Rate versus Risk

1
Rate versus Risk

• Two basic measures of the occurrence of new
  events (disease)
• Cumulative incidenceRiskProbability
• Incidence rateRateevents per time units
• Last week we discussed the concept of cumulative
  incidence
• Commonly calculated by the Kaplan-Meier method
  when different follow-up times exist
• Incidence rate of disease is somewhat less
  intuitive but is the more fundamental measure

2
Main Points to be Covered

• Cumulative incidence and person-time incidence
  rate different but related
• Hazard
• Calculating a person-time incidence rate
• Uses of person-time incidence rates
• STATA commands for rates
• Assumptions of survival and person-time analyses

3
The Three Elements in Measures of Disease
Incidence

• E an event a disease diagnosis or death
• N number of at-risk persons in the population
  under study
• T time period during which the events are
  observed

4
Two Measures of Incidence

• The proportion of individuals who experience the
  event in a defined time period (E/N during some
  time T) cumulative incidence
• The number of events divided by the amount of
  person-time observed (E/NT) incidence rate

5
Person-Time Incidence Rates

• The numerator is the same as incidence based on
  proportion of persons events (E)
• The denominator is the sum of the follow-up times
  for each individual
• The resulting ratio of E/NT is not a
  proportion--may be greater than 1
• Value depends on unit of time used

6
Incidence rate value depends on the time units
used

• Incidence rate of 8 cases per 100 person-years
• 0.67 cases per 100 person-months
• 0.15 cases per 100 person-weeks

7
Assumption of Person-Time Incidence Estimation

• A time units of follow-up on B persons is the
  same as B time units on A persons
• Observing 20 deaths in 200 persons followed for
  50 years gives the same incidence rate as 20
  deaths in 10,000 persons followed 1 year
• The rate is constant for the time period during
  which it is calculated
• Rates calculated over long time periods may be
  less meaningful

8
Understanding the Difference between a Rate and
Cumulative Incidence

• Rate can be thought of as how likely an event is
  to happen at any moment in time
• Cumulative incidence is the result of applying
  that rate to a defined population for a specified
  period of time
• A rate is calculated by using data from a time
  period, but the rate is assumed constant during
  that period (i.e., at any moment in time during
  the period the rate is the same)

9
Illustration of Rate versus Cumulative Incidence

• The mortality rate in the U.S. population in 2001
  was 855 per 100,000 person-years (or 0.855 per
  100 person-years)
• If everyone alive at the beginning of the period
  were followed for 5 years, the cumulative
  incidence of death (if the rate held constant)
  would be 4.2 at 5 years at 10 years it would be
  8.2.

10
Relationship between Incidence Rate and
Cumulative Incidence

• A constant rate produces an exponential
  cumulative incidence (or survival) distribution
• If know the instantaneous incidence rate, can
  derive the cumulative incidence/survival function
  or vice-versa
• where F(t) cumulative incidence and
• 1 - F(t) cumulative survival
• e 2.71828 ? rate t time units

11
Constant Rate
Increasing Rate
12
Effect of high and low constant incidence rates
on cumulative incidence
13
Hazard

• Hazard is an instantaneous incidence rate
• h(t) P(event in interval between t and t?t
  alive at t)
• ?t
• Hazard function
• Shape of the relationship between time and
  hazard.
• Constant rate, or constantly increasing rate,
  shown in previous slides are particular examples
• Can take on any shape

14
Hazard function for mortality in general
population
Years
15
Note on Person-Time Rates

• Person-time concept may seem unfamiliar because
  often described as annual rate or annual rate
  per 100,000 persons or per 100,000 persons
  (i.e., person-time denominator is not made
  explicit)
• Example The incidence of Pediatric
  Cardiomyopathy in two regions of the United
  States (NEJM, 2003)
• 467 cases of cardiomyopathy in registry of 38
  centers (New England, Southwest) 1996 - 1999
• denominator population estimates1990 census
  with an in- and out-migration algorithm ages 1 -
  18
• overall annual incidence of 1.13 per 100,000
  children
• Better to make person-time explicit incidence
  among children was 1.13 per 100,000 person-years

16
How to Calculate a Person-Time Rate Obtaining
the Denominator

• Method 1 If have exact entry, censoring, and
  event times for each person, can sum person-time
  for each person for denominator
• Method 2 If no individual data but have the
  time interval and average population size, can
  take their product as denominator
• Some datasets may only have the average
  population size at risk

