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# Analysis of Lifetime Data STAT 567

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This 265 page Class Notes was uploaded by Kamren McLaughlin on Monday October 26, 2015. The Class Notes belongs to STAT 567 at University of Tennessee - Knoxville taught by Ramon Leon in Fall. Since its upload, it has received 21 views. For similar materials see /class/229889/stat-567-university-of-tennessee-knoxville in Statistics at University of Tennessee - Knoxville.

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Date Created: 10/26/15

Unit 19 Analyzing Accelerated Life Test Data Ram on V Leon Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 10232004 Unit 19 Stat 567 Ram n Le n Unit 19 Objectives Describe and illustrate nonparametric and graphical methods of analyzing and presenting accelerated life test data Describe and illustrate maximum likelihood methods of analyzing and making inferences from accelerated life test data Illustrate different kinds of data and ALT models Discuss some specialized applications of accelerated testing 10232004 Unit 19 Stat 567 Ram n Le n Example TemperatureAccelerated Life Test on DeviceA from Hooper and Amster 1990 Data Sineg right censored observations from temperature accelerated life test Purpose To determine if the device would meet its hazard function objective at 10000 and 30000 hours at operating temperature of 10 C We will show how to fit an accelerated life regression model to these data to answer this and other questions 10232004 Unit 19 Stat 567 Ram n Le n Hours Versus Temperature Data from a TemperatureAccelerated Life Test on DeviceA Subexperunent Number Temperature In C U mts Fauures Hours Status of Devrces 5000 Censored 30 10 30 030 1298 FaHed 1 40 100 10100 1390 FaHed 1 40 5000 Censored 90 40 581 FaHed 60 20 920 5 Faued 60 1432 FaHed 60 5000 Censored 11 60 283 FaHed 1 80 15 1415 361 FaHed 1 80 515 Faued 1 80 6 23 Fatted 1 80 5000 Censored 1 80 10232004 Umt 19 r Stat 567 r Ramon Leo39n DeviceA Hours Versus Temperature Hooper and Amster 1990 50000 7 20000 10000 0 30 WUMOVJ 9120 HMS m 5000 A A 8 o I 2000 7 g 1000 7 g 500 7 39 I c a 200 0 20 40 00 80 100 10232004 Umt 19 7 8151567 7 Ramon Leon ALT Data Plot Examine a scatter plot of lifetime versus stress data Use different symbols for censored observations Note Heavy censoring makes it difficult to identify the form of the lifestress relationship from this plot 10232004 Unit 19 Stat 567 Ram n Le n Weibull Multiple Probability Plot Giving Individual Weibull Fits to Each Level of Temperature for DeviceA ALT Data 5 92 i 0 Dec C mm I l I l l l l l l 100 200 599 1000 2000 was 10000 29mm warm Hollis 10232004 Unit 19 7 8151567 7 Ramon Leo39ri Lognormal Multiple Probability Plot Giving Individual Lognormal Fits to Each Level of Temperature for DeviceA ALT Data Proporlioii Failing I l I l l l l l l 100 200 500 1000 20m 51m 10000 2mm 5mm Houis 10232004 Unit 19 7 8151567 7 Ramon Leon 8 ALT Multiple Probability Plot of Nonparametric Estimates at Individual Levels of Acceleration Variables Compute nonparametric estimates of F for each level of accelerating variable plot on a single probability plot Try to identify a distributional model that fits the data well at all of the stresslevels Note Eitherthe lognormal orthe Weibull distribution model provides a reasonable description for the deviceA data But the lognormal distribution provides a better fit to the individual subexperiments 10232004 Unit 19 Stat 567 Ram n Le n 9 ALT Multiple Probability Plot of ML Estimates at Individual Levels of Accelerating Variable For each individual level of accelerating variable compute the ML estimates Let 1 be the failure time at temperature Temp For the lognormal T N LOGNORum assumed model gt Compute ML estimates 07161 gt Plot the LOGNORmLfn cdfs i 1 2 on same plot Assess the commonly used assumption that a does not depend on Temp and that Temp only affects in Note There are some small differences among the slopes that could be due to sampling error 10232004 Unit 19 r Stat 567 r Ramon Leon DeviceA ALT Lognormal ML Estimation Results at Individual Temperatures 95 Approximate Standard Confidence Interval Parameter Estimate Error Lower Upper 40quot C i 981 42 89 U 10 27 59 172 60 C 1L 864 35 80 93 T 119 32 70 20 80 C 1L 708 21 67 75 U 80 16 55 117 The individual Ioglikelihoods were 40 711546 450 78972 and 80 711558 The confidence intervals are based on the normal approximation method Total likelihood 32076 10232004 Unit 19 7 8151567 7 Ramon Leon Strategy for Analyzing ALT Data For ALT data consisting of a number of sub experiments each having been run at a particular set of conditions Examine the data graphically Scatter and probability plots Analyze individual subexperiment data Examine a multiple probability plot Fit an overall model involving a lifestress relationship Perform residual analysis and other diagnostic checks Assess the reasonableness of using the ALT data to make the desired inferences 10232004 Unit 19 Stat 567 Ram n Le n 12 The ArrheniusLognormal Regression Model 10232004 The Arrheniuselognormal regression model is Iogltti U i Pr739temp S t n0r Where it 330 23112 I 11605temp K 11605temp 5C 27315 331 EU is the activation energy and LT assumed to be constant Unit 19 r Stat 567 r Ramon Leon Lognormal Multiple Probability Plot of the ArrheniusLognormal LogLinear Regression Model Fit to the DeviceA ALT Data mopanou Fgmug 39 w Deg c 0091 I W I W W W 100 200 590 1000 20m 5000 How 10232004 Umt 19 r Stat 567 r Ramon Leo39n Scatter Plot Showing the ArrheniusLognormal LogLinear Regression Model Fit to the DeviceA ALT Data 50000 7 1 20000 10000 5000 g 2000 7 c I1000 500 7 10 100 39 0 20 40 80 100 Degrees c on Arrhemus some 10232004 Umt 19 7 8151567 7 Ramon Leon ML Estimation Results for the DeviceA ALT Data and the ArrheniusLognormal Regression Model 95 Approximate Standard Confidence Intervals Parameter Estimate Error Lower Upper 30 7135 29 8 31 63 08 47 79 U 98 13 75 128 The Ioglikelinoocl is L 73217 The confidence intervals are based on the normal approximation method iEIZSZEIEM Unit 1 a 7 5131567 7 Ramc m Lenin Analytical Comparison of Individual and Arrhenius Lognormal Model ML Estimates of DeviceA Data Distributions fit to individual levels of temperature can be viewed 35 an unconstrained model The Arrheniuselognormal regression model can be viewed as a constrained model it linear in r and a constant Use likelihood ratio test to check for lack of fit with respect to the Constraints 39unconst 40 50 80 i32076 cor1st 3217 39 2 const Eunconst 23217 32076 188 lt V5753 41 indicating that there is no evidence of in adequacy of the constrained model relative to the uncone strained model 10232004 Unit 19 r Stat 567 r Ramon Leon l7 ALT Multiple Probability Plot of ML Estimates with an Assumed LifeStress Relationship To make inferences at levels of accelerating variable not used in the ALT use a lifestress relationship to fit all the data Let T139 be the failure time at 1391 11605Temp1 27315 For the MIA m LOGNORQI 30 1317 lognormal SAFT assumed model gt Compute ML estimates 3 18 gt Plot the LOGNORDEL o 11398 cdfs z 12 on same plot gt PIOI 7 9X9 v o E11 334108 for various values of p and a range of values of 139 lEIZSZEIEM Unit l a 7 5131567 7 Ramc m Lenin ML Estimation forthe DeviceA Lognormal Distribution F30000 at 10 C 17 o311 7 713469 6279 x 1160510 27315 122641 3 Iogt 7 018 Iog3oooo 7 1226419778 72000 3oooo mmm dmr72000 02281 Using Mi firm mzm 7 287 048 Formula quot7 Cov j vamg 048 0176 39 1711inthe textbook l Vatm 25c vm 8 2v3rltagtij t7 1 72000 1 7 286 2 x 72000 x 047 720002 x 01762 0225 10232004 Unit 19 Stat 567 Ram n Le n Con dence Interval for the DeviceA Lognormal Distribution F30000 at 10 C 5 normaleapproximation confidence interval based on the assumption that Z Og mg NORO 1 is 55 lquotltle lt1 7 I X w I ltii MI 02231 02281 1 02281 1 7 02281 x w39 02281 1 7 02281 0032 where w eXDZleq2SA fF1 H exp196 x 0225022811 7 02281 7232 This wide interval reflects sampling uncertainty when activu tion energy is unknown The interval does not reflect model uncertainty With given activation energy the confidence intervals would be much narrower 10232004 Unit 19 Stat 567 Ram n Le n Checking Model Assumptions It is important to check model assumptions by using residual analysis and other model diagnostics I Define Standardized residuals BS 0 t exp groan I U 73111 Residuals corresponding to censored observations are called censored standardized residuals where tr is a failure time at 1 Plot residuals versus the fitted values given by exp 30 31 Do a probability plot of the residuals Note For the DeviceeA data these plots do not conflict with the model assumptions 10232004 Unit 19 r Stat 567 r Ramon Leon 21 Plot of Standardized Residuals Versus Fitted Values for the Arrhenius Lognormal LogLinear Regression Model Fit to the DeviceA ALT Data Sta nziamIzezi Resldmls lEIZSZEIEM 500 A 200 I lquot c 050 0207 010 l 0115 002 A I I I I I I ii I I V 103 2x103 5x103 10 2x10 5x10 10 2x105 5x10 Fitted Values um i a r Stat 5B7 r Ramc m Lenin Probability Plot of the Residuals from the ArrheniusLognormal Log 39 D t 10232004 Inear Regression Model Fit tothe DeviceA ALT a a I 005 010 929 050 mu 299 SEN Staudmmzed Resmnms Umt 19 7 8151567 7 Ramon Leo39n Some Practical Suggestions Build on previous eltperience with similar products and materials Use pilot experiments evaluate the effect of stress on degradation and life Seek physical understanding of causes of failure Use results from failure mode analysis Seek physical justification for lifestress relationships Design tests to limit the amount of extrapolation needed for desired inferences See Nelson 1990 10232004 Unit 19 Stat 567 Ram n Le n 24 Inferences from AT Experiments Inferences or predictions from ATS require important assumptions about Focused correctly on relevant failure modes Adequacy of AT model for extrapolation AT manufacturing testing processes can be related to actual manufacturinguse of product Important sources of variability usually ovedooked Deming would call ATs analytic studies See Hahn and Meeker 1993 American Statistician 10232004 Unit 19 Stat 567 Ram n Le n 25 Breakdown Times in Minutes of a MylarPolyurethane Insulating Structure from Kalkanis and Rosso 1989 Minutes Mylarpolyid I I I I I I I WOO 150 200 250 300 350 400 kVmm 10232004 Unit 19 r Stat 567 r Ramon Leo39ri 26 Accelerated Life Test of a MylarPolyurethane Laminated Direct Current High Voltage Insulating Structure Data from Kalkanis and Rosso 1989 Time to dielectric breakdown of units tested at 1003 1224 1571 2190 and 3614 kVmm Needed to evaluate the reliability of the insulating structure and to estimate the life distribution at system design voltages eg 50 kVmm Except for the highest level of voltage the relationship between log life and log voltage appears to be approximately linear Failure mechanism probably different at 3614 kVmm 10232004 Unit 19 Stat 567 Ram n Le n 27 Lognormal Probability Plot ofthe Individual Tests is the MylarPolyurethane ALT Minuies 10232004 Unit 19 r Stat 567 r Ramon Leo39i i Inverse Power RelationshipLognormal Model The inverse power reIationshipilognormaI model is lo I 7 Fr PrTvolt g p mm M I where u 80 811 and 1 IogVoltage Stress I 039 assumed to be constant mzaznm Umt 19 7 5131567 7 Ram n Latin 29 Lognormal Probability Plot ofthe Inverse Power RelationshipLognormal Model Fitted to the Mylar Polyurethane Data Including 3614 kVmm tagWm t w a 105 u Mmutes 10232004 Umt 19 r Stat 567 r Ramon Leo39n 30 Plot of Inverse Power RelationshipLognormal Model Fitted to the MylarPolyurethane Data Including 3614 lemm 103 107i 105g 1051 ES 104 h nut 103i 102 10H I 3910 1 10391 06 f1 13 pk 10232004 I quotlquotquotl quotI 20 ED 100 EDD kMWnnwmnlegscam Un 19Swt567Ram nLe n 31 Lognormal Plot of the Standardized Residuals versus expo for the Inverse Power Relationship Lognormal Model Fitted t0 the Mylar7Polyurethane Data with the 3614 kVmm Data 200 7 l 00 7 