6535 1.9993 4.5611 25.19 <.0001
AGE 1 -0.0182 0.0110 -0.0397 0.0033 2.74 0.0976
SEX 1 0.1028 0.2572 -0.4014 0.6069 0.16 0.6895
DBT 1 -0.5736 0.3238 -1.2082 0.0609 3.14 0.0764

Standard
PTX 1 -0.1584 0.3342 -0.8133 0.4966 0.22 0.6355
MIS 1 -0.0618 0.1533 -0.3622 0.2386 0.16 0.6866
ALG 1 1.1756 0.3406 0.5079 1.8432 11.91 0.0006
Scale 1 2.6282 0.1455 2.3580 2.9293
Shape 1 -1.0678 0.2335 -1.5255 -0.6101
We rejectsex
Standard
Intercept 1 3.3702 0.6187 2.1576 4.5828 29.68 <.0001
AGE 1 -0.0190 0.0108 -0.0402 0.0022 3.10 0.0785
DBT 1 -0.5885 0.3222 -1.2200 0.0431 3.34 0.0678
PTX 1 -0.1638 0.3343 -0.8191 0.4915 0.24 0.6242
MIS 1 -0.0595 0.1536 -0.3606 0.2416 0.15 0.6986
ALG 1 1.1823 0.3404 0.5152 1.8494 12.07 0.0005
Scale 1 2.6304 0.1453 2.3604 2.9311
Shape 1 -1.0502 0.2174 -1.4763 -0.6242
We rejectMIS
Standard
Intercept 1 3.2396 0.5946 2.0742 4.4050 29.68 <.0001
AGE 1 -0.0191 0.0108 -0.0403 0.0022 3.10 0.0785
DBT 1 -0.5815 0.3212 -1.2111 0.0481 3.28 0.0702
PTX 1 -0.1444 0.3281 -0.7875 0.4986 0.19 0.6598
ALG 1 1.1802 0.3408 0.5123 1.8481 11.99 0.0005
Scale 1 2.6291 0.1460 2.3580 2.9313
Shape 1 -1.0905 0.2471 -1.5748 -0.6062
We rejectPTX
Standard
Intercept 1 3.2692 0.5712 2.1498 4.3886 32.76 <.0001
AGE 1 -0.0197 0.0107 -0.0407 0.0014 3.36 0.0669
DBT 1 -0.5758 0.3218 -1.2065 0.0549 3.20 0.0736
ALG 1 1.1947 0.3403 0.5276 1.8618 12.32 0.0004
Scale 1 2.6330 0.1455 2.3628 2.9341
Shape 1 -1.0523 0.2170 -1.4777 -0.6269

infittingthe model.
Log logisticdistribution-
Standard
Intercept 1 4.8836 0.5809 3.7451 6.0221 70.68 <.0001
AGE 1 -0.0300 0.0108 -0.0512 -0.0089 7.76 0.0053
SEX 1 0.1009 0.2663 -0.4210 0.6227 0.14 0.7048
DIALY 1 -0.0001 0.0003 -0.0006 0.0005 0.11 0.7400
DBT 1 -0.6366 0.3362 -1.2955 0.0224 3.58 0.0583
PTX 1 0.0587 0.3906 -0.7069 0.8242 0.02 0.8806
BLOOD 1 -0.0085 0.0094 -0.0269 0.0100 0.81 0.3694
MIS 1 -0.1580 0.1652 -0.4818 0.1657 0.92 0.3387
ALG 1 1.3047 0.3317 0.6545 1.9549 15.47 <.0001
Scale 1 1.3746 0.0832 1.2209 1.5476
We rejectPTX
Standard
Intercept 1 4.9011 0.5690 3.7860 6.0163 74.20 <.0001
AGE 1 -0.0302 0.0107 -0.0512 -0.0092 7.92 0.0049
SEX 1 0.1018 0.2662 -0.4199 0.6234 0.15 0.7022
DIALY 1 -0.0001 0.0003 -0.0006 0.0005 0.11 0.7419
DBT 1 -0.6387 0.3359 -1.2970 0.0196 3.62 0.0572
BLOOD 1 -0.0080 0.0090 -0.0257 0.0096 0.80 0.3719
MIS 1 -0.1611 0.1638 -0.4822 0.1599 0.97 0.3253
ALG 1 1.2983 0.3289 0.6537 1.9428 15.58 <.0001
Scale 1 1.3744 0.0831 1.2207 1.5473
We rejectDialy
Standard
Intercept 1 4.8988 0.5693 3.7829 6.0147 74.04 <.0001
AGE 1 -0.0311 0.0104 -0.0514 -0.0108 8.98 0.0027

