The life table
• LT statistics: rates, probabilities, life
expectancy (waiting time to event)
• Period life table
• Cohort life table
Life table from
observational data
22 respondents
Namboodiri and Suchindran, 1987, Chapter 4
A. Exact time
Table 1
Hypothetical survey data for construction of life table (1986)
CASE ID
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Date
of entry
in observation
Jan-02
Jan-17
Jan-18
Jan-22
Feb-10
Jan-30
Apr-04
Apr-29
May-18
May-20
May-15
Feb-05
Feb-05
Feb-06
Feb-26
Mar-10
Mar-11
Mar-28
Mar-15
Apr-13
Apr-04
Apr-25
- indicates censoring
Date
Date
of event of interview
Feb-11
May-04
Feb-28
May-17
Feb-12
Feb-25
Apr-18
May-18
May-08
Mar-23
May-09
May-16
May-25
May-17
May-10
May-13
May-23
May-15
May-06
May-27
May-29
May-31
May-18
May-19
May-10
May-28
May-22
May-25
May-12
May-29
May-10
May-20
May-11
May-31
Duration
Duration
Duration
to event to interview Exposure
Days
Days
Days
40
143
40
107
120
107
112
112
37
111
37
96
102
96
13
105
13
32
32
28
28
11
11
11
11
3
3
20
103
20
72
94
72
101
111
101
85
85
76
76
58
62
58
62
62
8
56
8
37
37
35
37
35
12
36
12
Table 2
Estimation of survival function
CASE ID
11
19
9
10
22
6
12
8
7
21
4
20
1
17
18
13
16
15
5
14
2
3
Date
of entry
in observation
May-15
Mar-15
May-18
May-20
Apr-25
Jan-30
Feb-05
Apr-29
Apr-04
Apr-04
Jan-22
Apr-13
Jan-02
Mar-11
Mar-28
Feb-05
Mar-10
Feb-26
Feb-10
Feb-06
Jan-17
Jan-18
Date
Date
of event of interview
Mar-23
May-16
Feb-12
Feb-25
May-09
Feb-28
Feb-11
May-08
Apr-18
May-17
May-18
May-04
-
May-18
May-10
May-29
May-31
May-31
May-15
May-19
May-27
May-06
May-11
May-13
May-20
May-25
May-12
May-29
May-10
May-25
May-22
May-23
May-28
May-17
May-10
Duration
Duration
Duration
At tisk
Event/
to event to interview Exposure at beginning censoring
Days
Days
Days
of day*
8
12
13
20
35
37
40
58
72
96
101
107
-
3
56
11
11
36
105
103
28
32
37
111
37
143
62
62
94
76
85
102
111
120
112
USE EXCEL DATA/SORT
Number of persons at risk decreases over time because of occurrence of event or censoring
ASS: population is homogeneous and censoring is independent of event of interest
* Assumes censoring NOT at beginning of interval (day)
3
8
11
11
12
13
20
28
32
35
37
37
40
58
62
72
76
85
96
101
107
112
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
0
0
1
1
1
0
0
1
1
0
1
1
0
1
0
0
1
1
1
0
Table 2
Estimation of survival function, cont
1/21
Day
At tisk
at beginning
of day*
Event Censored
3
8
11
11
12
13
20
28
32
35
37
37
40
58
62
72
76
85
96
101
107
112
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
1
0
0
1
1
1
0
0
1
1
0
1
1
0
1
0
0
1
1
1
0
1
0
1
1
0
0
0
1
1
0
0
1
0
0
1
0
1
1
0
0
0
1
Total
12
10
Prob
of event
0.0476
Observed
survival
proportions
0.9524
0.0556
0.0588
0.0625
0.9444
0.9412
0.9375
0.0769
0.0833
0.9231
0.9167
0.1000
0.1111
0.9000
0.8889
0.1429
0.8571
0.2500
0.3333
0.5000
0.7500
0.6667
0.5000
Survival
function
1
0.9524
0.9524
0.9524
0.8995
0.8466
0.7937
0.7937
0.7937
0.7326
0.6716
0.6716
0.6044
0.5372
0.5372
0.4605
0.4605
0.4605
0.3454
0.2302
0.1151
0.1151
0.9524 = 1-1/21
=1 - failure/number at risk
before failure
0.8995 = 0.9524*[1-1/18]
Kaplan-Meier
Discrete time interval (30 days)
B. Discrete time interval: completed months
Duration
Month
0
1
2
3
Duration
Days
Events
Censoring
4
3
1
1
4
2
3
1
0-30
31-59
60-89
90-119
Event+censoring in same month:
Events+
censoring
0
1
0
2
case 21 (event on May 9 and interview on May 11)
Case 5 (event on May 17 and interview on May 23)
Case 14 (event on May 18 and interview on May 28)
A. Censoring at beginning of month
Duration
Risk set
Events
Censored
in months
0
18
4
4
1
11
3
3
2
5
1
3
3
1
1
3
4
NOTE: Case 21 is censored instead of event
NOTE: 11 = 18 - 4- 3
Prob of
event
0.2222
0.2727
0.2000
1.0000
Prob
surviving
0.7778
0.7273
0.8000
0.0000
Survival function
S intermed
1.0000
0.0000
0.7778
0.0123
0.5657
0.0319
0.4525
0.0707
0.0000
0.0000
s.e.
0.0000
0.0864
0.1190
0.1534
0.0000
B. Censoring at end of month
Duration Risk set
Events Censored Prob of
in months
event
0
22
4
4
0.1818
1
14
4
2
0.2857
2
8
1
3
0.1250
3
4
3
1
0.7500
4
NOTE: Case 21 experiences event and is not censored
Prob
surviving
0.8182
0.7143
0.8750
0.2500
Survival function
S intermed
1.0000
0.0000
0.8182
0.0083
0.5844
0.0249
0.5114
0.0267
0.1278
0.0774
s.e.
0.0000
0.4236
0.2979
0.2478
0.0583
C. Censoring in middle of month (half at beginning and half at end of month)
Duration Risk set
Events Censored Prob of
in months
event
0
20
4
4
0.2000
1
13
4
2
0.3077
2
6.5
1
3
0.1538
3
3.5
3
1
0.8571
4
NOTE: Case 21 experiences event and is not censored
Prob
surviving
0.8000
0.6923
0.8462
0.1429
Survival function
S intermed
1.0000
0.0000
0.8000
0.0100
0.5538
0.0296
0.4686
0.0427
0.0669
0.0484
s.e.
