MST 564 Statistical Reliability [Kebolehpercayaan Statistik]

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1 UNIVERSITI SAINS MALAYSIA Firs Semeser Examinaion 2009/2010 Academic Session November 2009 MST 564 Saisical Reliabiliy [Kebolehpercayaan Saisik] Duraion : 3 hours [Masa : 3 jam] Please check ha his examinaion paper consiss of THIRTEEN pages of prined maerial before you begin he examinaion. [Sila pasikan bahawa keras peperiksaan ini mengandungi TIGA BELAS muka sura yang berceak sebelum anda memulakan peperiksaan ini.] Insrucions: [Arahan: Answer all four [4] quesions. Jawab semua empa [4] soalan.] In he even of any discrepancies, he English version shall be used. [Sekiranya erdapa sebarang percanggahan pada soalan peperiksaan, versi Bahasa Inggeris hendaklah diguna pakai]. 2/-

2 2 [MST 564] 1. (a) There are four relaed probabiliy funcions in describing lifeime disribuions. Each of hese funcions can be used o compue reliabiliies, bu hey offer four differen perspecives. Specifying any one of hese funcions will uniquely and compleely characerize he failure or survival process. Name he four funcions. Describe each of he funcions. Give an example o illusrae he funcions. The bahub curve characerizes human moraliy and engineering failures. Summarize he disinguishing feaures of he bahub curve. Draw a picure o illusrae he curve. A simplified form of he bahub curve is based upon linear and consan hazard raes: () c c0 c 1 0 c c 0 0 c1 c ( ) Find R ( ). [100 marks] 2. (a) For a hree-parameer Weibull disribuion wih 1.54 hr, 8500, and 0 50 hr, find he following: R (150 hr) MTTF median (iv) The design life for a reliabiliy of /-

3 3 [MST 564] 1. (a) Terdapa empa fungsi kebarangkalian yang berkaian dalam menghuraikan aburan-aburan masahaya. Seiap fungsi ini dapa mengira kebolehpercayaan, eapi mereka memberi perspekif yang berbeza. Memperincikana sebarang fungsi ini akan mencirikan proses gagal aau hidup secara unik dan lengkap. Namakan empa fungsi ini. Huraikan seiap fungsi ini. Berikan sau conoh unuk mengilusrasi fungsi ini. Lengkungan bahub mencirikan moralii manusia dan kegagalan kejurueraan. Perihalkan ciri-ciri pening bagi lengkungan bahub. Lukis suau gambar unuk mengilusrasi lengkungan iu. Sau benuk yang mudah bagi lengkungan bahub adalah berdasarkan pada kadar-kadar bahaya linear dan raa: () Cari R ( ). c c0 c 1 0 c c 0 0 c1 c ( ) [100 markah] 2. (a) Bagi suau aburan Weibull iga-parameer dengan 1.54 jam, 8500, dan 0 50 jam, cari yang beriku: R (150 jam) MTTF median (iv) Haya rekabenuk bagi suau kebolehpercayaan /-

4 4 [MST 564] The failure rae on a new brake drum design is esimaed o be 6 4 ( ) exp(10 ) per se, where is in kilomeers of normal driving. Fory vehicles are each es-driven for 15,000 km. How many failures are expeced, assuming ha he vehicles wih failed drives are removed from he es? Wha is he probabiliy ha more han wo vehicles will fail? Skech he survival funcion for an exponenial random variable wih mean ime o failure of 10 hours. Deermine he probabiliy ha a componen having his survival funcion will operae a leas 20 hours before failure. [100 marks] 3. (a) The survival imes (in days afer ransplan) for he original n 69 members of he Sanford Hear Transplan Program (see Crowley and Hu, 1977) were as follows: Table 2. Survival Time Afer Hear Transplan (Days) (Source: Crowley, J. and Hu, M. (1977). The Covariance Analysis of Hear Transplan Daa. Journal of he American Saisical Associaion, 72, 27-36) Obain he bes fi parameers for he fied disribuion. Wha are your conclusions abou he survival disribuion of hese paiens?...5/-

