Lecture 2: Time, Rates and Rate Ratios

1
Lecture 2Time, Rates and Rate Ratios

• 513-668 Statistical Models in Epidemiology

2
Outline

• Cumulative survival/failure
• KM curve
• Redistributed to the right algorithm
• Rates
• Poisson as a limit of Binomial
• Relation between failure rate and probability of
  cumulative survival
• Nelson-Aalen curve
• Competing risks
• Inference
• Estimation of single rate and confidence interval
• Rate-ratio likelihood, profile likelihood,
  conditional likelihood

3
Ch 4 Consecutive Follow-up Intervals

• Conditional probability at t1 Estimate
  probability (of death or survival) in interval
  (t1, t2) based on patients who survive up to t1
• Probability of failing at t1 Multiply survival
  probability in consecutive intervals up to t1 by
  probability of death in (t1,t2)

4
Example Cumulative survival (No censoring)

• Fig 4.3, page 30.
• Likelihood ?
• Bin(10100,?1) ? Bin(1590, ?2) ? Bin(875, ?3)
• Sample size person-time units
  265 units

5
Table 4.1, page 32 Cohort life table of data on
stage I cancer of the cervix (censoringexact
time of failure not known)
6
Likelihood Exact time of censoring not known

• (assuming study stops after year 3)
• Likelihood ?
• Bin(5107.5,?1) ? Bin(796.5, ?2) ? Bin(782.5,
  ?3)
• Sample size person-time units
  286.5 units (9.5 units
  lost due to censoring)
• What about Bin(5100,?1) (P(Tgt1))5 etc?

7
Likelihood (if exact failure/censor times were
available)

• Time divided into small intervals so that
  instantaneous rate of failure/success can be
  estimated
• Let T denote the survival time,
• S(t)P(Tgtt), 1-survival distribution
• f(t), survival density
• ?i1 if not censored, 0 if censored
• L ? ?i f(ti)?i S(t)1-?i (for right-censored
  data)

8
Using exact failure times for each subject
Kaplan-Meier method

• Non-parametric method.
• Assumptions Censoring time is unrelated to event
  time, no confounding
• If last observation censored, cumulative survival
  cannot become 0
• Why does the step size change with time? How can
  we estimate the step size?

9
Table 4.2 K-M for non-melanoma deaths
10
Kaplan-Meier Self consistency

• Alternative way to derive K-M estimates (Efron,
  1967)
• Each patient contributes 1/N to overall failure
  rate. Censored subjects pass weight to others
  because they still have a probability of failure
  beyond the time of censoring.
• In melanoma example
• at start weight of subjects at risk 1/N 0.02
• at month 8 weight of subjects at risk 1/N(1/N
  ? 1/(N-6)) 0.0205
• Better approach for KM estimate when competing
  risks are present

11
Ch 5 Rates

• By splitting follow-up period into sufficiently
  small intervals (clicks) we minimize arbitrary
  assumptions about losses
• As interval length -gt 0, probability-gt 0,
  probability/unit-time-gtprobability rate
• also called instantaneous probability rate,
  hazard rate or force of mortality

12
Rate parameter vs Observed rate

• prob rate refers to a single individual but
  observed rate, the estimated value (i.e. the
  MLE), is based on the group at risk at t
• Observed rate

13
Poisson distribution

• Ex 5.2 30 subjects, infinite follow-up
• Total person years 261.9
• Failure rate 30/261.9 114.5/1000 P-Yrs
• Qs How were these data generated?
• Ex 5.3 30 subjects, 5-year follow-up (largest
  follow-up is 36.5 yrs)
• Failure rate 14/115.8 120.9/1000 P-Yrs
• See JH notes for more on Poisson.

14
Likelihood for a rate Limiting value of the
binomial likelihood

• D failures, N intervals of length h. N large, h
  small.P(failure in each interval) ?. Prob
  rate ? ?/h
• ? likelihood ?D (1-?)N-D (? h)D (1-? h)N-D
• ?
• Also, cumulative survival

15
Rates that vary with time

• Cumulative rate ? log(cumulative survival
  probability)
• Ex 5.8

16
Rates varying continuously with time

• Aalen-Nelson estimator equivalent to the KM
  estimator but gives cumulative failure (hazard)
  rates rather than cumulative survival
• Instantaneous failure rate 1/Nh
• Step size (1/Nh) ? h 1/N
• Plot useful for comparing hazard functions to
  determine if they are proportional
• Preferable to KM curve, particularly in case of
  competing risks (See 7.4, Pg 66)

17
Table 4.2 K-M for non-melanoma deaths
18
Ch 7 Competing risks

• When failures occur due to multiple causes we
  have a multinomial model
• This is based on the assumption that the
  cumulative survival is independent across causes,
  which may not be true

19
Competing risks Selection bias

• Censoring is non-informative if those lost to
  follow-up have the same probability of survival
  as those remaining in the study
• Informative censoring, ex. of severely ill
  subjects in RCT, would lead to bias.
  Intention-to-treat approach would involve
  continuing follow-up after treatment ceases
• Late entry is a problem in cohort studies
• Dynamic vs closed cohort

20
Likelihood ? vs log(?)

• Ex D7, Y500 (90 CI 7.0/1000-24.6/1000)

21
Ch 13 Likelihood for the rate ratio

• Ex. 13.1. Comparison of CIs of individual rates
  not appropriate
• Likelihood of rate ratio
• D0 log(?0)-?0Y0 D1 log(?1)-?1Y1
• D0 log(?0)-?0Y0 D1 log(??0)-??0Y1,
• Thus ? is a nuisance parameter

22
Profile likelihood

• Likelihood maximized conditional on MLE of
  nuisance parameter for each ?
• Cautions Not a real likelihood, cannot be used
  to calculate p-values etc

23
Conditional likelihood

• Treat total number of failures (D) as fixed. What
  is the probability that D splits into D1 failures
  in exposed and D0 in unexposed?
• Calculating a conditional likelihood depends on
  finding a statistic conditional on which the
  likelihood is independent of the nuisance
  parameter.
• Not always possible, ex. the rate difference is
  not possible

24
Bayesian inference

• Likelihood
• X1 ? Poisson(?1), X2 ? Poisson(?2)
• Prior
• log(?1) ? N(0,0.0001)
• log(?2) ? N(0,0.0001)
• Draw a sample from ?1/?2
• Alternatively, let log(?2) log(?1)logRR and
  place prior on logRR with appropriate limits to
  ensure ?2gt0
• Essentially, under the Bayesian approach you can
  integrate out the nuisance parameter from the
  joint posterior to obtain the marginal posterior
  of interest

25
Ch 6 Lexis diagrams

• Next lecture