what is survival analysis

  • modeling “time to event” data
  • event of interest not always death
  • family of methods, 2 most common = KM, COX
  • events e.g. death (specific vs non-specific cause), relapse, adverse events
  • if non-fatal outcomes, need to deal with people that die before non-fatal outcomes

vs logistic regression

  • LR doesn’t care WHEN event happens
  • SA cares about WHAT and WHEN
  • example:
    • recruit cohort, record diet/ exercise, time of death
    • wait for deaths to accumulate
    • end study in 10 years
  • SA accounts for those “lost to follow-up”
    • don’t know if event occured
    • LR need to know outcomes for all subjects, or else biased

KM plot and log-rank test

KM plot

  • answers 2 questions
    • how long you have to live
    • chances you will be alive in next 1, 5 years
  • allows comparison of survival between patient groups (e.g. treatment vs control)
  • estimates probability of surviving AT LEAST to a given time point
  • probability = survival function (plotting over time = KM plot)
  • example: TNM classification and survival
  • KM plot of M0 vs M1 over time
    • KM plot
    • at time 0, cumulative survival probability = 1,
    • as time progress, probability falls

log-rank test

  • statistical test to test if differences btw survival curves significant
  • p-value of 0.0001 is highly statistically significant

censoring

  • censoring is another central idea in survival analysis
  • patient dropped out of study
  • different types of censoring
    • drop out of study
    • people still alive (or did not have event) at end of study

life tables

  • measures:
    • probability of death at a given age, and
    • life expectancy at varying/different ages
  • 2 different kind of life tables
    • cohort/ generational life table
    • period/ current / static life table

cohort life table

  • take a set of people born at the same time, follow them up to death
  • e.g. Millenium cohort study in UK
  • might not be relevant for people born at other time points
  • longitudinal, same group of people moving through time

period life table

  • take a hypothetical cohort of people born at the same time,
  • assumption = all subject to age-specific mortality rates of a region/country
  • mortality rate taken from age-specific death rates during census year + a year on either side

  • cross sectional, population is different age groups at a moment in time >

    wikipedia
- life table = mortality table = actuarial table
- shows for each age, probability of death before his/her next birthday
- i.e. survivorship of people from certain population
- period life table
    - mortality rates during specific time, of certain population
    - for people of different ages, in the current year
- cohort life table
    - overall mortality rate (probably of death),
      over course of their lifetime, of certain population
- period and cohort life table same if
    - population in equilibrium (constant number of people each year)
    - environment does not change

cohort study

  • cohort of patients enrolld at time zero,
  • followed up to log when outcome of interest happens (e.g. death)

constructing life table

time(t) in days #patients alive at time t #patients died at time t probability of survival past time t
0 (study starts) 100 0 1
1 100 2 0.98
2 98 ?? ??
3 ?? ?? ??
  • start with 100 patients
  • all makes it past time 0
  • prob of surviving at least to t=0 is 1
  • 2 people die at t=1, P(Surv) = 0.98

  • this assumes eeryone enters study at the same time (t=0),
  • and no one leaves study, except death

calc plot KM by hand

  • KM can estimate survival curve empirically from observed survival times,
    • without assumption of underlying probability distribution
  • (other survival analysis do require an underlying distribution of survival times)
  • KM has no assumptions (no need to test if assumptions are valid)

  • proportion = pts alive at end of time t/ pts alive at begining of time t
  • probability = prob of surviving past day t-1 * proportion of pts alive at time t
  • censoring
    • we know pt alive at time t, but do not know when event occured
    • censored pt not classified as survived or died
    • simply deduct censored pts from #alive
    • when there’s both pt died and censored, count num_death first
      • calculate proportion surviving first,
      • then deduct num_censored from num@risk to arrive at num@risk for next time point
      • e.g. t=12, n.risk=10, 3 died and 1 censored
        • proportion surviving @ t=12 = (10-3)/10
        • t=13, n.risk = 10-3-1 = 6
  • if no one dies at time t
    • proportion = 1
    • P(Surv) unchanged
    • KM plot stays flat
time(t) in days #pts alive at time t #pts died at time t proportion of pts surviving (at) past time t probability of survival past time t
0 (study starts) 8 pts recruited 0 (8-0)/8=1 1
1 8 2 (8-2)/8=0.75 1*0.75 = 0.75
4 6 1 (6-1)/6=0.83 0.75*0.83 = 0.623
5 5 1 (5-1)/5=0.8 0.623*0.8 = 0.498
6+ 4 0 (1pt leaves) 4/4=1 0.498*1 = 0.498
9 3 1 (3-1)/3=0.667 0.498*0.667 = 0.332
9+ 2 1pt leaves 2/2=1 0.332*1 = 0.332
22 1 1 0/1=0 0

final KM plot

EOF