The life table

• LT statistics: rates, probabilities, life expectancy (waiting time to event)

• Period life table

• Cohort life table

Life table fromobservational data

22 respondents

Namboodiri and Suchindran, 1987, Chapter 4

A. Exact time

CASE ID Date Date Date Duration Duration Durationof entry of event of interview to event to interview Exposure

in observation Days Days Days1 Jan-02 Feb-11 May-25 40 143 402 Jan-17 May-04 May-17 107 120 1073 Jan-18 - May-10 - 112 1124 Jan-22 Feb-28 May-13 37 111 375 Feb-10 May-17 May-23 96 102 966 Jan-30 Feb-12 May-15 13 105 137 Apr-04 - May-06 - 32 328 Apr-29 - May-27 - 28 289 May-18 - May-29 - 11 11

10 May-20 - May-31 - 11 1111 May-15 - May-18 - 3 312 Feb-05 Feb-25 May-19 20 103 2013 Feb-05 Apr-18 May-10 72 94 7214 Feb-06 May-18 May-28 101 111 10115 Feb-26 - May-22 - 85 8516 Mar-10 - May-25 - 76 7617 Mar-11 May-08 May-12 58 62 5818 Mar-28 - May-29 - 62 6219 Mar-15 Mar-23 May-10 8 56 820 Apr-13 - May-20 - 37 3721 Apr-04 May-09 May-11 35 37 3522 Apr-25 May-16 May-31 12 36 12

- indicates censoring

Table 1Hypothetical survey data for construction of life table (1986)

CASE ID Date Date Date Duration Duration Duration At tisk Event/of entry of event of interview to event to interview Exposure at beginning censoring

in observation Days Days Days of day*

11 May-15 - May-18 - 3 3 22 019 Mar-15 Mar-23 May-10 8 56 8 21 19 May-18 - May-29 - 11 11 20 0

10 May-20 - May-31 - 11 11 19 022 Apr-25 May-16 May-31 12 36 12 18 16 Jan-30 Feb-12 May-15 13 105 13 17 1

12 Feb-05 Feb-25 May-19 20 103 20 16 18 Apr-29 - May-27 - 28 28 15 07 Apr-04 - May-06 - 32 32 14 0

21 Apr-04 May-09 May-11 35 37 35 13 14 Jan-22 Feb-28 May-13 37 111 37 12 1

20 Apr-13 - May-20 - 37 37 11 01 Jan-02 Feb-11 May-25 40 143 40 10 1

17 Mar-11 May-08 May-12 58 62 58 9 118 Mar-28 - May-29 - 62 62 8 013 Feb-05 Apr-18 May-10 72 94 72 7 116 Mar-10 - May-25 - 76 76 6 015 Feb-26 - May-22 - 85 85 5 05 Feb-10 May-17 May-23 96 102 96 4 1

14 Feb-06 May-18 May-28 101 111 101 3 12 Jan-17 May-04 May-17 107 120 107 2 13 Jan-18 - May-10 - 112 112 1 0

USE EXCEL DATA/SORTNumber of persons at risk decreases over time because of occurrence of event or censoringASS: population is homogeneous and censoring is independent of event of interest* Assumes censoring NOT at beginning of interval (day)

Table 2Estimation of survival functionsort

Day At tisk Event Censored Prob Observed Survivalat beginning of event survival function

of day* proportions

3 22 0 1 18 21 1 0 0.0476 0.9524 0.9524

11 20 0 1 0.952411 19 0 1 0.952412 18 1 0 0.0556 0.9444 0.899513 17 1 0 0.0588 0.9412 0.846620 16 1 0 0.0625 0.9375 0.793728 15 0 1 0.793732 14 0 1 0.793735 13 1 0 0.0769 0.9231 0.732637 12 1 0 0.0833 0.9167 0.671637 11 0 1 0.671640 10 1 0 0.1000 0.9000 0.604458 9 1 0 0.1111 0.8889 0.537262 8 0 1 0.537272 7 1 0 0.1429 0.8571 0.460576 6 0 1 0.460585 5 0 1 0.460596 4 1 0 0.2500 0.7500 0.3454

101 3 1 0 0.3333 0.6667 0.2302107 2 1 0 0.5000 0.5000 0.1151112 1 0 1 0.1151

Total 12 10

Table 2Estimation of survival function, cont

0.9524 = 1-1/21

=1 - failure/number at risk before failure

0.8995 = 0.9524*[1-1/18]

Kaplan-Meier

1/21

Discrete time interval (30 days)

B. Discrete time interval: completed months

Duration Duration Events Censoring Events+Month Days censoring0 0-30 4 4 01 31-59 3 2 12 60-89 1 3 03 90-119 1 1 2

Event+censoring in same month: 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 Prob of Probin months event surviving S intermed s.e.

0 18 4 4 0.2222 0.7778 1.0000 0.0000 0.00001 11 3 3 0.2727 0.7273 0.7778 0.0123 0.08642 5 1 3 0.2000 0.8000 0.5657 0.0319 0.11903 1 1 3 1.0000 0.0000 0.4525 0.0707 0.15344 0.0000 0.0000 0.0000

NOTE: Case 21 is censored instead of eventNOTE: 11 = 18 - 4- 3

Survival function

B. Censoring at end of month

Duration Risk set Events Censored Prob of Probin months event surviving S intermed s.e.