0.0000
0.2892
0.1924
0.1415
0.0147
Survival probability
Survival function
1.00
0.80
Censoring begin
0.60
censoring end
0.40
Censoring middle
0.20
0.00
0
1
2
Duration (months)
3
4
Survival probability
Survival function and 95% interval
1.00
0.80
Censoring middle
0.60
S-1.96*s.e.
0.40
S+1.96*s.e.
0.20
0.00
1
2
3
4
Duration (months)
5
Standard error of survival function: Greenwood formula
 i  t  1 qi 
SE ( S )  S  

p
i

1
i Ri 

t
1/ 2
t
With St = survival function
Rt = risk set
qt = probability of event
pt = survival probability (probability of NO event)
Assume 100 respondents (R = risk set)
Probability of event (q): 0.10 => survival probability (p): 0.90
Var(p) = pq/R = 0.9*0.1/100 = 0.0009 SQRT(0.0009) = 0.03
p2q/(pR) = 0.81*0.1/(0.9*100) = 0.0009
SQRT(0.0009) = 0.03
15 survivors experience event. Prob of event is 15/90 = 0.1667
Risk set: 90 (= 100*0.9)
Prob of surviving second interval: S2 = 0.9*0.833=0.75
Var(S2) = 0.752 * [0.1/(0.9*100) + 0.1667/(0.8333*90)]
= 0.00188
Standard error of (hazard) rate:
1/ 2
SE (r t ) 
2

r 1 r t  
q R   2  
t
t
t
With rt = hazard rate
Life table with grouped data
Remarriage of divorced women, aged 25 to 34, USA, 1975
Source: 1975 US Current Population Survey
Namboodiri and Suchindran, 1978, pp. 63 ff
Remarriage life table, Unites States 1975
Number Number of
Divorced
Interval
divorced remarriages at interview
at beginning
(censored)
(n')
(d)
(c)
(1)
(2)
(3)
(4)
[0,1)
1298
238
101
[1,2)
959
177
65
[2,3)
717
107
51
[3,4)
559
64
42
[4.5)
453
50
34
[5,6)
369
34
18
[6,7)
317
27
21
[7,8)
269
21
11
[8,9)
237
14
4
[9,10)
219
15
11
[10,11)
193
16
14
[11,12)
163
8
8
[12,13)
147
6
8
[13,14)
133
6
7
[14,15)
120
5
5
[15,16)
110
2
10
[16,17)
98
5
5
[17,18)
88
5
2
[18,19)
81
2
2
[19,20)
77
2
9
[20,+)
66
12
54
Total
816
482
Probability
Survival
Risk set
of event probability
[3/5]
[1-3/5]
(n)
(q)
(p)
(5)
(6)
(7)
1247.5
0.1908
0.8092
926.5
0.1910
0.8090
691.5
0.1547
0.8453
538.0
0.1190
0.8810
436.0
0.1147
0.8853
360.0
0.0944
0.9056
306.5
0.0881
0.9119
263.5
0.0797
0.9203
235.0
0.0596
0.9404
213.5
0.0703
0.9297
186.0
0.0860
0.9140
159.0
0.0503
0.9497
143.0
0.0420
0.9580
129.5
0.0463
0.9537
117.5
0.0426
0.9574
105.0
0.0190
0.9810
95.5
0.0524
0.9476
87.0
0.0575
0.9425
80.0
0.0250
0.9750
72.5
0.0276
0.9724
39.0
0.3077
0.6923
Survival
function
(S)
(8)
1.0000
0.8092
0.6546
0.5533
0.4875
0.4316
0.3908
0.3564
0.3280
0.3085
0.2868
0.2621
0.2489
0.2385
0.2274
0.2178
0.2136
0.2024
0.1908
0.1860
0.1809
[q/pR]
SE(S)
[6/(7*5)] Greenwood
(9)
0.0000
0.0002
0.0003
0.0003
0.0003
0.0003
0.0003
0.0003
0.0003
0.0003
0.0004
0.0005
0.0003
0.0003
0.0004
0.0004
0.0002
0.0006
0.0007
0.0003
0.0004
Risk set: assuming censoring in MIDDLE of interval
[ T,T+1) Greater or equal to T, but less than T+1
Greenwood formula: see also Blossfeld and Rohwer, 1995, p. 53
SE(rate): see Blossfeld and Rohwer, 1995, p. 54
Conditional density varies with duration
(10)
0.0000
0.0111
0.0138
0.0147
0.0151
0.0153
0.0154
0.0154
0.0154
0.0153
0.0152
0.0151
0.0150
0.0150
0.0150
0.0149
0.0149
0.0150
0.0150
0.0150
0.0150
h(t)
h(t)
Linear Exponential
(11)
0.2109
0.2112
0.1677
0.1265
0.1217
0.0991
0.0922
0.0830
0.0614
0.0728
0.0899
0.0516
0.0429
0.0474
0.0435
0.0192
0.0538
0.0592
0.0253
0.0280
-
(12)
0.2117
0.2120
0.1681
0.1267
0.1218
0.0992
0.0922
0.0831
0.0614
0.0728
0.0899
0.0516
0.0429
0.0474
0.0435
0.0192
0.0538
0.0592
0.0253
0.0280
-
SE[h(t)]
(13)
0.0136
0.0158
0.0162
0.0158
0.0172
0.0170
0.0177
0.0181
0.0164
0.0188
0.0224
0.0182
0.0175
0.0194
0.0194
0.0136
0.0240
0.0265
0.0179
0.0198
-
0.2109 = 238/(0.5*(1298+959))
0.2117 = -ln(0.8092)
1.0000
0.2500
0.8000
0.2000
0.6000
0.1500
0.4000
0.1000
0.2000
0.0500
0.0000
0.0000
0
2
4
6
8
10 12 14 16
Duration since divorce
18 20
Rate
Probability
Estimated survival function and hazard function
S(t)
h(t)
Leaving parental home,
The Netherlands, Birth cohort 1961
Survey Sept. 87 - Febr. 88
Source: W. Dijkstra ed. (1989) Het proces van sociale integratie van jong-volwassenen:
De gegevensverzameling voor de eerste hoofdmeting (The process of social integration of
young adults. Data collection for first measurement). VU Uitgeverij, Amsterdam.
Jong Gierveld, J. de, Liefbroer, A.C. & Beekink, E. (1991a). The effect
of parental resources on patterns of leaving home among young adults
in the Netherlands. European Sociological Review, 7, 55-71.
Thanks to Jenny Gierveld and Aat liefbroer, NIDI, The Hague, for providing the data.
