Estimating and Comparing Rates

1
Estimating and Comparing Rates

• Incidence Density
• Incidence Rate Difference and Ratio
• Confidence Intervals
• Standardized Rates and Their Comparison

2
A Definition

• Kleinbaum, Kupper and Morgenstern, Epidemiologic
  Research Principles and Quantitative Methods
  (1982), p.97
• A true rate is a potential for change in one
  quantity per unit change in another quantity,
  where the latter quantity is usually time. ()
  Thus, a rate is not dimensionless and has no
  finite upper bound i.e., theoretically, a rate
  can approach infinity.

3
Rates

• A well-known example of rate is velocity, i.e.,
  change of distance per unit of time (given, e.g.,
  in km/h).
• In practice, it does (should?) have an
  upper-bound ?
• We can talk about instantaneous and average
  rates.
• Example instantaneous your car velocity at a
  particular time-point (can depend on the
  time-point, e.g., city and highway).
• Example average your average speed after
  travelling a particular distance (assumed
  constant across the whole trip).
• In epidemiology, we usually talk about average
  rates.

4
Incidence/Mortality Rate

• Kleinbaum, Kupper and Morgenstern, Epidemiologic
  Research Principles and Quantitative Methods
  (1982), p.100
• The incidence rate of disease occurrence is the
  instantaneous potential for change in disease
  status (i.e., the occurrence of new cases) per
  unit of time at time t, or the occurrence of
  disease per unit of time, relative to the size of
  the candidate (i.e., disease-free) population at
  time t.
• We could similarly define the mortality rate.

5
Incidence Rate

• Other terms
• an instantaneous risk (or probability)
• a hazard (especially for mortality rates)
• a person-time incidence rate
• a force of morbidity.
• It is expressed in units of 1/ time.
• It is sometimes confused with risk.

6
Rates and Risks

• Assume that the incidence rate is constant over
  time (?), and the same for all individuals.
• The risk (probability) of developing disease in
  time T will then be equal to 1-e-?T.
• Risk is sometimes called a cumulative incidence.
• In a disease-free (at time 0) cohort of N
  individuals, you would thus expect N(1-e-?T) new
  cases after time T.
• Similarly, we could talk about the risk of death.
• Thus, formally these are two different quantities.

7
Estimating Rates

• Rates require observations of incidence in time.
  Thus, they are estimated from cohort studies.
• Instantaneous rates are seldom obtained. Rather,
  the average rates are computed.
• The most basic estimator is the incidence density
  (ID)

• PT is expressed in person-years, person-days etc.

8
Incidence Density

• A hypothetical cohort of 12 subjects.
• Followed for the period of 5.5 years.
• 7 withdrawals among non-cases
• three (7,8,12) lost to follow-up
• two (3,4) due to death
• two (5,10) due to study termination.

• PT 2.53.51.5 26.
• ID5/260.192 per (person-) year
• or
• 1.92 per 10 (person-)years.

9
Population-Time Without Individual Data

• E.g., population-based registries.
• Person-years computed using the mid-year
  population.
• For rare events, periods of several years may be
  used.
• Ideally, one would like to use mid-year
  populations for each year.
• Alternatively, one can use information for
  several time-points, or the mid-period population
  (these are less accurate solutions).
• One may face the problem of removing those not at
  risk (e.g., women for prostate cancer incidence).

10
Population-Time Without Individual Data Example
11
Incidence Density Remarks

• It is an estimate of an average rate.
• So we will sometimes refer to it as an incidence
  rate.
• Any fluctuations in the instantaneous rate are
  obscured and can lead to misleading conclusions.
  E.g.,
• 1000 persons followed for 1 year
• 100 persons followed by 10 years
• produce the same number of person-years.
• If the average time to disease onset is 5 years,
  ID in the first cohort will be lower.

12
Incidence Density Remarks

• If applied to the whole cohort/population,
  sometimes called crude rate.
• However, sex, age, race etc. can have substantial
  influence on the incidence of disease.
• Comparing crude rates for two populations, which
  differ w.r.t., e.g., age, can be misleading
  (confounding!).
• Therefore, usually standardized rates are
  compared.
• E.g., for cancer, age- and sex-standardized rates
  are used.
• They will be discussed later.

13
Confidence Interval for Incidence Density

• By using a Poisson model, standard error of IDI
  / PT can be estimated by

• Thus, an approximate 95 CI for ID is given by
• ID 1.96SE(ID).
• 99 CI ID 2.58SE(ID) .