0 22 4 4 0.1818 0.8182 1.0000 0.0000 0.00001 14 4 2 0.2857 0.7143 0.8182 0.0083 0.42362 8 1 3 0.1250 0.8750 0.5844 0.0249 0.29793 4 3 1 0.7500 0.2500 0.5114 0.0267 0.24784 0.1278 0.0774 0.0583

NOTE: Case 21 experiences event and is not censored

C. Censoring in middle of month (half at beginning and half at end of month)

Duration Risk set Events Censored Prob of Probin months event surviving S intermed s.e.

0 20 4 4 0.2000 0.8000 1.0000 0.0000 0.00001 13 4 2 0.3077 0.6923 0.8000 0.0100 0.28922 6.5 1 3 0.1538 0.8462 0.5538 0.0296 0.19243 3.5 3 1 0.8571 0.1429 0.4686 0.0427 0.14154 0.0669 0.0484 0.0147

NOTE: Case 21 experiences event and is not censored

Survival function

Survival function

Survival function

0.00

0.20

0.40

0.60

0.80

1.00

0 1 2 3 4

Duration (months)

Sur

viva

l pro

babi

lity

Censoring begin

censoring end

Censoring middle

Survival function and 95% interval

0.00

0.20

0.40

0.60

0.80

1.00

1 2 3 4 5

Duration (months)

Surv

ival

pro

babi

lity

Censoring middle

S-1.96*s.e.

S+1.96*s.e.

Standard error of survival function: Greenwood formula

SE i

i ii

i t

t tS Sq

p R( )

/

1 2

1

1

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 = 0.1667Risk set: 90 (= 100*0.9)Prob of surviving second interval: S2 = 0.9*0.833=0.75Var(S2) = 0.752 * [0.1/(0.9*100) + 0.1667/(0.8333*90)]

= 0.00188

Standard error of (hazard) rate:

21

2

)(

2/1

rRq

rr tSE

tt

tt

With rt = hazard rate

Life table with grouped data

Remarriage of divorced women, aged 25 to 34, USA, 1975Source: 1975 US Current Population Survey

Namboodiri and Suchindran, 1978, pp. 63 ff

Number Number of Divorced Probability Survival Survival [q/pR] SE(S) h(t) h(t) SE[h(t)]Interval divorced remarriages at interview Risk set of event probability function [6/(7*5)] Greenwood Linear Exponential

at beginning (censored) [3/5] [1-3/5](n') (d) ( c ) (n) (q) (p) (S)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)[0,1) 1298 238 101 1247.5 0.1908 0.8092 1.0000 0.0000 0.0000 0.2109 0.2117 0.0136[1,2) 959 177 65 926.5 0.1910 0.8090 0.8092 0.0002 0.0111 0.2112 0.2120 0.0158[2,3) 717 107 51 691.5 0.1547 0.8453 0.6546 0.0003 0.0138 0.1677 0.1681 0.0162[3,4) 559 64 42 538.0 0.1190 0.8810 0.5533 0.0003 0.0147 0.1265 0.1267 0.0158[4.5) 453 50 34 436.0 0.1147 0.8853 0.4875 0.0003 0.0151 0.1217 0.1218 0.0172[5,6) 369 34 18 360.0 0.0944 0.9056 0.4316 0.0003 0.0153 0.0991 0.0992 0.0170[6,7) 317 27 21 306.5 0.0881 0.9119 0.3908 0.0003 0.0154 0.0922 0.0922 0.0177[7,8) 269 21 11 263.5 0.0797 0.9203 0.3564 0.0003 0.0154 0.0830 0.0831 0.0181[8,9) 237 14 4 235.0 0.0596 0.9404 0.3280 0.0003 0.0154 0.0614 0.0614 0.0164[9,10) 219 15 11 213.5 0.0703 0.9297 0.3085 0.0003 0.0153 0.0728 0.0728 0.0188[10,11) 193 16 14 186.0 0.0860 0.9140 0.2868 0.0004 0.0152 0.0899 0.0899 0.0224[11,12) 163 8 8 159.0 0.0503 0.9497 0.2621 0.0005 0.0151 0.0516 0.0516 0.0182[12,13) 147 6 8 143.0 0.0420 0.9580 0.2489 0.0003 0.0150 0.0429 0.0429 0.0175[13,14) 133 6 7 129.5 0.0463 0.9537 0.2385 0.0003 0.0150 0.0474 0.0474 0.0194[14,15) 120 5 5 117.5 0.0426 0.9574 0.2274 0.0004 0.0150 0.0435 0.0435 0.0194[15,16) 110 2 10 105.0 0.0190 0.9810 0.2178 0.0004 0.0149 0.0192 0.0192 0.0136[16,17) 98 5 5 95.5 0.0524 0.9476 0.2136 0.0002 0.0149 0.0538 0.0538 0.0240[17,18) 88 5 2 87.0 0.0575 0.9425 0.2024 0.0006 0.0150 0.0592 0.0592 0.0265[18,19) 81 2 2 80.0 0.0250 0.9750 0.1908 0.0007 0.0150 0.0253 0.0253 0.0179[19,20) 77 2 9 72.5 0.0276 0.9724 0.1860 0.0003 0.0150 0.0280 0.0280 0.0198[20,+) 66 12 54 39.0 0.3077 0.6923 0.1809 0.0004 0.0150 - - -Total 816 482