Standard
SEX 1 0.0930 0.2649 -0.4262 0.6123 0.12 0.7255
DBT 1 -0.6319 0.3353 -1.2890 0.0253 3.55 0.0595
BLOOD 1 -0.0085 0.0089 -0.0259 0.0088 0.93 0.3357
MIS 1 -0.1635 0.1638 -0.4846 0.1576 1.00 0.3183
ALG 1 1.3038 0.3284 0.6602 1.9475 15.76 <.0001
Scale 1 1.3748 0.0831 1.2212 1.5478
We rejectsex
Standard
Intercept 1 4.9432 0.5556 3.8542 6.0321 79.16 <.0001
AGE 1 -0.0313 0.0104 -0.0516 -0.0109 9.08 0.0026
DBT 1 -0.6363 0.3351 -1.2931 0.0206 3.60 0.0576
BLOOD 1 -0.0080 0.0088 -0.0252 0.0091 0.84 0.3582
MIS 1 -0.1657 0.1635 -0.4862 0.1548 1.03 0.3110
ALG 1 1.3044 0.3285 0.6606 1.9483 15.77 <.0001
Scale 1 1.3752 0.0832 1.2215 1.5482
We rejectblood
Standard
Intercept 1 4.8255 0.5407 3.7658 5.8852 79.66 <.0001
AGE 1 -0.0313 0.0104 -0.0516 -0.0109 9.09 0.0026
DBT 1 -0.6190 0.3343 -1.2743 0.0362 3.43 0.0641
MIS 1 -0.1598 0.1632 -0.4797 0.1601 0.96 0.3275
ALG 1 1.2998 0.3283 0.6563 1.9433 15.67 <.0001
Scale 1 1.3751 0.0831 1.2215 1.5480
We rejectMIS
Standard
Intercept 1 4.6074 0.4917 3.6437 5.5711 87.81 <.0001
AGE 1 -0.0318 0.0104 -0.0521 -0.0114 9.35 0.0022
DBT 1 -0.5949 0.3343 -1.2501 0.0604 3.17 0.0752
ALG 1 1.2934 0.3294 0.6477 1.9390 15.41 <.0001
Scale 1 1.3795 0.0833 1.2255 1.5528

infittingthe model.
Goodness of fit test for the selected covariates and selction of model
In thissectionwe performthe goodnessof fittestonce againforthe three covariatesselectedinthe
above section(AGE,DBTandALG).
We selectthe Gammadistributionasourappropriate model withthe lowestAICvalue comparedtothe
others.
Followingtablesshowsthe detailedAICandBICvaluesforeachand everydistributions.
SASoutput-
Exponential distribution-
Fit Statistics
Weibull distribution-
Fit Statistics
Lognormal distribution-
Fit Statistics

Gamma distribution-
Fit Statistics
Log logisticdistribution-
Fit Statistics
Interpretation-Gammadistributionisselectedwiththe lowest AICvalue of 1179.171 comparedtothe
others.
Fitting the appropriate model
Gamma AFT model hasbeenselectedasthe appropriate model forthe givendataset.
For the gamma distributionwe have the followingtable-
Standard
Intercept 1 3.2692 0.5712 2.1498 4.3886 32.76 <.0001
AGE 1 -0.0197 0.0107 -0.0407 0.0014 3.36 0.0669
DBT 1 -0.5758 0.3218 -1.2065 0.0549 3.20 0.0736
ALG 1 1.1947 0.3403 0.5276 1.8618 12.32 0.0004
Scale 1 2.6330 0.1455 2.3628 2.9341
Shape 1 -1.0523 0.2170 -1.4777 -0.6269
Hence if T be definedasthe survival time we canwrite the appropriate model as-
Log (Ti)= 3.2692 - .0197(AGE) - .5758(DBT) + 1.1947(ALG) + 2.6330(Error)

