Measures of disease frequency (II)

Calculation of incidenceStrategy #2

ANALYSIS BASED ON PERSON-TIME

CALCULATION OF PERSON-TIME AND INCIDENCE RATES

Example 1 Observe 1st graders, total 500 hours

Observe 12 accidents

Accident rate (or Accident density):

hour-personper0.024500

12R

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (12)

6 (3)

Follow-up time (years)

CALCULATION OF PERSON-TIME AND INCIDENCE RATES

Example 2

Person ID

No. of person-years in

Total FU1st FU year 2nd FU year

6

2

5

4

3

1

3/12=0.25

6/12=0.50

12/12=1.00

12/12=1.00

12/12=1.00

12/12=1.00

0

0

0

3/12=0.25

6/12=0.50

12/12=1.00

0.25

0.25

1.00

1.25

1.50

2.00

Total 4.75 1.75 6.50

Step 1: Calculate denominator, i.e. units of time contributed by each individual, and total:

Step 2: Calculate rate per person-year for the total follow-up period:

year-personper0.466.5

3R

It is also possible to calculate the incidence rates per person-years separately for shorter periods during the follow-up:

For year 1:

For year 2:

year-personper0.424.75

2R

year-personper0.571.75

1R

Person ID

No. of person-years in

Total FU1st FU year 2nd FU year

6

2

5

4

3

1

3/12=0.25

6/12=0.50

12/12=1.00

12/12=1.00

12/12=1.00

12/12=1.00

0

0

0

3/12=0.25

6/12=0.50

12/12=1.00

0.25

0.25

1.00

1.25

1.50

2.00

Total 4.75 1.75 6.50

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (12)

6 (3)

Follow-up time (years)

Notes:

• Rates have units (time-1). • Proportions (e.g., cumulative incidence) are unitless.• As velocity, rate is an instantaneous concept. The choice

of time unit used to express it is totally arbitrary. Depending on this choice, the value of the rate can range between 0 and .

E.g.:0.024 per person-hour = 0.576 per person-day

= 210.2 per person-year

0.46 per person-year = 4.6 per person-decade

Notes:

• Rates can be more than 1.0 (100%):– 1 person dies exactly after 6 months:

• No. of person-years: 1 x 0.5 years= 0.5 person-years

R ate per P Y per P Y s 10 5

2 0 2 0 0 1 0 0.

.

Confidence intervals and hypothesis testing Assume that the number of events follow a Poisson distribution (use next page’s table).

Example:

95% CL’s for accidental falls in 1st graders:

– For number of events: Lower= 120.517=6.2

Upper=121.750=21.0

– For rate: Lower= 6.2/500=0.0124/hr

Upper=21/500=0.042/hr

TABULATED VALUES OF 95% CONFIDENCE LIMIT FACTORSFOR A POISSON-DISTRIBUTED VARIABLE.*

Observednumber onwhich estimateis based

LowerLimitFactor

UpperLimitFactor

Observednumber onwhichestimate isbased

LowerLimitFactor

UpperLimitFactor

Observednumber onwhichestimate isbased

LowerLimitFactor

UpperLimitFactor

123456789

1011121314151617181920

.00253

.121

.206

.272

.324

.367

.401

.431

.458

.480

.499

.517

.532

.546

.560

.572

.583

.593

.602

.611

5.573.612.922.562.332.182.061.971.901.841.791.751.711.681.651.621.601.581.561.54

212223242526272829303540455060708090

100

.619

.627

.634

.641

.647

.653

.659

.665

.670

.675

.697

.714

.729

.742

.770

.785

.798

.809

.818

1.531.511.501.481.481.471.461.451.441.431.391.361.341.321.301.271.251.241.22

120140160180200250300350400450500600700800900

1000

.833

.844

.854

.862

.868

.882

.892

.899

.906

.911

.915

.922

.928

.932

.936

.939

1.2001.1841.1711.1601.1511.1341.1211.1121.1041.0981.0931.0841.0781.0721.0681.064

*Source: Haenszel W, Loveland DB, Sirken MG. Lung cancer mortality as related to residence andsmoking histories. I. White males. J Natl Cancer Inst 1962;28:947-1001.

Assigning person-time to time scale categories

• One time scale, e.g., age:

25 30 35 40 45 50Age

Number of person-years between 35-44 yrs of age: 30

Number of events between 35-44 yrs of age: 3

years-personofNumber

eventsofNumberrateIncidence 44yrs34

/py1.030

3

1980 1985 199081 82 83 84 86 87 88 89

4

3

2

1

Wom

en

When exact entry/event/withdrawal time is not known, it is usually assumed that the (average) contribution to the entry/exit period is half-the length of the period.