Risk set: assuming censoring in MIDDLE of interval[ T,T+1) Greater or equal to T, but less than T+1Greenwood formula: see also Blossfeld and Rohwer, 1995, p. 53SE(rate): see Blossfeld and Rohwer, 1995, p. 54

Remarriage life table, Unites States 1975

0.2109 = 238/(0.5*(1298+959))

0.2117 = -ln(0.8092)Conditional density varies with duration

Estimated survival function and hazard function

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

0 2 4 6 8 10 12 14 16 18 20

Duration since divorce

Pro

bab

ilit

y

0.0000

0.0500

0.1000

0.1500

0.2000

0.2500

Rat

e

S(t)

h(t)

Source: W . D ijkstra ed . (1989) H et proces van sociale integratie van jong-volw assenen:De gegevensverzameling voor de eerste hoofdm eting (The process of social integration ofyoung adults . D ata collection for first m easurement) . VU U itgeverij, Am sterdam .

Jong G ierveld, J . de , Liefbroer, A .C . & Beekink, E . (1991a). The effectof parental resources on patterns of leaving hom e am ong young adultsin the Netherlands . European Sociological Review , 7 , 55-71 .

Thanks to Jenny G ierveld and A at liefbroer , N ID I, The H ague, for providing the data .

Leaving parental home, The Netherlands, Birth cohort 1961

Survey Sept. 87 - Febr. 88

Leaving parental home, 1961 cohort, micro-data, 583 respondents (first 30 respondents)# Sex Father Month Reason # Sex Father Month Reason1 2 2 268 2 16 1 2 251 3 Sex2 1 3 268 2 17 2 2 212 1 1 Female3 1 2 202 1 18 2 2 320 2 2 Male4 2 2 320 4 19 1 2 221 35 1 1 237 1 20 2 2 322 4 Father status6 1 1 295 2 21 2 1 221 2 1 Low7 1 1 272 2 22 2 3 308 2 2 Middle8 2 1 231 1 23 1 2 233 1 3 High9 2 1 312 3 24 1 1 273 2

10 1 2 289 2 25 1 1 208 1 Reason11 1 1 316 2 26 1 2 219 1 1 Educ/work12 2 1 321 4 27 1 1 261 2 2 Marriage/cohabit13 2 1 260 1 28 2 2 270 3 3 Freedom14 2 2 281 2 29 2 2 277 2 4 Censored15 2 1 273 2 30 1 1 290 3 at interview

Data

AgeLiving Exposure Living

at home Total Censored Leave (months) at home Total Censored Leave15 291 4 0 4 3480 288 4 0 416 287 7 0 7 3412 284 7 0 717 280 23 0 23 3279 277 23 0 2318 257 53 0 53 2761 254 53 0 5319 204 48 0 48 2181 201 48 0 4820 156 41 1 40 1652 153 40 0 4021 115 31 0 31 1223 113 31 0 3122 84 26 0 26 855 82 26 0 2623 58 23 0 23 596 56 23 0 2324 35 16 1 15 325 33 15 0 1525 19 8 2 6 183 18 7 1 626 11 11 9 2 42 11 11 9 227 0 0 0 0 0 0

291 13 278 19989 288 10 278Early 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 1987There are females censored in months 244, 297 and 301. IMPOSSIBLE (see HOME.XLS)

AgeLiving Exposure

at home Total Censored Leave (months)15 292 2 0 2 349516 290 5 0 5 345717 285 15 0 15 334718 270 24 1 23 311219 246 30 1 29 279120 216 35 2 33 240821 181 28 0 28 204322 153 33 0 33 164323 120 30 0 30 128324 90 30 0 30 91525 60 25 5 20 62626 35 35 31 4 19627 0 0 0 0 0

292 40 252 25316

The first male censored is in month 310. Hence no early censoring.

Data

Attrition

Females, excluding early censoringAttrition

Females, including early censoring

Males, including early censoring

Attrition

d:\s\teach\98\lessn\lt_obs\jenny

Age At home Leave Censored Risk set Exposure Probab Probab Survival q/(p*R) SE(S(t))(months) leaving surviving function (9/(10*5)

(3/5) 1-(7)Column 1 2 3 4 5 6 7 8 9 10

15 291 4 0 291.0 3480 0.0137 0.9863 1.0000 0.0000 0.000016 287 7 0 287.0 3412 0.0244 0.9756 0.9863 0.0001 0.006917 280 23 0 280.0 3279 0.0821 0.9179 0.9622 0.0003 0.011518 257 53 0 257.0 2761 0.2062 0.7938 0.8832 0.0010 0.020519 204 48 0 204.0 2181 0.2353 0.7647 0.7010 0.0015 0.033820 156 40 1 155.5 1652 0.2572 0.7428 0.5361 0.0022 0.038221 115 31 0 115.0 1223 0.2696 0.7304 0.3982 0.0032 0.038722 84 26 0 84.0 855 0.3095 0.6905 0.2908 0.0053 0.036523 58 23 0 58.0 596 0.3966 0.6034 0.2008 0.0113 0.034124 35 15 1 34.5 325 0.4348 0.5652 0.1212 0.0223 0.031825 19 6 2 18.0 183 0.3333 0.6667 0.0685 0.0278 0.026426 11 2 9 6.5 42 0.3077 0.6923 0.0457 0.0684 0.018827 0 0 0 0.0 0 0.0316 0.0173

