2 Survival Analysis - Ku
Survival analysis Types of response dataSurvival analysisBirthe Lykke ThomsenH. Lundbeck A/S( Event-time analysis) continuous data [simple plots; t-test,ANOVA, regression . . . ] Characteristics of event-time data discrete data [histograms, plots; tables,χ2 -test, logistic regression, proportionalodds, Poisson regression . . . ] Randomized studies: The Intention-to-treat principle censored data event-time data (failure-time data,survival data) [Kaplan-Meier curves;Cox regression] data with detection limit(s) Example Non-parametric estimation (Kaplan-Meier,Nelson-Aalen) Comparison of to groups (log rank test) Regression model for event-time data (the Cox-model ) Test in the Cox model123
Characteristics of event-time data Explanatory variables in the Cox model covariates strati cation interaction Transformation of explanatory variables Model checking for the Cox model Choice of time scale/time zero Delayed entry Competing risks Comparison with dichotomous response Matched case-control designs4 response: time to the occurrence of acertain event (death, recurrence, pregnancy,. . . ); makes speci c models relevant censoring: for some of the individuals weonly know that the event has not yetoccurred (e.g., alive and healthy at the endof the study) left truncation/delayed entry: some ofthe individuals are not at risk in the studyfrom time zero (if the event had occurredbefore a speci c time point, it would nothave been counted as an event in the study)No assumptions about the form of thestatistical distribution of the event times, butcensoring must be independent of futurefailure given the covariates.5ExampleRandomized study of the e ect of sclerotherapyA study of 187 patients with bleedingoesophageal varices (due to liver cirrhosis).During the hospital admission for the rstvariceal bleeding, the patient were randomizedinto one of two groups:1. (standard) medical treatment (n 94)2. medical treatment supplemented withsclerotherapy (n 93)(EVASP, 1984)We wish to investigate whether sclerotherapya ects the risk for rebleeding.6
Rebleeding in the two groupsIf the event studied is all-cause mortality thenthe Kaplan-Meier curve estimates the survivalprobability as a function of time ( the survivalcurve ).Intention-to-treat(Randomized studies)The mathematical relation between survivalprobability and the cumulative rateSome patients do not receive sclerotherapyalthough they were randomized to thesclerotherapy group how should thesepatients be treated in the analyses?Sg (T ) exp( Rg (T ))Rg (T ) ln(Sg (T ))All persons randomized to sclerotherapymust be included in the sclerotherapygroup in the analyses to avoid biasInterpretation of the e ect of the treatment :E ect of the treatment regime(Piecewise) constant rate gives (piecewise)linear cumulative rate (and a Poisson modelwould make better use of the available data)78The rate is number of events per time unit but the cumulative rate has no immediateinterpretation, it is not equal to the probabilitythat the event has occurred at or before thegiven time point (but for small values it is agood approximation)9
Calculations of survival curveand cumulative rateNon-parametric estimationOn a given day t we observe the following ineach group g (denoted stratum/strata )1. ng (t) individuals in total2. mg (t) individuals starting to rebleedwhich gives the daily rebleeding ratesrg (t) mg (t)ng (t)The Kaplan-Meier curve Sg (T ) for group g isobtained by multiplying the terms 1 rg (t) forall days t before and including day T .The Nelson-Aalen estimate for the cumulativerebleeding rate Rg (T ) for group g is obtainedby adding the daily rebleeding rates for all dayst before and including day T .1011Estimation of a Kaplan-Meier curve withpoint-wise 95 % con dence intervalsResponse: The time when the event occurs .But it does not occur for everyone, so 2variables are needed to describe the response: time and what happened .The data set SCL contains (among other things)DAY: time of exit from the studyBLD: 1 if rebleeding occurs, 0 if censoredSCLERO: 1 for the sclerotherapy group,0 for the medically treated groupPROC PHREG DATA scl NOPRINT;TITLE 'Kaplan-Meier curves';MODEL day*bld(0) ;STRATA sclero;BASELINE OUT km SURVIVAL kmcurvesLOWER lowerb UPPER upperb/ CLTYPE LOGLOG;RUN;12
