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April 18

April 18. Intro to survival analysis Le 11.1 – 11.2 Not covered in C & S. Intro to Survival Data. Our voyage so far… Continuous outcome data T-tests, linear regression, ANOVA Categorical data Odds ratios, relative risk, chi-square tests, logistic regression

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April 18

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  1. April 18 • Intro to survival analysis • Le 11.1 – 11.2 • Not covered in C & S

  2. Intro to Survival Data Our voyage so far… • Continuous outcome data • T-tests, linear regression, ANOVA • Categorical data • Odds ratios, relative risk, chi-square tests, logistic regression • New scenario; time to event data • Categorical outcome (yes/no) • Follow-up time

  3. Rational • Want to take into account not just whether a patient has an event of interest but the amount of time from some starting point until the event. • Patient who dies 2 weeks after diagnosis of cancer should be considered differently than a patient who dies 2 years after diagnosis

  4. Goals • Describe the rate (probability) of the event over time • Called the survival function • Compare survival function among groups • Examine risk factors for having the event taking into consideration the time of the event

  5. Kaplan-Meier survival curve Survival After Diagnosis of Lung Cancer S (t) is the probability of surviving to at least t S (200) = 0.37

  6. Comparing Two Survival Curves

  7. Time To ? • Death after diagnosis of cancer • CVD event after enrolled in a study • Re-arrest after release from prison • Divorce after marriage Survival analyses better described as “Time to Event” analyses Note: The event does not have to inevitable

  8. Kaplan-Meier Life Curves

  9. Nature of Data • Definitive starting point (become “at risk”) • Definitive ending point • If had event then date of event • If did not have event then date last know not to have had the event • Analyses based on two factors: • Had event or did not have event (0/1 variable) • Length of time followed (ending – starting date)

  10. Examples • Death after diagnosis of cancer • Starting point: date of diagnosis • Ending point: date of death or date last know to be alive • Divorce after marriage • Starting point: date of marriage • Ending point: date of divorce or date last know to be still married

  11. Censoring • After a certain period of time the patient does not have the event but it is unknown as to whether the patient had the event after this time. • Called right censoring

  12. Reasons for Censoring • Patient no longer followed (thus event status not know after a certain date) • Patient has a different event that make the primary event not possible • Primary event: death from cancer but patient dies from CHD • Primary event: divorce but one spouse dies Patient no longer “at risk” for study purposes

  13. Censoring example • Follow-up for study is 365 days • Patient survives 245 days then is lost • At that point, we KNOW that they survived 245 days but we do NOT KNOW whether they survived between days 246 and 365 • If we exclude them from any end-point calculations we ignore 245 days worth of information

  14. Types of censoring • Uninformative • “lost” status not related to outcome • Those lost similar to those not lost (usually not true) • Informative • “lost” status is related to outcome • Those who are lost are more likely to be dead than those not lost • Most methods assume we have uninformative censoring

  15. Example of Follow-up Times C O U P E S Divorced after 6 years D C Has been married 10 years at time of analyses C One spouse dies after 3 yrs C No contact with couple after 5 years 0 5 10 Years Since Marriage

  16. Survival Function Estimation • Patients are followed for different length of time • Like to use all the data to estimate the survival function • Patients followed 1-year can help estimate survival function in first year • Patients followed 2-years can help estimate survival function in first 2-years

  17. Life Table Calculation 100 couples married in 2002 followed 2 years 100 couples married in 2003 followed 1 year Follow-up through 2004 Year 1 of follow-up Year 2 of follow-up 10 D (5 each from 2002 and 2003 marriages) 200 8 D 95 C 190 95 S (1) = 190/200 = .95 87 S (2) = S (1) * S (2| S>1) = .95 * .92 = .870 Note: S (1) is estimated with more precision than S(2)

  18. Estimating Survival Curves Kaplan-Meier Method • Also called Product-Limit or Life-table curve • For each time where 1 or more events occur, calculate number who die at that point over number who survived to that point (di/ni) • Multiply all these quantities;

  19. Calculating Kaplan-Meier estimates Number at risk • SAS calculates these automatically 0.8571 x 0.9412 x 0.9375 x 0.8571

  20. Questions • What is the survival rate over time for persons diagnosed with lung cancer? • Is the survival rate over time different for different types of cancer? • Are patient characteristics related to survival

  21. Comparing Two Survival Curves

  22. How do we describe this data? • Logistic regression? • Model risk of death • Would ignore the amount of follow-up time • Linear regression? • Model survival time • How do you handle those who died vs. those who survived? • Survival times not normally distributed (all >0) • Need new methods that incorporate follow-up time information • Survival or time-to-event analyses

  23. Comparing survival curves • For any time point, can see probability of survival for either group • Median survival time; point where probability surviving = 50% • Rank Tests – Compare entire curves

  24. Estimating survival curves • Survival curve estimates less precise over time • SAS can produce confidence intervals for the survival curve • 95% CI of form;