278 13 19989

Risk set = number initially at home - 0.5*censoring during year. Assume UNIFORM distributionExposure in YEARS = risk set - 0.5*number leaving home during year. Assume: UNIFORM distribution

Females

d:\s\teach\98\lessn\lt_obs\jenny

Probability of living at home, Females, (+95% interval)

0.0000

0.2000

0.4000

0.6000

0.8000

1.0000

15 16 17 18 19 20 21 22 23 24 25 26 27

Age

Pro

bab

ilit

y

d:\s\teach\98\lessn\lt_obs\jenny

Age At home Leave Risk set Exposure Event SE(rate) Waiting Expectedin years Rate time waiting

(T) time (yrs)Column 1 2 3 5 11 12 13 14

15 291 4 291.0 289.0 0.0138 0.0069 1651.5 5.6816 287 7 287.0 283.5 0.0247 0.0093 1362.5 4.7517 280 23 280.0 268.5 0.0857 0.0178 1079.0 3.8518 257 53 257.0 230.5 0.2299 0.0314 810.5 3.1519 204 48 204.0 180.0 0.2667 0.0381 580.0 2.8420 156 40 155.5 135.5 0.2952 0.0462 400.0 2.5721 115 31 115.0 99.5 0.3116 0.0553 264.5 2.3022 84 26 84.0 71.0 0.3662 0.0706 165.0 1.9623 58 23 58.0 46.5 0.4946 0.0999 94.0 1.6224 35 15 34.5 27.0 0.5556 0.1378 47.5 1.3825 19 6 18.0 15.0 0.4000 0.1600 20.5 1.1426 11 2 6.5 5.5 0.3636 0.2528 5.5 0.8527 0 0 0.0 0.0

278 1651.5

d:\s\teach\98\lessn\lt_obs\jenny

Leaving home: occurrences and exposures

d:\s\teach\98\lessn\lt_obs\jenny

Females Males TotalTiming Early (LT 20) 135 74 209

Late (GE 20) 143 178 321Censored at interview 13 40 53Total 291 292 583

Timing Early (LT 20) 15113 16202 31315Late (GE 20) 4876 9114 13990Total 19989 25316 45305

Females Males TotalTiming Early (LT 20) 135 74 209

Late (GE 20) 143 178 321Total 278 252 530

Timing Early (LT 20) 14333 13826 28159Late (GE 20) 3998 6339 10337Total 18331 20165 38496

Exposure

Event count

Exposure

A. Events and exposure including censored observations

B. Events and exposure excluding censored observationsSex

SexEvent count

d:\s\teach\99pisa\leaveh\expos1.xls

SPSS SURVIVAL Females 583 observationsNumber Number Number Number Cumul

Intrvl Entrng Wdrawn Exposd of Propn Propn Propn Proba-Start this During to Termnl Termi- Sur- Surv bility HazardTime Intrvl Intrvl Risk Events nating viving at End Densty Rate------ ------ ------ ------ ------ ------ ------ ------ ------ ------

15 291 0 291.0 4 0.0137 0.9863 0.9863 0.0137 0.013816 287 0 287.0 7 0.0244 0.9756 0.9622 0.0241 0.024717 280 0 280.0 23 0.0821 0.9179 0.8832 0.0790 0.085718 257 0 257.0 53 0.2062 0.7938 0.7010 0.1821 0.229919 204 0 204.0 48 0.2353 0.7647 0.5361 0.1649 0.266720 156 1 155.5 40 0.2572 0.7428 0.3982 0.1379 0.295221 115 0 115.0 31 0.2696 0.7304 0.2908 0.1073 0.311622 84 0 84.0 26 0.3095 0.6905 0.2008 0.0900 0.366223 58 0 58.0 23 0.3966 0.6034 0.1212 0.0796 0.494624 35 1 34.5 15 0.4348 0.5652 0.0685 0.0527 0.555625 19 2 18.0 6 0.3333 0.6667 0.0457 0.0228 0.400026 11 9 6.5 2 0.3077 0.6923 0.0316 0.0141 0.3636

The median survival time for these data is 20.26

SPSS output

d:\s\teach\98\lt_obs\jenny\lt.doc

d:\s\teach\98\lt_obs\jenny\spss.xls

SPSS SURVIVAL Males 583 observationsNumber Number Number Number Cumul

Intrvl Entrng Wdrawn Exposd of Propn Propn Propn Proba-Start this During to Termnl Termi- Sur- Surv bility HazardTime Intrvl Intrvl Risk Events nating viving at End Densty Rate------ ------ ------ ------ ------ ------ ------ ------ ------ ------

