Lfe Tables (Tmes) Summary... 1 Data Input... 2 Analyss Summary... 3 Survval Functon... 5 Log Survval Functon... 6 Cumulatve Hazard Functon... 7 Percentles... 7 Group Comparsons... 8 Summary The Lfe Tables (Tmes) procedure s desgned to analyze data contanng lfetmes or tmes untl falure, where the value of each lfetme s measured on a contnuous scale. The data may nclude censorng, n whch some falure tmes are not known exactly due to wthdrawals of tems from the test before they fal. Nonparametrc estmates of the survval and hazard functons are obtaned and plotted. Percentles are also calculated. If desred, the data for more than one group may be specfed. In such cases, a separate estmate of the survval functon for each group wll be derved. Sample StatFolo: lfetable tmes.sgp Sample Data: The fle absorbers.sgd contans the data from a lfe test on n = 38 shock absorbers, reported by Meeker and Escobar (1998). A porton of the data s shown below: Dstance Censored 6700 0 6950 1 7820 1 8790 1 9120 0 9660 1 9820 1 11310 1 11690 1 11850 1 11880 1 12140 1 28100 1 The Dstance column represents the number of klometers of use for each shock absorber when t was nspected. The Censored column contans a 0 for each absorber that had faled at the tme of nspecton and a 1 for each absorber that had not faled. All of the data contan nformaton about 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 1
the tme untl falure of the shock absorbers. For those that had faled, the observaton s a true falure tme. For those that had not faled, the observaton s a rght-censored falure tme, wth the actual tme untl falure known to be greater than the value ndcated. Data Input The data nput dalog box requests nformaton about the falure tmes and ther status: Data: numerc column wth the n observed tmes. Censored: numerc column of 0 s and 1 s. A 0 ndcates that the observaton s not censored and therefore represents a true tme untl falure. A 1 ndcates that the observaton s rght-censored, wth the falure tme known only to be greater than that ndcated. Select: subset selecton. 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 2
Analyss Summary The Analyss Summary dsplays a table showng the estmated survval and hazard functons. Lfe Tables (Tmes) Dstance Data varable: Dstance Censorng: Censored Product-Lmt (Kaplan-Meer) Estmates Number at Cumulatve Standard Cumulatve Row Tme Status Rsk Survval Error Hazard 1 6700.0 FAILED 37 0.9737 0.0260 0.0267 2 6950.0 WITHDRAWN 36 3 7820.0 WITHDRAWN 35 4 8790.0 WITHDRAWN 34 5 9120.0 FAILED 33 0.9450 0.0378 0.0565 6 9660.0 WITHDRAWN 32 7 9820.0 WITHDRAWN 31 8 11310.0 WITHDRAWN 30 9 11690.0 WITHDRAWN 29 10 11850.0 WITHDRAWN 28 11 11880.0 WITHDRAWN 27 12 12140.0 WITHDRAWN 26 13 12200.0 FAILED 25 0.9087 0.0509 0.0957 14 12870.0 WITHDRAWN 24 15 13150.0 FAILED 23 0.8708 0.0613 0.1383 16 13330.0 WITHDRAWN 22 17 13470.0 WITHDRAWN 21 18 14040.0 WITHDRAWN 20 19 14300.0 FAILED 19 0.8273 0.0720 0.1896 20 17520.0 FAILED 18 0.7838 0.0803 0.2437 21 17540.0 WITHDRAWN 17 22 17890.0 WITHDRAWN 16 23 18450.0 WITHDRAWN 15 24 18960.0 WITHDRAWN 14 25 18980.0 WITHDRAWN 13 26 19410.0 WITHDRAWN 12 27 20100.0 FAILED 11 0.7184 0.0966 0.3307 28 20100.0 WITHDRAWN 10 29 20150.0 WITHDRAWN 9 30 20320.0 WITHDRAWN 8 31 20900.0 FAILED 7 0.6286 0.1192 0.4642 32 22700.0 FAILED 6 0.5388 0.1317 0.6184 33 23490.0 WITHDRAWN 5 34 26510.0 FAILED 4 0.4311 0.1428 0.8415 35 27410.0 WITHDRAWN 3 36 27490.0 FAILED 2 0.2874 0.1511 1.2470 37 27890.0 WITHDRAWN 1 38 28100.0 WITHDRAWN 0 Mean survval tme = 22874.7 Standard error = 1304.1 The table shows each of the observatons n the data set, sorted n ncreasng order. In cases where there are both censored and uncensored observatons equal to the same value, the uncensored observatons are lsted frst. The table ncludes: Row - the row number of the tem n the fle. 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 3
