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## Chapter 11 Survival Analysis Part 1

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**Our Voyage So Far…**Where we have been: • Continuous Outcome Data • T-tests, ANOVA, Linear Regression • Binary Outcome Data • Odds ratios, Logistic Regression • Count Outcome Data • Poisson Regression (briefly!) Where we are now: • Binary Outcome (yes/no) with Follow-up time • Survival Analysis**Creating Rational**• A randomized clinical trial compared radiation vs. surgery + radiation for treatment of cancers that had spread to the brain. (N Engl J Med 1990;322:494-500). After two years, nearly all the patients had died: • Chi-square probability 0.1868 => No association between death and type of treatment. Is this a reasonable conclusion? • What happens to the rates if we follow the patients for 100 years?**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 (censored) • Analyses based on two factors: • Had event or did not have event (0/1 variable) • Length of time followed (ending – starting date)**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**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 (other types of censoring exist, but this is most common).**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**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**Example of Censoring**Begin time 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**Plot of time to event data**• Suppose we hatch a large number of mosquitoes in a container, and measure how long they live in days. Credits: AP Photo/Jim Newman, University of Florida/IFAS**Survival Function EstimationS(t)=P(T>t)**• 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**Estimating Survival Curves**Kaplan-Meier Method • Sometimes called the product-limit method • 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;**Calculating Kaplan-Meier estimates**Consider Survival Data: 6, 6, 6, 6+, 7, 10, 11+, 13, 13, 15, 17+, 18, 20, 21, 22, 22, 23+ , 24, 24+, 25, 28 • n=number at risk • d=number of events • d/n=chance of an event • 1-d/n = chance of no event • S(t) = Kaplan Meier Estimate**Calculating Kaplan-Meier estimates**Consider Survival Data: 6, 6, 6, 6+, 7, 10, 11+, 13, 13, 15, 17+, 18, 20, 21, 22, 22, 23+ , 24, 24+, 25, 28 Number at risk • SAS calculates these automatically 0.8571 x 0.9412 x 0.9375 x 0.8571**Survival Analysis**• Describe the rate (probability) of the event over time • Called the survival function S(t) • Compare survival function among groups • log rank, Wilcoxon significance tests • Examine risk factors for having the event taking into consideration the time of the event • Cox proportional hazard regression**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**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**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**Estimating survival curves**• Survival curve estimates less precise over time • SAS can produce confidence intervals for the survival curve • 95% CI of form;**Comparing 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 (log-rank assumes constant risk – called hazard ratio)-long term • Wilcoxon more powerful for detecting short term risk**SAS : The form of the data**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 (100+)**PROCLIFETESTPLOTS = (s);**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**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**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**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 ………..more….. Survtime: time t Survival: Kaplan-Meier Estimate of the survivor function S(t) Failure: The Kaplan –Meier Estimate of cumulative mortality (1-S(t)) Survival Standard Error: The point-wise standard error of the estimate S(t) Number Failed: the total number of events and censored observations, NOT the number of events. Number Left: the number still under observation and at risk for an event.**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**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 (100%) 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**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**Crossing Survival curves**• Validity of comparison 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 • If they cross, use alternative method to compare (beyond the scope of this class)**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**Statistical Techniques for censored data**• Kaplan-Meier • Survival curves • log rank, Wilcoxon significance tests • Log-rank test more powerful when relative risk is constant • Wilcoxon more powerful for detecting short term risk • Cox proportional hazards regression • Relate covariates to survival