  25. Testing survival curves • Formal statistical tests exist • Log-rank test and Wilcoxon test • Both assess whether survival distributions are equal • Null hypothesis: survival distributions (curves) are equal • Alternative hypothesis: survival distributions (curves) are not equal; one greater/less than other • Each compares survival distributions in a slightly different way • Log-rank test more powerful when relative risk is constant • Wilcoxon more powerful for detecting short term risk

  26. USING SAS Patient died 72 days after diagnosis Obs Age Cell death SurVTime 1 69 squamous 1 72 2 64 squamous 1 411 10 70 squamous 0 100 11 81 squamous 1 42 12 63 squamous 1 8 13 63 squamous 1 144 14 52 squamous 0 25 15 48 squamous 1 11 23 41 large 1 200 24 66 large 1 156 25 62 large 0 182 26 60 large 1 143 Patient alive after 100 days but status after that time is unknown

  27. PROCLIFETESTPLOTS = (s); WHERE cell in('squamous','large'); TIME survtime*death(0); STRATA cell; Tells SAS to draw life table plot Tells SAS that values of 0 are censored observations Tells SAS to compute life table estimates separately for each cell type

  28. RUNNING ON SATURN (UNIX) GOPTIONS DEVICE = png htext=0.8 htitle=1 ftext=swissb gsfmode=replace PROCLIFETESTPLOTS = (s); WHERE cell in('squamous','large'); TIME survtime*death(0); STRATA cell; Creates a file called sasgraph.png FTP over to PC and insert file into word insert/ picture/ from file

  29. PROC LIFETEST OUTPUT Summary of the Number of Censored and Uncensored Values Percent Stratum Cell Total Failed Censored Censored 1 large 27 26 1 3.70 2 squamous 35 31 4 11.43 --------------------------------------------------------------- Total 62 57 5 8.06

  30. Test of Equality over Strata Pr > Test Chi-Square DF Chi-Square Log-Rank 0.8226 1 0.3644 Wilcoxon 0.0520 1 0.8197 -2Log(LR) 1.0218 1 0.3121 Tests equality of 2 survival functions

  31. Stratum 1: Cell = large Product-Limit Survival Estimates Survival Standard Number Number SurvTime Survival Failure Error Failed Left 0.000 1.0000 0 0 0 27 12.000 0.9630 0.0370 0.0363 1 26 15.000 0.9259 0.0741 0.0504 2 25 19.000 0.8889 0.1111 0.0605 3 24 43.000 0.8519 0.1481 0.0684 4 23 49.000 0.8148 0.1852 0.0748 5 22 52.000 0.7778 0.2222 0.0800 6 21 53.000 0.7407 0.2593 0.0843 7 20 100.000 0.7037 0.2963 0.0879 8 19 103.000 0.6667 0.3333 0.0907 9 18 105.000 0.6296 0.3704 0.0929 10 17 111.000 0.5926 0.4074 0.0946 11 16 133.000 0.5556 0.4444 0.0956 12 15 143.000 0.5185 0.4815 0.0962 13 14 156.000 0.4815 0.5185 0.0962 14 13 162.000 0.4444 0.5556 0.0956 15 12 164.000 0.4074 0.5926 0.0946 16 11 177.000 0.3704 0.6296 0.0929 17 10 182.000* . . . 17 9 200.000 0.3292 0.6708 0.0913 18 8 X-Y points for life table graph First death after 12 days

  32. Stratum 1: Cell = large Product-Limit Survival Estimates Survival Standard Number Number SurvTime Survival Failure Error Failed Left 0.000 1.0000 0 0 0 27 12.000 0.9630 0.0370 0.0363 1 26 15.000 0.9259 0.0741 0.0504 2 25 19.000 0.8889 0.1111 0.0605 3 24 S(0) = 1 S(12) = .9630 (26/27) S(15) = .9259 (25/27) which is also 26/27 * 25/26 S(19) = .8889 (24/27) What is S(17) ? Estimated survival function is a step function

  33. Stratum 2: Cell = squamous Product-Limit Survival Estimates Survival Standard Number Number SurvTime Survival Failure Error Failed Left 0.000 1.0000 0 0 0 35 1.000 . . . 1 34 1.000 0.9429 0.0571 0.0392 2 33 8.000 0.9143 0.0857 0.0473 3 32 10.000 0.8857 0.1143 0.0538 4 31 11.000 0.8571 0.1429 0.0591 5 30 15.000 0.8286 0.1714 0.0637 6 29 25.000 0.8000 0.2000 0.0676 7 28 25.000* . . . 7 27 30.000 0.7704 0.2296 0.0713 8 26 2 patients died after 1 day

  34. Crossing Survival curves • Validity of tests require risk in one group always greater than risk in other group • When survival curves cross, terms used in calculating test statistic cancel out • Give test statistic value near zero • P-value is larger than it should be • Graph survival curves to check for crossing • Use alternative method

  35. Censoring vs. missing data • Censoring is a special case of having missing data • Missing; don’t know whether or not person had outcome • Censoring; don’t know whether or not person had outcome, but know they didn’t have outcome after being followed for some time

  36. Statistical Techniques for censored data • Kaplan-Meier (life table analysis) • Survival curves • log rank, wilcoxon significance tests • Tests to compare survival curves • Cox proportional hazards regression • Relate covariates to survival

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