The data set KM after the changesThe data set KM generated by PHREGObs sclero day100200301402:::370 245380 444390 589401041124213:::641 308651 330661 90.97480:0.544010.522690.50087Data modi cationsData modi cations are necessary if we want thepoint-wise con dence limits in the gure:DATA km; SET km;IF sclero 0 THEN DO;type 1; curve kmcurves; OUTPUT;type 2; curve lowerb; OUTPUT;type 3; curve upperb; OUTPUT;END;IF sclero 1 THEN DO;type 4; curve kmcurves; OUTPUT;type 5; curve lowerb; OUTPUT;type 6; curve upperb; OUTPUT;END;RUN;14sclero000000000000:111111day kmcurves lowerb upperb type curve0 1.00000 .1 1.0000 1.00000 .2 .0 1.00000 .3 .0 0.98936 0.928 0.9981 0.9890 0.98936 0.928 0.9982 0.9280 0.98936 0.928 0.9983 0.9981 0.96809 0.905 0.9891 0.9681 0.96809 0.905 0.9892 0.9051 0.96809 0.905 0.9893 0.9892 0.94388 0.871 0.9761 0.9442 0.94388 0.871 0.9762 0.8712 0.94388 0.871 0.9763 0.976::::::330 0.40466 0.283 0.5234 0.405330 0.40466 0.283 0.5235 0.283330 0.40466 0.283 0.5236 0.523340 0.38086 0.260 0.5014 0.381340 0.38086 0.260 0.5015 0.260340 0.38086 0.260 0.5016 0.50115
Log rank testPlot of the estimated curvesExample of the GPLOT procedurePROC GPLOT DATA km;PLOT curve*day type/ HAXIS AXIS1 VAXIS AXIS2;SYMBOL1SYMBOL2SYMBOL3SYMBOL4R 1R 2R 1R 2V NONEV NONEV NONEV NONEI STEPLJI STEPLJI STEPLJI STEPLJL 1 W 2 C BLACK;L 33 C BLACK;L 1 W 2 C GRAYAA;L 35 C BLACK;AXIS1 LABEL ('Days from randomization');AXIS2 LABEL (A 90 R 0'Kaplan-Meier curves');RUN;16Groups may be compared using a log rank test.Principle (for 2 groups):We assume ( the null hypothesis ), that there isno di erence between the 2 groups andcondition for each time of death by the observed number of deaths in totalm(ti ) ( m1 (ti ) m2 (ti )) the number of individuals (presently) atrisk in each of the groups n1 (ti ) andn2 (ti )( n(ti ))For group 1 we then calculate, for each time ofdeath ti the expected number of deaths 1 (ti )E1 (ti ) m(ti ) · nn(ti) the variance of the number of deaths (ti )·m(ti )(n(ti ) m(ti ))V1 (ti ) n1 (ti )n2(n(t2i )) (n(ti ) 1)17Log rank testThe expected number of deaths E1 (ti ) andthe variance V1 (ti ) are added for all deathtimes ti to give E1 and V1 , respectively.Furthermore, we count the total number of deaths O1 in group 1.The log rank test statisticχ2log rank (O1 E1 )2,V1is χ2 -distributed with 1 degree of freedom.The result does not depend on whichgroup we decide to label 1!Approximation which may also be applied formore than 2 groups, here G groups:χ2log rank GX(Og Eg )2g 1Eg,which is χ2 -distributed with G 1 degrees offreedom (note that all groups contribute).18
Output from PROC PHREGLog rank test using PROC PHREGCalculation of log rank test statistic as ascore test using PROC PHREG:PROC PHREG DATA scl;MODEL day*bld(0) sclero/ TIES DISCRETE;RUN;Here we compare the group with SCLERO 1(the sclerotherapy group) to the group withSCLERO 0 (the medically treated group).If we have more groups, we need a 0-1 variablefor each group except one, which then becomesthe reference group (the log rank test doesnot depend on the choice of reference group)19The PHREG ProcedureModel InformationData SetWORK.SCLDependent VariabledayCensoring VariablebldCensoring Value(s)0Ties HandlingDISCRETESummary of the Number of Event and Censored 4Model Fit -2 LOG L738.406737.488:::Testing Global Null Hypothesis: BETA 0TestChi-Square DFPr ChiSqLikelihood 39520Output from PROC PHREG continuedAnalysis of Maximum Likelihood EstimatesParameter StandardVariable DF Estimate Error Chi-Sq. Pr ChiSqsclero1 -0.20261 0.21212 0.9124 0.3395Analysis of Maximum Likelihood EstimatesHazardVariable Ratiosclero0.817The log rank test is not suitable for detectingtime-dependent di erences like a bettershort-term prognosis for one group and a betterlong-term prognosis for the other group.21