Data
Leaving parental home, 1961 cohort, micro-data, 583 respondents (first 30 respondents)
# Sex Father Month Reason
# Sex Father Month Reason
1
2
2
268
2
16
1
2
251
3
2
1
3
268
2
17
2
2
212
1
3
1
2
202
1
18
2
2
320
2
4
2
2
320
4
19
1
2
221
3
5
1
1
237
1
20
2
2
322
4
6
1
1
295
2
21
2
1
221
2
7
1
1
272
2
22
2
3
308
2
8
2
1
231
1
23
1
2
233
1
9
2
1
312
3
24
1
1
273
2
10
1
2
289
2
25
1
1
208
1
11
1
1
316
2
26
1
2
219
1
12
2
1
321
4
27
1
1
261
2
13
2
1
260
1
28
2
2
270
3
14
2
2
281
2
29
2
2
277
2
15
2
1
273
2
30
1
1
290
3
Sex
1 Female
2 Male
Father status
1 Low
2 Middle
3 High
Reason
1
2
3
4
Educ/work
Marriage/cohabit
Freedom
Censored
at interview
Data
Females, including early censoring
Females, excluding early censoring
Living
Attrition
Exposure
Living
Attrition
at home
Total Censored
Leave (months) at home
Total Censored
Leave
15
291
4
0
4
3480
288
4
0
4
16
287
7
0
7
3412
284
7
0
7
17
280
23
0
23
3279
277
23
0
23
18
257
53
0
53
2761
254
53
0
53
19
204
48
0
48
2181
201
48
0
48
20
156
41
1
40
1652
153
40
0
40
21
115
31
0
31
1223
113
31
0
31
22
84
26
0
26
855
82
26
0
26
23
58
23
0
23
596
56
23
0
23
24
35
16
1
15
325
33
15
0
15
25
19
8
2
6
183
18
7
1
6
26
11
11
9
2
42
11
11
9
2
27
0
0
0
0
0
0
291
13
278
19989
288
10
278
Early censoring: Person born at the end of December 1961 and interviewed in September 1987 reaches 25
in December 1986 and is 25 + 9 months = 309 months early September 1987
There are females censored in months 244, 297 and 301. IMPOSSIBLE (see HOME.XLS)
Age
Age
15
16
17
18
19
20
21
22
23
24
25
26
27
Males, including early censoring
Living
Attrition
Exposure
at home
Total Censored
Leave (months)
292
2
0
2
3495
290
5
0
5
3457
285
15
0
15
3347
270
24
1
23
3112
246
30
1
29
2791
216
35
2
33
2408
181
28
0
28
2043
153
33
0
33
1643
120
30
0
30
1283
90
30
0
30
915
60
25
5
20
626
35
35
31
4
196
0
0
0
0
0
292
40
252
25316
The first male censored is in month 310. Hence no early censoring.
d:\s\teach\98\lessn\lt_obs\jenny
Age
At home
Column 1
15
16
17
18
19
20
21
22
23
24
25
26
27
2
291
287
280
257
204
156
115
84
58
35
19
11
0
Females
Leave Censored
3
4
7
23
53
48
40
31
26
23
15
6
2
0
278
4
0
0
0
0
0
1
0
0
0
1
2
9
0
13
Risk set Exposure
(months)
5
291.0
287.0
280.0
257.0
204.0
155.5
115.0
84.0
58.0
34.5
18.0
6.5
0.0
6
3480
3412
3279
2761
2181
1652
1223
855
596
325
183
42
0
19989
Probab
leaving
(3/5)
7
0.0137
0.0244
0.0821
0.2062
0.2353
0.2572
0.2696
0.3095
0.3966
0.4348
0.3333
0.3077
Probab
surviving
1-(7)
8
0.9863
0.9756
0.9179
0.7938
0.7647
0.7428
0.7304
0.6905
0.6034
0.5652
0.6667
0.6923
Survival
q/(p*R)
function (9/(10*5)
9
1.0000
0.9863
0.9622
0.8832
0.7010
0.5361
0.3982
0.2908
0.2008
0.1212
0.0685
0.0457
0.0316
0.0000
0.0001
0.0003
0.0010
0.0015
0.0022
0.0032
0.0053
0.0113
0.0223
0.0278
0.0684
SE(S(t))
10
0.0000
0.0069
0.0115
0.0205
0.0338
0.0382
0.0387
0.0365
0.0341
0.0318
0.0264
0.0188
0.0173
Risk set = number initially at home - 0.5*censoring during year. Assume UNIFORM distribution
Exposure in YEARS = risk set - 0.5*number leaving home during year. Assume: UNIFORM distribution
d:\s\teach\98\lessn\lt_obs\jenny
Probability of living at home, Females, (+95% interval)
Probability
1.0000
0.8000
0.6000
0.4000
0.2000
0.0000
15
16
17
18
19
20
21
22
23
24
25
26
27
Age
d:\s\teach\98\lessn\lt_obs\jenny
Age
At home
Leave
Column 1
15
16
17
18
19
20
21
22
23
24
25
26
27
2
291
287
280
257
204
156
115
84
58
35
19
11
0
3
4
7
23
53
48
40
31
26
23
15
6
2
0
278
Risk set Exposure
in years
5
291.0
287.0
280.0
257.0
204.0
155.5
115.0
84.0
58.0
34.5
18.0
6.5
0.0
289.0
283.5
268.5
230.5
180.0
135.5
99.5
71.0
46.5
27.0
15.0
5.5
0.0
1651.5
Event
Rate
SE(rate)
11
0.0138
0.0247
0.0857
0.2299
0.2667
0.2952
0.3116
0.3662
0.4946
0.5556
0.4000
0.3636
12
0.0069
0.0093
0.0178
0.0314
0.0381
0.0462
0.0553
0.0706
0.0999
0.1378
0.1600
0.2528
Waiting Expected
time
waiting
(T) time (yrs)
13
14
1651.5
5.68
1362.5
4.75
1079.0
3.85
810.5
3.15
580.0
2.84
400.0
2.57
264.5
2.30
165.0
1.96
94.0
1.62
47.5
1.38
20.5
1.14
5.5
0.85
d:\s\teach\98\lessn\lt_obs\jenny
Leaving home: occurrences and exposures
A. Events and exposure including censored observations
Sex
Event count
Females
Males
Timing
Early (LT 20)
135
74
Late (GE 20)
143
178
Censored at interview
13
40
Total
291
292
Timing
16202
9114
25316
31315
13990
45305
B. Events and exposure excluding censored observations
Sex
Event count
Females
Males
Timing
Early (LT 20)
135
74
Late (GE 20)
143
178
Total
278
252
Total
209
321
530
Timing
Early (LT 20)
Late (GE 20)
Total
Exposure
15113
4876
19989
Total
209
321
53
583
Early (LT 20)
Late (GE 20)
Total
Exposure
14333
3998
18331
13826
6339
20165
28159
10337
38496
d:\s\teach\98\lessn\lt_obs\jenny
d:\s\teach\99pisa\leaveh\expos1.xls
