Survival Analysis

Question 1

What is survival analyis?

Answer 1

• set of statistical methods for analyzing data where the outcome variable is the time until the occurrence of an event of interest
• events could be death, occurrence of disease, married, divorced, etc.
• time can be measured in days, weeks, years

Question 2

When would “time-to-event” be particularly interesting?

Answer 2

• time until tumour recurrence
• time until cardiovascular death after treatment
• time until AIDS for HIV+ patients
• time until medical school admissions, etc. (other life events)

Question 3

How are time-to-event studies normally carried out?

Answer 3

• prospective cohort studies
• participants followed over a specified period of time and the focus is on the time at which the event of interest occurs

Question 4

What is time-to-event?

Answer 4

-variable measuring the elapsed time from a particular starting point to a particular event

Question 5

What makes survival analysis different?

Answer 5

• survival times are positive numbers (always skewed)
• the probability of surviving past a certain point in time may be of more interest than the “expected” time of event
• the hazard function, used for regression in survival analysis, lends more insight into the “picture” of failure mechanism (can see how people move through the study)
• censoring (some people will reach the end and not experience the event, drop out, or withdraw- “censored”)

Question 6

What is censoring?

Answer 6

• a particular type of missing data
• observations are called “censored” when the information about their survival time is incomplete
• typically we see “right censoring”
• information about how long they truly survived since we started watching them is unknown
• means that participant does not have event before study ends, lost to follow up, withdraws
• means they did not have the event while observed but we do not know what happened afterwards

Question 7

What is right censoring?

Answer 7

• people who make it without experiencing the event
• assume that this person’s survival is at least to be as long as the duration of the study

Question 8

What is the dependent variable in survival analysis?

Answer 8

-time-to-event and event status (event or no event)

Question 9

What is survival function?

Answer 9

• for every time, the probability of surviving (not experiencing the event) up to that time
• probability of surviving past a given time period

of subjects survivng past a given time period/# of subjects in the study at the start of the time period

• as t ranges from 0 to infinity, survival function never increases (survival can’t be more than 1)
• at t=0, probability of survival is 1 S(t)=1
• at t=infinity, S(infinity)=0

Question 10

What is the hazard function?

Answer 10

-the potential that the event will occur, per time-unit, given that an individual has survived up to that specified time

Question 11

Estimate the survival functions and interpret them

Answer 11

S(t)2001= 0.9: 0.9 probability of surviving through 2001

S(t)2002= 0.83: if you survive up until beginning of 2002, 0.83 probability of surviving until the end of 2002

S(t)2003= 0.63 “ “

S(t)2004= 0.11

S(t)2005= 0

Question 12

If you have a 90% chance of surviving to the end of 2001 and an 83% chance of surviving to the end of 2002, what is your cumulative probability of surviving to the end of 2002?

Answer 12

0. 9 x 0.83= 0.747
   - conditional probability; in order to survive until end of 2002 you have to survive until end of 2001

-cumulative gives probability of surviving both time periods

Question 13

What are assumptions and limitations of a survival curve?

Answer 13

1. assumes everyone is recruited at the start of 2001 and followed-up until they die
   - in reality, people are recruited at different times so we calculate survival time from recruitment until event occurs or subject is censored
   - reframe starting points to be the same and see how long people survived past year 0 even if they were recruited at year 2
2. if a subject survives to the end of a period then she is considered to survive past that period (ex: calendar year of 2007 ends on Dec 31st 2007- subject is assumed to survive past the period and be alive in 2008)

Question 14

How are survival times adjusted for people entering the study late and only having for example 1 year to be followed up?

Answer 14

-when calculating probability of surviving past 1 year follow up, add all participants who have at least 1 year in study

S(t) 1 year= 0.525

-for year 2, remove people who only entered with 1 year of possible follow up

S(t) 2 years= 0.461

S(t) 3 years= 0.655

Question 15

How can this data be interpreted in terms of time-to-event?

Answer 15

• event: pregnancy
• outcome: time to pregnancy
• non-smoking women get pregnant quicker so after any cycle more smokers will not be pregnant

Question 16

What is used to determine if two curves are different?

Answer 16

• log rank test
• p value less than 0.05 means statistically significant

Question 17

What do survival curves tell us?

Answer 17

• at any given point in time, what is the difference in event rate between two groups of interest
• if the curves lie on top of each other, there is no difference in rate of events between groups
• use this to calculate hazard ratio (interpreted the same was as other ratios)

Question 18

How can we determine if the two survival curves are different from each other?

Answer 18

• statistically compare them using log rank test
• if p value is less than 0.05 then the curves are different

Question 19

What is cox regression? Hazard function?

Answer 19

• way to estimate the association of independent variables and time-to-event
• hazard function: probability of an event occuring at time t, given that the individual has survived to time t

Question 20

What are the num and denom for hazard function?

Answer 20

• instantaneous rate of occurrence of an event
• numerator: conditional probability that the event will occur in the interval given that it has not occured before
• denominator: the width of the interval
• dividing one by the other gives us rate of event occurrence per unit of time

Question 21

What are assumptions made for cox regression?

Answer 21

• hazard functions are proportional; at any point in time, the difference between the curves are proportional (have to say the risk is true across all points in time)
• when the curves cross, violation of proportional hazards model (then it can’t be used)
• when there is an effect of time, violates proportional hazards model (over time, the width of difference of curves increases- can test for this using Schoenfeld residuals)

Question 22

What happens if we violate proportional hazards?

Answer 22

• conduct regression stratified by time
• conduct regression separately by group
• include an interaction term for time x group

Question 23

How are drop outs accommodated for?

Answer 23

• life table: based on predetermined study periods (eg. per cycle, per year)
• KM: each event starts a new period (KM maximizes use of information because the data are used to define the periods)
• KM has become preferred method in literature

**go through slideshow examples to calculate parts of each table

Question 24

What can be observed in this survival curve?

Answer 24

• the two surgery options typically follow the same curve until approx 2 years after diagnosis
• minimally invasive surgery survival is worse than open surgery
• HR is 1.65: 65% more likely to die within 4 years of diagnosis with the minimally invasive procedure