Proportional hazards(hazard instantaneous rate)Quanti cation of treatment e ect:r(t; sclero) r(t; medical) · BE ect of ascites:r(t; ascites) r(t; without ascites) · ACombined:r(t; sclero, ascites) r(t; medical, ascites) · B r(t; medical, without ascites) · A · B r(t; medical, without ascites) · ea bwith a ln(A) and b ln(B ).without ascitesSet X1 {0 1 with ascitesmedical aloneand X2 {0 1 sclerotherapythenr(t; sclero, ascites) r(t; X1 1, X2 1) r0 (t) · eaX1 bX222Cox's regression modelThis model is denoted Cox's regression model,generally formulated:r(t; X1 , X2 , . . . , Xk ) r0 (t)·eb1 X1 b2 X2 . bk XkIf we log-transform and use a(t) for log(r0 (t)),we get something that looks more like otherregression models:log(r(t; X1 , X2 , . . . , Xk )) a(t) b1 X1 b2 X2 . . . bk XkThe covariates X1 , X2 , . . . , Xk may becontinuous like serum bilirubin.A positive value of bj means that large valuesof the covariate Xj increases the rate: Forunwanted events, large values worsen theprognosis (be cautious with positive/negativee ect , use, e.g., bene cial/harmful).23Example with several covariatesPROC PHREG DATA scl;MODEL day*bld(0) ascites bilirub sclero/ -----------Summary of the Number of Event and Censored bscleroParameter StandardDF EstimateError Chi-Sq. Pr ChiSq10.18072 0.227210.6326 0.426410.00476 0.00112 18.1500 .00011 -0.21924 0.218011.0113 0.3146HazardRatio1.1981.0050.80395% Hazard RatioConfidence Limits0.7681.8701.0031.0070.5241.23124
Estimation in a Cox modelTest of covariates in the Cox modelWald's test:Wald's test of a single covariate appears in theoutput, like p .4264 for ascites.Wald's test for several covariatessimultaneously may be performed by adding aTEST-statement, for instance:PROC PHREG DATA scl;MODEL day*bld(0) ascites bilirub sclero/ RISKLIMITS;Asc bili: TEST ascites 0, bilirub 0;RUN;which gives some extra lines of output:Linear Hypotheses Testing ResultsWaldLabelChi-Square DF Pr ChiSqAsc bili21.2800 2 .000125Explanatory variables in event-time analysesA variable may enter the model in two verydi erent ways: as a covariate continuous variables may enter in theusual fashion for categorical variables we have tosupply dummy variables to be used ascovariates in PROC PHREG as a strati cation variableThese possibilities are basically di erent andthe choice has consequences for the modellingas well as the interpretation of the variable inquestion and for the estimation of the e ect ofother variables.26For the particular day on which patient jrebleeds, we calculate the probability that thishappens precisely for this patient j, given thata rebleeding occurs among the patients in thestratum where patient j belongs:exp(bXj )Pi in j exp(bXi )where j denotes all those patients (the i's)who were at risk of rebleeding in the samestratum as j, when j started to rebleed.These contributions are multiplied together forall rebleeding time points, and b is estimatedby the value, b̂, which maximizes this totalproduct called Cox's partial likelihood .If there are more than one patient rebleedingon the same day, we have ties in the data. Tiesmay be handled in several ways: TIES EXACT(the correct method) TIES DISCRETE (as in thelog rank test), or TIES BRESLOW (the quickestand SAS's standard)27
InteractionsExplanatory variables in event-time analysesCovariates enter only in the exponent, thus therates are assumed proportional for di erentvalues of X :r(t; X 1) r0 (t) · eb andr(t; X 2) r(t; X 1) · eb r0 (t) · e2bConsequences:1. the e ect is described using a single number2. but this quantity can only be interpreted ifthe assumption of proportional rates holds(approximately)For strati cation variables we let theunderlying rate depend upon the value of thevariable, thus the di erence betweenindividuals with X 1 and individuals withX 2 may change over time:r(t; X 1) r1 (t) and r(t; X 2) r2 (t)Consequences:1. we do not get a simple measure of the e ect2. the strati cation variable(s) must becategorical with only few values28Strati cation must not mistaken forinteraction! The e ects of the remainingvariables are assumed to be identical in thedi erent strata in contrast to theepidemiological use