15 292 0 292.0 2 0.0068 0.9932 0.9932 0.0068 0.006916 290 0.0 290.0 5 0.0172 0.9828 0.9760 0.0171 0.017417 285 0.0 285.0 15 0.0526 0.9474 0.9247 0.0514 0.054118 270 1.0 269.5 23 0.0853 0.9147 0.8457 0.0789 0.089119 246 1.0 245.5 29 0.1181 0.8819 0.7458 0.0999 0.125520 216 2.0 215.0 33 0.1535 0.8465 0.6314 0.1145 0.166221 181 0.0 181.0 28 0.1547 0.8453 0.5337 0.0977 0.167722 153 0.0 153.0 33 0.2157 0.7843 0.4186 0.1151 0.241823 120 0.0 120.0 30 0.2500 0.7500 0.3139 0.1046 0.285724 90 0.0 90.0 30 0.3333 0.6667 0.2093 0.1046 0.400025 60 5.0 57.5 20 0.3478 0.6522 0.1365 0.0728 0.421126 35 31.0 19.5 4 0.2051 0.7949 0.1085 0.0280 0.2286

The median survival time for these data is 22.29

SPSS output

• 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 witha. Location of drop is random (time at event)

b. Size of drop depends on censoring

The Kaplan-Meier estimator

Kaplan-Meier estimator

-->

Survival Aalen-NelsonMonth N Death Censored of death of survival function (Cum. Rate)

1.00000 50 2 0.0400 0.9600 0.9600 0.04001 48 1 0.0208 0.9792 0.9400 0.06082 47 2 0.0426 0.9574 0.9000 0.10343 45 1 1 0.0222 0.9778 0.8800 0.12568 43 1 0.0233 0.9767 0.8595 0.1489

10 42 1 0.0238 0.9762 0.8391 0.172712 41 1 1 0.0244 0.9756 0.8186 0.197113 39 1 0.0256 0.9744 0.7976 0.222715 38 1 0.0263 0.9737 0.7766 0.249018 37 1 0.0000 1.0000 0.7766 0.249019 36 1 0.0278 0.9722 0.7551 0.276821 35 1 0.0000 1.0000 0.7551 0.276827 34 2 0.0000 1.0000 0.7551 0.276830 32 1 0.0000 1.0000 0.7551 0.276833 31 1 1 0.0323 0.9677 0.7307 0.309134 29 1 0.0345 0.9655 0.7055 0.343538 28 1 0.0000 1.0000 0.7055 0.343540 27 1 0.0000 1.0000 0.7055 0.343541 26 1 0.0385 0.9615 0.6784 0.382043 25 1 0.0000 1.0000 0.6784 0.382044 24 1 0.0000 1.0000 0.6784 0.382046 23 1 0.0000 1.0000 0.6784 0.382054 22 1 0.0000 1.0000 0.6784 0.382055 21 1 0.0476 0.9524 0.6461 0.429656 20 1 0.0500 0.9500 0.6138 0.479657 19 2 0.0000 1.0000 0.6138 0.479660 17 1 0.0000 1.0000 0.6138 0.4796

Time from diagnosis to death from melanomaor loss to follow-up, 50 subjects (Clayton and Hills, 1993, p. 36)

Conditional probability

Kaplan-Meier estimator

as surviving one monthSubjects dying during first month, are recorded

Hence probability of falure is 1/42 = 0.02381and the probability of surviving is 0.8595*(1-0.02381) = 0.8391

Number at risk is 42.

For 2 subjects, diagnosis took place at death,hence time recorded as zero

In 11th month (10 months completed), 1 death occurs.

Time in complete months

Exact time of failure and censoring are known

tiX

^

)Xi(/Di - 1 (t)S YWhere 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)

Survival function by Kaplan-Meier method

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0 5 10 15 20 25 30 35 40 45 50 55 60

Time

Su

rviv

al p

rob

ab

ility

Kaplan-Meier estimatorTime from diagnosis to death

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)

• 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 witha. 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

The Nelson-Aalen estimator

A(t) = -lnS(t)

tiX

^

)Xi(/Di (t)A Y

Cumulative rates using Aalen-Nelson method

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0 5 10 15 20 25 30 35 40 45 50 55 60

Time (months)

Cu

mu

lati

ve r

ate

Clayton and Hills, 1993, p. 50

Nelson-Aalen estimator (rates)Time from diagnosis to death

Duration of job episodes (Blossfeld and Rohwer, 1995) (sorted)

ID Job Starting Ending Duration ID Job Starting Ending Duration ID Job Starting Ending Duration number time time (p. 46) number time time number time time

33 3 981 982 2 27 5 971 977 7 9 3 634 642 9110 3 672 673 2 100 4 588 594 7 27 4 773 781 9202 2 845 846 2 100 5 595 601 7 44 7 974 982 923 1 855 857 3 106 2 837 843 7 110 4 673 681 981 3 604 606 3 107 1 934 940 7 161 1 625 633 9

145 2 980 982 3 127 3 873 879 7 176 2 565 573 9175 3 981 983 3 135 2 834 840 7 194 1 844 852 9185 2 817 819 3 172 3 940 946 7 194 2 853 861 9193 5 969 971 3 193 6 976 982 7 27 2 703 712 10203 1 935 937 3 194 4 897 903 7 58 5 942 951 1044 1 802 805 4 21 1 689 696 8 59 3 973 982 1087 3 915 918 4 21 2 697 704 8 70 2 973 982 10