17
c
18
Rate 6/9.583 0.626 per person-year 62.6 per
100 person-years
19
Method 2 Using average number of persons at risk
during time interval
10 persons at baseline 1 person at end of 2
years (6 deaths 3 censored before 2 years 9
losses) Formula Average number of persons at
risk N baseline N end / 2 11 / 2
5.5 Rate 6/5.5 over 2 years 0.545 per
person-year or 54.5 per 100 person-years
20
Person-time incidence based on grouped vs.
individual data

• Szklo and Nieto use incidence rate when based on
  group data (average population at risk) and
  incidence density when based on individual data
• This terminology distinction is not followed by
  most
• Average population method assumes uniform
  occurrence of events and of censoring during the
  interval (like life table)

21
Waiting Time Property of Incidence Rates

• Waiting time to an event is reciprocal of the
  incidence rate (1/rate)
• Eg, if rate 300 per 100 person-years, reciprocal
  is 1
• (300/100 person-years)
• (1/3) person-year
• Average waiting time between events is 0.33
  person-year 4 person-months

22
Why Use Incidence Rates?

• To calculate incidence from population-based
  disease registries

23
(1) Calculating a rate from population-based
registry of diagnoses

• Research question What is the incidence rate for
  first diagnoses of breast cancer in Marin County
  and how does it compare with rates from other
  counties?
• Nearly all new breast cancer diagnoses are
  reported to the SEER cancer registry
• How to obtain a denominator for a rate?

24
Large Population Person-Time Rates
Since the production of stable rates for cancers
at most individual sites requires a population
of at least one million subjects, the logistic
and financial problems of attempting to maintain
a constant surveillance system of everyone
in the population are usually prohibitive.
Breslow and Day, Statistical Methods in Cancer
Research Solution Do surveillance of all the
cancer diagnoses and estimate the population
denominator to get person-time at risk. To get
an incidence rate person-time denominator by the
group method requires only an estimate of the
average population size during the year (the
population at mid-year).
25
Average Population (Group data) rates versus
individual data rates

• If losses are perfectly uniform, total
  person-time calculation for the denominator (and
  thus the rate) is the same whether based on
  average population size or individual follow-up
• For large populations the rate will be nearly
  identical calculated by either method

26
Potential Weakness of Using Census Data

• Calculating rates from census population data is
  very useful but caution is required as a full
  census is only done every 10 years
• Interim estimates of population change are made
  by the Census but over 10 years denominators may
  become inaccurate

27
Invasive Breast Cancer Rates for Marin County
versus Other California, 1995-2000
Rates per 100,000 person-years Excluding 5 Bay
Area Counties
28
Census Denominators for Incidence
Rates are Estimates
The estimates of breast cancer incidence (number
of new cancers per year) most recently reported
for Marin and other areas of the country were
based on 1990 census information. Data from
Census 2000 have enabled researchers to
recalculate rates for Marin. Preliminary results
show that revised incidence rates for Marin
County based on the 2000 census are substantially
lower than the rates calculated using 1990 census
information. The discrepancy between using the
1990 and 2000 census data is due to projected
population growth differing considerably from
actual population growth.
29
Why Use Incidence Rates?

• To calculate incidence from population-based
  disease registries
• To compare disease incidence in a cohort
  (individual-level data) with rate from the
  general population OR to compare incidences
  between 2 or more general populations

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(2) Comparing a rate from a cohort to the rate in
the general population

• A cohort study of petroleum refinery workers
  followed up subjects for mortality for 36 years
  and found 765 deaths.
• Research question Was the cohort mortality
  incidence high, low, or just average for those
  calendar years?
• How would you calculate the mortality incidence
  in the cohort?

31
Example of Using Person-Time Rates for Cohort
Analysis

• Cohort of petrochemical workers
• 6,588 white male employees of Texas plant
• Mortality determined from 1941-1977
• 137,745 person-years of follow-up time
• 765 deaths
• Overall death rate 765 / 137,745 person-years
  
• 5.6 per 1000 person-years
• Question Is this a high death rate?
• Austin SG, et al., J Occupat Med, 1983

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Cohort of petrochemical workers

• Could calculate KM estimate of cumulative
  incidence (for 36 years of follow-up), but what
  is the comparison group?
• Using the person-time rate, the observed rate can
  be compared to the rate that would be expected if
  the person-time rate from a reference population
  (eg, U.S. population) is applied to the cohort

33
Standardized Mortality Ratio

• If U.S. death rates for age-sex-race-calendar
  period groups applied to the cohort, 924 deaths
  were expected in the cohort versus the 765
  observed.
• Ratio of 765 observed/924 expected 0.83. This
  is called a Standardized Mortality Ratio (SMR).