SPSS output
Intrvl
Start
Time
-----15
16
17
18
19
20
21
22
23
24
25
26
SPSS SURVIVAL
Females
Number Number Number
Entrng Wdrawn
Exposd
this
During
to
Intrvl
Intrvl
Risk
---------------291
0
291.0
287
0
287.0
280
0
280.0
257
0
257.0
204
0
204.0
156
1
155.5
115
0
115.0
84
0
84.0
58
0
58.0
35
1
34.5
19
2
18.0
11
9
6.5
583 observations
Number
of
Termnl
Events
-----4
7
23
53
48
40
31
26
23
15
6
2
Propn
Terminating
-----0.0137
0.0244
0.0821
0.2062
0.2353
0.2572
0.2696
0.3095
0.3966
0.4348
0.3333
0.3077
Propn
Surviving
-----0.9863
0.9756
0.9179
0.7938
0.7647
0.7428
0.7304
0.6905
0.6034
0.5652
0.6667
0.6923
Cumul
Propn
Surv
at End
-----0.9863
0.9622
0.8832
0.7010
0.5361
0.3982
0.2908
0.2008
0.1212
0.0685
0.0457
0.0316
Probability
Densty
-----0.0137
0.0241
0.0790
0.1821
0.1649
0.1379
0.1073
0.0900
0.0796
0.0527
0.0228
0.0141
Hazard
Rate
-----0.0138
0.0247
0.0857
0.2299
0.2667
0.2952
0.3116
0.3662
0.4946
0.5556
0.4000
0.3636
The median survival time for these data is 20.26
d:\s\teach\98\lt_obs\jenny\spss.xls
d:\s\teach\98\lt_obs\jenny\lt.doc
SPSS output
Intrvl
Start
Time
-----15
16
17
18
19
20
21
22
23
24
25
26
SPSS SURVIVAL
Males
Number Number Number
Entrng Wdrawn
Exposd
this
During
to
Intrvl
Intrvl
Risk
---------------292
0
292.0
290
0.0
290.0
285
0.0
285.0
270
1.0
269.5
246
1.0
245.5
216
2.0
215.0
181
0.0
181.0
153
0.0
153.0
120
0.0
120.0
90
0.0
90.0
60
5.0
57.5
35
31.0
19.5
583 observations
Number
of
Termnl
Events
-----2
5
15
23
29
33
28
33
30
30
20
4
Propn
Terminating
-----0.0068
0.0172
0.0526
0.0853
0.1181
0.1535
0.1547
0.2157
0.2500
0.3333
0.3478
0.2051
Propn
Surviving
-----0.9932
0.9828
0.9474
0.9147
0.8819
0.8465
0.8453
0.7843
0.7500
0.6667
0.6522
0.7949
The median survival time for these data is 22.29
Cumul
Propn
Surv
at End
-----0.9932
0.9760
0.9247
0.8457
0.7458
0.6314
0.5337
0.4186
0.3139
0.2093
0.1365
0.1085
Probability
Densty
-----0.0068
0.0171
0.0514
0.0789
0.0999
0.1145
0.0977
0.1151
0.1046
0.1046
0.0728
0.0280
Hazard
Rate
-----0.0069
0.0174
0.0541
0.0891
0.1255
0.1662
0.1677
0.2418
0.2857
0.4000
0.4211
0.2286
The Kaplan-Meier estimator
• A method for the nonparametric estimation
of the survival function (1958)
• Also called product-limit estimator
• The risk set is calculated at every point in
time where at least one event occurred.
• Hence all episodes must be sorted according to their ending
times.
• It is a staircase function with
a. Location of drop is random (time at event)
b. Size of drop depends on censoring
Kaplan-Meier estimator
Time from diagnosis to death from melanoma
or loss to follow-up, 50 subjects (Clayton and Hills, 1993, p. 36)
Conditional probability
Survival Aalen-Nelson
Month
N
Death Censored of death of survival function
(Cum. Rate)
1.0000
0
50
2
0.0400
0.9600
0.9600
0.0400
1
48
1
0.0208
0.9792
0.9400
0.0608
2
47
2
0.0426
0.9574
0.9000
0.1034
3
45
1
1
0.0222
0.9778
0.8800
0.1256
8
43
1
0.0233
0.9767
0.8595
0.1489
10
42
1
0.0238
0.9762
0.8391
0.1727
12
41
1
1
0.0244
0.9756
0.8186
0.1971
13
39
1
0.0256
0.9744
0.7976
0.2227
15
38
1
0.0263
0.9737
0.7766
0.2490
18
37
1
0.0000
1.0000
0.7766
0.2490
19
36
1
0.0278
0.9722
0.7551
0.2768
21
35
1
0.0000
1.0000
0.7551
0.2768
27
34
2
0.0000
1.0000
0.7551
0.2768
30
32
1
0.0000
1.0000
0.7551
0.2768
33
31
1
1
0.0323
0.9677
0.7307
0.3091
34
29
1
0.0345
0.9655
0.7055
0.3435
38
28
1
0.0000
1.0000
0.7055
0.3435
40
27
1
0.0000
1.0000
0.7055
0.3435
41
26
1
0.0385
0.9615
0.6784
0.3820
43
25
1
0.0000
1.0000
0.6784
0.3820
44
24
1
0.0000
1.0000
0.6784
0.3820
46
23
1
0.0000
1.0000
0.6784
0.3820
54
22
1
0.0000
1.0000
0.6784
0.3820
55
21
1
0.0476
0.9524
0.6461
0.4296
56
20
1
0.0500
0.9500
0.6138
0.4796
57
19
2
0.0000
1.0000
0.6138
0.4796
60
17
1
0.0000
1.0000
0.6138
0.4796
-->
Kaplan-Meier estimator
Exact time of failure and censoring are known
Time in complete months
Subjects dying during first month, are recorded
as surviving one month
For 2 subjects, diagnosis took place at death,
hence time recorded as zero
In 11th month (10 months completed), 1 death occurs.
Number at risk is 42.
Hence probability of falure is 1/42 = 0.02381
and the probability of surviving is 0.8595*(1-0.02381) = 0.8391
S(t)   1 - Di / Y (Xi)
X t
^
i
Where Y(xi) is the risk set (individuals at risk just before
time t (time at event))and Di is the failure indicator (1 in
case of failure)
Kaplan-Meier estimator
Time from diagnosis to death
Survival function by Kaplan-Meier method
Survival probability
1.00
0.95
0.90
0.85
0.80
0.75
0.70
0.65
0.60
0
5
10
15
20
25
30
35
40
45
50
55
60
Time
Clayton and Hills, 1993, p. 37
References: Kaplan-Meier
• Good introduction: Clayton and Hills, 1993,
pp. 35ff
• Technical: Andersen and Keiding, 1996, pp.