Tme - the falure or removal tme for the -th tem n the lst. Note: s the row number n the dsplayed table, not the orgnal fle. Status - whether the tme represents an tem that faled or one that was wthdrawn from the test. In cases where there were both falures and wthdrawn tems at the same tme, the falures are assumed to occur mmedately before the tems are wthdrawn. Number at Rsk - the number of tems at rsk r mmedately followng the falure or removal of the -th tem. Ths number begns at n and decreases by one after each falure or removal. Cumulatve Survval - the estmated survval functon mmedately followng the falure of tem. The estmate s gven by Sˆ n j 1 j j1 n j 1 (1) where j = 1 f the -th tem faled, or j = 0 f the -th tem was wthdrawn. Ths estmator s commonly known as the product-lmt or Kaplan-Meer estmator of the survval functon. Standard Error - the standard error of the estmated cumulatve survval functon, calculated by s. e.( Sˆ ) Sˆ j n j n j j1 1 Cumulatve Hazard - the cumulatve hazard functon computed from (2) Hˆ ln( ˆ ) (3) S Mean survval tme - the estmated mean tme to falure, gven by 1 1 1 ˆ Sˆ t t (4) n where S ˆ0 1 and t 0 = 0. Standard error - the standard error of the estmated mean tme to falure, gven by where n 2 L s. e.( ˆ) (5) j n j n j 1 j1 L n j S 1 t t 1 j ˆ (6) 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 4
survval probablty The last estmate of the survval functon shown n the above table equals 28.74%, whch corresponds to the last uncensored falure tme. The functon s undefned beyond the last censored observaton. Of consderable nterest s the estmated mean tme to falure, whch equals 22,874.7 km. An approxmate 95% confdence nterval for the mean s gven by or 22,874.7 1.96(1304.1) 22,875 2,556. Ths value s consderably larger that the average of the 38 observatons, whch equals only 16,447.4 km and does not account for the censorng. Survval Functon The Survval Functon plots the estmated probablty that an tem wll survve untl tme t: 1 Estmated Survval Functon 0.8 0.6 0.4 0.2 0 0 5000 10000 15000 20000 25000 30000 Dstance It decreases accordng to a step functon, changng mmedately after each uncensored falure tme. Pane Optons Include confdence ntervals: f selected, confdence ntervals wll be added to the plot. 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 5
log survval probablty survval probablty Confdence Level: percentage of confdence for the ntervals. Example: Survval Functon wth 95% Confdence Intervals 1 Estmated Survval Functon 0.8 0.6 0.4 0.2 0 0 5000 10000 15000 20000 25000 30000 Dstance The confdence ntervals are calculated from: Sˆ z s. e.( Sˆ ) (7). 025 Log Survval Functon The Log Survval Functon s the natural logarthm of the survval functon: Estmated Log Survval Functon 0.2-0.1-0.4-0.7-1 -1.3 0 5000 10000 15000 20000 25000 30000 Dstance 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 6
cumulatve hazard Cumulatve Hazard Functon The cumulatve hazard functon at t s the ntegral of the hazard functon from 0 to t: 1.5 Estmated Cumulatve Hazard Functon 1.2 0.9 0.6 0.3 0 0 5000 10000 15000 20000 25000 30000 Dstance Percentles The Percentles pane shows a table of estmated tmes at whch gven percentages of the tem wll stll be operatng: Percentle Table Standard Percentle Estmate Error 90.0 13150.0 3181.66 75.0 20100.0 3209.75 50.0 26510.0 4054.94 25.0 Percentles are computed by fndng the frst pont at whch the estmated survval functon s less than or equal to the percentage desred. For example, the 50-th percentle equals 26,510 hours snce the estmated survvor functon falls below 0.50 at row 34. Because the estmated survval functon never falls below 0.2874, percentles below 28.74% cannot be estmated (the survval functon s undefned after 28,100 hours, when the last tem was wthdrawn from the test). Pane Optons Percentles: percentages at whch to calculate the percentles. 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 7
Group Comparsons If more than one group of data has been entered, the above tables and plots wll show separate estmates for each group. In addton, the Group Comparsons pane wll summarze the data n each group: Comparson of Groups Proporton Group Total Faled Wthdrawn Wthdrawn 1 38 11 27 0.7105 Total 38 11 27 0.7105 The table shows the total number of tems n each group, the number n each group that faled, the number n each group that were wthdrawn (censored), and the proporton of wthdrawn tems. In the current example, there s only one group. Of the n = 38 tems n that group, approxmately 71% were wthdrawn before they faled. 2013 by StatPont Technologes, Inc. Lfe Tables (Tmes) - 8