What is survival analysis? Survival Analysis. Survival Analysis SA - 1. If the event is: If the event is:

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1 What is survival analysis? Survival Analysis Analysis of Time-to-event data The event may be: Death of a person Failure of a piece of equipment Development (or remission) of symptoms Health code violation (or compliance) Modified from Dr. R. Steiner s Lecture Materials 1 2 If the event is death, the time-to-event (T) is the length of time that the individual survives, that is the person s survival time. If the event is death, the time-to-event (T) is the length of time that the individual survives, that is the person s survival time. Survival Analysis can sometimes studied through modeling the function of survival time, S(t), or the hazard function (or risk function), h(t). 3 S(t): Probability of survive beyond time t. h(t): Probability of event just after time t, given survival to time t. 4 If the event is: Death of a person T = Survival time Failure of a piece of equipment T = Failure time Development of symptoms T = Time until symptoms develop Incubation period If the event is: Remission of symptoms T = Time until symptoms disappear Duration of disease Time in acute phase Health code violation T = Time in compliance Health code compliance T = Time in violation 5 6 SA - 1

2 General Survival Analysis Techniques Life Table Methods (Actuarial Methods) c Population Life Table c Clinical Life Table Product-limit Method or Kaplan-Meier Method of estimating a survival function. Cox Regression Analysis (Relative Risk) Life Table Methods Population Life tables Describe the survival experience for an entire population Clinical Life Tables Describe the survival experience of individuals following a medical or health intervention 7 8 Population Life Tables Column 1 x is the age of individuals entering each interval n is the length of the interval Notice that all intervals are not the same length [0, 1) [1, 5) [5, 10) Abridged life table uneven age intervals 9 10 Column 2 q = number dying in the interval n x number alive at begining of interval = proportion alive at beginning of interval that die before the end of the interval Column 3 Number alive at beginning of interval Based on a hypothetical cohort of 100,000 live births l x = no. alive at beginning of previous interval - no. dying in previous interval For example, l 1 = 100, Can be determined from age-specific death rates Related to the hazard function Hazard function is adjusted for length of interval SA - 2

3 Column 4 n d x = l x n q x This is the number alive at beginning of interval times the death rate for the interval Columns 5 & 6 Based on concept of a stationary population nl x is number in interval or it can be viewed as no. of person-years lived in interval T x is number living beyond age x Column 7 Average remaining lifetime Computed as T x / l x For the first interval, this is the life expectancy. In clinical study, life tables comparisons for results from different clinical procedures. Complete Life Table Like abridged life table, but uses 1-year intervals Clinical Life Tables Based on shorter follow-up period Problem is (not a problem in population life table since it usually keeps all detail information) The subject is followed, but the event is not observed. Data is tracked but no complete information. This could be due to: Subject withdraws from study Subject moves Subject dies from another cause Study Ends SA - 3

4 19 20 This is called Right Assumption on censoring in the analysis: Non-informative (Random censoring) This means that someone who is censored at time t is no more or no less likely to have an event than someone who is still under observation at time t. Censor is not related to the chance of the event. This is called Right Type I : - Study stopped at the end of the 3 rd year Comparison of Survival Distributions Methods for Comparing Survival Distributions Logrank Test Breslow Test (Modified Wilcoxon Test, Gehan s Test) Tarone-Ware Tes Hypotheses H 0 : The populations have the same survival distributions H A : The populations have the different survival distributions Comparison of Survival Distributions Methods for Comparing Survival Distributions Logrank Test Emphasizes longer survival times Breslow Test Emphasizes shorter survival times (Modified Wilcoxon Test, Gehan s Test) Tarone-Ware Test Can be set to emphasize long, short, or intermediate survival times by choosing different values of a tuning parameter, k, between 0 and 1. k = 0 (Logrank test) k = 1 (Breslow test) k =.5 (Intermediate survival time) SA - 4

5 Comparison of Survival Distributions Example: Recidivism in Smoking Cessation Program. Subjects had quit smoking at the beginning of the study. The variables are: AGE (1 = 40 or younger; 2 = > 40) and ABSTAIN (days to recidivism, starting to smoke again). 25 ID Age Variable Name Gender Cig CO MinAfter Smoking Cessation Study Description Identification Number Age in years Cigarettes smoked /day Carbon Monoxide level in blood (x10) Minutes elapsed since last cigarette Values 1=Male, 2=Female logcoadj CO adjusted for minutes since last cigarette Abstain Days abstinent Time-to event AgeGrp Age group 1 = 40 or younger Groups to compare 2 = >40 Status indicator 1 = Started smoking again (event) Indicator of event or censoring 0 = Not smoking at end of study (censored) 26 Data for Smoking Cessation Study Set-up for Comparison of Recidivism between Age Groups Kaplan-Meier Dialog Time Variable and Variable Selected SA - 5