180ff (includes several references)
The Nelson-Aalen estimator
• A method for the nonparametric estimation
of the cumulative hazard function
• The risk set is calculated at every point in
time where at least one event occurred.
• Hence all episodes must be sorted according to their ending
times.
• It is a staircase function with
a. Location of drop is random (time at event)
b. Size of drop is 1/risk set (number at risk: count of persons alive
before the death
• Easier to generalise to multistate situations
The Nelson-Aalen estimator
• Nelson (1969) and Aalen (1978)
• Clayton and Hills, 1993, p. 48
• Andersen and Keiding, 1996, p. 181
^
A(t)   Di / Y (Xi)
X t
i
A(t) = -lnS(t)
Nelson-Aalen estimator (rates)
Time from diagnosis to death
Cumulative rates using Aalen-Nelson method
Cumulative rate
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0
5
10
15
20
25
30
35
40
45
50
55
60
Time (months)
Clayton and Hills, 1993, p. 50
Duration of job episodes (Blossfeld and Rohwer, 1995) (sorted)
ID
33
110
202
23
81
145
175
185
193
203
44
87
100
100
110
110
127
170
194
27
106
161
177
7
8
40
44
44
44
76
84
153
203
Duration of job episodes
Job Starting
Ending
number
time
time
3
981
982
3
672
673
2
845
846
1
855
857
3
604
606
2
980
982
3
981
983
2
817
819
5
969
971
1
935
937
1
802
805
3
915
918
2
579
582
3
583
586
1
664
667
2
668
671
2
869
872
1
552
555
6
939
942
6
978
982
1
832
836
2
732
736
1
700
704
3
730
735
1
838
843
3
772
777
3
826
831
4
832
837
5
856
861
1
838
843
1
634
639
1
849
854
2
938
943
Duration of job episodes
Duration
ID
Job Starting
Ending
Duration
(p. 46)
number
time
time
2
27
5
971
977
7
2
100
4
588
594
7
2
100
5
595
601
7
3
106
2
837
843
7
3
107
1
934
940
7
3
127
3
873
879
7
3
135
2
834
840
7
3
172
3
940
946
7
3
193
6
976
982
7
3
194
4
897
903
7
4
21
1
689
696
8
4
21
2
697
704
8
4
28
3
702
709
8
4
31
4
975
982
8
4
40
2
764
771
8
4
49
1
975
982
8
4
73
3
937
944
8
4
95
4
975
982
8
4
135
1
826
833
8
5
5
5
5
6
201 respondents
6
6
600 job episodes
6
6
6
6
6 Duration: 'To avoid zero durations, we have added
6 one months to the job duration, or the observed
6 duration if the episode is right censored.' (p. 46)
ID
9
27
44
110
161
176
194
194
27
58
59
70
79
96
123
167
171
173
188
194
9
24
127
167
3
3
3
15
15
24
40
Duration of job episodes
Job Starting
Ending
number
time
time
3
634
642
4
773
781
7
974
982
4
673
681
1
625
633
2
565
573
1
844
852
2
853
861
2
703
712
5
942
951
3
973
982
2
973
982
2
973
982
2
772
781
2
877
886
1
532
541
6
831
840
1
654
663
2
834
843
8
973
982
1
591
601
4
938
948
1
839
849
2
542
552
1
688
699
3
730
741
5
817
828
1
820
831
2
832
843
1
700
711
1
752
763
Duration
9
9
9
9
9
9
9
9
10
10
10
10
10
10
10
10
10
10
10
10
11
11
11
11
12
12
12
12
12
12
12
Product-limit estimate of survival function (Kaplan-Meier)
Index
Time
0
1
2
3
4
5
6
7
8
9
10
0
2
3
4
5
6
7
8
9
10
11
Number Number
of events censored
E
0
0
2
0
5
1
9
2
3
0
10
1
9
0
6
1
7
3
8
1
4
4
Exposed
to risk
(episodes)
R
600
600
597
590
581
577
567
557
548
540
528
Survivor
Sum
function E/(R(R-E))
1.00000
0.99667
0.98832
0.97324
0.96822
0.95144
0.93634
0.92625
0.91442
0.90087
0.89405
5.574E-06
1.415E-05
2.626E-05
8.933E-06
3.057E-05
2.845E-05
1.955E-05
2.361E-05
2.785E-05
1.446E-05
Std
error
0.00000
0.00235
0.00439
0.00660
0.00717
0.00880
0.00999
0.01070
0.01146
0.01225
0.01262
Median duration: 43.03 months
Number censored = number of censored episodes
with ENDING TIMES less than time shown in TIME column.
0.98832 = 0.99667 * (1-5/597)
S(t-1)*[1-E/R]
0.00001415 = 5/(597*(597-5)) = E/(R*(R-E)) .
Std error: S(t) * SQRT{sum [ E/R(R-E))]} GREENWOOD FORMULA (Blossfeld and Rohwer, 95, p. 67)
Table: see Blossfeld and Rohwer, p. 69.
Pisa99/blossfeld/rrdat_sort.xls
Product-limit estimate of survival function (Kaplan-Meier)
and 95% interval
Survival function (men+women)
1.00
0.95
0.90
0.85
0.80
0.75
0
2
3
4
5
S
6
7
Lower
8
Upper
9
10
11
12
Plot of survival function, generated by TDA
Duration up to 428 months (shown up to 300 months)
Blossfeld and Rohwer, 1995, p. 70 (ehc6_1.cf => ehc6_1.ps)
Blossfeld and Rohwer, 1995, p. 73 (ehc8.ps)
Product-limit estimate of survival function (Kaplan-Meier)
#
ehc5.cf
nvar(
dfile
ID
SN
TS
TF
SEX
TI
TB
TE
TMAR
PRES
PRESN
EDU
Kap
=
rrda
[3.0]
[2.0]
[3.0]
[3.0]
[2.0]
[3.0]
[3.0]
[3.0]
[3.0]
[3.0]
[3.0]
[2.0]
#
define
add
DES
[1.0]
=
TFP
[3.0]
=
);
edef(
ts
=
0,
tf
=
TFP,
org
=
0,
des
=
DES,
);
ple
=
ehc5.ple;
Product-limit estimate of survival function (Kaplan-Meier): output
D:\S\TEACH\99PISA\BLOSSF\OEF>d:\s\software\tda\62b\tda\tda_nt
cf=ehc5.cf
Creating new single episode data. Max number of transitions: 100.