14
Estimating Incidence Densities Example

• Postmenopausal Hormone and Coronary Heart Disease
  Cohort Study
• Stampfer et al., NEJM (1985).
• Involving female nurses

• ID1 30/54308.7 0.00055 SE(ID1)
  (30/54308.72)1/2 0.00010
• 95 CI for ID1 0.00055 1.960.0001
  (0.00035, 0.00075)
• ID0 60/51477.5 0.00116 SE(ID0)
  (60/51477.52)1/2 0.00015
• 95 CI for ID0 0.00116 1.960.00015
  (0.00086, 0.00145)

15
Comparing Two Incidence Densities

• Assume data from a cohort study

• We get two estimates for non- and exposed
  subjects
• ID0I0/PT0 and ID1I1/PT1.
• To compare them, we can look at
• Incidence rate difference IRD ID1 - ID0 .
• Incidence rate ratio IRR ID1 / ID0 .

16
Comparing Two Incidence Densities Example

• Postmenopausal Hormone and Coronary Heart Disease
  Cohort Study
• Stampfer et al., NEJM (1985).
• Involving female nurses

• ID1 30/54308.7 0.00055 ID0 60/51477.5
  0.00116
• IRD ID1 - ID0 -0.00061
• IRR ID1 / ID0 0.474

17
Comparing Two Incidence Densities Poisson Model
Method

• By using a Poisson model, standard error of IRD
  can be estimated by

• Standard error of ln IRR can be estimated by

• Thus, an approximate 95 CI for IRD is given by
• IRD 1.96SE(IRD).
• 99 CI IRD 2.58SE(IRD)

• Thus, an approximate 95 CI for IRR is given by
• exp ln IRR 1.96SE(ln IRR)
• 99 CI exp ln IRR 2.58SE(ln IRR)

18
Comparing Two Incidence Densities Example

• 95 CI for
• IRD -0.00061 1.960.00018 (-0.00096,
  -0.00025)
• ln IRR ln(0.474) 1.960.22 (-1.178, -0.315)
• IRR (e-1.178, e-0.315) (0.308, 0.729)
• Both CIs allow to reject the null hypothesis of
  no difference.

19
Comparing Two Incidence Densities Test-Based
Method

• 95 test-based CI for IRD can be computed as
• IRD 1.96 SE(IRD),
• where SE(IRD) IRD / ? and

• Similarly, SE(ln IRR) (ln IRR) / ?
• 95 test-based CI for ln IRR is
• ln IRR 1.96 (ln IRR) / ?
• Can be written as
• ( 1 1.96 / ? ) ln IRR
• 95 CI for IRR is thus
• exp ( 1 1.96 / ? ) ln IRR

• Can be re-expressed as
• (1 1.96 / ?) IRD
• 99 CI (1 2.58 / ?) IRD

20
Comparing Two Incidence Densities Example

• 95 test-based CI for
• IRD (1 1.96/3.41) (-0.00061) (-0.001,
  -0.0002)
• Close to the one based on the Poisson
  approximation (not in general).
• ln IRR (1 1.96/3.41) ln(0.474) (-1.176,
  -0.317)
• IRR (e-1.176, e-0.317) (0.309, 0.728)

21
Exact Confidence Interval for IRR

• The presented CIs for ln IRR (and IRD) assume
  that the estimates of ln IRR vary according to
  the normal distribution.
• Hence their form, e.g., ln(IRR) 1.96 SE(ln
  IRR).
• The use of the normal distribution is an
  approximation.
• Can be problematic, especially in small samples.
• It is possible to construct a CI for ln IRR using
  the exact distribution (i.e., without
  approximating it by the normal).
• The CI is valid in all samples in large samples,
  it is close to the approximate CIs.
• Computation is a bit more difficult (but easily
  handled by computers).

22
Standardized Rates

• We will introduce the standardization w.r.t. age.
  
• We will assume that our population is stratified
  by age (i.e., subdivided into age-groups).
• One needs to define age-groups (e.g., 0-4,
  5-9,).
• One needs to compute age-specific rates (ID).
• Population-time and no. of cases for each
  age-group are required.
• There are two methods of standardization
• Direct
• Indirect.

23
Standardization

• Direct method
• Age-specific rates of the study population are
  applied to the age-distribution of the standard
  population
• (rates study ? age standard)
• Theoretical rate that would have occurred if the
  rates observed in the study population applied to
  the standard population.
• Indirect method
• Age-specific rates from the standard population
  are applied to the age-distribution of the study
  population.
• (rates standard ? age study)

24
Direct Standardization

• Crude Rate in study population It / PTt .
• Directly Standardized Rate (DSR)
• DSR (I1/PT1)N1 (I2/PT2)N2 (I3/PT3)N3 /
  Nt
• (I1/PT1)(N1/Nt) (I2/PT2)(N2/Nt)
  (I3/PT3)(N3/Nt).
• Make sure units are consistent!!!

25
Direct Standardization

• If there is no confounding, crude rate is
  adequate.
• DSR by itself is not meaningful it makes sense
  only when comparing two or more populations.
• If possible, compare age-specific rates.
• The rates should exhibit more or less similar
  trends (also in the standard).
• DSR depends on the choice of the standard
  population.
• The age-distribution of the latter should not be
  radically different from the compared
  populations.
• There are several standard populations (e.g., for
  the world, continents etc.).