5 5 [MST 564] Kadar kegagalan bagi rekabenuk brake drum yang baru dianggar 6 4 ( ) exp(10 ) bagi sau se, di mana adalah dalam kilomeer bagi pemanduan biasa. Empa puluh buah kenderaan dipandu uji unuk 15,000 km. Berapa banyak kegagalan dijangka, diberi andaian bahawa kenderaan yang gagal dikeluarkan dari ujian? Apakah kebarangkalian bahawa lebih daripada dua kenderaan akan gagal? Lakar fungsi survival bagi pemboleh ubah rawak eksponen dengan min masa kegagalan (MTTF) 10 jam. Dapakan kebarangkalian bahawa sebuah komponen yang mempunyai fungsi survival ini akan berfungsi sekurang-kurangnya 20 jam sebelum kegagalan. [100 marks] 3. (a) Masa-masa survival (dalam hari selepas pemindahan) bagi n 69 pesaki yang asal unuk Program Transplan Janung Sanford (liha Crowley and Hu, 1977) adalah seperi yang beriku: Jadual 2. Masa Survival Selepas Transplan Janung (Hari) (Sumber: Crowley, J. and Hu, M. (1977). The Covariance Analysis of Hear Transplan Daa. Journal of he American Saisical Associaion, 72, 27-36) Dapakan parameer yang paling sesuai bagi aburan yang disuai. Apakah kesimpulan anda enang aburan survival bagi pesaki-pesaki ini?...6/-

6 6 [MST 564] The daa given below are survival imes T i (in monhs) afer an operaion: 8, 1, 7, 7, 4, 5, 7, 5, 6, 3 Calculae by hand he Kaplan-Meier esimae of survival S ˆ( ). Skech he Kaplan-Meier curve and mark he censored daa. According o he Kaplan-Meier esimae, how large is he probabiliy ha a paien survives a leas half a year? When do you use non-parameric or parameric mehods o esimae he survival and hazard funcions? Wha are he srenghs and weaknesses of hese wo mehods? [100 marks] 4. (a) Wha are he reasons for modeling survival daa? Describe he Proporional Hazards Model which is a basic model for survival daa. (d) Are here differences beween he Acceleraed Failure Time (AFT) Model and he usual Linear Regression Model? If here are differences, wha are hey? Discuss wo AFT models and heir implicaions for hazard funcions. Surgical reamen of ovarian cancer may be followed by a course of chemoherapy. Consider a clinical rial which involved 26 women wih minimal residual disease and who had experienced surgical excision of all umour masses greaer han 2 cm in diameer. Following surgery, he paiens were furher classified according o wheher he residual disease was compleely or parially excised. The age of he paien and heir performance saus were also recorded a he sar of he rial. The response variable was he survival ime in days following randomizaion o one or oher of he wo chemoherapy reamens: cyclophosphamide alone (single) and cyclophosphamide combined wih adriamycin (combined). The variables in he daa se are herefore as follows: TIME: Survival ime in days CENS: Censoring indicaor (0=censored, 1=uncensored) TREAT: Treamen (1=single, 2=combined) AGE: Age of paien in years RDISEASE: Exen of residual disease (1=incomplee, 2=complee) PERF: Performance saus (1=good, 2=poor) The daa are given in Table 4(d)....7/-