Definition: org=0, des=DES, ts=0, tf=TFP
Mean
SN Org Des Episodes Weighted Duration TS Min TF Max
Excl
---------------------------------------------------------------------------1 0 0
142
142.00
128.18
0.00
428.00
1 0 1
458
458.00
49.30
0.00
350.00
Sum
600
600.00
Number of episodes: 600
ple=ehc5.ple
Product-limit estimation. Current memory: 367814 bytes.
Sorting episodes according to ending times.
Product-limit estimation.
1 table(s) written to: ehc5.ple
---------------------------------------------------------------------------Current memory: 311232 bytes. Max memory used: 387781 bytes.
End of program. Mon Mar 27 23:49:40 2000
Product-limit estimate of survival function (Kaplan-Meier): output
# SN 1. Transition: 0,1 - Product-Limit Estimation ehc5.ple
#
# ID Index
0
0
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
9
0
10
0
11
0
12
0
13
0
14
0
15
0
16
0
17
0
18
0
19
Time
0.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00
10.00
11.00
12.00
13.00
14.00
15.00
16.00
17.00
18.00
19.00
Number
Events
0
2
5
9
3
10
9
6
7
8
4
24
8
10
6
4
9
6
8
20.00
9
Number
Censored
0
0
1
2
0
1
0
1
3
1
4
0
1
3
1
0
0
0
0
1
Exposed
to Risk
600
600
597
590
581
577
567
557
548
540
528
524
499
488
477
471
467
458
452
443
Survivor
Std.
Function Error
1.00000 0.00000
0.99667 0.00235
0.98832 0.00439
0.97324 0.00660
0.96822 0.00717
0.95144 0.00880
0.93634 0.00999
0.92625 0.01070
0.91442 0.01146
0.90087 0.01225
0.89405 0.01262
0.85310 0.01455
0.83942 0.01510
0.82222 0.01574
0.81188 0.01610
0.80498 0.01633
0.78947 0.01681
0.77913 0.01711
0.76534 0.01749
Cum.
Rate
0.00000
0.00334
0.01175
0.02712
0.03230
0.04978
0.06578
0.07661
0.08947
0.10439
0.11200
0.15888
0.17504
0.19575
0.20841
0.21694
0.23640
0.24958
0.26744
0.74979 0.01789 0.28797
Blossfeld and Rohwer, 1995, p. 69
Deaths in first years of life, Kerala. Source: NFHS, 1992-93
Table 1. Life table results for deaths during first four years of life, 1988-92 birth cohort, Kerala (S. Padmadas, dissertation)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------Age
Number Number Number Number Prob of Prob of
Survival
Prob den Hazard Standard Standard Standard
interval
entering censored exposed
of
dying
surviving function
function
rate
error
error
error
in months
to risk
deaths
(x)
(lx)
(cx)
(lx’)
(dx)
(qx)
(px)
(sx)
(fx)
( x)
(sx)
(fx)
( x)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------0-1
2026.0
8
2022.0
32
0.0158
0.9842
1.0000
0.0158
0.0160
0.0028
0.0028
0.0028
1-2
1986.0
33
1969.5
4
0.0020
0.9980
0.9842
0.0020
0.0020
0.0030
0.0010
0.0010
2-3
1949.0
46
1926.0
2
0.0010
0.9990
0.9822
0.0010
0.0010
0.0030
0.0007
0.0007
3-4
1901.0
33
1884.5
5
0.0027
0.9973
0.9812
0.0026
0.0027
0.0032
0.0012
0.0012
4-5
1863.0
37
1844.5
1
0.0005
0.9995
0.9786
0.0005
0.0005
0.0033
0.0005
0.0005
5-6
1825.0
32
1809.0
0
0.0000
1.0000
0.9780
0.0000
0.0000
0.0033
0.0000
0.0000
6-7
1793.0
28
1779.0
0
0.0000
1.0000
0.9780
0.0000
0.0000
0.0033
0.0000
0.0000
7-8
1765.0
32
1749.0
1
0.0006
0.9994
0.9780
0.0006
0.0006
0.0033
0.0006
0.0006
8-9
1732.0
42
1711.0
1
0.0006
0.9994
0.9775
0.0006
0.0006
0.0034
0.0006
0.0006
9-10
1689.0
35
1671.5
0
0.0000
1.0000
0.9769
0.0000
0.0000
0.0034
0.0000
0.0000
10-11
1654.0
32
1638.0
2
0.0012
0.9988
0.9769
0.0012
0.0012
0.0035
0.0008
0.0009
11-12
1620.0
28
1606.0
0
0.0000
1.0000
0.9757
0.0000
0.0000
0.0035
0.0000
0.0000
12-13
1592.0
30
1577.0
1
0.0006
0.9994
0.9757
0.0006
0.0006
0.0035
0.0006
0.0006
13-14
1561.0
32
1545.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
14-15
1529.0
25
1516.5
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
15-16
1504.0
32
1488.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
16-17
1472.0
39
1452.5
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
17-18
1433.0
35
1415.5
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
18-19
1398.0
34
1381.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
19-20
1364.0
35
1346.5
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
20-21
1329.0
31
1313.5
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
21-22
1298.0
34
1281.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
22-23
1264.0
36
1246.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
23-24
1228.0
32
1212.0
0
0.0000
1.0000
0.9751
0.0000
0.0000
0.0035
0.0000
0.0000
continued
Deaths in first years of life, Kerala. Source: NFHS, 1992-93
Age
Number Number Number Number Prob of Prob of
Survival
Prob den Hazard Standard Standard Standard
interval
entering censored exposed
of
dying
surviving function
function
rate
error
error
error
in months
to risk
deaths
(x)
(lx)
(cx)
(lx’)
(dx)
(qx)
(px)
(sx)
(fx)
( x)
(sx)
(fx)
( x)
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------24-25
1196.0
37
1177.5
4
0.0034
0.9966
0.9751
0.0033
0.0034
0.0039
0.0017
0.0017
25-26
1155.0
27
1141.5
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
26-27
1128.0
26
1115.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
27-28
1102.0
30
1087.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
28-29
1072.0
34
1055.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
29-30
1038.0
30
1023.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
30-31
1008.0
22
997.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
31-32
986.0
43
964.5
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
32-33
943.0
29
928.5
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
33-34
914.0
39
894.5
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
34-35
875.0
38
856.0
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
35-36
837.0
37
818.5
0
0.0000
1.0000
0.9718
0.0000
0.0000
0.0039
0.0000
0.0000
36-37
800.0
32
784.0
1
0.0013
0.9987
0.9718
0.0012
0.0013
0.0041
0.0012
0.0012
37-38
767.0
26
754.0
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
38-39
741.0
44
719.0
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
39-40
697.0
38
678.0
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
40-41
659.0
30
644.0
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
41-42
629.0
25
616.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
42-43
604.0
23
592.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
43-44
581.0
31
565.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
44-45
550.0
26
537.0
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
45-46
524.0
45
501.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
46-47
479.0
29
464.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
47-48
450.0
39
430.5
0
0.0000
1.0000
0.9705
0.0000
0.0000
0.0041
0.0000
0.0000
48+
411.0
411
205.5
0
0.0000
1.0000
-------------------
Table 7. Estimated life table functions for the first year (in months) of life by place of residence
for 1988-92 birth cohort, Kerala.