of the term strati edanalyses !!Interaction means that the e ect of onevariable, e.g., bilirubin, depends on the value ofanother variable, e.g., the treatment. We thenhave to estimate di erent associations withbilirubin in the two treatment groups. Thisrequires dummy variables!Some SAS-procedures can make these dummyvariables automatically (CLASS and '*'), but PHREGcannot (yet). It has to be done in a DATA-step:DATA scl; SET scl;IF sclero 1 THEN DO;scl bili bilirub; med bili 0;END;IF sclero 0 THEN DO;scl bili 0; med bili bilirub;END;RUN;29Interactions cont.As always, the interpretation of parameterestimates depends on which other covariatesare included in the model:SCL BILI together with MED BILI:Variable Parm.Est. Std.Err. Chi-Sq. Pr ChiSqsclero -0.08500 0.26087 0.10620.7445med bili 0.00578 0.00146 15.6066 .0001scl bili 0.00423 0.00155 7.4938 0.0062Here, we estimate separate linear relations inthe two treatment groups. The SCLEROcovariate must be included, otherwise thebilirubin-relations are forced to meet in 0 (nodi erence between the treatment groups forbilirubin 0).SCL BILI together with BILIRUB:Variable Parm.Est. Std.Err. Chi-Sq. Pr ChiSqsclero -0.08500 0.26087 0.10620.7445bilirub 0.00578 0.00146 15.6066 .0001scl bili -0.00155 0.00207 0.5621 0.4534Here, SCL BILI estimate the di erence betweenthe e ects of BILIRUB in the two groups! Thisversion may be used to test the interaction.30
Need for transformationof explanatory variablesCriteria for choice ofparametrisation/transformation Biologic/medical justi cation (best, butmost often not possible). The rate increasesexponentially with untransformedcovariates, while a logarithmictransformation of a covariate means thatthe rate is increased by a xed factorwhenever the covariate increases with e.g.10%.Need for transformationof explanatory variablesCriteria for choice ofparametrisation/transformation, cont. A few extreme values of the explanatoryvariable may have to much in uence on theresult unless the variable is transformed[a few extremely large log2 (x), a fewextremely small exp(x/c) (rarely used)]Trick: The best possible transformation of thepresent data take care, when evaluatingthe signi cance and interpreting the e ect(the signi cance will be exaggerated, donot put too much emphasis on the chosentransformation).By choosing log2 we get eb̂ (Hazard Ratio) toestimate the factor by which the rate ismultiplied for every time x is doubled, sincelog2 (x) 1 log2 (x) log2 (2) log2 (2 · x). Iflog2 is not directly available (it is in SAS), itmay be calculated as log2 (x) log(x)/ log(2).Likewise, we may estimate the factorcorresponding to a 10% increase directly byusing XX log(x)/ log(1.1) as the covariate.3132 Transformations used by others(comparability).Transformation of serum bilirubinPROC UNIVARIATE DATA scl PLOT;VAR --------Histogram# Boxplot525 *1*.*3*.*1*.*3*.275 .*3*.**60.***70.****************46 -- -- 25 ************************************ 107 *-----*---- ---- ---- ---- ---- ---- ---- * may represent up to 3 counts33
Transformation of serum bilirubinDATA scl; SET scl; log2bili LOG2(bilirub); RUN;Plot of a linear splineSimple numerical evaluations:PROC UNIVARIATE DATA scl PLOT;VAR ---------Stem Leaf#9 0128 566789687 55677967 122233476 5556689986 00000000001111122233344444265 55556666666666667777888889999295 00000000000011222222222333444444444354 55556666777777788899999234 00011122233334444173 66778888999113 000342 612 032---- ---- ---- ---- ---- ---- ---- Need for transformationof explanatory variablesBoxplot00 ----- *-----* ----- 00 De ne the squared term X2 X**2; andinclude both X and X2 in order to testwhether X2 gives a signi cant improvement(test for curvature/linearity, although notvery powerful). Include both the untransformed and thetransformed variable simultaneously to seewhether there is a clear-cut answer as towhich is the better predictor (requires areasonable alternative).Graphical evaluation with correspondingtest:Linear splines (Greenland 1995, Epidemiology,p. 356-365)343536
Plot of the covariates neededConstruction of the linear splinePlot of the covariates needed373839
