2. Calculation of incidence
Strategy #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
-
person
per
0.024
500
12
R
3. 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 FU
1st 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:
4. Step 2: Calculate rate per person-year for the total follow-up
period:
year
-
person
per
0.46
6.5
3
R
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
-
person
per
0.42
4.75
2
R
year
-
person
per
0.57
1.75
1
R
Person ID
No. of person-years in
Total FU
1st 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)
5. 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
6. 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
Rate per PY per PYs
1
05
2 0 200 100
.
.
7. 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
8. TABULATED VALUES OF 95% CONFIDENCE LIMIT FACTORS
FOR A POISSON-DISTRIBUTED VARIABLE.*
Observed
number on
which estimate
is based
Lower
Limit
Factor
Upper
Limit
Factor
Observed
number on
which
estimate is
based
Lower
Limit
Factor
Upper
Limit
Factor
Observed
number on
which
estimate is
based
Lower
Limit
Factor
Upper
Limit
Factor
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
.00253
.121
.206
.272
.324
.367
.401
.431
.458
.480
.499
.517
.532
.546
.560
.572
.583
.593
.602
.611
5.57
3.61
2.92
2.56
2.33
2.18
2.06
1.97
1.90
1.84
1.79
1.75
1.71
1.68
1.65
1.62
1.60
1.58
1.56
1.54
21
22
23
24
25
26
27
28
29
30
35
40
45
50
60
70
80
90
100
.619
.627
.634
.641
.647
.653
.659
.665
.670
.675
.697
.714
.729
.742
.770
.785
.798
.809
.818
1.53
1.51
1.50
1.48
1.48
1.47
1.46
1.45
1.44
1.43
1.39
1.36
1.34
1.32
1.30
1.27
1.25
1.24
1.22
120
140
160
180
200
250
300
350
400
450
500
600
700
800
900
1000
.833
.844
.854
.862
.868
.882
.892
.899
.906
.911
.915
.922
.928
.932
.936
.939
1.200
1.184
1.171
1.160
1.151
1.134
1.121
1.112
1.104
1.098
1.093
1.084
1.078
1.072
1.068
1.064
*Source: Haenszel W, Loveland DB, Sirken MG. Lung cancer mortality as related to residence and
smoking histories. I. White males. J Natl Cancer Inst 1962;28:947-1001.
9. Assigning person-time to
time scale categories
• One time scale, e.g., age:
25 30 35 40 45 50
Age
Number of person-years between 35-44 yrs of age: 30
Number of events between 35-44 yrs of age: 3
years
-
person
of
Number
events
of
Number
rate
Incidence 44yrs
34
/py
1
.
0
30
3
10. 1980 1985 1990
81 82 83 84 86 87 88 89
4
3
2
1
Women
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 surgery
Age at menopause
Event
Date of event
1983
54
Death
1989
1985
46
Loss
1988
1980
47
Censored
1990
1982
48
Death
1984
13. 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
.
3
2
1
6
2
end)
at the
n
(Populatio
)
population
(Initial
population
(midpoint)
Average
14. 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
-
2
per
86
.
0
5
.
3
3
rate
year
-
2
year
per
43
.
0
2
5
.
3
3
years
of
Number
population
Midpoint
events
of
Number
rate
Yearly
15. 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
16. 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: /py
308
.
0
19.5
6
Rate
Based on midpoint population: year
per
308
.
0
3
6.5
6
Rate
Note:
time
-
person
per
Rate
time
-
person
Total
events
of
Number
years(t)
of
Number
(n)
population
Midpoint
events(x)
of
Number
rate
Yearly
t
n
x
17. 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
C
N
x
q
2
1
x
-
C
N
x
Rate
2
1
2
1
In actuarial
life-table:
18. Use of person-time to account for changes in
exposure status (Time-dependent exposures)
Example:
Is menopause a risk factor for myocardial infarction?
1
2
3
4
5
6
Number of PY in each group
ID 1 2 3 4 5 6 7 8 9 10
No. PY
PRE meno
No. 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 4
0 5
6 0
0 1
5 5
3 3
17 18
Year of follow-up
Note: Event is assigned to exposure status when it occurs
20. 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
22. Odds
The ratio of the probabilities of an event to that of
the non-event.
Prob
1-
Prob
Odds
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
4
1:
0.80
0.20
0.20
1-
0.20
Odds
23. 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.).