Interval in
Months
0
1
2
3
4
5
6
7
8
9
10
11
12
Survival function S(x)
All
Urban Rural
1.0000 1.0000 1.0000
0.9842 0.9942 0.9807
0.9822 0.9942 0.9780
0.9812 0.9942 0.9767
0.9786 0.9922 0.9739
0.9780 0.9922 0.9732
0.9780 0.9922 0.9732
0.9780 0.9922 0.9732
0.9775 0.9922 0.9724
0.9769 0.9922 0.9716
0.9769 0.9922 0.9716
0.9757 0.9898 0.9708
0.9757 0.9898 0.9708
Prob density function f(x)
All
Urban Rural
0.0159 0.0058 0.0193
0.0020 0.0000 0.0027
0.0010 0.0000 0.0014
0.0026 0.0020 0.0028
0.0005 0.0000 0.0007
0.0000 0.0000 0.0000
0.0000 0.0000 0.0000
0.0006 0.0000 0.0007
0.0006 0.0000 0.0008
0.0000 0.0000 0.0000
0.0012 0.0024 0.0008
0.0000 0.0000 0.0000
0.0006 0.0000 0.0008
Hazard rate
All
Urban
0.0160 0.0059
0.0020 0.0000
0.0010 0.0000
0.0027 0.0021
0.0005 0.0000
0.0000 0.0000
0.0000 0.0000
0.0006 0.0000
0.0006 0.0000
0.0000 0.0000
0.0012 0.0024
0.0000 0.0000
0.0006 0.0000
Note: Number of children entering the interval 0-1; All: 2026; Urban: 519; Rural: 1507
(x)
Rural
0.0195
0.0027
0.0014
0.0029
0.0007
0.0000
0.0000
0.0008
0.0008
0.0000
0.0008
0.0000
0.0008
Table 8. Estimated life table results for the interval between first and second birth (in months),
Kerala, January 1988-February 1993
Interval
in
months
Women
Women Number Prob.of Propn.
entering Women exposed of
second surviv
the intv censored to risk births births
ing
9-10
10-11
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
22-23
23-24
24-25
25-26
26-27
27-28
28-29
29-30
30-31
31-32
32-33
33-34
34-35
35-36
36-37
37-38
38-39
39-40
40-41
41-42
42-43
43-44
44-45
45-46
46-47
47-48
48+
1408
1389
1368
1350
1327
1296
1274
1249
1210
1175
1152
1113
1081
1046
1005
954
906
869
828
799
765
738
704
674
646
614
585
562
542
516
488
460
444
419
394
375
349
335
324
303
16
15
11
13
11
6
11
18
11
8
15
9
8
10
14
16
9
13
6
9
3
5
12
8
9
12
9
4
7
8
10
5
3
6
7
7
8
5
8
54
1400.0
1381.5
1362.5
1343.5
1321.5
1293.0
1268.5
1240.0
1204.5
1171.0
1144.5
1108.5
1077.0
1041.0
998.0
946.0
901.5
862.5
825.0
794.5
763.5
735.5
698.0
670.0
641.5
608.0
580.5
560.0
538.5
512.0
483.0
457.5
442.5
416.0
390.5
371.5
345.0
332.5
320.0
276.0
3
6
7
10
20
16
14
21
24
15
24
23
27
31
37
32
28
28
23
25
24
29
18
20
23
17
14
16
19
20
18
11
22
19
12
19
6
6
13
249
0.0021
0.0043
0.0051
0.0074
0.0151
0.0124
0.0110
0.0169
0.0199
0.0128
0.0210
0.0207
0.0251
0.0298
0.0371
0.0338
0.0311
0.0325
0.0279
0.0315
0.0314
0.0394
0.0258
0.0299
0.0359
0.0280
0.0241
0.0286
0.0353
0.0391
0.0373
0.0240
0.0497
0.0457
0.0307
0.0511
0.0174
0.0180
0.0406
---
0.9979
0.9957
0.9949
0.9926
0.9849
0.9876
0.9890
0.9831
0.9801
0.9872
0.9790
0.9793
0.9749
0.9702
0.9629
0.9662
0.9689
0.9675
0.9721
0.9685
0.9686
0.9606
0.9742
0.9701
0.9641
0.9720
0.9759
0.9714
0.9647
0.9609
0.9627
0.9760
0.9503
0.9543
0.9693
0.9489
0.9826
0.9820
0.9594
---
Cum.
propn
surv
1.0000
0.9979
0.9935
0.9884
0.9811
0.9662
0.9543
0.9437
0.9277
0.9093
0.8976
0.8788
0.8606
0.8390
0.8140
0.7838
0.7573
0.7338
0.7100
0.6902
0.6685
0.6474
0.6219
0.6059
0.5878
0.5667
0.5509
0.5376
0.5222
0.5038
0.4841
0.4661
0.4549
0.4323
0.4125
0.3998
0.3794
0.3728
0.3661
---
Prob.
density
function
Hazard
rate
0.0021
0.0043
0.0051
0.0074
0.0148
0.0120
0.0105
0.0160
0.0185
0.0116
0.0188
0.0182
0.0216
0.0250
0.0302
0.0265
0.0235
0.0238
0.0198
0.0217
0.0210
0.0255
0.0160
0.0181
0.0211
0.0158
0.0133
0.0154
0.0184
0.0197
0.0180
0.0112
0.0226
0.0197
0.0127
0.0204
0.0066
0.0067
0.0149
---
0.0021
0.0044
0.0052
0.0075
0.0152
0.0125
0.0111
0.0171
0.0201
0.0129
0.0212
0.0210
0.0254
0.0302
0.0378
0.0344
0.0315
0.0330
0.0283
0.0320
0.0319
0.0402
0.0261
0.0303
0.0365
0.0284
0.0244
0.0290
0.0359
0.0398
0.0380
0.0243
0.0510
0.0467
0.0312
0.0525
0.0175
0.0182
0.0415
---
The fetal life table
Goldhaber and Fireman (1991)
• 9564 pregnancies identified retrospectively from urine tests as well as
first prenatal care visits at three Kaiser Permanente clinics in San
Francisco Bay area during 10-month period in 1981-1982. Twin and
triplet pregnancies, pregnancies with less than 2 days follow-up, and
few other pregnancies were omitted => 9055 pregnancies. Of these,
103 withdrew during follow-up (pregnancy outcome not known), 6629
resulted in live births, 549 in spontaneous fetal loss (including 27
ectopic pregnancies), and 1774 induced abortion. 2-day lag was used
to avoid bias arising when women selectively report for medical care
because of threatened abortion. Many of these women miscarry within
2 days (selection!). Inclusion would overestimate the risk of abortion!
