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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.).