Rates 2007

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Estimating and Comparing

RatesIncidence Density

Incidence Rate Difference and RatioConfidence Intervals

Standardized Rates and Their Comparison

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A Definition

Kleinbaum, Kupper and Morgenstern, EpidemiologicResearch: Principles and Quantitative Methods (1982),p.97:

A true rate is a potential for change in one quantity per unitchange in another quantity, where the latter quantity isusually time. () Thus, a rate is not dimensionless andhas no finite upper bound i.e., theoretically, a rate can

approach infinity.

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Rates

A well-known example of rate is velocity, i.e., change ofdistance 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.

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Incidence/Mortality Rate

Kleinbaum, Kupper and Morgenstern, EpidemiologicResearch: Principles and Quantitative Methods (1982),p.100:

The incidence rate of disease occurrence is the instantaneouspotential for change in disease status (i.e., the occurrence of

new cases) per unit of time at time t, or the occurrence ofdisease 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.

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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.

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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 Twillthen be equal to 1-e-T.

Risk is sometimes called a cumulative incidence.

In a disease-free (at time 0) cohort ofNindividuals, 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.

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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):

timepopulationaccrued

)t,(tperiodcalendarin thecasesnewofno. 10==PT

IID

PTis expressed in person-years, person-days etc.

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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= 2.5+3.5++1.5 = 26.ID=5/26=0.192 per (person-) year

or

1.92 per 10 (person-)years.

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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 themid-period population (these are less accurate solutions).

One may face the problem of removing those not at risk (e.g., women forprostate cancer incidence).

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Population-Time Without IndividualData: Example

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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 andcan 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 firstcohort will be lower.

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Incidence Density: Remarks

If applied to the whole cohort/population, sometimes calledcrude rate.

However, sex, age, race etc. can have substantial influenceon 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.

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Confidence Interval for IncidenceDensity

By using a Poisson model, standard error ofID=I / PTcanbe estimated by:

( ) 2)SE(

PT

IID =

Thus, an approximate 95% CI forID is given by:

ID 1.96SE(ID).

99% CI: ID 2.58SE(ID) .

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Estimating Incidence Densities: Example

Postmenopausal Hormone and Coronary Heart Disease Cohort Study: Stampferet al., NEJM(1985). Involving female nurses:

Hormone use

105786.251477.554308.7Person-years

906030CHDTotalNoYes

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)

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Comparing Two Incidence Densities

Assume data from acohort study:PTPT0PT1Pop.-time

II0I1Cases

TotalUnexposedExposed

We get two estimates for non- and exposed subjects:ID0=I0/PT0 and ID1=I1/PT1.

To compare them, we can look at Incidence rate difference: IRD = ID1- ID0.

Incidence rate ratio: IRR= ID1/ID0.

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Comparing Two Incidence Densities:Example

Postmenopausal Hormone and Coronary Heart Disease Cohort Study: Stampferet al., NEJM(1985). Involving female nurses:

Hormone use

105786.251477.554308.7Person-years

906030CHD

TotalNoYes

ID1 = 30/54308.7 = 0.00055; ID0= 60/51477.5 = 0.00116

IRD = ID1- ID0= -0.00061

IRR= ID1/ID0= 0.474

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Comparing Two Incidence Densities:Poisson Model Method

PTPT0PT1Pop.-time

II0I1Cases

TotalUnexposedExposed By using a Poisson model,standard error ofIRD can beestimated by:

( ) ( ) 21

1

2

0

0

)SE( PT

I

PT

IIRD +=

10

11)SE(ln

IIIRR +=

Standard error of ln IRRcan beestimated by:

Thus, an approximate 95% CIforIRD is given by:

IRD 1.96SE(IRD). 99% CI: IRD 2.58SE(IRD)

Thus, an approximate 95% CI forIRRis given by:

exp{ ln IRR 1.96SE(ln IRR) }

99% CI: exp{ ln IRR 2.58SE(ln IRR) }

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00018.054308.7

30

51477.5

60)SE(

22=+=IRD

Hormone use

105786.251477.554308.7Person-years

906030CHD

TotalNoYes

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.