3 50 7 g 20 7 S 5 I g 1 O I l n E x g 05 7 39 02 7 01 w w w w w 1 5 20 100 500 2000 FMed Vames 10232004 Unit 19 Stat 567 Ram n Le n 32 Lognormal Probability Plot of the Inverse Power Relationship Lognormal Model Fitted to the MylarPolyurethane Data Without the 3614 kVImm Data mopomou Famng w Wm A D 10232004 Umt 19 r Stat 567 r Ramon Leo39n 33 Plot of Inverse Power RelationshipLognormal Model Fitted to the MylarPolyurethane Data also showing 3614 kVmm Data Omitted from the ML Estimation 108 1071 logUP ampCDrpamp 105 1051 1041 Minutes 1031 1 r33 1 a 1 E 1 U 1 502 1091 1 0393 1 104 quot 2E 50 1 EH 2130 500 kit n m on log scale 10232004 Unit 19 Stat 567 Ram n Le n 34 Inverse Power RelationshipLognormal Model ML Estimation Results for the MylarPolyurethane ALT Data 95 Approximate Standard Confidence Intervals ML Parameter Estimate Err r Upper do 30 216 Si 7429 60 7546 7311 r 105 12 83 132 The loglikelihoocl is E 72714 The confidence intervals are based on the normal approximation method 10232004 Unit 19 7 8151567 7 Ramon Leon Lognormal Plot of the Standardized Residuals versus expm for the Inverse Power RelationshipLognormal Model Fitted t0 the Mylar Polyurethane Data without the 3614 kVmm Data 100 7 50 E 20 7 2 E 39 g 1 g 05 7 2 F7 I 02 39 01 7 50 100 200 500 1000 2000 5000 Fmed Vames 10232004 Unit 19 Stat 567 Ram n Le n Analysis of Interval ALT Data on a NewTechnology lC Device Test were run at 150 175 200 250 and 300 C Developers interested in estimating activation energy of the suspected failure mode and the longlife reliability Failure had been found only at the two higher temperatures After early failures at 250 and 3000 C there was some concern that no failures would be observed at 175 C before decision time Thus the 200 C test was started later than the others 10232004 Unit 19 Stat 567 Ram n Le n 37 NewTechnology IC Device ALT Data Hours Number of Iempera ure Lower Upper Status Devices C 1536 Right Censored 50 150 1536 Right Censored 50 175 96 Right Censored 50 200 384 788 Failed 1 250 788 1536 Failed 3 250 1536 2304 Failed 5 250 2304 Right Censored 41 250 192 384 Failed 4 300 384 788 Failed 27 300 788 1536 Failed 16 300 1536 Right Censored 3 300 cdevice02ld 10232004 Unit 19 7 8151567 7 Ramon Leon Lognormal Probability Plot ofthe Failures at 250 and 300 C for the NewTechnology Integrated Circuit Device ALT Experiment Piopoiiiou ngiiiig I 100 249 509 100 2000 5ku was Houis 10232004 Unit 19 Stat 567 Ram n Le n 39 Individual Lognormal ML Estimation Results forthe NewTechnology IC Device 95 Approximate ML Standard Confidence Intervals Parameter Estimate Error Lower Upper 2509C 854 33 9 a 87 26 48 157 300 C p 656 07 64 67 a 46 05 36 58 The Ioglikeiihood were 250 73216 and 1300 75385 The confidence intervals are based on the normal approximation method 10232004 Unit 19 r Stat 567 r Ramon Leon SPLIDA LogLikeIihood Output Maximum likelihood estimation results Response units Hours Lognormal Distribution Log likelihood l lEDDagraasE HA 2 l5DagraasE HA 3 EUDDagraasC HA 4 ESDDEgraasE 5 EUDDagraasE 5385 Total log likelihood 86 39H I 10232004 mu HA HA HA samu HA HA HA sigma sasigma HA HA NA 3215 8540 0333TT 08Tl 6553 DDTDTE 045T2 Un 19Swt567Ram nLe n HA HA HA 026219 00548 41 Lognormal Probability Plot Showing the Arrhenius Lognormal Model ML Estimation Results for the New Technology IC Device Hours 10232004 UmHQVStat567rRamo39n Leo39ri ArrheniusLognormal Model ML Estimation Results for the NewTechnology lC Device 95 Approximate Standard Confidence Intervals Error Lower pper ML Para meter Estimate 7 02 to 7132 772 31 83 07 68 97 a 52 06 42 64 The Ioglikelihood is LT 78836 Comparing the constrained and unconstrained models L39ucomst 250L300 78601 and for the constrained model Liconst 78836 The comparison has just one degree of freedom and 72l78836 8601 47 gt V2 95 L 384 again indicating that there is some lack of fit in the constantm Arrheniuse Iognormal model 10232004 Unit 19 Stat 567 Ram n Le n SPLIDA Output Lag likelihaad at maximum paint 0535 Parameter Approx Canf Interval HLE StdErr 95 Lamer 95 Upper Intercept 101T15 15259 13154T T190 giDegreeaE 00255 00T319 05531 09200 sigma 05155 00524 04153 05424 10232004 Unit 19 Stat 567 Ram n Le n Arrhenius Plot Showing ALT Data and the ArrheniusLognormal Model ML Estimation Results for the NewTechnology IC Device log it 0 I ltIgt pm Hours 1 A A i o3 7 5m m 1 A we 60 o 250 300 350 Degiees c on Armanms scaie mZaZDDA Unit 19 r Stat 5B7 r Ramc m Latin 45 Lognormal Probability Plot Showing the Arrhenius Lognormal Model ML Estimation Results for the m i c 3 o 1 E 10232004 Hows Unit 19 r Stat 567 r Ramon Leo39n I could not nd how to do this plot in the SPLIDA GUI Can use echapter 19 New Techno ogy Devwce ALT Mode MLE DegreesCArrhemus Dwst Lognorma Lognorma Probabmty P ot Frachnn Fa ng snoegyeesc mu Degveesc 1 nzaznm mm mm mm mm mm Umt 19 7 5131567 7 Ram n Lean mm mm Pitfall 1 Multiple Unrecognized Failure Modes High levels of accelerating factors can induce failure modes that would not be observed at normal operating conditions or othenNise change the lifeacceleration factor relationship Other failure modes if not recognized in data analysis can lead to incorrect conclusions Suggestions Determine failure mode of failing units Analyze failure modes separately 10232004 Unit 19 Stat 567 Ram n Le n 48 Pitfall 2 Failure to Properly Quantify Uncertainty There is uncertainty in statistical estimates Standard statistical confidence intervals quantify uncertainty arising from limited data Confidence intervals ignore model uncertainty which can be tremendously amplified by extrapolation in Accelerated Testing Suggestions Use confidence intervals to quantify statistical uncertainty Use sensitivity analysis to assess the effect of departures from model assumptions and uncertainty in other inputs 10232004 Unit 19 Stat 567 Ram n Le n 49 Pitfall 3 Multiple Time Scales Composite material Chemical degradation over time changes material ductility Stress cycles during use lead to initiation and growth of cracks Incandescent light bulbs Filament evaporates during burn time Onoff cycles induce thermal and mechanical shocks that can lead to fatigue cracks lnkjet pen Real time corrosion Characters or pages printed ink supply resistor degradation Onoff cycles of a print operation thermal cycling of substrate and print head lamination 10232004 Unit 19 Stat 567 Ram n Le n 50 Dealing with Multiple Time Scales Suggestions Need to use the appropriate time scales for evaluation of each failure mechanism With multiple time scales understand ratio or ratios of time scales that arise in actual use Plan ATs that will allow effective prediction of failure time distributions at desired ratio or ratios of time scales 10232004 Unit 19 Stat 567 Ram n Le n 51 Possible Results for a Typical Temperature Acceleration Failure Mode on an IC Device Hours 40 so so 100 120 140 Degrees c 10232004 Umt 19 7 8151567 7 Ramon Leo39n 52 Unmasked Failure Mode with Lower Activation Energy 105 1057 m 1047 3 0 Made 2 I 103 NW M2 Madii 1m 10 7 i 40 so so i 00 120 i 40 Degrees c 10232004 Unit 19 7 8151567 7 Ramon Leo39n Pitfall 4 Masked Failure Mode Accelerated test may focus on one known failure mode masking another Masked failure modes may be the first one to show up in the field Masked failure modes could dominate in the field Suggestions Know anticipate different failure modes Limit acceleration and test at levels of accelerating variables such that each failure mode will be observed at two or more levels of the accelerating variable Identify failure modes of all failures Analyze failure modes separately 10232004 Unit 19 Stat 567 Ram n Le n 54 Hours 10232004 Comparing of Two Products I Simple Comparison 105 1057 10 103 mm 102 10quot 10quot vim 10 y 40 so so 100 120 140 Degrees c Umt 19 7 8151567 7 Ramon Leo39n 55 Hours 10232004 Comparison of Two Products Questionable Comparison mam w Vendar2 an i i i 40 so so 100 120 140 Degrees o Unit 19 7 8151567 7 Ramon Leon 56 Pitfall 5 Faulty Comparison It is sometimes claimed that accelerated testing is not useful for predicting reliability but is use ul for comparing alternatives Comparisons can however also be misleading Beware of comparing products that have different kinds of failures Suggestions Know anticipate different failure modes Identify failure modes of all failures Analyze failure modes separately Understand the physical reason for any differences 10232004 Unit 19 Stat 567 Ram n Le n 57 Pitfall 6 Acceleration Factors Can Cause Deceleration Increased temperature in an accelerated circuitpack reliability audit resulted in fewer failures than in the field because of lower humidity in the accelerated test Higher than usual use rate of a mechanical device in an accelerated test inhibited a corrosion mechanism that eventually caused serious field problems Automobile air conditioners failed due to a material drying out degradation lack of use in winter not seen in continuous accelerated testing lnkjet pens fail from infrequent use Suggestion Understand failure mechanisms and how they are affected by experimental variables 10232004 Unit 19 Stat 567 Ram n Le n 58 Pitfall 7 Untested DesignProduction Changes Leadacid battery cell designed for 40 years of service New epoxy seal to inhibit creep of electrolyte up to the positive post Accelerated life test described in published article demonstrated 40 year life under normal operating conditions 200000 units in service after 2 years of manufacturing First failure after 2 years of service third and fourth failures followed shortly thereafter Improper epoxy cure combined with chargedischarge cycles hastened failure Entire population had to be replaced with a redesigned cell 10232004 Unit 19 Stat 567 Ram n Le n 59 TemperatureVoltage ALT Data on Tantalum Electrolytic Capacitors Twofactor ALT Nonrectangular unbalanced design Much censoring The Weibull distribution seems to provide a reasonable model for the failures at those conditions with enough failures to make a judgment Temperature effect is not as strong Data analyzed in SingpunNalla Castellino and Goldschen 1975 10232004 Unit 19 Stat 567 Ram n Le n 60 10232004 Unit 19 Stat 567 Ram n L e n Table 16 Temperature Electrolytic Capacium Stalm and Voll dge ccclcmted IAil39e Number 01 Dcvvcas l 1 l 006 1 Test Data for Tanlalum Ismpcmnnc 3 r r mch NFAPI cccmm Tantalum Capacitors ALT Design Showing Fraction Failing at Each Point Degrees C 10232004 100 41000 4200 250 453 0 I a o 80 i 60 6502 150 I I 40 e 20 1175 18M 74 O I 0 I I I I I I 35 40 45 50 55 60 65 Volts Unit 19 Stat 567 Ram n Le n Scatter Plot of Failures in the Tantalum Capacitors ALT Showing Hours to Failure Versus Voltage with Temperature Indicted hv Different Svmhols ltgt 5 Degrees c O 45 Degrees C om Gen III 85 Degrees c e 995 cen 196 can me new 49 new 49 cen l66 new 105 D D 9 4 10 D O D 0 g D D D O 3 10 7 g quot I g E I 102 e D O ltgt 0 1 El El 0 10 100 e r r I 20 30 40 50 60 70 Volts 10232004 Unit 19 Stat 567 Ramon Leon Could not get this plot in SPLIDA Weibull Probability Plot for the Individual Voltage and Temperature Level Combination for the Tantalum Capacitors ALT with ML Estimate of Weibull cdfs 2 O asoegcasv S asoegcmav ii x BSDegCJGSV o 5099 l 05 X J 03 r g 02 r E K at r o E g 005 r 8 I1 003 r 001 0005 r 0003 l l l l I 10 50 200 500 2000 5000 20000 50000 How rs 10232004 Unit 19 Stat 567 Ramon Leon Tantalum Capacitors ALT WeibullArrheniuslnverse Power Relationship Models Model 12 p 30 81371 532 E2 Model 22 p 30 3111 i322 631F112 where m1 Ogvolt 12 2 11605temp K and 332 Ea o Coefficients of the regression model are higth sensitive to whether the interaction term is included in the model or not because of the nonrectangular design with higth Lin balanced allocation Data provide no evidence of interaction 0 Strong evidence for an important voltage effect on life 10232004 Unit 19 Stat 567 Ramon Leon 65 Tantalum Capacitor ALT WeibuIIInverse Power Relationship Regression ML Estimation Results 95 Approximate M Standard Confidence