Main Points to be Covered

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Main Points to be Covered

• Difference between cumulative incidence based on
  proportion of persons at risk and incidence rate
  based on person-time
• Calculating person-time incidence rates
• Uses of person-time incidence rates
• Relation of prevalence to incidence
• Odds versus probability

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The Three Elements in Measures of Disease
Incidence

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

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E/N
E/T
E/NT
E
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Two Measures Described as Incidence in the Text

• 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 or
  density (not a proportion)

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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

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c
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rates year 1 3/7.083 42.4/100 person-years
year 2 3/2.50 120/100 person-years both
yrs 6/9.583 62.6/100 person-years
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We have been calculating average rates rate
is often instantaneous change in one measure
with respect to a second measure as interval 0
death rate
Population size
time
In disease, the occurrence rate is often called
a hazard or the force of morbidity (mortality)
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Rates

• We are used to rates being change in a measure
  with respect to time but time does not have to be
  involved
• Accidents per passenger-mile, for example, is
  often used in transportation
• Economics often uses rates in which time is not
  an element (eg, energy use per unit of gross
  national product)

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Comparison of cumulative incidence and incidence
rate (density)

• Kaplan-Meier cumulative incidence estimate for
  these data was (1 - 0.18) 0.82 (ie, 82 of
  persons will experience event in a two-year
  period)
• Two-year incidence density is 62.6 / 100
  person-years or 0.626 per person-year
• Not a proportion--if calculated per person-days,
  rate would be 0.17 / 100 person-days

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Incidence rate (density) value depends on the
time units used

• An incidence rate of 100 cases per 1 person-yr
• 100 cases/person-year
• 10,000 cases/person-century
• 8.33 cases/person-month
• 1.92 cases/person-week
• 0.27 cases/person-day
• Note time period during which rate is measured
  can differ from the units used

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Relationship of cumulative incidence and
incidence rate

• Our example E is large compared to N and the
  cumulative incidence and the rate differ
• If E very small in relation to N (ie, incidence
  and losses to follow-up are low), then they will
  be approximately the same
• Can make them nearly the same by choosing a small
  enough time interval so that no more than 1/N
  leaves pop. at risk
• This is true if E is a non-recurrent event

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Person-time incidence based on grouped vs.
individual data

• Szklo and Nieto use rate when based on group data
  and density when based on individual data (not
  followed by most)
• Total person-time for grouped data is based on
  the time interval x the average population at
  risk during the interval
• Assumes uniform occurrence of events and of
  censoring during the interval (like life table)

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Calculating person-time incidence using grouped
data

• Use average number of persons at risk
• In the text example, start with 10 persons, 6 die
  and 3 are lost to follow-up
• Subtract 0.5 x (6 3) from 10 5.5
• uniformity assumption as in life tables
• Total person-time is 5.5 x 2-years 11
  person-years. 6 events, so rate 6/11 0.545
  54.5 per 100 person-years (compare to 62.6 when
  calculated using individual data)

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Incidence from grouped data

• Most commonly used for large secondary data sets
  where precise information on occurrence of events
  and on persons leaving and entering population
  are not available
• eg, annual cancer mortality rates per 100,000
  population ( per 100,000 person-years)
• If times of events and of censoring available,
  would normally use individual level data

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Group data rates versus individual data rates

• How much they differ depends on how close events
  and losses are to being uniform throughout the
  follow-up
• Total person-time calculation for the denominator
  (and thus the rate) is the same whether based on
  average population size or individual follow-up
  if losses are perfectly uniform

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Individual calculation 2 deaths / 5 pers-yrs
0.4 per pers-yr Group data average population
(4 1) / 2 2.5 rate 2 / 2.5 x 2 0.4
pers-yr
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Rates based on group data

• Uniformity of events and losses likely to be
  approximately true for large secondary data sets
• Rates using secondary data sets on free-living
  populations assume new members and losses balance
  out ( approx. stable)
• Important for the use of population reference
  rates (eg, expected mortality in U.S. population)

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Assumption of Person-Time Incidence Estimation

• T time units of follow-up on N persons is the
  same as N time units on T persons
• Observing 2 deaths in 2 persons followed for 50
  years gives the same incidence rate as 2 deaths
  in 100 persons followed 1 year
• Assumption is not reasonable if sample sizes and
  follow-up times differ greatly (as example above)
  

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Assumption of Person-Time Incidence Estimation

• If looking at relationship between exposure and
  outcome rate, one rate for a follow-up period
  implies exposure does not have cumulative effect
  on probability of event over time
• Clearly false for exposures with cumulative
  effects like length of time smoking

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Why use person-time rather than cumulative
incidence?

• Rates using group data can be calculated in open
  populations from a variety of data sources where
  population sizes are estimated
• Incidence rates from a cohort study can be
  compared to standardized rates from the general
  population to obtain ratio measures called
  standardized mortality ratio (SMR) or
  standardized incidence ratio (SIR)

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Why use person-time rather than cumulative
incidence?

• If E is a recurrent event, rate may seem more
  natural.
• For example, cumulative incidence of episodes of
  the common cold, would have to be done separately
  for each (ie, proportion with 1st cold,
  proportion with 2nd cold given that you have had
  1, etc.).

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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
• eg, ethnicity is fixed smoking is a behavior
  that can change over time
• Total person-time in an exposure category is one
  way to deal with risk factors that change over
  time

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Relation of Prevalence and Incidence

• Prevalence is a function of incidence and
  duration of disease by the equation
• point prevalence incidence x duration x (1 -
  point prevalence) P I x D (1 - P)
• For many typically low prevalence diseases
  prevalence becomes approximately I x D since (1 -
  P) is close to 1 if P is very low

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Prevalence and Etiology

• Because prevalence depends both on incidence and
  duration of disease, it is not a good measure for
  etiological studies
• Cannot examine the determinants of occurrence
  alone when you have to account for determinants
  of duration (Rx, etc.)
• Etiologic study designs should avoid sampling
  prevalent cases of disease

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Odds versus Probability

• Odds based on probability expresses probability
  (p) as ratio odds p / (1 - p)
• odds is always gt p because divided by lt 1
• For example, if probability of dying 1/5, then
  odds of dying 1/5 / 4/5 1/4
• Thinking of odds as 2 outcomes, the numerator is
  the of times of one outcome and the denominator
  the of times of the other
• P odds / (1 odds), so 1/4 / 1 1/4 1/5

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Odds versus Probability

• Less intuitive than probability (probably
  wouldnt say my odds of dying are 1/4)
• No less legitimate mathematically, just not so
  easily understood
• Used in epidemiology primarily because the ratio
  of two odds is given by the coefficients in
  logistic regression equations

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Summary Points

• Person-time incidence rate or density is not
  equivalent to cumulative incidence and is not a
  proportion
• Person-time incidence rate can be calculated with
  group or individual data
• Allows comparison with population reference rates
  from other data sources
• Allows accumulation of time at risk for different
  strata

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Calculating stratified person-time incidence
rates in cohorts

• Assumption of no temporal trends may not be true
  and therefore need to stratify the follow-up by
  calendar time
• Person-time can be accumulated in strata defined
  by calendar periods or by age or by both
• Graphically represented by Lexis diagram

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Lexis Diagram
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