( )224.0

30

1

60

1lnSE =+=IRR

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Comparing Two Incidence Densities:Test-Based Method

PTPT0PT1Pop.-time

II0I1Cases

TotalUnexposedExposed 95% test-based CI forIRD canbe computed as

IRD 1.96 SE(IRD),

where SE(IRD)= IRD / and

Similarly, SE(ln IRR)= (ln IRR) /

95% test-based CI for ln IRRis

ln IRR 1.96 (ln IRR)/

Can be written as

( 1 1.96 / ) ln IRR

95% CI forIRRis thus

exp{ ( 1 1.96 / ) ln IRR}

2

10

1

1

PT

PTPTI

PT

PTII

=

Can be re-expressed as(1 1.96 / ) IRD

99% CI: (1 2.58 / ) IRD

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Hormone use

105786.251477.554308.7Person-years

906030CHDTotalNoYes

PTPT0PT1Pop.-time

II0I1Cases

TotalUnexposedExposed

2

10

11

PT

PTPTI

PTPTII

=

95% test-based CI forIRD: (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)

41.3

2.105786

5.514777.5430890

2.1057867.543089030

2

=

=

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Exact Confidence Interval forIRR

The presented CIs for ln IRR(and IRD) assume that theestimates of ln IRRvary 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 IRRusing the exactdistribution (i.e., without approximating it by the normal).

The CI is valid in all samples; in large samples, it is close to theapproximate CIs.

Computation is a bit more difficult (but easily handled by computers).

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Standardized Rates

We will introduce the standardization w.r.t. age.

We will assume that our population is stratifiedby 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.

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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 ratesobserved in the study population applied to the standard

population.

Indirect method

Age-specific rates from the standard population are applied to theage-distribution of the study population.

(rates standard age study)

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Direct Standardization

Crude Rate in study population = It/ PTt .

Directly Standardized Rate (DSR):DSR= { (I

1/PT

1)N

1+ (I

2/PT

2)N

2+ (I

3/PT

3)N

3} /Nt= (I

1/PT

1)(N

1/N

t) + (I

2/PT

2)(N

2/N

t) + (I

3/PT

3)(N

3/N

t).

Make sure units are consistent!!!

Study Population Standard Population(e.g., USA 1990)

AgeGroup

Observed Person-years Rate Observed Population Rate

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Direct Standardization

If there is no confounding, crude rate is adequate.

DSRby itself is not meaningful it makes sense only whencomparing two or more populations. If possible, compare age-specific rates.

The rates should exhibit more or less similar trends (also in thestandard).

DSRdepends 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.).

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Indirect Standardization

Direct standardization requires age-specific rates for allcompared populations.

If these are not available, or they are imprecise, theindirect method is preferred.

Both should lead to similar conclusions; if they do not, thereason should be investigated.

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Indirect Standardization

Standardized (Incidence or Mortality) Ratio (SIRorSMR):

SIR = It / Ej = Observed/ Expected.

Take Indirectly Standardized Rate (ISR) as:

ISR = SIR (crude rate for the standard population).

Make sure units are consistent!!!

Study Population Standard Population

(e.g., USA 1990)

Age

GroupObs Person-

yearsRate Obs Population Rate

Expected

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Standardization of Rates: Example

Infant deaths (for children less than 1 year of age) in Colorado andLouisiana 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.

USA, 1987Race LifeBirths

%LifeBirths

InfantDeaths

Rate(x1000)

Black 641567 16.8 11461 17.9

White 2992488 78.6 25810 8.6

Other 175339 4.6 1137 6.5

Total 3809394 100 38408 10.1

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The distribution of race of new-born children is different in thetwo states.

Infant mortality rates depend onrace.

Race is a confounder.

Compare race-specific infantmortality rates.

Unclear (differences in variousdirections).

Colorado LouisianaRace

Life Birth % Life Births %

Black 3166 5.9 29670 40.1

White 48805 90.7 42749 57.8

Other1837 3.4 1548 2.1

Total 53808 100 73967 100

Colorado LouisianaRace

Rate(x1000) Rate(x1000)

Black 16.4 17.7White 9.6 8.0Other 3.3 1.9

Total 9.8 11.8

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Direct standardization: apply state- and race-specific rates to thestandard race distribution (US, 1987).