Interval Parameter Estimate Error Upper Model 1 844 136 578 111 31 4201 44 7288 4114 32 33 19 704 69 7 233 36 172 316 Model 2 60 7786 1090 72923 1351 31 199 267 7325 7235 52 513 33 7135 116 33 7117 80 728 40 a 233 36 172 316 Loglikelihoods 1 753963 and Q 753840 10232004 UanQyStat567yRamo l l Leon 10232004 Tantalum data Maximum likelihood estimation results Response units Hours Heibull Distribution variable Relationship l Volts Log 2 DegreesC Arrhenius eV 3920 Model formula Location gtVolts gtDegreesC Log likelihood at maximum point 5395 Parameter Approx Conf Interval HLE StdErr 95 Lower 95 Upper Intercept 844501 1359442 5380550 111094 goltsj 200941 444373 2351153 ll3T5T g DegreesE 03251 013351 004153 05935 sigma 3332quot 035943 1T3451 31552 meihullheta 0425 00550 031594 05399 Unit 19 Stat 567 Ram n Le n 67 10232004 Tantalum data Maximum likelihood estimation results Response units Hours Ueihull Distribution Variable Relationship g 1 0015 Log 2 DegreesC Arrhenius a Model formula Location go1ts g DegreesE g o1tsgDegreesC Log likelihood at maximum point 5304 Parameter Approx Conf Interve 1 HLE 5tdErr 95 Lower 95 Upper Intercept T06244 10902310 2923136 1350640 gtvoltsl 199150 265549 325240 323540 gEDegreesCJ 51265 330309 13569 116090 gvo1tsgDegreesC 11363 000306 27510 03993 sigma 23316 036100 1202 31604 meihu11heta 04209 006655 03164 05013 Unit 19 Stat 567 Ram n Le n 68 Weibull Multiple Probability Plot of the Tantalum Capacitor ALT Data ArrheniusInverse Power Relationship Weibull Model with no Interaction ltIOXO 30 Deg 2 v 10232004 Unit 19 Stat 567 Ram n Le n 69 Hours ML Estimates of mforthe Tantalum Capacitor Life Using the ArrheniusInverse Power Relationship Weibull Model 10 0 109 IgtO 2 Could not get 109 this plot in 1071 1061 1051 lo 1037 1021 10 20 30 40 50 60 70 Volts iDZaZum Unit l a r Stat 5B7 r Ramc m Leon 7D Likelihood for Life Data i d 2 number of observations interval censored in t and t j gt number of l observations lelt i censored at t 7 7 j r 2 number of J observations right 1 y i39 censored at t BZZEIEIZ Stat 5B7 r Ramc m v Lenin Likelihood for Life Table Data I For a life table the data are the number Of failures ll right Censored n and left censored 1 units on each or the i39lonovel39iapping interval r1fl l39 1in1 tn The likelihood probability of the data fov a Single obson vatlen data in 39 L1rldata rim Iv r Ln Assuming that tile censoring is at I Type or Characteristh Number Likelihood ur Censorlng o asss Responsa LTr data Left at l 1 lt l i F1quot Interval l a I x i ll Fr 7 I i Right at t 1 gt r l 1 r I39tlquot BZZEIEIZ Stat 5B7 r Ramc m v Lenin Likelihood Probability of the Data 0 The total likelihood or joint probability of the DATA for n independent Observations iS 77 L739r DATA C H L7r data 39 1 m i l I c H POI1 iFf7Ff171i 17Ftlquot i1 where n 2711 d r l and C is a constant depend ing on the sampling inspection scheme but not on 7r So we can take C l o Want to find 7r so that Lrr is large 922002 Stat 567 Ramon V Leon Likelihood for Arbitrary Censored Data 0 In general the the 1th observation consists of an interval lgFri i 1 n tiL lt K that contains the time event T for the ith individual The intervals fl9n may overlap and their union may not cover the entire time line 090 In general 1 131 o Assuming that the censoring is at it Type of Characteristic Likelihood of a single R Censorihg esponse L7r data Left at t T g t Ft Interval tZL lt T g t Ft 7 FOIL Right at t T gt t 1 7 Fr 922002 Stat 567 Ram nV Leon Likelihood for General Reliability Data I The total likelihood TO the DATA WIth u Independent OI scrvatlons ls L7r DATA L7r data 1 Some of the observations may have multiple occurrences Let rigj J 1 I he the distinct intervals in me DATA and let it be the frequency of observation of t y r Then quot Lm DATA 11 lLrdatal 1 39 o In this case the nonparametric parameters 7r correspond to 39 abilities of a partition of D x determined bv the data Examples later BZZEIEIZ Stat 5B7 r Ramon v Leon 5 Other Topics in Chapter 2 Random censoring Likelihood with censoring in the intervals How to determine C BZZEIEIZ Stat 5B7 r Ramon v Leon E Unit 3 Nonparametric Estimation Ram n V Le n Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 922002 Stat 567 Ramon V Le n Unit 3 Objectives Show the use ofthe binomial distribution to estimate Ft from interval and singly right censored data without assumptions on Ft This is called nonparametric estimation Explain and illustrate how to compute standard error for a and approximate con dence intervals for Ft Show how to extend nonparametric estimation to allow for multiply rightcensored data Illustrate the KaplanMeier nonparametric estimator for data with observations reported as exact failures Describe and illustrate a generalization that provides a nonparametric estimator of Ft with arbitrary censoring 922002 Stat 567 Ramon V Le n Data for Plant 1 of the Heat Exchanger Tube Crack Data 39 l ilL39Lk39tl iulxm Inn libuN tn izul Ytin I Ych 1 Yam 3 1iml lml W i limi i l I l 3 l 1 95 Failure Probability N 1 71 1 4 Likelihood L71 C gtltrr11 x 722 gtlt n32 X 4195 4 TTI39 1 21 922002 Stat 567 Ramon v Leon 9 A Nonparametric Estimator of Fz Based on Binomial Theory for Interval SinglyCensored Data We consider the nonparametric estimate of Ftv for data situations as illustrate by Plant 1 of the Heat Exchanger Tube Crack 0 The data are n sample size rJ of failures deaths in the ith interval o Simple binomial theory gives of failures up to time 1 231 1 n n FOE SAeA Foalle w F f39 922002 Stat 567 Ramon V Leon Plant 1 Estimate of CDF o For Plant 1 n 100d1 ld2 213 2 one gets g 2 1100 H2 2 3100 F3 2 5100 020 015 E E LL 5 no 1 8 n u 005 o I I 00 i i i 0 0 0 5 I 0 1 5 2 0 25 3 0 Years 922002 Stat 567 Ramon V Leon 11 Comments on the Nonparametric Estimate of Fz o 150 is only defined at the upper ends of the intervals flil rJ39 o Ft is the ML estimator of Fti o The increase in F at each value of 2 is Fan 7 Fuel mm 922002 Stat 567 Ramon V Leon 12 Confidence Intervals A point estimate can be misleading quantify uncertainty in point estimates It is important to Confidence intervals are very useful in quantifying uncer tainty in point estimates due to sampling error arising from limited sample sizes In general confidence intervals do not quantify possible de viations arising from incorrectly specified model or model assumptions 922002 Stat 567 Ramon V Leon Some Characteristic Features of Confidence Intervals The level of confidence expresses one s con dence not probability that a speCI c interval contains the quantity of interest The actual u r the procedure will resu quantity of interest hat ning the l 39 39 quot39 is the r It In an Interval contai of con dence is not equal to the actual coverage probability V th censored data most con dence intervals are approximate Better approximations require more computations 922002 Stat 567 Ramon V Leon A confidence interval is approximate ifthe specified level Pointwise Binomial Based Confidence Interval for Ft n A 1001 7 a conservative confidence interval for Frv based on binomial sampling see Chapter 6 of Hahn and Meeker 1991 is A 71 n7nF1f WM 7 A v a F 1 1 22 2n 2n1 N nF A 71 7 F Fm 1 7 quotF 1f17a2 Enf2t2ni27i j where F 2 1567 and akaQWWQ is the 1001 7 a2 quantile of the F distribution with 121112 degrees of free7 dom This confidence interval is conservative in the sense that the actual coverage probability is at least equal to 1 7 0 922002 Stat 567 Ramon v Leon 15 Pointwise NormalApproximation Confidence Interval for Ft o For a specified value of ti an approximate 1001 7 u confidence interval for Ft is E1 FT E1H2SA where ably2 is the 17n2 quantile of the standard normal distribution and self l39 Ft 1 71300 n is an estimate of the standard error of 1301 o This confidence interval is based on 2P a NORO 1 5e 922002 Stat 567 Ramon V Leon 16 Plant 1 Heat Exchanger Tube Crack Nonparametric Estimate with Conservative Pointwise 95 Confidence Intervals Based on Binomial Theory 020 015 U E E LL v E 010 5 v Q 9 EL 005 V o o A o 0 0 A A 1 1 1 0 0 05 10 15 20 25 30 Years 922002 Stat 567 Ramon v Leon 17 Calculation ofthe Nonparametric Estimate of Ft for Plant 1 from the Heat Exchanger Tube Crack Data Year 11 1 fi sAel Pointwise Confidence Interval 150 F0 0 7 1 1 1 001 00995 95 Confidence Intervals for F1 ed on Binomial Theory 00030545 Based on Zf amp NOR01 700950295 1 7 2 2 2 003 01706 95 Confidence Intervals for F2 Based on Binomial Theory 0062 0852 Based on 2 4 NOR01 70034 0634 273 3 2 005 002179 95 Confidence Intervals for 1 3 Based on Binomial Theory 0164 1128 Based on Z 39 NOR01 0073092 922002 Stat 567 Ramon V Leon 18 Integrated Circuit IC Failure Times in Hours Data from Meeker 1987 10 10 15 60 80 80 120 250 300 400 400 600 1000 1000 1250 2000 2000 4300 4300 4800 4800 5400 7400 8400 9400 16800 26300 59300 When the test ended at 1370 hours there were 28 observed failures and 4128 Linfailed units Note Ties in the data Reason 922002 Stat 567 Ram n V Le n 19 Nonparametric Estimator of Ft Based on Binomial Theory for Exact Failures and Singly Right Censored Data When the number of inspections increases the width of the intervals l1l approaches zero and the failure times are exact o For the integrated circuit life test data we have n 4156 Wllli 28 exact failures in 1370 hours For any particular I 0amp1 lt 1370 simple binomial theon glvcs it of failures up to time t n 39l 1 l1 2 Fi n so u Methods to obtain con dence intervals for FI are the some as the methods described for the interval date 922002 Stat 567 Ram n v Leon 20 Hours l n1 2 D1 3 u15 4 n5 5 us s as 7 12 a 25 a a 1n 0 11 4 12 s 1a 1a 14 1m 15 125 15 m 17 zu 1a 43 12 u 20 as 21 as 22 54 23 74 24 a 25 94 m 155 27 252 25 592 29 1370 922002 Frequamv 22222222222222222222222222222 Cansm sagaaaaaaannasaaaanmaaaaaanus JMP Analysis awn rum V1sw wmuw Ham 9M3Ebk1 1 39 1 Mbdehng Mumvmate Methods gt r m1 quot quot E F1tPavamatr1c Survwa1 1 E F1tPrnpunwn51Hazavds 1 Ratununca Ana1v 1s 1 ms 1 us 1 1 emf cwsgwcmmm 1 Fiyemm 39 a 7 m Fa11uve1nstead a1 Suvvwa1 131567 Ram n V Le n o 15 Exganential Plot Exponentia1 Fit Weibull Plut e1hu1 Fit LDgNormal Plot 922002 Show Points Show Cumbmed 0V r111dIn 1 1 1 1 1 1 1 1 1 100 200 300 400 500 600 700 800900 H on rs 1 1 1 1 1100 1300 131567 Ram n V Le n jJMP Analysis Delta Method and Derivative of the Logit CDF Delta Method VarfX f X2 VarX Derivative of the Logit Function A g A fx 10g j logx log1 x selogim w1 F w 3 1 1 x EE x1 x 922002 Stat 567 Ramon v Leon 23 Pointwise NormalApproximation Con dence Interval for Ft Based on the Logit Transformation o Generally better confidence intervals can be obtained by using the logit transformation logitp logp1 7p and basing the confidence intervals on Io itFtv lo itFtv g lt3 g mimomw Z selogitif logiu o A pointwise normal approximation 1001 70 confidence interval for logitFtZ is logitF IogiE Iogit i 217ul252elogit logitF i akaESEFFG 7 F sincesTe s7e F17Fi logit jquot 922002 Stat 567 Ramon V Leon Pointwise NormalApproximation Confidence Interval for Ft Based on the Logit Transformation The confidence interval for Fox is obtained from the intere val for ogitF and using the inverse logit transformation 7 1 3909quot 1 m o Then Em FM IOQit 1logit ilt1n2gt 1 logitl 1 exp ilogit zlt1 hr2gt mgitltp 13 F 7 1i xui I1ilui where w egtltpzltlw2 eI IA 1 7 17 n The endpoints 1301 and FM will always lie between 0 and 1 922002 Stat 567 Ramon V Leon 25 Nonparametric Estimate for the IC Data with Normal Approximation Pointwise 95 Confidence Interval Based on the Logit Transformation 0012 0010 o o o o o o m or I l 39on Failing pom Pro p o o p i 9 o C N I I l l l I l I I 0 200 400 600 800 1000 1200 1400 Hours 922002 Stat 567 Ramon V Leon 26 Comments on the Nonparametric Estimate of Ft Ft is defined for all t in the interval 01 where t is the singly censoring time o 130 is the ML estimator of Ft The estimate PI is a step up function with a step of size 171 at each exact failure time Sometimes the step size is a multiple of 1n because there are