12/20091 EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009 Dr. N....
-
Upload
martina-scott -
Category
Documents
-
view
215 -
download
0
Transcript of 12/20091 EPI 5240: Introduction to Epidemiology Incidence and survival December 7, 2009 Dr. N....
12/2009 1
EPI 5240:Introduction to Epidemiology
Incidence and survivalDecember 7, 2009
Dr. N. Birkett,Department of Epidemiology & Community
Medicine,University of Ottawa
12/2009 3
12/2009 4
Survival curve (1)
• Previous graph has a problem– What if some people were lost to follow-up?– Plotting the number of people still alive would
effectively say that the lost people had all died.
• Instead– True survival curve plots the probability of
surviving.
12/2009 5
12/2009 6
Time ‘0’ (1)
• Survival (or incidence) measures time of events from a starting point– Time ‘0’
• No best time ‘0’ for all situations– Depends on study objectives and design
• RCT of Rx– ‘0’ = date of randomization
• Prognostic study– ‘0’ = date of disease onset– Inception cohort– Often use: date of disease diagnosis
12/2009 7
Time ‘0’ (2)
• Effect of ‘point source’ exposure– ‘0’ = Date of event– Hiroshima atomic bomb– Dioxin spill, Seveso, Italy
• Chronic exposure– ‘0’ = date of study entry OR Date of first exposure– Issues
• There often is no first exposure (or no clear data of 1st exposure)
• Recruitment long after 1st exposure– Immortal person time– Lack of info on early events.
12/2009 8
12/2009 9
12/2009 10
Survival Curves (1)
• Primary outcome is ‘time to event’• Also need to know ‘type of event’
Person Type Time
1 Death 100
2 Alive 200
3 Lost 150
4 death 65
And so on
12/2009 11
Survival Curves (2)
• People who do not have the targeted outcome (death), are called ‘censored’
• For now, assume no censoring
• How do we represent the ‘time’ data.– Histogram of death times - f(t)– Survival curve - S(t)– Hazard curve - h(t)
• To know one is to know them all
12/2009 12
t
dxxftF0
)()(
Histogram of death time-Skewed to right-pdf or f(t)-CDF or F(t)
-Area under pdf from ‘0’ to ‘t’
t
12/2009 13
Survival curves (3)
• Plot % of group still alive (or % dead)
S(t) = survival curve
= % still surviving at time ‘t’
= P(survive to time ‘t’)
Mortality rate = 1 – S(t)
= F(t)
= Cumulative incidence
12/2009 14
‘Rate’ of dying
• Consider these 2 survival curves
• Which has the better survival profile?– Both have S(3) = 0
12/2009 16
Survival curves (4)
• ‘A’ is better.– Death rate is lower in first two years.– Will live longer than in pop ‘B’
• Concept is called:– Hazard: Survival analysis/stats– Force of mortality: demography– Incidence rate/density: Epidemiology
• DEFINITION– h(t) = rate of dying at time ‘t’ GIVEN that you have
survived to time ‘t’• Slight detour and then back to main theme
12/2009 17
)(
)()|(
BP
BAPBAP
)()(
)(
)(
)(
)(
)(
)|(
1)|(
*
0
0
0
0
0
00
tStS
tS
ttosurviveP
tsurvivalP
ttosurviveP
ttosurviveANDtsurvivalP
ttosurvivetsurvivalP
ttifttosurvivetS
Survival Curves (5)Conditional Probability
h(t0) = rate of failing at ‘t0’ conditional on surviving to t0
Requires the ‘conditional survival curve’S(t|survive to t0) = 1 if t ≤ t0
= P(survival ≥ t | survive to t0)
Essentially, you are re-scaling S(t) so that S*(t0) = 1.0
12/2009 18
12/2009 19
)(
)(1
0tS
tS
)(
)()(
)(
)(1
)()(
0
0
0
*
0
0
tS
tSth
tS
tS
dt
d
tCIdt
dth
t
t
S*(t) = survival curve conditional on surviving to ‘t0‘
CI*(t) = failure/death/cumulative incidence at ‘t’ conditional on surviving to ‘t0‘
= 1 - S*(t)
Hazard at ‘t0‘is defined as: ‘the slope of CI*(t) at t0
Hazard (instantaneous)Force of MortalityIncidence rateIncidence density
Range: 0 ∞
12/2009 20
Hazard curves (1)
12/2009 21
Hazard curves (2)
12/2009 22
Hazard curves (3)
12/2009 23
Some relationships
h(t) = ‘instantaneous’ incidence density at ‘t’Cumulative hazard = H(t) = = area under h(t) (or ID(t))
from ‘0’ to ‘t’
If the rate of disease is small: CI(t) ≈ H(t)If we assume h(t) is constant (= ID): CI(t)≈ID*t
t
dtth0
)(
)(1)( tHetCI
12/2009 24
The real world
• So much for theory• In real world, you can’t measure time to
infinite precision– Often only know year of event– Or, perhaps even just the event happened– Standard Epi formulae make BIG
assumptions• We can do better• More advanced statistics can use discrete survival
models• We won’t go there
12/2009 25
Key Concept to estimate CI
• Divide the follow-up period into smaller time units– Often, use 1 year intervals– Can be: days, months, decades, etc.
• Compute an incidence measure in each year
• Combine these into an overall measure
Year # at start # dying
0 1000 100
1 900 90
2 810 81
What is CI over 3 years? 100+90+81Standard Epi formula: ------------------- = 0.27 1000
Another view:P(die in 3 years) = 1 – P(not dying in 3 years)
How can you still be alive after 3 years?Don’t die in year 1 andDon’t die in year 2 andDon’t die in year 3
Basis for alternate analysis method
12/2009 27
12/2009 28
12/2009 29
12/2009 30
12/2009 31
12/2009 32AND SO ON
12/2009 33
12/2009 34
12/2009 35
12/2009 36
12/2009 37
12/2009 38
12/2009 39
12/2009 40
1990-1 10,000 6,750 2,025 6,625 0.306 0.694 0.694 0.306
12/2009 41
12/2009 42
12/2009 43
• These are the main ways to estimate CI directly
• If cohort is not ‘fixed’, require assumptions about losses.
• Dynamic population– Don’t know who is in cohort– Can not know who dropped out and when.– Can not know who joined population and
when– These formulae don’t work well
• Instead– Estimate ID and the convert to CI
12/2009 44
12/2009 45
12/2009 46
Computes ID for each year of follow-up- 0.22 in each year ID is constant at 0.22
1990-4 10,000 7,652 2,296 5,026 25,130 0.091 0.366
12/2009 47
12/2009 48
12/2009 49
12/2009 50
12/2009 51
12/2009 52
12/2009 53
• These methods can be applied with dynamic populations– Only need to estimate PY’s
• If rate is small (<0.01), then – CI(1 interval) ≈ ID
EXAMPLE• Apply these methods to the example I’ve been
working with.• Target is to estimate: Prob(dying in 5 years)
12/2009 55
12/2009 56
12/2009 57
12/2009 58