Example:

Women 1 Women 2 Women 3 Women 4

Date of surgeryAge at menopauseEventDate of event

198354

Death1989

198546

Loss1988

198047

Censored1990

198248

Death1984

1980 1985 199081 82 83 84 86 87 88 89

4

3

2

1

Wom

en

Calendar time Person-years Events Rate (/py)1980-841985-89

(1990-94)

812.5(0.5)

11

(0)

0.1250.080

(0)

Assigning person-time to time scale categories

• Two time scales (Lexis diagram)

Source: Breslow & Day, 1987.

Approximation: Incidence rate based on mid-point population

(usually reported as “yearly” average)

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (12)

6 (3)

Follow-up time (years)

Midpoint population

Midpoint population: estimated as the average population over the time period

Example:

5.32

162

end) at then (Populatio)population(Initialpopulation(midpoint)Average

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (12)

6 (3)

Follow-up time (years)

Midpoint population

This approach is used when rates are calculated from aggregate data(e.g., vital statistics)

years-2per 86.05.3

3rateyear-2

yearper 43.02

5.33

years ofNumber populationMidpoint

events ofNumber

rateYearly

Correspondence between individual-based and aggregate-based incidence rates

When withdrawals and events occur uniformly, average (midpoint)-rate per unit time (e.g., yearly rate) and rate per person-time (e.g., per person-year) tend to be the same.

Example: Calculation of mortality rate

12 persons followed for 3 years

Number of person-years of observation Person Follow-up

(Months) Year 1 Year 2 Year 3 Total

Outcome

1 2 3 4 5 6 7 8 9

10 11 12

3 6 9

12 15 18 21 24 27 30 33 36

3/12 6/12 9/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12

0 0 0 0

3/12 6/12 9/12

12/12 12/12 12/12 12/12 12/12

0 0 0 0 0 0 0 0

3/12 6/12 9/12 12/12

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

D D C D C C D C D C C D

Total 10.50 6.50 2.50 19.5

Number of person-years of observation Person Follow-up

(Months) Year 1 Year 2 Year 3 Total

Outcome

1 2 3 4 5 6 7 8 9

10 11 12

3 6 9

12 15 18 21 24 27 30 33 36

3/12 6/12 9/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12 12/12

0 0 0 0

3/12 6/12 9/12

12/12 12/12 12/12 12/12 12/12

0 0 0 0 0 0 0 0

3/12 6/12 9/12 12/12

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00

D D C D C C D C D C C D

Total 10.50 6.50 2.50 19.5

Based on individual data: /py308.019.5

6Rate

Based on midpoint population: yearper 308.036.5

6Rate

Note:

time-personper Ratetime-person Total

events ofNumber

years(t) ofNumber (n)populationMidpoint

events(x) ofNumber

rateYearly

tn

x

Person ID

0 1 2

4

1 (24)

2 (6)

3 (18)(15)

5 (12)

6 (3)

Follow-up time (years)

SUMMARY OF ESTIMATES

Method Estimate Value

Life-table

Kaplan-Meier

q (2 years) 0.60

0.64

Person-year

Midpoint pop’n

Rate (per year) 0.46/py

0.43 per year

CN

xq

21

x-CN

xRate

21

21

In actuarial life-table:

Use of person-time to account for changes in exposure status (Time-dependent exposures)

Example:Is menopause a risk factor for myocardial infarction?

123456

Number of PY in each group

ID 1 2 3 4 5 6 7 8 9 10No. PY

PRE menoNo. PY

POST meno

C

C

: Myocardial Infarction; C: censored observation.

Rates per person-year:Pre-menopausal = 1/17 = 0.06 (6 per 100 py)Post-menopausal = 2/18 = 0.11 (11 per 100 py)

Rate ratio = 0.11/0.06 = 1.85

3 40 56 00 15 53 317 18

Year of follow-up

Note: Event is assigned to exposure status when it occurs

PREVALENCE

Prevalence“The number of affected persons present at the population at a specific time divided by the number of persons in the population at that time”Gordis, 2000, p.33

Relation with incidence --- Usual formula:

Prevalence = Incidence x Duration* P = I x D

* Average duration (survival) after disease onset. It can be shown to be the inverse of case-fatality

ODDS

OddsThe ratio of the probabilities of an event to that of the non-event.

Prob1-

ProbOdds

Example: The probability of an event (e.g., death, disease, recovery, etc.) is 0.20, and thus the odds is:

That is, for every person with the event, there are 4 persons without the event.

0.25) (or 41:0.80

0.20

0.201-

0.20Odds

Notes about odds and probabilities:

• Either probabilities or odds may be used to express “frequency”

• Odds nearly equals probabilities when probability is small (e.g., <0.10). Example:

– Probability = 0.02

– Odds = 0.02/0.98 = 0.0204

• Odds can be calculated in relation to any kind of probability (e.g., prevalence, incidence, case-fatality, etc.).