ties on the failure times a When there is no censoring 521 is the well known empirical Stat 567 r Ramon V Leon 27 Notation Example cdf 922002 i 7 7 39 7 quot39 gt 4 7 W 7 gtI A74 7 Va 922002 l i l l r l l n 13 sample size a 3 of failures in the 1quot interval 2 ofright censored observation at t 4 ti n 7Iiskset attH nizdj 72 l 1 1 p 3 estimate of the probability of failing in the 1 interval given that item has survived to the begining of the interval Stat 567 r Ramon V Leon 28 A Nonparametric Estimate of Fz Based on Interval Data and Multiple Right Censoring The combined data from the heat exchanger tube crack are multiply censored and the simple binomial method to estie mate Ft cannot be used Here we describe a more general method to compute 3 none parametric estimator of Ft if 1 Em C A d where 50 H 1 7137 With 1 1 39 39 quot397 n sample size d of failures deaths in the ith interval let iil n n 7 Z 1 7 Z 7quot the risk set at Ikl o 70 7 7 r of right censored obs at t 922002 Stat 567 Ramon V Le n 29 Pooling of the Heat Exchanger Tube Crack Data m m 3 g l mm M Us us Plum ll l m l l W 95 Hmwmi mm ll I39lilllh llnlm l39mlmbrlun 1 uu trmm my 7 C 74n3573n415nfw 5nin rm 922002 Stat 567 Ramon V Le n 30 Calculation of the Nonparametric Estimate of F2 for the Heat Exchanger Tube Crack Data Year I II 739I 17 1 7 IT M 0471 1 300 1472 2 197 27m 3 97 4 99 4300 5197 297 296300 192197 9597 95 95 9867 9616 9418 0133 0384 0582 922002 Stat 567 RamOn V Le n Nonparametric Estimate for the Heat Exchanger Tube Crack Data 020 o 15 107 Proportion Faiiing o 0 05 922002 I 05 I I I 10 15 20 Years Stat 567 RamOn V Le n Approximate Variance of Estimated CDF Recall 154 1 301 the Var ag Var 34 Also H11 Hia j and Sui Hiaqj Then a Taylor series rstorder approximation 013201 is x a 3 zS A a ANZH 6 q q 61 139 St A StiZJ1 qiqi 11 922002 Stat 567 Ramon V Latin 33 Approximate Variance of Estimated CDF Then it follows that Z Z A i St A i S t q 7 VarSt ZH 5 VarqjZJI l 1 j f f 1 because the 1 are approximately uncorrelated binomial proportions the 1 values are asymtotically as n a 00 uncorrelated Var t St Z2lnp sg f 922002 Stat 567 Ramon V Latin 34 Estimating the Standard Error of the Estimated CDF 0 Using the variance formula one gets 7A7 A K A A I 1339 Var 141 Var 5r s392t r i i i i i E W l which is known as Greenwood39s formula 0 An estimate of the standard error 5e17 is A A 17 iVav irmi m 13 922002 Stat 567 Ram n V Le n 35 Standard Errors forthe Estimated CDF of the Heat Exchanger Tube Crack Data o Computation of standard errors A A 7 s2 39 73 Var Fm 7 5 WEN17m VEPUQ 98672 OOOO438 0 33 3009867 OOOO438 00662 gem 0133 0254 3009867 1979746 VAai39 t2 96162 0001639 quote 00016390128 922002 Stat 567 Ram n V Le n 36 NormalApproximation Pointwise Con dence Intervals of the Heat Exchanger Tube Crack Data For r 1 Wm 170 0133 5 0000438 00662 7 R11 7 mune7 e NORO1 11 0133 1 196OO662 0003 0263 1501 Based on 2mm 1ognrn 7 ogc11 e om 19 4 NORO1 N 0133 0133 F 1quot U 0 lt0 0133 1 7 0133 A 11 0133 1 7 0133 1 exp196006620133l 7 0133 2687816 0050 0350 For r2 with P02 0384 sAe M v0001639 0128 Based on Z3 02 70211133 0635 1502 175 0198 0730 Based on Mung 922002 Slal 567 Ramon V Le n 37 Results of Calculations for Nonparametric Pointwise Con fidence Intervals for F0 for the Heat Exchanger Tube Crack Data Year I PU gt m 6f Pointwse Con dence Intervals D 7 1 l 0133 00662 95 Confidence Intewms for 391 Based on 39 y y s N l 0050 0350 Based on 39 N0R01 0003 0263 1 7 2 2 0384 0128 95 Con dence Intenvob for I 2 Based on 7 WE x NOR01 0198 0730 Based on 39 7 NORM 1 0133 0635 2 3 3 0532 0137 95 Confidence Intew0s for I 3 Based on lef NORO1 0307 1076 Based on 2 x NORM 1 0216 0949 922002 Slal 567 Ramon V Le n 38 Heat Exchanger Tube Crack Nonparametric Estimate With POiI ItWiSB 95 Confidence Intervals Based on Z ogr tf39 020 015 7 E E 010 7 v EL v Q 005 v 0 A A 00 A 0 0 05 10 15 2 0 25 30 Years 922002 Stat 567 Ramon V Le n 39 J M P AnaIySIS r F a Year Le Year Right Freqency 1 El 1 4 2 1 99 3 1 2 5 4 2 95 5 2 3 2 S 3 0 95 922002 Stat 567 Ramon V Le n Shock Absorber Failure Data First reported in O Connor 1985 Failure times in number of kilometers of use of vehicle shock absorbers Two failure modes denoted by M1 and M2 One might be interested in the distribution of time to failure for mode M1 mode M2 or the overall failure time distribution of the part Here we do not differentiate between mode M1 and M2 We will estimate the distribution of time to failure by either mode M1 or M2 BZZEIEIZ Stat 5B7 r Ramc m v Lenin Failure Pattern in the Shock Absorber Data Failure Mode Ignored Vehiclc 4 in n lllh H u w u llmuimid omimm BZZEIEIZ Stat 5B7 r Ramc m v Lenin Nonparametric Estimation of Ft with Exact Failures Kaplan Meier Estimator In the limit as the number of inspections increases and the width of the inspection intervals approaches zero we get the product limit or Kaplan Meier estimator o Failures are concentrated in a small number of intervals of infinitesimal length 0 Fa will be constant over all intervals that have no failures FOE is a step function with jumps at each reported failure time Note The binomial estimator for exact failures and singly right censored data is a special case of the Kaplan Meier estimate 922002 Stat 567 Ramon V Leon 43 Nonparametric Estimates for the Shock Absorber Data up to 12220 km Conditional Unconditional Il km S39I FI 6700 38 6950 37 133 3738 097368 002632 134 3334 094505 005495 0 1 1 1 O 1 1 1 1 l 1 1 O D O M 0 L0 to quot b OODOOOOD OOOH 126 2526 090870 009130 922002 Stat 567 Ramon V Leon 44 Nonparametric Estimate for Shock Absorber Data with PointWIse 95 Confidence Intervals Based on Z OQRI 0 0 B i W E E o s 7 5 c 8 o 4 S n r 7 0 2 0 0 i g 0 5000 10000 15000 20000 25000 Ki umeiers 922002 Stat 5B7 r Ram ln V LEE In 45 a smutmm 6 E780 Censuvsd C saved 1 6 9550 Cenwved Cwsmzd 1 7 Wm mm Yank mew Wmdaw Hub 5 a 39 Dwsmhuum 9 Mg 3 r39 Htvhyx be 1 Matched Paws kg a ma Mn my gt Mu t vanale Methads D J M P amp 6700 m mm Marlm Mme E 4tPaiamaszuwwa An alySIS 2 50 c nsursd Cansmg E ammmwa vdi 3 7320 Camrw Censnv Remnants Anarysxs 822002 Stat 5B7 r Ramc m v Latin 46 m Famve mm at Suvvwa Ce WW V Failule Plot E r f JMP Analysis Suwiva Plat v Fa Plot rAdmn 7 Show Points Show Cumbvned M In mm Wmcn Exponentwa Plot Exponentat Flt Wewbull Pint mm t t t 1 5 00 10000 15000 20000 25000 300 0 Distance a Mdstep Quantile Points 3 unnent Quantile Paints 922002 Stat 567 Ram n V Le n 47 Nonparametric Estimate for SHOCK Absorber Data With Simultaneous 95 Con dence Bands Based on Z Og tu o 7 O 8 a E 06 LL c g a 04 r o E 02 i 7 00 u t u x t I O 5000 10000 15000 20000 25000 KHometers 922002 Stat 567 Ram n V Le n 48 Need for Nonparametric Simultaneous Confidence Bands for Fr o Pointwise confidence intervals for 131 are useful for making a statement about 171 at one particular value of r o Simultaneous confidence bands for F1 are necessary to quantify the sampling uncertainty over a range of values of 1 922002 Stat 567 Ramon V Le n 49 Nonparametric Simultaneous Confidence Bands for 171 Approximate 1001 70 simultaneous confidence bands for F can be obtained from 50 Pm ftien 1n2sAepi for all te warmth where tlta fvb is a complicated function of the censoring pattern in the data Comments The approximate factors Mhlimrz can be computed from a large sample approximation given in Nair 1984 ELlH2 is the same for all values oft The factors ew b liL Q are greater than the corresponding 21702 922002 Stat 567 Ramon V Latin 50 Factors emu 2 for Computing the EP Nonparametric Simultaneous Approximate Confidence Bands Limits Confidence Levei a D 80 90 95 9 005 999 01 999 05 999 001 995 005 995 01 995 05 995 001 99 005 99 1 9 05 99 001 95 005 95 01 95 05 95 001 9 005 9 01 9 05 9 922002 Stat 567 Ram n V Le n Theory of Simultaneous Confidence Bands The approximate 100171000 simultaneous confidence bands UM Fm f39unrm msAe u fornii if punyquotm are based on the the approximate distribution of Z max 111 i 131 max E11 i 1 sAelzm 922002 Stat 567 Ram n V Le n Better Nonparametric Simultaneous Confidence Bands f0 o It is generally better to compute the Simultaneoug confi lence bands based on the Iogit transformation of F This gives Fm Hr r a A m M rm 1 7 1 gt m 1 7 rm where w explr mlm 15 f1 rm These are based on the approximate distribution of 7 A 7 max logitfl ogitl J 3llF v r A Ill X m l 39Lrll 1 wuwwt 922002 Slat 567 Ram n V Le n 53 Nonparametric Estimate Heat Exchanger Tube Crack Data with Simultaneous 95 Confidence Bands Based on Zmax logitF 020 V 015 7 m g S v E 010 7 E 9 V L 005 a a A A o 0 A 0 0 05 10 15 20 25 30 Years 922002 Slat 567 Ramon V Leon 54 Nonparametric Estimation of F with Arbitrary Censoring o The methods described so far works only for some kinds of censoring patterns multiple right censoring interval cen soring with intervals that lo not overlap and some other very special censoring patterns a The nonparametric maximum likelihood generalizations pro vided by the PetoTurnbull estimator can be used for gt Arbitrary censoring eg both left and right gt Censoring with overlapping intervals gt Truncated data 922002 Stat 567 Ramon V Leon 55 Turbine Wheel Inspection Data Summary looehours of Cracked Not Cracked Proportion Cracked Exposure Left Censored Right Censored Crude Estimate of i m 4 O 39 10 4 49 14 2 31 18 7 66 22 5 25 26 9 3O 30 9 33 34 6 7 38 22 12 42 21 19 46 21 15 Data from Nelson 1982 page 409 o The analysts did not know the initiation time for any of the wheels 0 All they knew about each wheel was its exposure time and whether a crack had initiated or not Units grouped by exposure time 922002 Stat 567 Ramon V Leon 56 Plot of Proportions Failing Versus Hours of Exposure for the Turbine Wheel Inspection Data 08 I m 06 g 0 S o E 04 t 8 9 L 02 0 I o 39 00 s I 0 10 40 50 Hundreds oillows 922002 Slat 567 Ram n V Le n 57 Basic Parameters Used in Computing the Nonparametric ML Estimate of FI for the Turbine Wheel Data ll if V V V39vvvvvvvv TII TI M M N5 n m s o n HII I I I I I I I I I U 4 10 18 20 34 42 4o TEI Hundreds 01 Hours 922002 Slat 567 Ram n V Le n 5B Nonparametric Estimation of PM with Arbitrary Censoring General Approach I Basic idea write the likelihood and maximize this likelie hood to obtain 13 or 7 from which one gets FM Peto 1973 Illustration the likelihood for the turbine wheel inspection datais Llt7T L7T DATA C x M0 X W2 tquot quot1213g X W1 724 gtlt 7T3 W1249 X W1W32 gtlt W4W1231 X W1 M1121 xW1215 where 7712 1 7 221 m The values of lmmnl that maximize L7r gives 7 the ML estimator of 7139 Then FIlZQl71 m 922002 Stat 567 Ram n V Le n 59 Nonparametric ML estimate for the turbine wheel data with 95 Pointwise Confidence Intervals for FM Based on Z logltF 087 v v v 067 I m o o g S A A A C 047 g v s v V 9 a I 02 0 A v V V o A A o I z A 00 A A l l l l o 10 20 so 40 50 Hundreds of Hours 922002 Stat 567 Ramon V Le n 60 Unit 3 Nonparametric Estimation Ram en V Le n Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 8292004 Stat 567 Unit 3 Ram n V Le n Unit 3 Objectives Show the use of the binomial distribution to estimate Ft from interval and singly right censored data without assumptions on Ft This is called nonparametric estimation Explain and illustrate how to compute standard error for z and approximate confidence intervals for Ft Show how to extend nonparametric estimation to allow for multiply rightcensored data Illustrate the KaplanMeier nonparametric estimator for data with observations reported as exact failures Describe and illustrate a generalization that provides a nonparametric estimator of Ft with arbitrary censoring 8292004 Stat 567 Unit 3 Ram n V Le n 2 Data for Plant 1 of the Heat Exchanger Tube Crack Data HM tubcx ut turt C lacde ubm mm x l Untondmonal Fmturc Probability I Likelihood L7r 4 Z 71 1 8292004 Yam t Wm 1 NH Uncmckcd tutm l 2 l l 95 71 II 3 Ti 3 TE 4 C X W111 X 7722 X 77312 X M95 l Stat 567 Unit 3 Ram n V Le n A Nonparametric Estimator of Fz Based on Binomial Theory for Interval SinegCensored Data We consider the nonparametric estimate of