6 Set-up for Variable Set-up for Variable Grouping Variable (Factor) Selected Set-up for Age Groups Comparison Set-up for Age Groups Comparison Smoking Cessation Study Results SA - 6

7 Smoking Cessation Study Results Smoking Cessation Study Results Survival Analysis for ABSTAIN Days Abstinent Survival Analysis for ABSTAIN Days Abstinent Factor AGEGRP = 40 or younger Survival Time Standard Error 95% Confidence Interval Mean: ( 53.44, ) Median: ( 11.01, ) Total Number Number Percent Events Censored Censored AGEGRP 40 or younger AGEGRP > Overall Survival Analysis for ABSTAIN Days Abstinent Test Statistics for Equality of Survival Distributions for AGEGRP Factor AGEGRP = > 40 Survival Time Standard Error 95% Confidence Interval Mean: ( 59.59, ) Median: ( 5.99, ) Statistic df Significance Log Rank Breslow Tarone-Ware P-values for the Logrank test, the Breslow test, and the Tarone-Ware test are greater than 0.05, so we cannot conclude a difference in the recidivsm between the two age groups Cox Proportional Hazards Regression Model Cox Proportional Hazards Regression Model Model: βx (t ) h(t;x) = h o (t)e Assesses the effects of covariates on survival Based on the Hazard Function h(t) = Probability of event just after time t, given survival to time t 39 where h(t; x) = the hazard function (the probability of event just after time t, given survival up to time t ) for individuals with the covariate X(t) = x. h 0 (t) = the baseline hazard for individuals with the covariate X(t) = 0 X(t) = the value of the covariate at time t. β = the regression coefficient that indicates the effect of the covariate on the risk of event. 40 Cox Proportional Hazards Regression Model Cox Proportional Hazards Regression Model Model: βx (t ) h(t;x) = h o (t)e β = the regression coefficient that indicates the effect of the covariate on the risk of event: β = 0 means that the covariate has no effect on the risk of event. β < 0 means that increasing values of the covariate reduce the risk of event. β > 0 means that increasing values of the covariate increase the risk of event. 41 Hypotheses H 0 : β = 0 (The covariate has no effect on the risk of event.) H A : β 0 (The covariate has an effect on the risk of event.) 42 SA - 7

8 Cox Proportional Hazards Regression Model Results Example: VA Lung Cancer Study. The study examined survival of lung cancer patients. One of the covariates studied was KPS: (Karnofsky performance status) = completely hospitalized = partial confinement = able to care for self Estimate of β negative here, indicating that patients in better condition (higher KPS) had lower risk of death (lived longer). Variables in the Equation 95.0% CI for B SE Wald df Sig. Exp(B) Exp(B) Lower Upper perfstat P -value for the chisquare test of H0: β = 0 vs. HA: β 0. Here the P - va lue is le s s tha n 0.05, indicating the null hypothes is is re je cte d: KPS had a statistically s ignifica nt e ffe ct on survival. (1) eβ = e = β (0) eβ = e Hazard Ratio (Relative Risk) Variables in the Equation B SE Wald df Sig. Exp(B) 95.0% CI for Exp(B) Lower Upper perfstat RR = P(Event X = x 2) P(Event X = x 1 ) = h(t;x 2 ) h(t;x 1 ) = h 0 (t)eβ (x 2 ) h 0 (t)e β (x 1 ) = eβ (x 2 ) e β (x 1 ) = e β (1) e β (0) = eβ = e Hazard Ratio (Relative Risk) In this example, a 1-unit increase in Karnofsky performance is a very small change in physical condition and results in a rather small reduction in risk of death: RR = A more meaningful change might be an increase in Karnofsky performance of 10 units. Using the previous development, this the relative risk is easy to determine: RR = e β (x 2 x 1 ) = e 0.038(10) = e β (x 2 x 1 ) 45 = Results Survival Curve for KPS = 20 Results Survival Curve f or KPS = 50 Survival Curve for KPS = 50 Survival Curve f or KPS = SA - 8