Estimation and test of linear splineQuartiles among rebleeders: 26, 47, 73Coding in SAS(PROC UNIVARIATE PCTLDEF 3; WHERE bld 1;VAR bilirub; RUN;)Extra variables:DATA scl; SET scl;IF bilirub NE . THEN DO;b u26 MIN(bilirub-26,0);b o26 MAX(bilirub-26,0);b o47 MAX(bilirub-47,0);b 26 47 b o26-b o47;b o73 MAX(bilirub-73,0);b 47 73 b o47-b o73;END;RUN;are included in the model and tested:PROC PHREG DATA scl;MODEL day*bld(0) b u26 b 26 47 b 47 73 b o73 sclero/ RISKLIMITS;Testline: TEST b u26 b 26 47 b 47 73 b o73;RUN;4041Estimation and test of linear splineVariableb u26b 26 47b 47 73b o73scleroVariableb u26b 26 47b 47 73b .876Chi-Sq.0.18012.26186.08381.01890.3613cont.Pr ChiSq0.67120.13260.01360.31280.547895% Hazard RatioConfidence 01.348Linear Hypotheses Testing ResultsLabelWald Chi-Square DFPr ChiSqTestline15.781130.0013 Parameter Estimate are the slopes for ln(rateratio) within each of the intervals. HazardRatio is therefore a measure of theinterval-speci c dose-response relations.42
Plot of linear spline5th and 95th percentiles among rebleeders: 12and 177DATA plot;DO bili 12, 26, 47, 73, 177;b u26 MIN(bili-26,0);b o26 MAX(bili-26,0);b o47 MAX(bili-47,0);b 26 47 b o26-b o47;b o73 MAX(bili-73,0);b 47 73 b o47-b o73;pi -0.01390*b u26 0.03250*b 26 47 0.03483*b 47 73 0.00162*b o73;rr EXP(pi);OUTPUT;END;RUN;43Transformation of serum bilirubinPlot of linear splineSYMBOL1 V CIRCLE I JOIN L 1 C BLACK;AXIS1 LABEL (F CENTX 'Bilirubin') ;AXIS2LABEL (F CENTX A 90 R 0 'Rate ratio')LOGBASE 2MINOR (N 3) ;PROC GPLOT DATA plot;PLOT rr*bili/ HAXIS AXIS1 VAXIS AXIS2;RUN;44Inclusion of serum bilirubin untransformed aswell as transformed by the logarithm:PROC PHREG DATA scl;MODEL day*bld(0) sclero bilirub ---------Parameter StandardVariable DFEstimateError Chi-Sq. Pr ChiSqbilirub 1 -0.0001959 0.00231 0.0072 0.9325log2bili 10.48004 0.18152 6.9939 0.0082sclero1 -0.18290 0.21596 0.7172 0.3971The bilirubin-related estimates cannot bereadily interpreted ( change when doubling ofbilirubin for xed value of bilirubin . . . ). Ifthey are both signi cant, then the conclusion isbest illustrated in a graph.45
Estimation with log2 (serum bilirubin)PROC PHREG DATA scl;MODEL day*bld(0) sclero log2bili/ -----------Parameter StandardVariable DF Estimate Error Chi-Sq. Pr ChiSqsclero1 -0.18373 0.21575 0.7252 0.3944log2bili 1 0.46716 0.09706 23.1656 .0001Analysis of Likelihood Estimates95% Hazard RatioConfidence Limits0.5451.2701.3191.930The e ect of serum bilirubin: a twice as largeserum bilirubin value corresponds to approx.60% increased rate of rebleeding46Model control in the Cox modelThe Cox model is based on the assumption ofproportional rates, so R(t; X) R0 (t)ebX andln(R(t; X)) ln(R0 (t)) bXGraphical check of proportional rates: Stratifyfor each variable separately and plotln(Rstratum (t)) ln( ln(Sstratum (t))),the curves should be approximately parallel.TITLE 'Graphical check of proportionality';PROC PHREG DATA scl NOPRINT;MODEL day*bld(0) log2bili;STRATA sclero;BASELINE OUT checkLOGLOGS lcumrate/ METHOD CH;RUN;47Plot of log(cumulative rates)Example of code for GPLOTSYMBOL1 V NONE I STEPLJ L 1 C BLACK;SYMBOL2 V NONE I STEPLJ L 1 C GRAYAA;AXIS1LABEL (F CENTX 'Days from randomization')LOGBASE 7MINOR (N 5);AXIS2LABEL (F CENTX A 90 R 0);PROC GPLOT DATA check; WHERE 0 day 343;PLOT lcumrate*day sclero/ HAXIS AXIS1 VAXIS AXIS2 NOLEGEND;RUN;48
Numerical test of proportionalityusing time-dependent variablesPlot of log(cumulative rates)for graphical check of proportionalityChoose appropriate time points (here 14 and105 days), allow for di erent proportionalityfactors in each time interval through the use ofdummy variables (here SCLFR14 and SCLFR105)and test, whether they are signi cant:PROC PHREG DATA scl;MODEL day*bld(0) log2bilisclero sclfr14 sclfr105;IF sclero 1 AND day 14 THEN sclfr14 1;ELSE sclfr14 0;IF sclero 1 AND day 105 THEN sclfr105 1;ELSE sclfr105 0;Testprop: TEST sclfr14, sclfr105;RUN;The variables inside PHREG are calculated foreach rebleeding time point for all patients atrisk at that particular time. The time variable,here DAY, is equal to the rebleeding time pointfor the patient in the numerator ( j , slide 27),while all other variables refer to the currentpatient in the denominator ( i , slide 27).4950Part of output from PROC 1.04480.1979Pr ChiSq .00010.71760.30670.6564Linear Hypotheses Testing ResultsWaldLabelChi-SquareDFPr ChiSqTestprop1.944520.378251