Fr for data situations as illustrate by Plant 1 of the Heat Exchanger Tube Crack o The data are n sample size 11 of failures deaths in the ith interval o Simple binomial theory gives of failures up to time t1 31 61 77 77 Fan 1 7 Fm F 77 130139 SN 8292004 Stat 567 Unit 3 Ramon V Leon 4 o For 8292004 Plant 1 Estimate of CDF Plant 1 n 100dl 1r12 2113 2 one gets F11100 F23100 F35100 020 015 010 Proportion Failing 005 i O I l 15 20 Years Stat 567 Unit 3 Ram n V Le n Comments on the Nonparametric Estimate of Ft 15t is only defined at the upper ends of the intervals Grail o FM is the ML estimator of Ft The increase in F at each value of ti is PM 7 W124 llW 8292004 Stat 567 Unit 3 Ramon V Leon Confidence Intervals A point estimate can be misleading It is important to quantify uncertainty in point estimates Confidence intervals are very useful in quantifying uncer tainty in point estimates due to sampling error arising from limited sample sizes In general confidence intervals do not quantify possible de viations arising from incorrectly specified model or model assumptions 8292004 Stat 567 Unit 3 Ramon V Leon Some Characteristic Features of Confidence Intervals The level of confidence expresses one s confidence not probability that a specific interval contains the quantity of interest The actual coverage probability is the probability that the procedure will result in an interval containing the quantity of interest A confidence interval is approximate if the specified level of confidence is not equal to the actual coverage probability With censored data most confidence intervals are approximate Better approximations require more computations 8292004 Stat 567 Unit 3 Ram n V Le n Pointwise BinomialBased Confidence Interval for Ft o A 1001 7 a conservative confidence interval for Fr1 based on binomial sampling see Chapter 6 of Hahn and Meeker 1991 is 1 71 ninFJrUf 7 7 j 2 PM 1 1 Alt2 2n 2nI 2271 N 12F A 71 7 F FM 1 F 1f1rtt22nf392 27471 where F Fm and aka2M is the 1001 7 112 quantile of the 7quot distribution with 14112 degrees of free dom This confidence interval is conservative in the sense that the actual coverage probability is at least equal to 1 7039 8292004 Stat 567 Unit 3 Ramon v Leon Pointwise NormalApproximation Confidence Interval for Ft o For a specified value of t7 an approximate 1001 7 a confidence interval for Ft1 is 301 Foo M iz1o2s7e where 20702 is the liq2 quantile of the standard normal distribution and se VlFOI39 1 FOO n is an GStimate of the standard error of Fr o This confidence interval is based on FOG FOG Z19 a NORO 1 5e 8292004 Stat 567 Unit 3 Ramon V Leon 10 Plant 1 Heat Exchanger Tube Crack Nonparametric Estimate with Conservative Pointwise 95 Confidence Intervals Based on Binomial Theory 020 015 03 E i LL v E 010 5 v CL 9 D 005 V o o A o 00 A A 1 1 1 00 05 10 15 20 25 30 Years 3292004 Stat 567 Unit 3 Ramon v Leon Calculation of the Nonparametric Estimate of F2 for Plant 1 from the Heat Exchanger Tube Crack Data Year t d 1123 w Ael Pointwise Confidence Interval 0 7 1 1 1 001 00995 95 Confidence Intervals for F1 ed on Binomial Theory 0003 0545 Based on Z amp NOR01 700950295 172 2 2 003 01706 95 Confidence Intervals for F2 d on Binomial Theory 0062 0852 Based on Z amp NOR01 700340634 273 3 2 005 002179 95 Confidence Intervals for F3 Based on Binomial Theory 0164 1128 Based on Z amp NOR01 00730927 8292004 Stat 567 Unit 3 Ramon V Leon 12 Integrated Circuit IC Failure Times in Hours Data from Meeker 1987 10 10 15 60 80 80 120 250 300 400 400 600 1000 1000 1250 2000 2000 4300 4300 4800 4800 5400 7400 8400 9400 16800 26300 59300 When the test ended at 1370 hours there were 28 observed failures and 4128 unfailed units Note Ties in the data Reason P1370 8292004 8151567 Umt 3 r Ramon V Leon 13 Nonparametric Estimator of Ft Based on Binomial Theory for Exact Failures and Singy Right Censored Data When the number of inspections increases the width of the intervals ti11tl approaches zero and the failure times are exact o For the integrated circuit life test data we have n 4156 with 28 exact failures in 1370 hours For any particular ta 0 lt rt 3 1370 simple binomial theory gives of failures up to time t6 IL A Fte 1 71666 se F n Fte Methods to obtain confidence intervals for Fte are the same as the methods described for the interval data 8292004 Stat 567 Unit 3 Ram n V Le n 14 111113 Fyequency aw 1 m 1 n JMP AnaIySIs 2 n1 1 a a D15 1 u a as 1 n m Gvaph m1 V1zw 11mm Help 5 11 1 u r pumman 91 m 5 DB 1 H mm 39 39 g 1 6 Matched Paws s a 1 u 12 m Mnde 1n 6 1 n quotlodging 11 b 1 Mukwavlam Me fwds 12 a 1 n 1 1n 1 n 15 m 1 n 15 125 1 D 15 2D 1 u 17 20 1 U 111 as 1 u 19 a 1 a 2a 1a 1 n 21 45 1 n 22 51 1 n 23 u 1 a 21 a 1 n 1111 1112111111 1151111111 25 91 1 n 25 151 1 u 27 253 1 a 11 593 1 u 29 137 ma 1 8292004 Stat 567 Unit 3 Ram n V Le n 15 i lntlur l Iimii Eurmug Fit l Survival Plot 39 1 Failure Plat Exponential Plet Exponential Fit Weibull Plet Weibull Fit LngNarmal Plet JMP Analysis Show Points Show Combined Show IConFid Interval Maul I L39I n 39 L 0020 0018 0016 0014 C0012 E 0010 6 LL l l l 1100 1300 Hours 8292004 Stat 567 Unit 3 Ram n V Le n Comments on the Nonparametric Estimate of Ft 0 is defined for all t in the interval 0t where t is the singly censoring time o 0 is the ML estimator of Ft The estimate 130 is a step Lip function with a step of size 171 at each exact failure time Sometimes the step size is a multiple of 1n because there are ties on the failure times 0 When there is no censoring 131 is the well known empirical cdf 8292004 Stat 567 Unit 3 r Ramon V Leon i7 Delta Method and Derivative of the Logit of the CDF Delta Method Varf f39672 Varw Derivative of the Legit Function fxlogijzlogx log1 x l x A gt A S616 1 1 1 selogi fltxgt E xax F1F 8292004 Stat 567 Unit 3 Ram n V Le n 18 Pointwise NormalApproximation Con dence Interval for Ft Based on the Logit Transformation 0 Generally better confidence intervals can be obtained by using the logit transformation Iogitp ogp1 7p and basing the confidence intervals on Iogiti m 7 IogitiFmH Z A amp NOR 0 l V log39uf seogit13 o A pointwise normal approximation 1001 70 confidence interval for ogitFt7 is ogit IogiE Iogit i 21G2salogitf Iogit i ZU Z eFA u 7 F se FU 7F Sll39iCe selogm 8292004 Stat 567 Unit 3 Ramon V Leon 19 Pointwise NormalApproximation Confidence Interval for Ft Based on the Logit Transformation The confidence interval for 171 is obtained from the inter val for ogitF and using the inverse logit transformation 1 V 7 1 loglt 17 1 exmily 0 Then EM Flttgt IOQitT1IogitP i 21H2 ognltfl 7 4 1 l expflogit 121inl2 g ogitlt f f 7 f11gtltw39 17 u39 vvhere u39 exp1m2 efIA 1 7 1 o The endpoints 21 and 1170 will always lie between 0 and 1 8292004 Stat 567 Unit 3 Ramon V Latin 20 x 3911 1 exp 11r11 25610le 1 1 1 F 13 exp zselogil A F F 1 exp zselogit 11 8292004 Stat 567 Unit 3 Ram n V Le n 21 Nonparametric Estimate for the IC Data with Normal Approximation Pointwise 95 Confidence Interval Based on the Logit Transformation 00i2 0010 in 99 o o on i on Faii 0006 i pom Pro 0 o o b i i i i i i i i 0 200 400 600 800 1000 i200 i 400 Hours 8292004 Stat 567 Unit 3 Ram n V Le n Notation Example n 13 sample size d1 3 of failures in the I39m interval 1 2 of right censored observation at It 11 11 ml 7risk set at ti1 n Zdj er j0 j0 131 g estimate of the probability of failing in the I39m interval given that item has survived to the begining of the interval 8292004 Stat 567 Unit 3 Ram n V Le n 23 A Nonparametric Estimate of Ft Based on Interval Data and Multiple Right Censoring The combined data from the heat exchanger tube crack are multiply censored and the simple binomial method to estie mate Ft cannot be used Here we describe a more general method to compute 3 none parametric estimator foo where 5H 8292004 Stat 567 Unit 3 Ramon V Leon of Ft 1 7 Em 39 1 1717 with 17 4 11 n sample size of failures deaths in the 1th interval 1 1 1 n 7 2 if 7 Z 7 the risk set at tkl 70 10 of right censored obs at t Pooling of the Heat Exchanger Tube Crack Data W 5 mammalme K 7 7 mm my c n nzflnglim nmf inywn 8292004 Stat 567 Umt 3 r Ramon V Leo39n 25 Calculation of the Nonparametric Estimate of Ft for the Heat Exchanger Tube Crack Data Year t n d 7 13 1 i 171 r1 139 0 71 1 300 4 99 4300 296300 9867 0133 1 72 2 197 5 95 5197 192197 9616 0384 2 7 3 3 97 2 95 297 9597 9418 0582 00133 09867 00254 09746 00206 09794 EZaZum Stat 567 Umt 3 r Ramc m v Latin Nonparametric Estimate for the Heat Exchanger Tube Crack Data 020 Proportion Famng o 8 I I I I I I I 00 05 10 15 2C 25 30 Years 8292004 851567 UmtSyRamoer Leon Approximate Variance of Estimated CDF Recall 134 21 54 the Var 134 2 Var 54 Also 54 2 211 13 216 and 54 2 1 q Then a Taylor series rstorder approximation of 34 is 3rlSrlZ1 Aral 5tlZ1 j qj 8292004 Stat 567 Unit 3 Ram n V Le n 28 Approximate Variance of Estimated CDF Then it follows that 2 2 Var z 1 Varcj 1 11 11 quot1 because the c2 are approximately uncorrelated binomial proportions The c2 values are asymtotically as n gt oo uncorrelated Var ti SMDZ SQDZ 8292004 Stat 567 Unit 3 Ram n V Le n 29 Estimating the Standard Error of the Estimated CDF Using the variance formula one gets 7 13 1 quotJ0 j which is known as Greenwood s formula 7ar fag v3 5 2t1 J An estimate of the Standard error sef is A r A 7 s 39 5 sef VVaI F00 7 501 Ezaznm 5131567 Unit r Ramc mv Latin in Standard Errors for the Estimated CDF of the Heat Exchanger Tube Crack Data 0 Computation of Standard errors 00133 09867 00254 09746 quot e 2 7 73 WWW 39ltt n1i j 00206 09794 A A 0133 V Ft 9867 2 OOOO438 ar 1 lt gt 3009867 56M VOOOO438OO662 A A 0133 0254 v F f 9616 2 0001639 ar 2 lt 3009867 1979746 quotea7 0001639 0128 8292004 Stat 567 Unit 3 Ram n V Le n Recall Pointwise NormalApproximation Confidence Interval for Ft Based on the Logit Transformation Em Flttrgt1 Iogit llogim iaim3530911 1 1 exp 409M153 1quot2s mgitlt F 131713gtltuv39 17 w where u39 exp1 2s eI F l 7 117 EZaZum Stat 567 Umt 3 r Ramc m v Latin 32 NormalApproximation Pointwise Confidence Intervals of the Heat Exchanger Tube Crack Data For H1 with Rn 0133 5 Based on 2 17107 F1 5AEEamp NORO1 v0000438 0066 INK1 0133 i 19600662 0003 0263 Based on 7 A ogtf1 7xog1crltn MW 7 a NORO1 e ogm F N 1 0133 M 1 0050 0350 U1 0 0133 1 7 0133 A 1139 0133 1 7 0133u39 1 w exml960066201331 7 0133 2687816 For 12 with 1702 0384 sAe w Based on Z5 1302 F 0133 0635 V20001639 0128 Based on 7 MW IjUzJ 1702 0198 0730 8292004 Stat 567 Umt 3 r Ramon V Leo39n 33 Results of Calculations for Nonparametric Pointwise Con fidence Intervals for My for the Heat Exchanger Tube Crack Data Year I t sAeE Pomwse Con dence IntervaB 0 7 1 1 0133 00662 95 Con dence Inter39vaB for Iquot1 Based at z m e NORO 1 0050 0350 Based on 2 e NORO1 0003 0263 1 7 2 2 0384 0128 95 Con dence menab for 132 Based on Z WE s NORO1 0198 0730 Based on Z2 39 NORO 1 0133 0635 2 7 3 3 0582 0187 95 Con dence Intewas for ms Based on Z Omuf amp NORO1 0307 1076 Based on 1539s NORO1 0216 0949 8292004 Stat 567 Urnt 3 r Ramon V Leo39 Heat Exchanger Tube Crack Nonparametric Estimate with Pointwise 95 Confidence Intervals Based on Z ogit 020 015 7 E E E 010 7 v E 9 v Q 005 a V A A 00 A 00 05 10 15 20 25 30 Years 8292004 Stat 567 Unit 3 Ramon V Le n 8292004 JMP Analysis Uncmcked lube AH Hams Fm Pmbahvmy Year Len Year Right Frequency 1 u 7 A 2 1 as 3 1 2 s 4 2 as 5 2 a 2 5 3 95 8151567 Umt 3 r Ramon V Leo39n He I39 msmbuuun as m m K 1 r thy EH mm Madam Mu may awemg gtj m 7 59 m mm mm F2 Survival I Reliabilin Um Fame maze1 ar swvan av Censmn mm cansumd Mia El Venus v m 5 Event Ev Veer Len E Year Rim Veer mam F Ezaznm 5131567 Umt r Ramc mv Lean Recall F11 h 0 133 0384 0 582 m 2 so P 000043 0066 E 5 25 3 i y 3 3quot quot 1 1 Time to everrt Year LeftYear Right Frequency counts from Frequency 391 Eombinetl L Start Time End Time Survival Failure SurvSidErr 1 00000 100000 0955 00133 00055 200000 200000 09515 00384 00125 300000 300000 09415 00552 0015 8292004 8151567 UnitSyRamo nV Leo39ri 38 Shock Absorber Failure Data First reported in O Connor 