100 2 579 582 4 28 3 702 709 8 79 2 973 982 10100 3 583 586 4 31 4 975 982 8 96 2 772 781 10110 1 664 667 4 40 2 764 771 8 123 2 877 886 10110 2 668 671 4 49 1 975 982 8 167 1 532 541 10127 2 869 872 4 73 3 937 944 8 171 6 831 840 10170 1 552 555 4 95 4 975 982 8 173 1 654 663 10194 6 939 942 4 135 1 826 833 8 188 2 834 843 1027 6 978 982 5 194 8 973 982 10

106 1 832 836 5 9 1 591 601 11161 2 732 736 5 24 4 938 948 11177 1 700 704 5 127 1 839 849 11

7 3 730 735 6 167 2 542 552 118 1 838 843 6 3 1 688 699 12

40 3 772 777 6 3 3 730 741 1244 3 826 831 6 3 5 817 828 1244 4 832 837 6 15 1 820 831 1244 5 856 861 6 15 2 832 843 1276 1 838 843 6 24 1 700 711 1284 1 634 639 6 Duration: 'To avoid zero durations, we have added 40 1 752 763 12

153 1 849 854 6 one months to the job duration, or the observed203 2 938 943 6 duration if the episode is right censored.' (p. 46)

Duration of job episodes Duration of job episodesDuration of job episodes

201 respondents600 job episodes

Product-limit estimate of survival function (Kaplan-Meier)Exposed

Number Number to risk Survivor Sum StdIndex Time of events censored (episodes) function E/(R(R-E)) error

E R0 0 0 0 600 1.00000 0.000001 2 2 0 600 0.99667 5.574E-06 0.002352 3 5 1 597 0.98832 1.415E-05 0.004393 4 9 2 590 0.97324 2.626E-05 0.006604 5 3 0 581 0.96822 8.933E-06 0.007175 6 10 1 577 0.95144 3.057E-05 0.008806 7 9 0 567 0.93634 2.845E-05 0.009997 8 6 1 557 0.92625 1.955E-05 0.010708 9 7 3 548 0.91442 2.361E-05 0.011469 10 8 1 540 0.90087 2.785E-05 0.01225

10 11 4 4 528 0.89405 1.446E-05 0.01262 Median duration: 43.03 months

Number censored = number of censored episodeswith 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

Survival function (men+women)

0.75

0.80

0.85

0.90

0.95

1.00

0 2 3 4 5 6 7 8 9 10 11 12

S Lower Upper

Product-limit estimate of survival function (Kaplan-Meier) and 95% interval

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)

# ehc5.cf Kaplan-Meier estimationnvar( dfile = rrdat.1, # data file ID [3.0] = c1, # identification number SN [2.0] = c2, # spell number TS [3.0] = c3, # starting time TF [3.0] = c4, # ending time SEX [2.0] = c5, # sex (1 men, 2 women) TI [3.0] = c6, # interview date TB [3.0] = c7, # birth date TE [3.0] = c8, # entry into labor market TMAR [3.0] = c9, # marriage date (0 if no marriage) PRES [3.0] = c10, # prestige of current job PRESN [3.0] = c11, # prestige of next job EDU [2.0] = c12, # highest educational attainment

# define additional variables DES [1.0] = if eq(TF,TI) then 0 else 1, TFP [3.0] = TF - TS + 1,);

edef( # define single episode data ts = 0, # starting time tf = TFP, # ending time org = 0, # origin state des = DES, # destination state);ple = ehc5.ple; estimates are written to output file ehc5.ple

Product-limit estimate of survival function (Kaplan-Meier)

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 MeanSN 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.pleProduct-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

# Number Number Exposed Survivor Std. Cum. # ID Index Time Events Censored to Risk Function Error Rate 0 0 0.00 0 0 600 1.00000 0.00000 0.00000 0 1 2.00 2 0 600 0.99667 0.00235 0.00334 0 2 3.00 5 1 597 0.98832 0.00439 0.01175 0 3 4.00 9 2 590 0.97324 0.00660 0.02712 0 4 5.00 3 0 581 0.96822 0.00717 0.03230 0 5 6.00 10 1 577 0.95144 0.00880 0.04978 0 6 7.00 9 0 567 0.93634 0.00999 0.06578 0 7 8.00 6 1 557 0.92625 0.01070 0.07661 0 8 9.00 7 3 548 0.91442 0.01146 0.08947 0 9 10.00 8 1 540 0.90087 0.01225 0.10439 0 10 11.00 4 4 528 0.89405 0.01262 0.11200 0 11 12.00 24 0 524 0.85310 0.01455 0.15888 0 12 13.00 8 1 499 0.83942 0.01510 0.17504 0 13 14.00 10 3 488 0.82222 0.01574 0.19575 0 14 15.00 6 1 477 0.81188 0.01610 0.20841 0 15 16.00 4 0 471 0.80498 0.01633 0.21694 0 16 17.00 9 0 467 0.78947 0.01681 0.23640 0 17 18.00 6 0 458 0.77913 0.01711 0.24958 0 18 19.00 8 0 452 0.76534 0.01749 0.26744

0 19 20.00 9 1 443 0.74979 0.01789 0.28797

Blossfeld and Rohwer, 1995, p. 69

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

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 ---- ---- ---- ---- ---- ----

continued Deaths in first years of life, Kerala. Source: NFHS, 1992-93

Table 7. Estimated life table functions for the first year (in months) of life by place of residencefor 1988-92 birth cohort, Kerala.