Time-dependent treatment e ectWould we expect a time-dependent e ect?PROC PHREG DATA scl;MODEL day*bld(0) sclero scl ltid log2bili;IF sclero 1 THEN scl ltid LOG2(day (day 0));ELSE scl ltid --Parameter StandardVariable EstimateError Chi-Sq. Pr ChiSqsclero0.10443 0.46763 0.0499 0.8233scl ltid -0.06675 0.09608 0.4826 0.4872log2bili 0.46349 0.09679 22.9323 --Omitting SCLERO gives usParameter StandardVariable EstimateError Chi-Sq. Pr ChiSqscl ltid -0.04774 0.044471.1526 0.2830log2bili 0.46488 0.09675 23.0883 .000152Time scalesDi erent timescalesTime since randomisationExamples of time scales age calendar time time since beginning of a disease time from some other event of greatimportance for the rate (here time fromtermination of latest bleeding) time from randomization (oftenproblematic) (pseudo)time from operation (veryproblematic if the comparison group hasnot been operated)The only di erence for the single individual isthe de nition of time 0, but it may make a bigdi erence for the results, because it has anin uence on which individuals that areconsidered at risk when something happens.5354
Di erent timescalesTime since cessation of rst bleedingChoice of time scaleDelayed entryChoose a relevant time scale! The advantage of the Cox model is that itallows for an unspeci ed relation betweenthe rate and the underlying time scale.Reason: The individuals must experience aspeci c event before they are at risk in thestudy, and this happens at di erent time pointsfor the di erent individuals. The ratio between the rates for any twopatients at any particular time point is onlyallowed to depend upon the covariates. Characteristic of a relevant time scale:There must be a good reason to assumethat time since time 0 has a large (and identical ) e ect on the rate for allpatients otherwise a constant underlyingrate is the only meaningful possibility, andin that case, the data can be better utilizedby performing a Poisson regression.Other time scales may enter as covariates inthe Cox model. If the dependence on anothertime scale cannot be assumed to follow thepattern one year more always means the samething , then you must use time-dependentcovariates or stratify.5556Examples: for some patients the randomization isperformed later than time 0 for the chosentime scale (in the example some patientsare randomized several days after thetermination of the rst bleeding) some covariates require a specialexamination and some of the patients haveto wait for this examination to be included, the patients must be aliveand well at the start of the study cancer among siblings of children withcancer: siblings can enter the study onlyfrom the age they had when the proband(the child who was diagnosed with cancerrst) got the cancer diagnosis57
Delayed entry in SASPROC PHREG DATA scl;MODEL tnotbld*bld(0) log2bili sclero/ ENTRYTIME t ----------------Model InformationData SetWORK.SCLEntry Time Variablet entryDependent VariabletnotbldCensoring VariablebldCensoring Value(s)0Ties 9866342.28:Analysis of Maximum Likelihood EstimatesParameter StandardVariable EstimateError Chi-Sq. Pr ChiSqlog2bili 0.43431 0.09580 20.5534 .0001sclero-0.16470 0.21682 0.5770 0.447558Separate analyses of the two eventsCompeting risksPatients may exit for several reasons, hererebleeding or death.Consequences: Technical: Endpoints other than the eventin focus are treated as censorings Interpretation: The rate, and thereforealso the estimated e ects have the sameinterpretations as before BUT the Kaplan-Meier curve cannot beinterpreted as the probability of avoidingthe event in focusIf several types of events are of interest, theneach type of event must be analysed separatelytreating the other types of events as censorings.59PROC PHREG DATA scl;MODEL tnotbld*bld(0) sclero log2bili ascites/ ENTRYTIME t entry;RUN;Parameter StandardVariable DF EstimateError Chi-Sq. Pr ChiSqsclero1 -0.19124 0.220210.7542 0.3851log2bili 10.42240 0.09677 19.0542 .0001ascites 10.15762 0.227760.4789 --PROC PHREG DATA scl;MODEL tnotbld*dead(0) sclero log2bili ascites/ ENTRYTIME t entry;RUN;Parameter StandardVariable DF EstimateError Chi-Sq. Pr ChiSqsclero1 0.17358 0.351730.2435 0.6217log2bili 10.50353 0.14482 12.0890 0.0005ascites 10.93763 0.381666.0354 0.014060