1985 Failure times in number of kilometers of use of vehicle shock absorbers Two failure modes denoted by M1 and M2 One might be interested in the distribution of time to failure for mode M1 mode M2 or the overall failure time distribution of the part Data Table 02 in the Appendix page 630 Here we do not differentiate between mode M1 and M2 We will estimate the distribution of time to failure by either mode M1 or M2 8292004 Stat 567 Unit 3 Ram n V Le n 39 Table C2 Distance to Failure far 38 Vehicle Shock Absorbers Ful1un Mud Dlsumtc km Ful1urc Mada Dutunuc km 6700 M 1 17520 69511 NnnL 17541 7821 um 17891 87 10 None 1845 211 Ml 18060 9030 Nulm MW 320 None 19410 11310 one 2011111 11090 Nona 201110 1850 Nunr 70130 1 1880 None 20320 12140 None 20 300 12200 M 1 217110 172170 NnnL 21490 13150 M2 26510 1113 Non 17410 11470 Nana 27190 1 140 None 37890 141110 M 58100 mu 1mm U39Cunnor 1 WES page 85 8292004 Stat 567 Unit 3 Ram n V Le n Failure Pattern in the Shock Absorber Data Failure Mode Ignored X l 4 h 539 t i H In W I 3 21 339quot SH 3 Timu a nd of Kilometers 8292004 Stat 567 Unit 3 Ram n V Le n Nonparametric Estimation of Ft with Exact Failures Kaplan Meier Estimator In the limit as the number of inspections increases and the width of the inspection intervals approaches zero we get the product limit or Kaplan Meier estimator o Failures are concentrated in a small number of intervals of infinitesimal length 0 15r will be constant over all intervals that have no failures o Fe is a step function with jumps at each reported failure time Note The binomial estimator for exact failures and singly right censored data is a special case of the Kaplan Meier estimate 8292004 Stat 567 Unit 3 Ram n V Le n Nonparametric Estimates for the Shock Absorber Data up to 12220 km Conditional Unconditional fj km 39nj 15 1 Zaj 53961 Q g C 6700 38 6950 37 7820 36 8790 35 9120 34 9660 33 9820 32 11310 31 11690 30 11850 29 11880 28 12140 27 12200 26 0 138 3738 097368 002632 134 3334 094505 005495 1 O 1 1 1 1 1 1 1 O quotI OOOOOOOi OOOH 126 2526 090870 009130 8292004 Stat 567 Unit 3 Ram n V Le n 43 Nonparametric Estimate for Shock Absorber Data with Pointwise 95 Confidence Intervals Based on Z logitUE Proportion Failing l 15000 Kilometers I l O 5000 10000 8292004 Stat 567 Unit 3 Ram n V Le n I 20000 i 25000 m ShockAbsorber JMP Analysis 8292004 iShoelt ibsorloer 5quot 7 a a 1 Distance Mode Failure Censor 1 B39i39IZIEI Model Failure 7 IA 27 5350 Censored Censored 1 3V i BED Censored Censored 1 4 STEM Censored Censored 1 I V CD39UWSE WJ l 5 9120 lillode2 Failure 7 o D39Stance B SEED Censored Censored V 1 7 IE Mode T quot 1 IE Failure 37 i39Jinalvae Graph Tools 39u39iew Window Help Censor 9 i I Distribution swgtt klll Fit r bv x her V M Matched Pairs Fit Model 5 4 Modeling Multivariate Methods Surquot39ival and Felialiiit3939 EFEIEI 39Mode1 Failure Fit Parametric Survival EQSEI Censored Censor Fit Proportional Hazards F820 Censored Censore Recurrence Analvsis Stat 567 Unit 3 Ram n V Le n 45 JMP Analysis Expnnential Fit Waibuu Not Dis1ance 8292004 Stat 567 Unit 3 Ram n V Le n 46 8292004 Nonparametric Estimate for Shock Absorber Data Propnmon Famng With Simultaneous 95 Confidence Bands Based on Z ogn I I I I I 0 5000 10000 15000 20000 25000 KHometers Stat 567 Unit 3 Ram n V Le n Need for Nonparametric Simultaneous Confidence Bands for Ft Pointwise confidence intervals for Ft are useful for making a statement about Ft at one particular value of r Simultaneous confidence bands for Ft are necessary to quantify the sampling uncertainty over a range of values of t 8292004 8151567 Unit 3 r Ramon V Leo39ri Nonparametric Simultaneous Confidence Bands for F0 Approgtltlmate 1001 7a simultaneous confidence bands for F can be obtained from 50 Ho og Mwyg fm for all e tLattrb where tLatr b is a complicated function of the censoring pattern in the data Comments 0 The approximate factors eowliMQ can be computed from a largeesample approximation given in Nair 1984 o Elthb39l 392 is the same for all values of t o The factors eMM E are greater than the corresponding lam2 8292004 Slat 567 Unit 3 Ram n V Le n Factors emblioz for Computing the EP Nonparametric Simultaneous Approximate Confidence Bands Limits Confidence Levei a i 80 90 35 99 256 8292004 Stat 567 Unit 3 Ram n V Le n 50 Theory of Simultaneous Confidence Bands The approximate 100170gt simultaneous confidence bands an R0 fiic 102 e i for aii lg iLu imi are based on the the approximate distribution of g 7 Ft A 7 max max 7 t E 11401 fi39b 5 Z 8292004 Stat 567 Unit 3 r Ramon V Leon 5i Better Nonparametric Simultaneous Confidence Bands for F0 o It is generally better to compute the simultaneOL1 confie clence bands based on the logit transformation of F This 65 f0 I m Fi 1 171 x w39 F11 171 where w exp me1 nrimsAe UAKl 7 1 NU170 These are based on the approximate distribution of 7 mum1 7 iogitl 2 A max maXImuiIil i 7 le ILui if1 se mn m ut 8292004 Stat 567 Unit 3 Ram n V Le n 52 Nonparametric Estimate Heat Exchanger Tube Crack Data with Simultaneous 95 Confidence Bands Based 0quot Zmax iogim 020 V 015 7 E E v E 010 7 E 9 v Q 005 I a A A 00 A 00 05 10 15 20 25 30 Years 8292004 Stat 567 Unit 3 Ramon V Leon Nonparametric Estimation of Ft1 with Arbitrary Censoring The methods described so far works only for some kinds of censoring patterns multiple right censoring interval cene soring with intervals that do not overlap and some other very special censoring patterns 0 The nonparametric maximum likelihood generalizations prof Vided by the PetoTurnbull estimator can be used for gt Arbitrary censoring eg both left and right gt Censoring with overlapping intervals gt Truncated data 8292004 Stat 567 Unit 3 r Ramon V Leo39n 54 Turbine Wheel Inspection Data Summary IDOeiiours of Cracked Not Cracked Proportion Cracked Exposure Left Censored Right Censored Crude Estimate of i m 4 O 39 1O 4 49 14 2 31 18 7 66 22 5 25 26 9 3O 3O 9 33 34 6 7 38 22 12 42 21 19 46 21 15 Data f39om Neison 1982 page 409 o The analysts did not know the initiation time for any of the wheels o All they knew about each wheel was its exposure time and whether a crack had initiated or not Units grouped by exposure time 8292004 Stat 567 Unit 3 Ram n V Le n Plot of Proportions Failing Versus Hours of Exposure 8292004 Proponion Faiiing for the Turbine Wheel Inspection Data 08 I 06 o o 04 02 0 I o 39 00 7 i i i 10 40 50 Hundreds of Hours Stat 567 Unit 3 Ram n V Le n Basic Parameters Used in Computing the Nonparametric ML Estimate of F0 for the Turbine Wheel Data II I x V V vvvvvvvv 7527 III I 0 4 10 18 26 34 42 4G 714 7T5 7136717 Tis T59 7TH nll nll I I I I I Hundreds ofHours 8292004 Sta 567 Unit Sr Ramon V Leo39rI 57 Nonparametric Estimation of Ft with Arbitrary Censoring General Approach 0 Basic idea write the likelihood and maximize this likelie hood to obtain 1 or 7 from which one gets ltffgt Peto 1973 0 Illustration the likelihood for the turbine wheel inspection datais C X M0 X W2 M2139 X W1 W214 gtlt 3 7712149 X W1 32 gtlt W4W1231 X M77 L739r DATA W1 39 quot 11121 X 2115 m The values of W1 11 that maximize L7r gives 7 the ML estimator of 71 Then tzm7 1m where 712 7 21 8292004 Stat 567 Unit 3 Ram n V Le n Nonparametric ML estimate for the turbine wheel data with 95 Pointwise Confidence Intervals for Fltiggt Based on Z ogMFJ 087 v v v 067 I m o o E S A A A C 047 9 V v 25L v 9 Q I 02 0 A v V V o A A o I z A 00 A A w w w w o 10 2o 30 40 50 Hundreds of Hours 8292004 Stat 567 Unit 3 Ramon V Le n SPLIDA GRAPH Fraction Failing 8292004 08 06 04 02 Turbine Wheel Crack Initiation Data with Nonparametric Pointwise 95 Con dence Bands 20 30 40 50 Hundreds of HourS Sat Aug 23 223634 EDT 2003 Stat 567 Unit 3 Ram n V Le n 60 SPLIDA GRAPH with Nonparametric Simultaneous 95 Confidence Bands 1 08 g O O LL C 9 4g O I 04 39 o 02 9 Z 9 o 0 I 10 20 30 40 50 Hundreds of Hours Sat Aug 23 22231259 EDT 2003 8292004 Stat 5672 Unit 3 Ram n V Le n 61 JMP Analysis Len Rm frequech 1 1 39 2 1 3 19 3 14 o 31 M ram 1mm 1151111111 1 15 o as 5 22 z a 25 an 7 an 1 32 a 31 o 7 a as 1 12 1 u 2 1s 11 as 1 15 12 o u 11 1a a 1 1 14 2 15 1 1a 7 1s 22 5 17 25 a 13 3n 9 1 9 1 31 a 2n as 22 1 1 1 1 21 u 2 1o 20 an an 5 22 45 21 mm 8292004 Stat 567 Umt 3 r Ramon V Leo39n 62 Combined Start Time 8292004 100000 140000 180000 220000 260000 300000 340000 380000 420000 460000 End Time 100000 140000 180000 220000 260000 300000 340000 380000 420000 460000 Survival 09302 09302 09041 08333 07778 07778 05385 04190 04190 04165 Failure SurvStdErr 00698 00698 00959 01667 02222 02222 04615 05810 05810 05835 Stat 567 Unit 3 Ram n V Le n 00337 00473 00345 00680 00657 00650 01383 00865 00766 00822 63 Unit 4 LocationScaleBased Parametric Distributions Ram n V Le n Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 8262003 Stat 567 Ramon V Le n Unit 4 Objectives Explain the importance of parametric models in the analysis of reliability data Define important functions of model parameters that are of interest in reliability studies Introduce the locationscale family of distributions Describe the properties of the exponential distribution Describe the Weibull and lognormal distributions and the related underlying locationscale distributions 8262003 Stat 567 Ramon V Le n Motivation for Parametric Models Complement nonparametric techniques Parametric models can be described concisely with just a few parameters instead of having to report an entire curve It is possible to use a parametric model to extrapolate in time to the lower or uppertail of a distribution Parametric models provide smooth estimates of failuretime distributions In practice it is often useful to compare various parametric and nonparametric analysis of a data set 8262003 Stat 567 r Ramon V Leon Function of the Parameters Cumulative distribution function Cdf of T 1706 MO 3 I r gt 0 o The p quantile of T is the smallest value 1 such that Ft 9 3p Hazard function of T ff9 t390 7 t O K39 17mm gt 8262003 Stat 567 r Ramon V Leon Functions of the Parameters Continued The mean time to failure MTTF of 39139 also known as expectation of T X X ETIOif19dtVO l F19dl If 0 rfr6 It x we say that the mean of 1 does not exist no Remark 1 J1 Ftdt 0 if the last time is a failure time so that F t reaches 1 at that failure time 8262003 Stat 567 Ramon V Leon 5 a l V39 rProdlitrEimit Suwival Fit We Cmquot Vi Failure Plot 1 1 E n 2 2 1 j j 3 0339 Area above curve 5 4 i 0339 estimated mean 5 4 1 0 7 7 5 U 06 a 5 1 E 9 2G 0 0395 39 Hidden l 04 3 A 0 3 02 the New Meme m an em 0 1 i quotit n WW DamFaimmieamwwai WW Time m7 Freq Wimnarkrimei 7 Time to event Time By WEE Censored by Censor Vi Summary 7 Group N Failed N Censored Mean Std Error Com ined 3 3 955555 334351 8262003 Stat 567 Ramon V Leon Functions ofthe ParametersContinued The variance or the second central moment of T and the standard deviation VarT OXI7ET2ft9dI SDT VWarm Coefficient of variation quot2 I 7 Spa l2 ET 8262003 Stat 567 r Ramon V Leon LocationScale Distributions Y belongs to the locationescale family of distributions if the cdf of Y can be expressed as ilxltyltgto Form my 3 w lt1gt where 7x lt n lt ac is a location parameter and lt7 gt O is a scale parameter lt1 is the cdf of Y when p 0 and a 1 and lt1 does not depend on any unknown parameters Note The distribution of Z Y 7mm does not depend on any unknown parameters 8262003 Stat 567 r Ramon V Leon Importance of LocationScale Distributions Most widely used statistical distributions are either members of this class or closely related to this class of distributions exponential normal Weibull lognormal loglogistic logistic and extreme value distributions Methods of inference statistical theory and computer software generated for the general family can be applied to this large important class of models Theory for locationscale distributions is relative simple 8262003 Stat 567 Ramon V Leon 9 One Parameter Exponential Distribution Parametrized by the Hazard Rate f0 2M Ft Mixdx 1 ew St 2 e7 ht 2 xi for t 2 0 1 1 ET dV T an ar12 8262003 Stat 567 Ramon V Latin 