Interval in Months

Survival function S(x) Prob density function f(x) Hazard rate (x) All Urban Rural All Urban Rural All Urban Rural

0 1 2 3 4 5 6 7 8 9 10 11 12

1.0000 1.0000 1.0000 0.0159 0.0058 0.0193 0.0160 0.0059 0.01950.9842 0.9942 0.9807 0.0020 0.0000 0.0027 0.0020 0.0000 0.00270.9822 0.9942 0.9780 0.0010 0.0000 0.0014 0.0010 0.0000 0.00140.9812 0.9942 0.9767 0.0026 0.0020 0.0028 0.0027 0.0021 0.00290.9786 0.9922 0.9739 0.0005 0.0000 0.0007 0.0005 0.0000 0.00070.9780 0.9922 0.9732 0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.9780 0.9922 0.9732 0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.9780 0.9922 0.9732 0.0006 0.0000 0.0007 0.0006 0.0000 0.00080.9775 0.9922 0.9724 0.0006 0.0000 0.0008 0.0006 0.0000 0.00080.9769 0.9922 0.9716 0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.9769 0.9922 0.9716 0.0012 0.0024 0.0008 0.0012 0.0024 0.00080.9757 0.9898 0.9708 0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.9757 0.9898 0.9708 0.0006 0.0000 0.0008 0.0006 0.0000 0.0008

Note: Number of children entering the interval 0-1; All: 2026; Urban: 519; Rural: 1507

Table 8. Estimated life table results for the interval between first and second birth (in months),Kerala, January 1988-February 1993

Intervalinmonths

Women Women Number Prob.of Propn. Cum. Prob.entering Women exposed of second surviv propn density Hazardthe intv censored to risk births births ing surv function rate

9-1010-1111-1212-1313-1414-1515-1616-1717-1818-1919-2020-2121-2222-2323-2424-2525-2626-2727-2828-2929-3030-3131-3232-3333-3434-3535-3636-3737-3838-3939-4040-4141-4242-4343-4444-4545-4646-4747-48 48+

1408 16 1400.0 3 0.0021 0.9979 1.0000 0.0021 0.00211389 15 1381.5 6 0.0043 0.9957 0.9979 0.0043 0.00441368 11 1362.5 7 0.0051 0.9949 0.9935 0.0051 0.00521350 13 1343.5 10 0.0074 0.9926 0.9884 0.0074 0.00751327 11 1321.5 20 0.0151 0.9849 0.9811 0.0148 0.01521296 6 1293.0 16 0.0124 0.9876 0.9662 0.0120 0.01251274 11 1268.5 14 0.0110 0.9890 0.9543 0.0105 0.01111249 18 1240.0 21 0.0169 0.9831 0.9437 0.0160 0.01711210 11 1204.5 24 0.0199 0.9801 0.9277 0.0185 0.02011175 8 1171.0 15 0.0128 0.9872 0.9093 0.0116 0.01291152 15 1144.5 24 0.0210 0.9790 0.8976 0.0188 0.02121113 9 1108.5 23 0.0207 0.9793 0.8788 0.0182 0.02101081 8 1077.0 27 0.0251 0.9749 0.8606 0.0216 0.02541046 10 1041.0 31 0.0298 0.9702 0.8390 0.0250 0.03021005 14 998.0 37 0.0371 0.9629 0.8140 0.0302 0.0378 954 16 946.0 32 0.0338 0.9662 0.7838 0.0265 0.0344 906 9 901.5 28 0.0311 0.9689 0.7573 0.0235 0.0315 869 13 862.5 28 0.0325 0.9675 0.7338 0.0238 0.0330 828 6 825.0 23 0.0279 0.9721 0.7100 0.0198 0.0283 799 9 794.5 25 0.0315 0.9685 0.6902 0.0217 0.0320 765 3 763.5 24 0.0314 0.9686 0.6685 0.0210 0.0319 738 5 735.5 29 0.0394 0.9606 0.6474 0.0255 0.0402 704 12 698.0 18 0.0258 0.9742 0.6219 0.0160 0.0261 674 8 670.0 20 0.0299 0.9701 0.6059 0.0181 0.0303 646 9 641.5 23 0.0359 0.9641 0.5878 0.0211 0.0365 614 12 608.0 17 0.0280 0.9720 0.5667 0.0158 0.0284 585 9 580.5 14 0.0241 0.9759 0.5509 0.0133 0.0244 562 4 560.0 16 0.0286 0.9714 0.5376 0.0154 0.0290 542 7 538.5 19 0.0353 0.9647 0.5222 0.0184 0.0359 516 8 512.0 20 0.0391 0.9609 0.5038 0.0197 0.0398 488 10 483.0 18 0.0373 0.9627 0.4841 0.0180 0.0380 460 5 457.5 11 0.0240 0.9760 0.4661 0.0112 0.0243 444 3 442.5 22 0.0497 0.9503 0.4549 0.0226 0.0510 419 6 416.0 19 0.0457 0.9543 0.4323 0.0197 0.0467 394 7 390.5 12 0.0307 0.9693 0.4125 0.0127 0.0312 375 7 371.5 19 0.0511 0.9489 0.3998 0.0204 0.0525 349 8 345.0 6 0.0174 0.9826 0.3794 0.0066 0.0175 335 5 332.5 6 0.0180 0.9820 0.3728 0.0067 0.0182 324 8 320.0 13 0.0406 0.9594 0.3661 0.0149 0.0415 303 54 276.0 249 --- --- --- --- ---