The probability of beingalive without rebleedingIf the various events are combined, we get anassessment of the e ects of the covariates onthe time until the rst of these events occur:Competing eventsThe probability of rebleeding at a particulartime t equalsr(t; X) · S(t; X)DATA scl; SET scl;status bld 2*dead;RUN;PROC PHREG DATA scl;MODEL tnotbld*status(0) sclero log2bili ascites/ ENTRYTIME t ------Parameter StandardVariable DF EstimateError Chi-Sq. Pr ChiSqsclero1 -0.08715 0.185550.2206 0.6386log2bili 1 0.44557 0.08044 30.6819 .0001ascites 1 0.37333 0.193033.7405 0.053161where S(t; X) is the probability of being aliveat time t without having experiencedrebleeding yet.Consequence: Factors that do not a ect the rate for aparticular event may, even so, have ane ect on the probability of experiencingthe event through the in uence on the ratefor a competing event and thereby on theprobability of being at risk.62Comparison to dichotomous response a dichotomous response is only concernedwith status at a particular time(dead/alive; diseased/healthy):1. you do not need to know the precisetime for occurrence of death/illness; youonly focus on the speci c time pointand register whether the event hashappened (yet)2. the result will depend upon the chosentime point3. the comparison between studies may beproblematic if the study periods are toodi erent63
Individually matched case-control designsComparison to dichotomous responses cont.Comparison to dichotomous responses cont.4. it is impossible to utilize any knowledgeabout the order of events (like whetheror not the untreated died before thetreated), so there is less power to detectpossible e ects5. patients who leave the study early forother reasons (the censorings) cannot beused in the analyses, since we do notknow whether or not they died/got sickafter they left, but before the end of thestudy period if there are competing events, there may beproblems with the de nition as well as theinterpretation of dichotomized data64 modelling of the probability instead of therate can make it harder to evaluate thee ect of age: For a speci c individual, theprobability of having experienced a certainevent will always increase with age, eventhough the event is most common amongthe very young if we have proportional rates, the rate ratiowill be further away from the neutral valueof 1 than the odds ratio (from logisticregression) and the risk ratio (from alog-linear model) (rate ratio OR riskratio 1 or rate ratio OR risk ratio 1).If we know the order (on the relevant timescale) of the events in focus, then theevent-time analysis is preferable.65If we match individually in case-controldesigns, the analysis should be performed usingconditional logistic regression. This is donethrough the use of PROC PHREG strati ed by thevariable which denotes the case-controlpair/group, and using a dummy time variable:DATA matched; SET rawdata;if case 1 then dum time 1; * cases ;if case 0 then dum time 2; * controls ;RUN;PROC PHREG DATA matched NOSUMMARY;MODEL dum time*case(0) exposure;STRATA matchgrp;RUN;Here, the variable dum time is set to 1 for casesand 2 for controls to ensure, t
Log rank test using PROC PHREG Calculation of log rank test statistic as a score test using PROC PHREG : PROC PHREG DATA scl; MODEL day*bld(0) sclero / TIES DISCRETE; RUN; Here we compare the group with SCLERO 1 (the sclerotherapy group) to the group with SCLERO 0 (the medically treated group). If we have more groups, we need a 0-1 variable
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