• Measurement issues: onset of pregnancy (date of last menstrual period)
and pregnancy outcome. 459 women entered observation in week 5 of
gestation (days 0-6 after last menstrual period = week 0; days 35-41 =
week 5; days 308-314 = week 44)
Fetal life table by gestational weeks (LMP), 1981-1982, California, USA
Gestational Foetuses entering
week s (LMP)
during week
5
459
6
1313
7
1249
8
922
9
792
10
725
11
638
12
517
13
458
14
319
15
246
16
187
17
153
18
102
19
116
20
102
21
78
22
87
23
58
24
53
25
44
26
58
27
38
28
39
29
34
30
38
31
32
32
39
33
34
34
23
35
29
36
27
37
18
38
11
39
10
40
4
41
2
42
1
43
0
44
0
Total
9055
Spontaneous
foetal loss
1
6
16
34
44
62
60
68
42
27
26
24
12
11
16
10
8
7
1
2
8
3
2
4
0
4
2
3
3
1
2
0
2
1
3
5
2
0
0
0
522
Ectopic
Induced
pregnancy abortion
1
3
1
24
4
152
4
359
5
367
3
284
4
244
1
136
0
71
1
43
0
29
0
24
1
8
2
11
0
9
0
6
0
1
0
2
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
27
1774
Live
birth
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
0
1
3
2
3
3
2
5
7
7
14
19
34
43
75
162
304
727
1452
2101
1026
477
134
26
6629
Withdrawal Number of pregnancies still
in progress at end
0
454
0
1736
0
2813
0
3338
2
3712
0
4088
4
4414
1
4725
5
5065
5
5308
3
5496
6
5629
7
5754
5
5827
5
5912
2
5995
6
6058
3
6132
6
6180
3
6225
3
6255
2
6305
6
6333
3
6360
3
6384
3
6408
2
6422
4
6435
3
6429
3
6405
0
6357
2
6220
2
5930
2
5211
0
3766
2
1662
0
636
0
160
0
26
0
0
103
Risk set
459.0
1767.0
2985.0
3735.0
4129.0
4437.0
4724.0
4930.5
5180.5
5381.5
5552.5
5680.0
5778.5
5853.5
5940.5
6013.0
6070.0
6143.5
6187.0
6231.5
6267.5
6312.0
6340.0
6370.5
6392.5
6420.5
6439.0
6459.0
6467.5
6450.5
6434.0
6383.0
6237.0
5940.0
5221.0
3769.0
1664.0
637.0
160.0
26.0
Risk set = foetuses at risk at the beginning of the week minus half of withdrawals during week
Goldhaber, M.K. and B.H. Fireman (1991) The fetal life table revisited: spontaneous abortion rates in three Kaiser Permanente cohorts.
Epidemiology, 2:33-39
Fetal life table by gestational weeks (LMP), 1981-1982, California, USA
Risk set
459.0
1767.0
2985.0
3735.0
4129.0
4437.0
4724.0
4930.5
5180.5
5381.5
5552.5
5680.0
5778.5
5853.5
5940.5
6013.0
6070.0
6143.5
6187.0
6231.5
6267.5
6312.0
6340.0
6370.5
6392.5
6420.5
6439.0
6459.0
6467.5
6450.5
6434.0
6383.0
6237.0
5940.0
5221.0
3769.0
1664.0
637.0
160.0
26.0
prob of
spont. death
0.004
0.004
0.007
0.010
0.012
0.015
0.014
0.014
0.008
0.005
0.005
0.004
0.002
0.002
0.003
0.002
0.001
0.001
0.000
0.000
0.001
0.000
0.000
0.001
0.000
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.000
0.000
0.000
prob of
induced abortion
0.007
0.014
0.051
0.096
0.089
0.064
0.052
0.028
0.014
0.008
0.005
0.004
0.001
0.002
0.002
0.001
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
prob of
live birth
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.001
0.001
0.001
0.002
0.003
0.005
0.007
0.012
0.025
0.049
0.122
0.278
0.557
0.617
0.749
0.838
1.000
prob of
survival
0.989
0.982
0.942
0.894
0.899
0.921
0.934
0.958
0.978
0.986
0.990
0.991
0.996
0.995
0.995
0.997
0.998
0.998
0.999
0.999
0.998
0.999
0.999
0.998
0.999
0.998
0.997
0.996
0.994
0.993
0.988
0.974
0.951
0.877
0.721
0.441
0.382
0.251
0.163
0.000
sum of
probs
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
0.999
0.999
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
1.000
Application and confusion (discussion)
Miller and Homan (1994) “Determining transition probabilities: confusion
and suggestions”, Medical Decision Making, 14(1):??? (based on Kleinbaum
et al.). Terminology used in this paper is confusing (and wrong!)
a. Distinguish between rates and risk
Rate (incidence rate): occurrences (incidences; new cases) over
exposure. Exposure is measured by ‘summing each subject’s time exposed to the
possibility of transiting’ (includes censored cases).
. Instantaneous incidence rate (‘also known as the hazard function’)
. Average incidence rate (also known as the ‘incidence density’ [ID])
Density  rate
Application and confusion (discussion)
Risk: Risk used to denote probability.
Three methods for estimating risk:
1. Simple cumulative method: new cases / number of disease-free
individuals at beginning of interval (no censoring or withdrawal): I/N0
where I is number of new cases and N0 is the number of disease-free individuals
at t=0.
2. Actuarial (life-table) method: new cases / number of disease-free
individuals at beginning of interval minus half of the number of
withdrawals: I/[N0-W/2] where W is number of withdrawals
= risk set
3. Density method: uses age-specific incidence densities (e.g. rates) to
estimate the risk for given age or time interval:
P(0,t) = 1 - exp[-ID*t]
where ID is the average rate and t is the elapsed time. Rates are translated into
probabilities.