DSRfor Colorado: 10.45 (per 1000 life births; crude: 9.8).

DSRfor Louisiana: 9.35 (per 1000 life births; crude: 11.8).

9.3535628.710.4539828.213809394Total

0.09333.11.90.15578.63.30.046175339Other

6.2823939.98.07.5428727.99.60.7862992488White

2.9811355.717.72.7610521.716.40.168641567Black

Rate*Ni/N

t

(x1000)

Rate*Ni

Rate

(x1000)

Rate*Ni/N

t

(x 1000)

Rate*Ni

Rate

(x1000)

Ni/ N

tN

i

(Births)

LouisianaColoradoUS, 1987Race

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Indirect standardization: apply race-specific rates of a standardpopulation (US, 1987) to the race-distribution of the states.

US Colorado LouisianaRace

Rate(x1000)

Life Births(PTi)

Deaths(Obs.)

Rate*PTi(Exp. Deaths)

Life Births(PTi)

Deaths(Obs.)

Rate*PTi(Exp. Deaths)

Black 17.9 3166 52 56.7 29670 525 531.1

White 8.6 48805 469 419.7 42749 344 367.6Other 6.5 1837 6 11.9 1548 3 10.1

Total 10.1 53808 527 488.3 73967 872 908.8

SMRfor Colorado: 527/488.3 = 1.08 (8% higher than the US). ISR= SMRx 10.1 = 10.9 (race-adjusted infant mortality-rate).

SMRfor Louisiana: 872/908.8 = 0.96 (4% lower than the US).

ISR= SMRx 10.1 = 9.7 (race-adjusted infant mortality-rate).

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Is it reasonable to use theadjusted rates?

The plot of race-specific ratesshows similar trend(black>white>other).

The distribution of race in the USis similar to the two states(white>black>other).

Results for both standardizationmethods are similar.

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Comparison of Directly StandardizedRates

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

( )

=

2

21 )SE(SE kt

k IRDN

NDSRDSR

where IRDk is the stratum-specific intensity rate difference.

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Comparison of Directly StandardizedRates

Alternatively, we might look at the standardized rate ratio:

SRR=DSR1/DSR2.

95% CI forSRRcan be written as: SRR1 (1.96 / Z), where

)SE( 21

21

DSRDSR

DSRDSRZ

=

99% CI can be written as: SRR1 (2.58 / Z ).

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Comparison of Directly StandardizedRates: Example

US Colorado LouisianaRace

%Births(Ni/Nt) Births(PTi) Deaths(Ii) Rate(IDi) Births(PTi) Deaths(Ii) Rate(IDi) IRDi(x1000) SE

i(x1000)

Black 16.8 3166 52 16.4 29670 525 17.7 -1.3 2.4White 78.6 48805 469 9.6 42749 344 8.0 1.6 0.6Other 4.6 1837 6 3.3 1548 3 1.9 1.4 1.7

Total 100 53808 527 9.8 73967 872 11.8

DSR1 (Colorado): 0.01045 (10.45 per 1000 life births).

DSR2 (Louisiana): 0.00935 (9.35 per 1000 life births).

( ) ( ) ( ) ( ) 0006.00017.0046.00006.0786.00024.0168.0SE 22221 =++=DSRDSR

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Comparison of Directly StandardizedRates: 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: 1.121 (1.96 / 1.83)

= (0.99, 1.26). -

( ) ( ) ( ) ( ) 0006.00017.0046.00006.0786.00024.0168.0SE 22221 =++=DSRDSR

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Comparison of Indirectly StandardizedRates

In directly standardized rates, stratum specific-rates for different studypopulations 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(orSMR) is different

from 1.

To do that, one can construct a 95% CI, e.g., as follows:

SIR 1.96(observed events)/(expected events).

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Standardization of Rates

Standardization is a simple way to remove effect ofconfounding.

It can be extended to more than one confounder.

Similar techniques can be used for differences or ratios ofrates.

An alternative is a stratifiedanalysis (later).

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Rates 2007