10 TwoParameter Exponential Distribution For T m EXPtaq Ft9 liexplt777gt ailt exp 7 l 9n 1 Mpg 7 fgth 1 7 Ftr 63 9 where 6 gt O is a scale parameter and a is both a location and a threshold parameter When 1 0 one gets the welleknown oneeparameter exponential distribution Quantiles t 1r 7 9 log1 7p Moments For integer in gt O ET 7 1 m 9 quot Then ET a VarT 92 8262003 Stat 567 Ram n V Le n 11 Examples of Exponential Distributions Cumulative Distribution Function Probability Density Function 1 mi tit 71 m in in 7 in t 1 Hazard Function 0 7 0 0 O o u 11 29 3 u 12 l 8262003 Stat 567 Ramon V Leon Motivation for the Exponential Distribution gimplest distribution used in the analysis of reliability a a is constant does not depend on tIm Popular distribution for some kinds of electronic components eg capacitors or robust highquality integrated circuits This distribution would not be appropriate for a population of electronic components having failure causing qualitydefects Might be useful to describe failure time for components that exhibit physical wear out only after expected technological life ofthe system in which the component would be installed Has the important characteristic that its hazard function e 8262003 Stat 567 r Ramon V Leon 13 Normal Gaussian Distribution For 339 s NORUtJI Fwym 0 1 r r WWW quotnorlty X39CJX n n where mods lbE exp7232 and lt1gtnor fianoi 101er are pdf and cdf for a standardized normal t1 0r 7x p lt m is a location parameter rr gt O is a scale parame eter Quantiles y pnlttgt ol p where b ol p is the p quantlle for a standardized normal Moments For integer m gt 0 E Y 7mm 0 if m is odd and EX39 7mm mzr39quot2 72 7n2 if m is even Thus E0quot yt VarY r72 8262003 Stat 567 r Ramon V Leon l4 Examples of Normal Distributions Cumulative Distribution Function Probability Density Function 3 4 5 u 7 3 4 5 e 7 l 1 Hazard Function 6 u 03 5 05 5 08 5 1 4 5 a 7 8262003 8151567 7 Ramon V Leon Lognormal Distribution If 139 m LOGNOFNmr then loglt1 NOWH1 with lOCI e FUNJ7 bnm 1 39II39T quot nvjl 109m ll 7 rr 39 7 gt0 my and hm are pdf and czlf for a standardized normal EXIH is a scale parameter 1739 e O is a shape parameter Quantiles t exp 1 rrlt1gtl011p where mg 1 is the 1 quantlle for a standardized normal Moments For integer m t 0 EU 39l39 exp 771 7712 22 EU exp 1 03 392 VarU exp211 n2 aptT e 1 8262003 8151567 7 Ramon V Leon l6 Examples of Lognormal Distributions Cumulative Distribution Function Probability Density Function Hazard Function 4 6 ll hm 03 0 t 05 0 0 08 o t EZEZDDS Stat 5B7 r Ramon V LE IH l7 Motivation for Lognormal Distribution The lognormal distribution is a common model for failure times It can be justi ed for a random variable that arises from a product of a num er of identically distributed independent positive random quant39t39es It has been suggested as an appropriate model for failure times caused by a de radation process with combinations of random rates that combine multiplica ivel Mdely used to describe time to 39acture from fatigue crack growth in metals Useful in modeling failure time of a population of electronic componen s wi h a decreasing hazard function due to a small proportion of defects in the population Useful for describing the failuretime distribution of certain degradation processes EZEZEIEIS Stat 5B7 r Ramc m v Lenin lE Smallest Extreme Value Distribution For Y N SEVUL Ftiirm Sev a l 039 1 11y IfT exp ixi lt y lt x 1 1 3911 iisev r7 Sev 1 iexpk exi msevt egtltp iexp are cdf and pdf for standardized SEV 1 O 1 ext lt1 lt x is a location parameter and a gt O is a scale parameter Quantiles til 21 gepn H log 7 og1 7 0 7 Mean and Variance El39 1 7 m VarO39 n2n26 where 1 z 5772 w 2 31416 8262003 Stat 567 Ram n V Le n 19 Examples of Smallest Extreme Value Distributions Cumulative Distribution Function Probability Density Function 30 35 4O 45 50 55 60 30 35 40 45 50 55 60 t 1 Hazard Function 1 3 Ll mt 0 50 05 50 00 30 35 40 45 50 55 60 8262003 Stat 567 Ram n V Le n 20 Weibull Distribution Common Parameterization f i Ft PI39T g t 1 iexp 7 lt gt 7 11 1 11 1 l ltgt ltgt 77 71 n 1 t 1 111 77 1gt0 n 77 d gt O is shape parameter 7 gt O is scale parameter Quantiles 171Og1 imp1 Moments For integer in gt O ETW 71111r1m3I Then ET 1rlt1 VarT 772 r 1 7 r2 1 where I H A wquotquotlegtltp7wd1tv is the gamma function Note When I 1 then 1 EXPO 8262003 Stat 567 Ramon V Leon 21 Examples of Weibull Distributions Cumulatwe Distribution Function Probability Density Function 1 Hazard Function B n 08 1 10 1 15 1 8262003 Stat 567 Ramon V Leon 22 8262003 Alternative Weibull Parametrization Note If T m WEIBQLU then Y logT N SEVQt For T m WEIBOl U then 1 l7 lt3 i i grew l lt9 i if l Ft no a ffwr Where rr 13 It 7997 am easelz explz feXDUH pig2 1 7 expk 9432 Quantiles t1 739090 71ml exp 1 quotkg9 where dietp is the p quantile for a standardized SEV ie n 00 1 Stat 567 Ramon V Leon 23 8262003 Motivation for the Weibull Distribution The theory of extreme values shows that the Weibull distribution can be used to model the minimum ofa large number of independent positive random variables from a certain class of distributions Failure of the weakest link in a chain with many links with failure mechanisms eg creep or fatigue in each link acting approximately independent Failure of a system with a large number of components in series and with approximately independent failure mechanisms in each component The more common justi cation for its use is empirical the Weibull distribution can be used to model failuretime data with a decreasing or an increasing hazard rate Stat 567 Ramon V Leon 24 Largest Extreme Value Distribution When Y N LEVA a F1 l1 7 4gtiev 1 ye l KymJ 0iev r T l1239u lt7 71X1lt1ltwlt1 viexoiewrwrlr 39 quot where lt1gtiev exp7 exp7 and maz exp77egtltp7 are the Cdf and pdf for a standardized LEV 1 00 1 distribution 739 lt n lt x is a location parameter and a gt O is a scale parameter 8262003 Slat 5677Ramon v Leori 25 Largest Extreme Value Distribution Continued Quantiles y 1 7 alog ilog1 Mean and Variance E0 1m Var39 r2796 where a t 5772 w N 31416 Notes The hazard is increasing but is bounded in the sense that limu KlJug 117 1r3 N LEVpathen 7139wSEV71cr 8262003 Stat 567 r Ramon V Leon Examples ofthe Largest Extreme Value Distribution Cumuiative Distribution Fu notion Probebiiity Density Function 8262003 Stat 567 Ramon V Leon Logistic Distribution For Y N LOGISQIJI I LLilk 1 WitMT f iogislt 1 17 A I39M1quot iogis V lt 1 X n rr I lt is a scale parameter rr gt O is a shape parameter 00915 and dams are pclf and cdf for a standardized logistic distribution defined by exp quot iOgiS 3 1 exw2 EXI lt1 i0gislt 1exp 8262003 Stat 567 Ramon V Leon Logistic Distribution Continued Quantiles in i U g ls i Hlog l li p where 450 11 l091 710 is the p quantile for a standardized logistic distribution Moments For integer m gt 0 E39 7mm 0 if m is odd and 7 I 2a in l 7 12 1ZJ11 if m is eveiL Thus G272 Em i VarO 8262003 Stat 567 Ram n V Le n 29 Examples of Logistic Distributions Cuinuiative Distribution Function Probabii39ity Density Function Hazard Fu nation 2 G Ll mg 7 1 15 02 2 15 O 1 3 15 8262003 Stat 567 Ram n V Le n 30 Loglogistic Distribution If 339 7 Loelsqm then 139 exp 7 LOGLOGISQLU with log I 7 1 172 147 b og s T 1 log I 7 I mm 7 7OG S n cr 1 i 7 110410 iog 5 I O at n exptz is a scale parameter a gt O is a shape parameter moms and niogis are Cdf and pdf for a LOGISD 1 8262003 8151567 7 Ramon V Leon 3 Loglogistic Distribution Continued Quantiles 71 exp 1 Jd g b egtltD1 p1 7 10 Moments For integer in gt O ET39 XD771Lr1 7710rl 7 ma The m moment is not finite when ma 2 1 For I lt 1 ET expu r1 lt7 r1 7 a and for a lt 12 Varm egtltp2tt H1 47 2a r1 7 20 7 r21 a u 7 0 8262003 8151567 7 Ramon V Leo39n Unit 4 LocationScaleBased Parametric Distributions Ram n V Le n Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 8312004 Stat 567 Unit4 Ramon V Le n Unit 4 Objectives Explain the importance of parametric models in the analysis of reliability data Define important functions of model parameters that are of interest in reliability studies Introduce the locationscale family of distributions Describe the properties of the exponential distribution Describe the Weibull and lognormal distributions and the related underlying locationscale distributions 8312004 Stat 567 Unit4 Ramon V Le n Motivation for Parametric Models Complement nonparametric techniques Parametric models can be described concisely with just a few parameters instead of having to report an entire curve It is possible to use a parametric model to extrapolate in time to the lower or uppertail of a distribution Parametric models provide smooth estimates of failuretime distributions In practice it is often useful to compare various parametric and nonparametric analysis of a data set 8312004 Stat567 UnlUl rRarnO39l iV Leon Function of the Parameters Cumulative distribution function Cdf of T 1706 MO 3 I r gt 0 o The p quantile of T is the smallest value 1 such that Ft 9 3p Hazard function of T ff9 t390 7 t O K39 17mm gt 8312004 Stat567 UnlUl rRarnO39l iV Leon Functions of the Parameters Continued The mean time to failure MTTF of 39139 also known as expectation of T X X EmsO rimmingO l F19dl If 0 tfr6 It x we say that the mean of 1 does not exist no no Remark 1 H1 Ftdt JStdt 0 0 if the last time is a failure time so that S t reaches 0 at that time 8312004 Stat 567 Unit 4 Ramon V Leon 5 a l V39 rProdlitrEimit Suwival Fit We Cmquot Vi Failure Plot 1 1 B n 2 2 1 j j 3 0339 Area above curve 5 4 i 0339 estimated mean 5 4 1 0 7 7 5 U 06 a 5 1 E 9 2G 0 0395 39 Hidden l 04 3 A 03 02 the New mmeme m an em 0 1 quotit 7 n WW Dpimrammeawwwa WW Time m7 Freq Wimnarkrimei 7 Time to event Time By WEE Censored by Censor Vi Summary 7 Group N Failed N Censored Mean Std Error Com ined 3 3 955555 334351 8312004 Stat 567 Unit 4 Ramon V Leon Functions ofthe ParametersContinued The variance or the second central moment of T and the standard deviation VarT OXI7ET2ft9dI SDT VWarm Coefficient of variation quot2 I 7 Spa l2 ET 8312004 8151567 Unll4 r Ramon V Leon LocationScale Distributions Y belongs to the locationescale family of distributions if the cdf of Y can be expressed as ilxltyltgto Form my 3 w lt1gt where 7x lt n lt ac is a location parameter and lt7 gt O is a scale parameter lt1 is the cdf of Y when p 0 and a 1 and lt1 does not depend on any unknown parameters Note The distribution of Z Y 7mm does not depend on any unknown parameters 8312004 8151567 Unll4 r Ramon V Leon Importance of LocationScale Distributions Most widely used statistical distributions are either members of this class or closely related to this class of distributions exponential normal Weibull lognormal loglogistic logistic and extreme value distributions Methods of inference statistical theory and computer software generated for the general family can be applied to this large important class of models Theory for locationscale distributions is relative simple 8312004 Stat 567 Unit4 Ramon V Leon 9 One Parameter Exponential Distribution Parametrized by the Hazard Rate f0 2M Ft Mixdx 1 ew St 2 e7 ht 2 xi for t 2 0 1 1 ET dV T an ar12 8312004 Stat 567 Unit4 Ramon V Latin 10 TwoParameter Exponential Distribution For T m EXPtaq Ft9 liexplt7t7 Hullt 1exp 373 139 9m 1 Mpg 7 fgth 1 7 Ftr 63 9 where 6 gt O is a scale parameter and a is both a location and a threshold parameter When 1 0 one gets the welleknown oneeparameter exponential distribution Quantiles t qt 7 9 log1 7p Moments For integer in gt O ET 7 1 m 9 quot Then ET a VarT 92 8312004 Stat 567 Unit 4 Ram n V Le n 11 Examples of Exponential Distributions Cumulative Distribution Function Probability Density Function 1 mi in 7n t ii i n in 7 H39l t t Hazard Function 0 7 0 0 O o u m 29 3 u 12 l 8312004 Stat 567 Unit 4 Ramon V Leon Motivation for the Exponential Distribution gimplest distribution used in the analysis of reliability a a is constant does not depend on I Popular distribution for some kinds of electronic components eg capacitors or robust highquality