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Measures of disease
frequency (II)
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):
12
R
 0.024 per person - hour
500
CALCULATION OF PERSON-TIME AND INCIDENCE RATES
Example 2
Person ID
6
2
5
4
3
1
(3)
(6)
(12)
(15)
(18)
(24)
1
Follow-up time (years)
0
2
Step 1: Calculate denominator, i.e. units of time contributed by
each individual, and total:
No. of person-years in
Person ID
1st FU year
2nd FU year
Total FU
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
No. of person-years in
Person ID
6
2
5
4
3
1
(3)
(6)
(12)
(15)
(18)
(24)
0
1
2
Follow-up time (years)
Person ID
1st FU year
2nd FU year
Total FU
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 2: Calculate rate per person-year for the total follow-up
period:
3
R
 0.46 per person - year
6.5
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:
2
R
 0.42 per person - year
4.75
1
R
 0.57 per person - year
1.75
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
1
Rate   2.0 per PY  200 per 100 PYs
0.5
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= 120.517=6.2
Upper= 121.750=21.0
– For rate:
Lower= 6.2/500=0.0124/hr
Upper= 21/500=0.042/hr
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
21
22
23
24
25
26
27
28
29
30
35
40
45
50
60
70
80
90
100
Lower
Limit
Factor
Upper
Limit
Factor
Observed
number on
which
estimate is
based
120
140
160
180
200
250
300
350
400
450
500
600
700
800
900
1000
Lower
Limit
Factor
1
.00253
5.57
.619
1.53
.833
2
.121
3.61
.627
1.51
.844
3
.206
2.92
.634
1.50
.854
4
.272
2.56
.641
1.48
.862
5
.324
2.33
.647
1.48
.868
6
.367
2.18
.653
1.47
.882
7
.401
2.06
.659
1.46
.892
8
.431
1.97
.665
1.45
.899
9
.458
1.90
.670
1.44
.906
10
.480
1.84
.675
1.43
.911
11
.499
1.79
.697
1.39
.915
12
.517
1.75
.714
1.36
.922
13
.532
1.71
.729
1.34
.928
14
.546
1.68
.742
1.32
.932
15
.560
1.65
.770
1.30
.936
16
.572
1.62
.785
1.27
.939
17
.583
1.60
.798
1.25
18
.593
1.58
.809
1.24
19
.602
1.56
.818
1.22
20
.611
1.54
*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.
Upper
Limit
Factor
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
Assigning person-time to
time scale categories
• One time scale, e.g., age:
Age 25
30
35
40
45
Number of person-years between 35-44 yrs of age:
Number of events between 35-44 yrs of age:
Incidencerate3444yrs
Number of events


Number of person- years
50
30
3
3
 0.1 /py
30
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
1983
1985
1980
1982
Age at menopause
54
46
47
48
Event
Death
Loss
Censored
Death
Date of event
1989
1988
1990
1984
1980 81 82 83 84 1985 86 87 88 89 1990
Women
1
2
3
4
1980 81 82 83 84 1985 86 87 88 89 1990
Women
1
2
3
4
Calendar time Person-years
1980-84
8
1985-89
12.5
(1990-94)
(0.5)
Events
1
1
(0)
Rate (/py)
0.125
0.080
(0)
Assigning person-time to
time scale categories
• Two time scales (Lexis diagram)
Source: Breslow & Day, 1987.
Approximation: Incidence rate based on midpoint population
(usually reported as “yearly” average)
Midpoint population: estimated as the average population over the
time period
Example:
Midpoint
population
Person ID
6
(3)
(6)
2
5
4
3
1
(12)
(15)
(18)
(24)
0
1
Follow-up time (years)
2
(Initialpopulation)  (P opulation at theend)
2
6 1

 3.5
2
Average(midpoint)population
Midpoint
population
Person ID
6
(3)
(6)
2
5
(12)
(15)
4
3
1
(18)
(24)
0
1
Follow-up time (years)
2 - year rate 
2
3
 0.86 per 2 - years
3.5
Number of events
3
Midpointpopulation
Yearly rate 
 3.5  0.43 per year
Number of years
2
This approach is used when rates are calculated from aggregate data
(e.g., vital statistics)
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
Person
Followup
(Months)
1
3
2
6
3
9
4
12
5
15
6
18
7
21
8
24
9
27
10
30
11
33
12
36
Total
Number of person-years of observation
Year 1
Year 2
Year 3
Total
3/12
6/12
9/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
10.50
0
0
0
0
3/12
6/12
9/12
12/12
12/12
12/12
12/12
12/12
6.50
0
0
0
0
0
0
0
0
3/12
6/12
9/12
12/12
2.50
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
19.5
Outcome
D
D
C
D
C
C
D
C
D
C
C
D
Person
Followup
(Months)
1
3
2
6
3
9
4
12
5
15
6
18
7
21
8
24
9
27
10
30
11
33
12
36
Total
Number of person-years of observation
Year 1
Year 2
Year 3
Total
3/12
6/12
9/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
12/12
10.50
0
0
0
0
3/12
6/12
9/12
12/12
12/12
12/12
12/12
12/12
6.50
0
0
0
0
0
0
0
0
3/12
6/12
9/12
12/12
2.50
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
3.00
19.5
Based on individual data:
Outcome
D
D
C
D
C
C
D
C
D
C
C
D
6
Rate 
 0.308 /py
19.5
6
Based on midpoint population: Rate  6.5  0.308per year
3
Note:
Number of events(x)
x
Number of events
Midpointpopulation(n)
Yearly rate 


 Rate per person- time
Number of years(t)
n  t T otalperson- time
SUMMARY OF ESTIMATES
Person ID
6
2
5
4
3
1
(3)
(6)
(12)
(15)
(18)
(24)
0
Method
In actuarial
life-table:
1
Follow-up time (years)
2
Estimate
Value
Life-table
Kaplan-Meier
q (2 years)
0.60
0.64
Person-year
Midpoint pop’n
Rate (per year)
x
q
N  12 C
0.46/py
0.43 per year
x
Rate 
N  12 C - 12 x
Use of person-time to account for changes in
exposure status (Time-dependent exposures)
Example:
Is menopause a risk factor for myocardial infarction?
ID 1
1
2
3
4
5
6
2
Year of follow-up
3 4 5 6 7

8
9 10
C

C

Number of PY in each group
No. PY
PRE meno
No. PY
POST meno
3
0
6
0
5
3
17
4
5
0
1
5
3
18
: Myocardial Infarction; C: censored observation.
Note: Event is assigned to exposure status when it occurs
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
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=IxD
* Average duration (survival) after disease onset. It can be shown to be the
inverse of case-fatality
ODDS
Odds
The ratio of the probabilities of an event to that of
the non-event.
Prob
Odds 
1- Prob
Example: The probability of an event (e.g., death, disease,
recovery, etc.) is 0.20, and thus the odds is:
Odds 
0.20
0.20

 1: 4 (or 0.25)
1- 0.20 0.80
That is, for every person with the event, there
are 4 persons without the event.
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.).