26
Indirect Standardization

• Direct standardization requires age-specific
  rates for all compared populations.
• If these are not available, or they are
  imprecise, the indirect method is preferred.
• Both should lead to similar conclusions if they
  do not, the reason should be investigated.

27
Indirect Standardization

• Standardized (Incidence or Mortality) Ratio (SIR
  or SMR)
• SIR It / ? Ej Observed / Expected .
• Take Indirectly Standardized Rate (ISR) as
• ISR SIR (crude rate for the standard
  population).
• Make sure units are consistent!!!

28
Standardization of Rates Example

• Infant deaths (for children less than 1 year of
  age) in Colorado and Louisiana in 1987.
• Colorado 527 deaths out of 53808 life births
  crude rate 9.8 per 1000.
• Louisiana 872 deaths out of 73967 life births
  crude rate 11.8 per 1000.
• Crude infant mortality rate for Colorado is lower
  than for Louisiana.
• In the US, infant mortality depends on race.

29
Standardization of Rates Example

• The distribution of race of new-born children is
  different in the two states.
• Infant mortality rates depend on race.
• Race is a confounder.
• Compare race-specific infant mortality rates.
• Unclear (differences in various directions).

30
Standardization of Rates Example

• Direct standardization apply state- and
  race-specific rates to the standard race
  distribution (US, 1987).

• DSR for Colorado 10.45 (per 1000 life births
  crude 9.8).
• DSR for Louisiana 9.35 (per 1000 life births
  crude 11.8).

31
Standardization of Rates Example

• Indirect standardization apply race-specific
  rates of a standard population (US, 1987) to the
  race-distribution of the states.

• SMR for Colorado 527/488.3 1.08 (8 higher
  than the US).
• ISR SMR x 10.1 10.9 (race-adjusted infant
  mortality-rate).
• SMR for Louisiana 872/908.8 0.96 (4 lower
  than the US).
• ISR SMR x 10.1 9.7 (race-adjusted infant
  mortality-rate).

32
Standardization of Rates Example

• Is it reasonable to use the adjusted rates?
• The plot of race-specific rates shows similar
  trend (blackgtwhitegtother).
• The distribution of race in the US is similar to
  the two states (whitegtblackgtother).
• Results for both standardization methods are
  similar.

33
Comparison of Directly Standardized Rates

• If we have two standardized rates, we may want to
  compare them.
• For the direct method, assume we have DSR1 and
  DSR2.
• 95 CI can then be obtained using the normal
  approximation
• (DSR1 - DSR2) 1.96 SE(DSR1 - DSR2) .
• 99 CI (DSR1 - DSR2) 2.58 SE(DSR1 - DSR2) .
• The standard error is given by

where IRDk is the stratum-specific intensity rate
difference.
34
Comparison of Directly Standardized Rates

• Alternatively, we might look at the standardized
  rate ratio
• SRRDSR1/DSR2.
• 95 CI for SRR can be written as SRR 1 (1.96 /
  Z), where

• 99 CI can be written as SRR 1 (2.58 / Z ).

35
Comparison of Directly Standardized Rates Example

• DSR1 (Colorado) 0.01045 (10.45 per 1000 life
  births).
• DSR2 (Louisiana) 0.00935 (9.35 per 1000 life
  births).

36
Comparison of Directly Standardized Rates Example

• DSR1 0.01045 DSR2 0.00935.
• DSR1 - DSR2 0.0011.
• 95 CI 0.0011 1.960.0006 (-0.0002, 0.002).
• CI includes 0 - we cannot reject H0 of no
  difference.
• SRR DSR1 / DSR2 1.12.
• Z (DSR1 - DSR2) / SE 1.83.
• 95 CI 1.12 1 (1.96 / 1.83) (0.99, 1.26).
• CI includes 1 - we cannot reject H0 of no
  difference.

37
Comparison of Indirectly Standardized Rates

• In directly standardized rates, stratum
  specific-rates for different study populations
  are combined using the same weights (relative
  stratum-sizes in the standard population).
• In indirectly standardized rates, the weights
  (PTi / expected Ii) differ.
• Thus, technically speaking, ISRs (SIRs) should
  not be compared.
• On the other hand, it is valid to ask whether SIR
  (or SMR) is different from 1.
• To do that, one can construct a 95 CI, e.g., as
  follows
• SIR 1.96(vobserved events)/(expected events).

38
Standardization of Rates

• Standardization is a simple way to remove effect
  of confounding.
• It can be extended to more than one confounder.
• Similar techniques can be used for differences or
  ratios of rates.
• An alternative is a stratified analysis (later).