B. Cox’sPH Model
The non-parametricmethoddoesnotcontrol forcovariatesanditrequirescategorical
predictors.Whenwe have several prognosticvariables,we mustuse multivariate approaches.
But we cannot use multiplelinearregressionorlogisticregressionbecause theycannot deal
withcensoredobservations.We needanothermethodtomodel survival datawiththe presence
of censoring.One verypopularmodelinsurvival dataisthe Cox proportional
hazardsmodel.
Procedure-
We are goingto use proc phregto estimate the regressioncoefficient(parameterestimate) basedon
differentmethodsfortiesonthe givendata.
Thenwe are goingto compare those resultswiththe twoparametricmodels,exponential andWeibull
for furtherclarification.
Conclusion-
From the testresultswe can saythat the signsof regressioncoefficientof Age,Durationof
hemodialysis(DIALY),Diabetes(DBT),Bloodare all positive andthushave ahigherhazardrisk inthe
kidneytransplants.
The coefficientsof SEX,PTX(numberof priortransplants) andALG(animmune suppressiondrug) are
all negative indicatinglowhazardriskinthe kidneytransplantation.
It isalso to be notedthatif we compare these value withthatof exponential andWeibulltheygive the
opposite results.AccordingtoWeibullandexponential-SEX,PTXandALGhave higherhazardrisk
whereasthe Age,DIALY,DBT,BloodandMIS have low hazardrisk.
The followingSAStabularvaluesexplainsouroutcome indetails.
Breslow-
Analysis of Maximum Likelihood Estimates
ParameterDF
Parameter
Estimate
Standard
ErrorChi-Square Pr > ChiSq
Hazard
Ratio
AGE 1 0.01732 0.00611 8.0468 0.0046 1.017
SEX 1 -0.02983 0.14882 0.0402 0.8411 0.971
DIALY 10.00009310.0001556 0.3581 0.5495 1.000
DBT 1 0.35004 0.19236 3.3115 0.0688 1.419
PTX 1 -0.07341 0.21763 0.1138 0.7359 0.929

BLOOD 1 0.00372 0.00497 0.5592 0.4546 1.004
MIS 1 0.08376 0.09196 0.8295 0.3624 1.087
ALG 1 -0.60697 0.17033 12.6983 0.0004 0.545
Discrete-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01776 0.00619 8.2292 0.0041 1.018
SEX 1 -0.03055 0.15076 0.0411 0.8394 0.970
DIALY 10.00009620.0001583 0.3688 0.5437 1.000
DBT 1 0.35882 0.19528 3.3763 0.0661 1.432
PTX 1 -0.07548 0.22026 0.1174 0.7318 0.927
BLOOD 1 0.00378 0.00503 0.5652 0.4522 1.004
MIS 1 0.08623 0.09312 0.8576 0.3544 1.090
ALG 1 -0.62313 0.17320 12.9439 0.0003 0.536
Efron-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01741 0.00610 8.1433 0.0043 1.018
SEX 1 -0.03178 0.14888 0.0456 0.8310 0.969
DIALY 10.00009030.0001557 0.3364 0.5619 1.000
DBT 1 0.35510 0.19234 3.4086 0.0649 1.426
PTX 1 -0.07098 0.21839 0.1056 0.7452 0.931
BLOOD 1 0.00373 0.00497 0.5617 0.4536 1.004
MIS 1 0.08553 0.09205 0.8633 0.3528 1.089
ALG 1 -0.61692 0.17051 13.0902 0.0003 0.540
Exact-

ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01741 0.00610 8.1440 0.0043 1.018
SEX 1 -0.03178 0.14889 0.0456 0.8310 0.969
DIALY 10.00009030.0001557 0.3362 0.5620 1.000
DBT 1 0.35513 0.19235 3.4086 0.0649 1.426
PTX 1 -0.07099 0.21842 0.1056 0.7452 0.931
BLOOD 1 0.00373 0.00497 0.5617 0.4536 1.004
MIS 1 0.08554 0.09205 0.8636 0.3527 1.089
ALG 1 -0.61703 0.17054 13.0913 0.0003 0.540
Parametermodels-
Exponential-
Standard
Intercept 1 5.4088 0.3362 4.7500 6.0676 258.90 <.0001
AGE 1 -0.0224 0.0062 -0.0345 -0.0102 13.07 0.0003
SEX 1 0.0393 0.1494 -0.2535 0.3321 0.07 0.7924
DIALY 1 -0.0002 0.0002 -0.0005 0.0001 1.31 0.2516
DBT 1 -0.5263 0.1897 -0.8981 -0.1544 7.70 0.0055
PTX 1 0.0234 0.2159 -0.3998 0.4466 0.01 0.9137
BLOOD 1 -0.0018 0.0049 -0.0113 0.0078 0.13 0.7162
MIS 1 -0.1278 0.0906 -0.3053 0.0497 1.99 0.1582
ALG 1 0.5028 0.1688 0.1720 0.8335 8.87 0.0029
Scale 0 1.0000 0.0000 1.0000 1.0000
Weibull Shape 0 1.0000 0.0000 1.0000 1.0000