7 7 [MST 564] Daa yang diberi di bawah adalah masa-masa survival T i (dalam bulan) selepas suau pembedahan: 8, 1, 7, 7, 4, 5, 7, 5, 6, 3 Hiung secara manual anggaran Kaplan-Meier bagi survival ˆ( ). S Lakar lengkungan Kaplan-Meier dan andakan daa erapis. Mengiku anggaran Kaplan-Meier, berapa besar kebarangkalian bahawa seorang pesaki hidup sekurangkurangnya seengah ahun? Bilakah anda menggunakan kaedah-kaedah ak berparameer aau berparameer unuk menganggar fungsi-fungsi survival dan hazard? Apakah kebaikan dan kekurangan kedua-dua kaedah ini? [100 markah] 4. (a) Apakah sebab-sebab bagi pemodelan daa survival? Huraikan Model Bahaya Berkadaran yang merupakan model asas bagi daa survival. (d) Adakah erdapa perbezaan anara Model Masa Kegagalan Terpecu (AFT) dan Model Regresi Linear yang biasa? Jika ada, apakah perbezaannya? Bincangkan dua model (AFT) dan implikasinya bagi fungsi hazard. Rawaan surgeri bagi kanser ovari boleh diikui dengan sau kursus kemoerapi Perimbangkan suau cubaan klinikal yang melibakan 26 orang wania dengan penyaki reja minimal dan yang mengalami pemoongan surgeri bagi semua umor yang melebihi garis pusa 2 senimeer. Selepas pembedahan, pesaki-pesaki dikelaskan lagi mengiku sama ada penyaki reja iu dipoong secara lengkap aau idak lengkap. Umur pesaki dan presasinya dicaa pada awal percubaan. Pemboleh ubah sambuan ialah masa survival dalam hari mengiku proses rawakan bagi dua rawaan chemoerapi. Pemboleh ubah dalam se daa adalah seperi beriku: TIME: CENS: TREAT: AGE: RDISEASE: PERF: Masa survival dalam hari Peunjuk apis (0=erapis, 1=ak erapis) Rawaan (1=unggal, 2=kombinasi) Umur pesaki dalam ahun Sejauh mana penyaki reja merebak (1=idak lengkap, 2=lengkap) Saus presasi (1=baik, 2=lemah) Daa diberi dalam Jadual 4(d)....8/-

8 8 [MST 564] Table 4(d). Survival Times of 26 Ovarian Cancer Paiens. Paien TIME CENS TREAT AGE RDISEASE PERF (iv) In modelling he daa in Table 4(d), fi a range of models o hese daa. Which model is mos saisfacory? Give your jusificaion. Then idenify which prognosic facors are associaed wih he survival imes of he paiens. Wha is your conclusion? Is here a difference in he effec of he wo chemoherapy reamens on he hazard of deah? If here is a reamen difference, is i consisen over age? Obain he esimaed median survival ime for paiens of a given age on a given reamen. [100 marks]...9/-

9 9 [MST 564] Table 4(d). Masa Survival bagi 26 Pesaki Kanser Ovari. Paien TIME CENS TREAT AGE RDISEASE PERF (iv) Dalam pemodelan daa dalam Jadual 4(d), suaikan beberapa model kepada daa ini. Model manakah yang paling memuaskan? Beri jusifikasi anda. Kemudian kenalpasi fakor-fakor prognosik yang berkenaan dengan masa survival bagi pesaki. Apakah kesimpulan anda? Adakah erdapa perbezaan dalam kesan bagi dua rawaan chemoerapi pada hazard kemaian? Jika erdapa perbezaan rawaan, adakah ia konsisen dengan umur? Dapakan aggaran median masa survival bagi pesaki diberi umur dan rawaan. [100 markah]...10/-

10 10 [MST 564] APPENDIX Summary of Reliabiliy Formulae F( ) f ( ) d 0 R( ) 1 F( ) df( ) dr( ) f () d d f () h () R () H( ) h( ) d 0 R() e H () H( ) ln R( ) MTTF f ( ) d R( ) d /-

11 11 [MST 564] APPENDIX (cond) Summary of Reliabiliy Formulae (cond) Lifeime following an Exponenial Disribuion: f () e F( ) 1e R() e h () H() 1 MTTF Lifeime following a Weibull Disribuion: f 1 ( ) exp F( ) 1e R() e h() 1 H () 1 MTTF 1 Design Life ( ln R) R 1/...12/-

12 12 [MST 564] APPENDIX (cond) Summary of Reliabiliy Formulae (cond) Lifeime following a Normal Disribuion: ( ) f ( ) exp F () R ( ) 1 f () h () 1 Lifeime following a Lognormal Disribuion: f ( ) exp 2 ln 2 s 2s median 1 F ( ) ln s median 1 R ( ) 1 ln s median 2 s MTTF median exp 2 R median 1R exp sz...13/-

13 13 [MST 564] - ooo O ooo -

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