The fetal life table

• 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)

Goldhaber and Fireman (1991)

Fetal life table by gestational weeks (LMP), 1981-1982, California, USAGestational Foetuses entering Spontaneous Ectopic Induced Live Withdrawal Number of pregnancies still Risk set

weeks (LMP) during week foetal loss pregnancy abortion birth in progress at end5 459 1 1 3 0 0 454 459.06 1313 6 1 24 0 0 1736 1767.07 1249 16 4 152 0 0 2813 2985.08 922 34 4 359 0 0 3338 3735.09 792 44 5 367 0 2 3712 4129.0

10 725 62 3 284 0 0 4088 4437.011 638 60 4 244 0 4 4414 4724.012 517 68 1 136 0 1 4725 4930.513 458 42 0 71 0 5 5065 5180.514 319 27 1 43 0 5 5308 5381.515 246 26 0 29 0 3 5496 5552.516 187 24 0 24 0 6 5629 5680.017 153 12 1 8 0 7 5754 5778.518 102 11 2 11 0 5 5827 5853.519 116 16 0 9 1 5 5912 5940.520 102 10 0 6 1 2 5995 6013.021 78 8 0 1 0 6 6058 6070.022 87 7 0 2 1 3 6132 6143.523 58 1 0 0 3 6 6180 6187.024 53 2 0 1 2 3 6225 6231.525 44 8 0 0 3 3 6255 6267.526 58 3 0 0 3 2 6305 6312.027 38 2 0 0 2 6 6333 6340.028 39 4 0 0 5 3 6360 6370.529 34 0 0 0 7 3 6384 6392.530 38 4 0 0 7 3 6408 6420.531 32 2 0 0 14 2 6422 6439.032 39 3 0 0 19 4 6435 6459.033 34 3 0 0 34 3 6429 6467.534 23 1 0 0 43 3 6405 6450.535 29 2 0 0 75 0 6357 6434.036 27 0 0 0 162 2 6220 6383.037 18 2 0 0 304 2 5930 6237.038 11 1 0 0 727 2 5211 5940.039 10 3 0 0 1452 0 3766 5221.040 4 5 0 0 2101 2 1662 3769.041 2 2 0 0 1026 0 636 1664.042 1 0 0 0 477 0 160 637.043 0 0 0 0 134 0 26 160.044 0 0 0 0 26 0 0 26.0

Total 9055 522 27 1774 6629 103

Risk set = foetuses at risk at the beginning of the week minus half of withdrawals during weekGoldhaber, 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, USARisk set prob of prob of prob of prob of sum of

spont. death induced abortion live birth survival probs459.0 0.004 0.007 0.000 0.989 1.000

1767.0 0.004 0.014 0.000 0.982 1.0002985.0 0.007 0.051 0.000 0.942 1.0003735.0 0.010 0.096 0.000 0.894 1.0004129.0 0.012 0.089 0.000 0.899 1.0004437.0 0.015 0.064 0.000 0.921 1.0004724.0 0.014 0.052 0.000 0.934 1.0004930.5 0.014 0.028 0.000 0.958 1.0005180.5 0.008 0.014 0.000 0.978 1.0005381.5 0.005 0.008 0.000 0.986 1.0005552.5 0.005 0.005 0.000 0.990 1.0005680.0 0.004 0.004 0.000 0.991 0.9995778.5 0.002 0.001 0.000 0.996 0.9995853.5 0.002 0.002 0.000 0.995 1.0005940.5 0.003 0.002 0.000 0.995 1.0006013.0 0.002 0.001 0.000 0.997 1.0006070.0 0.001 0.000 0.000 0.998 1.0006143.5 0.001 0.000 0.000 0.998 1.0006187.0 0.000 0.000 0.000 0.999 1.0006231.5 0.000 0.000 0.000 0.999 1.0006267.5 0.001 0.000 0.000 0.998 1.0006312.0 0.000 0.000 0.000 0.999 1.0006340.0 0.000 0.000 0.000 0.999 1.0006370.5 0.001 0.000 0.001 0.998 1.0006392.5 0.000 0.000 0.001 0.999 1.0006420.5 0.001 0.000 0.001 0.998 1.0006439.0 0.000 0.000 0.002 0.997 1.0006459.0 0.000 0.000 0.003 0.996 1.0006467.5 0.000 0.000 0.005 0.994 1.0006450.5 0.000 0.000 0.007 0.993 1.0006434.0 0.000 0.000 0.012 0.988 1.0006383.0 0.000 0.000 0.025 0.974 1.0006237.0 0.000 0.000 0.049 0.951 1.0005940.0 0.000 0.000 0.122 0.877 1.0005221.0 0.001 0.000 0.278 0.721 1.0003769.0 0.001 0.000 0.557 0.441 1.0001664.0 0.001 0.000 0.617 0.382 1.000637.0 0.000 0.000 0.749 0.251 1.000160.0 0.000 0.000 0.838 0.163 1.00026.0 0.000 0.000 1.000 0.000 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 riskRate (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

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.

= risk set