Weibull-
Standard
Intercept 1 5.7554 0.5405 4.6961 6.8147 113.39 <.0001
AGE 1 -0.0287 0.0099 -0.0482 -0.0093 8.37 0.0038
SEX 1 0.0449 0.2424 -0.4301 0.5200 0.03 0.8529
DIALY 1 -0.0002 0.0003 -0.0007 0.0003 0.50 0.4814
DBT 1 -0.5754 0.3101 -1.1831 0.0323 3.44 0.0635
PTX 1 0.1449 0.3570 -0.5548 0.8447 0.16 0.6848
BLOOD 1 -0.0073 0.0081 -0.0232 0.0085 0.82 0.3663
MIS 1 -0.1537 0.1495 -0.4467 0.1393 1.06 0.3038
ALG 1 0.9736 0.2852 0.4146 1.5327 11.65 0.0006
Scale 1 1.6326 0.1013 1.4457 1.8436
Weibull Shape 1 0.6125 0.0380 0.5424 0.6917
Choice of Covariates
Like we didinthe AFTmodel inthissectionwe will be selectingthe significantcovariates only,using
Breslow’smethod(defaultprocedure) forties,the backward selectionmethod,andthe SASproc phreg.
Conclusion- AGE,DBTand ALG are the three covariatesthatare significantfromourtestresults.
FollowingSAStablesgive usthe detailedprocessof the backwardselectionprocedure.
UsingBreslowapproximationof tiesandbackwardselection-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01732 0.00611 8.0468 0.0046 1.017
SEX 1 -0.02983 0.14882 0.0402 0.8411 0.971
DIALY 10.00009310.0001556 0.3581 0.5495 1.000
DBT 1 0.35004 0.19236 3.3115 0.0688 1.419
PTX 1 -0.07341 0.21763 0.1138 0.7359 0.929
BLOOD 1 0.00372 0.00497 0.5592 0.4546 1.004

ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
MIS 1 0.08376 0.09196 0.8295 0.3624 1.087
ALG 1 -0.60697 0.17033 12.6983 0.0004 0.545
We rejectsex-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01731 0.00611 8.0406 0.0046 1.017
DIALY 10.00009100.0001551 0.3441 0.5574 1.000
DBT 1 0.35031 0.19234 3.3171 0.0686 1.420
PTX 1 -0.07537 0.21728 0.1203 0.7287 0.927
BLOOD 1 0.00359 0.00493 0.5305 0.4664 1.004
MIS 1 0.08476 0.09181 0.8524 0.3559 1.088
ALG 1 -0.60890 0.17001 12.8275 0.0003 0.544
We rejectPTX-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01753 0.00609 8.2854 0.0040 1.018
DIALY 10.00009500.0001542 0.3799 0.5376 1.000
DBT 1 0.35394 0.19211 3.3945 0.0654 1.425
BLOOD 1 0.00309 0.00475 0.4232 0.5154 1.003
MIS 1 0.08957 0.09078 0.9735 0.3238 1.094
ALG 1 -0.59912 0.16774 12.7568 0.0004 0.549
We rejectDialy-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01835 0.00593 9.5785 0.0020 1.019
DBT 1 0.34675 0.19167 3.2728 0.0704 1.414
BLOOD 1 0.00393 0.00454 0.7474 0.3873 1.004
MIS 1 0.09187 0.09122 1.0142 0.3139 1.096
ALG 1 -0.60177 0.16772 12.8728 0.0003 0.548

We rejectblood-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01824 0.00592 9.4984 0.0021 1.018
DBT 1 0.33636 0.19123 3.0940 0.0786 1.400
MIS 1 0.09052 0.09121 0.9848 0.3210 1.095
ALG 1 -0.59750 0.16753 12.7202 0.0004 0.550
We rejectMIS-
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01827 0.00591 9.5362 0.0020 1.018
DBT 1 0.32304 0.19083 2.8656 0.0905 1.381
ALG 1 -0.58475 0.16708 12.2492 0.0005 0.557
Thisis the table forfinal model withsignificantcovariates.
Fitting of the Model
ParameterDF
Parameter
Estimate
Standard
Hazard
Ratio
AGE 1 0.01827 0.00591 9.5362 0.0020 1.018
DBT 1 0.32304 0.19083 2.8656 0.0905 1.381
ALG 1 -0.58475 0.16708 12.2492 0.0005 0.557
Form the above table we can getour significantcovariatesandhence we canconstruct our final model
The final model withsignificant(p<.10) covariatesis-
Log[h(t)/ho(t)]=.01827 AGE + .32304 DBT - .58475 ALG
Interpretation-The positive signof the regressioncoefficientof AGEand DBT implieshighhazardrisk
rate inkidneytransplantwhereasthe negative coefficientof ALGdrug implieslow hazardriskrate.

Survival analysis on kidney failure of kidney transplant patients

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