34
Obtaining an expected rate for comparison
35
Cause Specific SMRs
Austin SG, et al., J Occupat Med, 1983
36
Example of using both cumulative incidence and
incidence rates in the same analysis for
different purposes
End stage renal disease Cumulative incidence
(survival) within cohorts defined by age at
diagnosis Ratios of mortality incidence rates
in renal disease children compared with national
child mortality rates
McDonald et al., NEJM 2004
37
Another example of SMR Is mortality higher after
a fracture?
Bluic et al. JAMA 2009
38
(2b) Comparing hip fracture incidence in
different populations
Per 100,000 person-years
e Standardized to 1990 non-Hispanic white US
population
39
Why Use Incidence Rates?

• To calculate incidence from population-based
  disease registries
• To compare disease incidence in a cohort with a
  rate from the general population OR to compare
  incidence in 2 or more populations
• To compare incidence from a time-varying exposure
  in persons while exposed and unexposed

40
(3) To compare incidence from a time-varying
exposure in persons while exposed and unexposed

• Research question In a Medicaid database is
  there an association between use of non-aspirin
  non-steroidal anti-inflammatory drugs (NSAID) and
  coronary artery disease (CAD)?
• How would you study the relationship between
  NSAID use and CAD?

41
Calculating stratified person-time incidence
rates in cohorts

• For persons followed in a cohort some potential
  risk factors may be fixed but some may be
  variable
• gender is fixed
• taking medications or getting regular exercise
  are behaviors that can change over time
• Adding up person-time in an exposure category to
  get a denominator of time at risk is a way to
  deal with risk factors that change over time

42
Analysis of changing exposure and disease
incidence

• Tennessee Medicaid data base, 1987-1998 are
  NSAIDs associated with CAD risk?
• Same person could both use and not use NSAIDs at
  different times over the 11 years
• Cant do cumulative incidence because would have
  to define groups by baseline characteristics
  without accounting for changes in subsequent
  behavior

Ray, Lancet, 2002
43
Analysis of changing exposure with person-time
rates

• Person-time totaled for using and not using
  NSAIDs MI or CAD death outcome
• 181,441 periods of new NSAIDS use in 128,002
  individuals 181,441 periods of non-use in
  134,642 individuals (matched by age, sex, and
  calendar date)
• A person can contribute to the denominator both
  for use and non-use but only after a 365 day
  wash out period between use and non-use

44
Analysis of changing exposure with person-time
rates

• Rate ratio 1.01
• Concluded no evidence that NSAIDS reduced risk of
  CHD events

Ray, Lancet, 2002
45
Calculating Rates in STATA
Declare data set survival data . stset timevar,
fail(failvar) .strate gives person-years
rate .strate groupvar gives rates within
groups Example Biliary cirrhosis time to death
data .use biliary cirrhosis data, clear .stset
time, fail(d) .strate D Y
Rate Lower Upper 96
747.04 0.1285 0.1052
0.1570 .strate treat Treat D Y
Rate Lower Upper Placebo
49 355.0 0.138 0.104
0.183 Active 47 392.0 0.120
0.090 0.160
46
Hazard function in Stata

• sts graph, hazard

K-M survival curve for same data
Incidence rate (from previous slide) 0.13
deaths per person-year
10 yr cum incidence 0.2375
47
Immediate Commands in STATA
STATA has an option to use it like a calculator
for various computations without using a data
set. Called immediate commands. Example, to
calculate the confidence interval around a
person-time rate . cii person-time units
events, poisson E.g. 6 events occur in 10
person-years of follow-up . cii 10 6,
poisson 95 CI 0.220 1.306
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49
Incidence rate
Cumulative incidence
50
Survival changing over calendar time
51
Summary Points

• Person-time incidence rate (or density)
• E/NT
• Related to cumulative incidence
• Not a proportion
• Person-time incidence rate can be calculated with
  individual or average population data
• Allows incidence estimates in large populations
  that are not completely enumerated
• Allows comparison with population reference rates
  from other data sources
• Allows accumulation of time at risk for different
  exposure strata