integrated circuits This distribution would not be appropriate for a population of electronic components having failure causing qualitydefects Might be useful to describe failure times for components that exhibit physical wearout only after expected technological life ofthe system in which the component would be installed Has the important characteristic that its hazard function me 8312004 Stat 567 UnlM r Ramon V Leon 13 Normal Gaussian Distribution For 339 s NORUtJI Mam 0 1 r r WWW quotnmlty xQ ix n n where mods lbE exp7z32 and lt1gtnor fianoi 101er are pdf and cdf for a standardized normal t1 0r 7x p lt m is a location parameter rr gt O is a scale parame eter Quantiles y pnlttgt ol p Where b ol p is the p quantlle for a standardized normal Moments For integer m gt 0 E Y 7mm 0 if m is odd and EX39 7mm mzr39quot2 72 7n2 if m is even Thus E0quot yt VarY r72 8312004 Stat 567 UnlM r Ramon V Leon l4 Examples of Normal Distributions Cumulative Distribution Function Probability Density Function 5 e 7 l 1 Hazard Function r u 03 5 05 5 08 5 1 4 5 a 7 8312004 8151567 Unlt4 r Ramon V Leon Lognormal Distribution If 139 m LOGNOFNmr then loglt1 NORl17Wlth lOCI l e FUNJ7 bnm 109m 1 39IIIT quot nvjl in a gt0 on and hm are pdf and czlf for a standardized normal EXIH is a scale parameter 1739 e O is a shape parameter Quantiles t exp 1 rrlt1gtl011p where mg 1 is the 1 quantlle for a standardized normal Moments For integer m t 0 EU 39l39 exp 771 7712 22 EU exp 1 03 392 VarU exp 2 quot2 expw2 e 1 8312004 8151567 Unlt4 r Ramon V Leon 16 Examples of Lognormal Distributions Cumulative Distribution Function Probability Density Function Hazard Function EBiZEIEM StatEB7 Unit4 rRamunV Lenin i7 Motivation for Lognormal Distribution The Iognormal distribution is a common model forfailure times It can be justified for a random variable that arises from a product of a number of identically distributed independent pOSItive random quantities It has been suggested as an appropriate model forfailure times caused by a degradation process With combinations of random rates that combine multiplicativer Widely used to describe time to fracture from fatigue crack growth in metals Useful in modeling failure time ofa population of electronic components with a decreasing hazard function due to a smal proportion of defects in the population Useful for describing the failuretime distribution of certain degradation processes EBiZEIEM StatEB7 Unit4 rRamunV Lenin i8 Smallest Extreme Value Gumbel Distribution For Y x SEI r7 1704 pm Uril 739 1 7 110 IfT exp ixi lt y lt x 1 l y Ill r1 visev r7 Sev 1 iexpk exp msevt egtltp iexp are cdf and pdf for standardized SEV 1 O 1 ext lt lt x is a location parameter and a gt O is a scale parameter Quantiles gl 21 gepn H log 7 og1 7 0 7 Mean and Variance El39 1 7 in VarO39 n2n26 where 1 z 5772 w 2 31416 8312004 Stat 567 Unit4 Ramon V Le n 19 Examples of Smallest Extreme Value Distributions Cumulative Distribution Function Probability Density Function 30 35 4O 45 50 55 60 30 35 40 45 50 55 on i 1 Hazard Function i5 G Ll nt 0 7 5 50 05 6 50 00 7 50 30 35 40 45 50 55 60 8312004 Stat 567 Unit4 Ramon V Le n 20 Weibull Distribution Common Parameterization f i Ft PI39T g t 1 iexp 7 lt gt 7 11 1 11 t l ltgt ltgt 77 71 n 1 1 1 111 77 1gt0 n 77 d gt O is shape parameter 7 gt O is scale parameter Quantiles 171Og1 imp1 Moments For integer in gt O ETW 71111r1m3I Then ET 1rlt1 VarT 772 r 1 7 r2 1 where I H DXLUH716XD71Ut39LU is the gamma function Note When I 1 then 1 EXPO 8312004 Stat 567 Unit 4 Ramon V Leon 21 Examples of Weibull Distributions Cumulatwe Distribution Function Probability Density Function 1 Hazard Function 15 n 08 1 10 1 15 1 8312004 Stat 567 Unit 4 Ramon V Leon 22 Alternative Weibull Parametrization Note If T m WEIBQLU then Y logT N SEVQt For T m WEIBOl U then 1 FltfL Ugt 1 iexp 7 J lt1gt3ev flit f 7 Mum i 5 exp 7 3 J iwsev 7mm l 7 r7 7 at 7 Where rr 18 h log7 and else2 explz iexp lt1gtsev2 1 7 expk exp Quantiles f1 77 7 mu 7 1011 eX31 a eld0 where dietp is the p quantile for a standardized SEV ie ti o a 1 8312004 Stat 567 Unit 4 Ramon V Leon 23 Motivation for the Weibull Distribution The theory of extreme values shows that the Weibull distribution can be used to model the minimum ofa large number of independent positive random variables from a certain class of distributions Failure of the weakest link in a chain with many links with failure mechanisms eg creep or fatigue in each link acting approximately independent Failure of a system with a large number of components in series and with approximately independent failure mechanisms in each component The more common justi cation for its use is empirical the Weibull distribution can be used to model failuretime data with a decreasing or an increasing hazard rate 8312004 Stat 567 Unit 4 Ramon V Leon 24 Largest Extreme Value Distribution When Y N LEVA a F1 l1 7 4gtiev 1 ye l KymJ 0iev r T l1139u lt7 71X1lt1ltwlt1 viexoiewrwrlr 39 quot where lt1gtiev exp7 exp7 and maz exp77egtltp7 are the Cdf and pdf for a standardized LEV 1 00 1 distribution 739 lt n lt x is a location parameter and a gt O is a scale parameter 8312004 8151567 UniM rRamonV Leori 25 Largest Extreme Value Distribution Continued Quantiles y 1 7 alog ilog1 Mean and Variance E0 1m Var39 r2796 where a t 5772 w N 31416 Notes The hazard is increasing but is bounded in the sense that limu KlJug 117 1r3 N LEVpathen 7139wSEV71cr 8312004 Stat 567 UnlM r Ramon V Leon Examples ofthe Largest Extreme Value Distribution Cumuiative Distribution Function Probabiiity Density Function 8312004 Stat 567 Unit4 Ramon V Latin 27 Logistic Distribution For 339 Loelsom Fzrr mgL 1 1 t WENL g iogis X 11 lt n K lt p lt x is a iocatiom parameter a gt O is a scaie param eter 5 and blows are pdf and cdf for a standardized logistic distribution defined by U Y z logis quot 1 egtltp2 wows eXI 1 exp39 8312004 Stat 567 Unit4 Ramon V Latin 28 Logistic Distribution Continued Quantiles in i U g ls i Hlog l li p where O i501gt lOgDIl 710 is the p quantile for a standardized logistic distribution Moments For integer m gt 0 E39 7mm 0 if m is odd and 7 I 2a in l 7 12 1ZJ11 if m is eveiL Thus G272 EY1 VarO 8312004 Stat 567 Unit 4 Ram n V Le n 29 Examples of Logistic Distributions Cuinuiative Distribution Function Probabii39ity Density Function Hazard Fu nation 2 G Ll mg 7 1 15 02 2 15 O 1 3 15 8312004 Stat 567 Unit 4 Ram n V Le n 30 Loglogistic Distribution If 339 7 Loelsqm then 139 exp 7 LOGLOGISQLU with Iogz 7 1 FTIrr ltD Og 5 T l 1090 7 I mm 7 7Og 5 n cr 1 lo 1 7 110410 iog 5 I O at n exptz is a scale parameter a gt O is a shape parameter moms and niogis are Cdf and pdf for a LOGISD 1 8312004 8151567 Unit4 r Ramon V Leo39n Loglogistic Distribution Continued Quantiles 71 exp 1 Jd g b egtltD1P1 7 10 Moments For integer in gt O ET39 XD771Lr1 7710rl 7 ma The m moment is not finite when ma 2 1 For I lt 1 ET expltu r1 lt7 r1 7 a and for a lt 12 Varm egtltp2tt H1 47 2a r1 7 20 7 r21 a u 7 0 8312004 8151567 Unit4 r Ramon V Leo39n Unit 18 Accelerated Test Models Ram on V Leon Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 10192004 Unit 18 Stat 567 Ram n Le n Review Log LocationScale Model Representation 10gTuaZ PZSt Igtt lt3 PTStP10gTSlogt PuaZSlogt P 23 109 0 qlogt uj 0 10192004 Unit 18 Stat 567 Ram n Le n Review Log LocationScale Families and the SAFT Model Baseline Distribution at x0 1 PTx0 g r Ogt MW 039 SAFT Model T x0 Tm AFx 10192004 Unit 18 Stat 567 Ram n Le n ReWew Log Loca on Scam Fan Hes andthe SAFT Model 10192004 PTx tP g AFltx j P Tx0 s tAFx logtAFltxgt yltxogtj 0 CD 10gt yx0 10g AFx 0 logt yxj 0 Q 00 XO10g AF x Unit 18 Stat 567 Ram n Le n Unit 18 Objectives Describe motivation and application of accelerated reliability testing Explain the connection between degradation physical failure and acceleration of reliability tests Examine the basis for temperature and humidity acceleration Examine the basis for voltage and pressure stress acceleration Show how to compute timeacceleration factors Review other accelerated test models and assumptions 10192004 Unit 18 Stat 567 Ram n Le n Accelerated Test Increasingly Important Today s manufacturers need to develop newer higher technology products in record time while improving productivity reliability and quality Important Issues Rapid product development Rapidly changing technologies More complicated products with more components Higher customer expectations for better reliability 10192004 Unit 18 Stat 567 Ram n Le n Need for Accelerated Tests Need timely information on high reliability products Modern products designed to last for years or decades Accelerated Tests ATs used for timely assessment of reliability of product components and materials Test at high levels of use rate temperature voltage pressure humidity etc Estimate life at use conditions Note Estimationprediction from ATs involves extrapolation 10192004 Unit 18 Stat 567 Ram n Le n Application of Accelerated Tests Applications of Accelerated Tests include Evaluating the effect of stress on life Assessing component reliability Demonstrating component reliability Detecting failure modes Comparing two or more competing products Establishing safe warranty times 10192004 Unit 18 Stat 567 Ram n Le n 8 Methods of Acceleration Three fundamentally different methods of accelerating a reliability test Increase the userate of the product eg test a toaster 400 timesday Higher use rate reduces test time Use elevated temperature or humidity to increase rate of failurecausing chemicalphysical process Increase stress eg voltage or pressure to make degrading units fail more quickly Use aphysicachemical preference or empirical model relating degradation or lifetime at use conditions 10192004 Unit 18 Stat 567 Ram n Le n 9 Change in Resistance Over Time of CarbonFilm Resistors Shiorni and Yanaqisawa 1979 a 5 0 7 173 name g m 9 9 f 133 mm C m g 10 7 m a 33 DEqrees c a 2000 mm saw ms was Hours iDiBZum Unit 1 E r Stat 5B7 r Ramc m Latin in Accelerated Degradation Tests ADTs Response Amount of degradation at points in time Model components Model for degradation over time A definition of failure as a function of degradation variable Relationships between degradation model parameters eg chemical process reaction rates and acceleration variables eg temperature or humidity 10192004 Unit 18 Stat 567 Ram n Le n 11 Breakdown Times in Minutes of a MylarPolyurethane Insulating Structure from Kalkanis and Rosso 1989 10quot x 3 z 39 10 7 I l u m 1027 L2 E 1 107 l 1007 10quot 100 150 200 250 300 350 400 kVmm 10192004 Umt 18 7 8151567 7 Ramon Leo39n 12 Accelerated Life Tests ALTs Response Failure time or interval for units that fail Censoring time for units that do not fail Model Components Constantstress timetofailure distribution Relationships between one or more of the constantstress model parameters and the accelerating variables 10192004 Unit 18 Stat 567 Ram n Le n 13 UseRate Acceleration Basic Idea Increase userate to accelerate failure causing wear or degradation Examples Running automobile engines or appliances continuously Rapid cycling of relays and switches Cycles to failure in fatigue testing Simple assumption Useful if life adequately modeled by cycles of operation Reasonable if cycling simulates actual use and if test units return to steady state after each cycle More complicated situation Ware rate or degradation rate depends on cycling frequency or product deteriorates in standby as well as during actual use 10192004 Unit 18 Stat 567 Ram n Le n 14 Elevated Temperature Acceleration of Chemical Reaction Rates The Arrhenius model Reaction Rate 39R39Kternp is e 1 71 7 11605 We where temp K temp C 27315 is temperature in degrees Kelvin and 13 111605 is Boltzmann s constant in units of electron volts per K The reaction activation energy E and we are characteristics of the product or material being tested The reaction rate Acceleration Factor is Rtemp Wrempu Rtemp 7 agexplt AfcemptempU II 11605 11605 exp 1 e tempp K temp K I When temp gt templj 4ftemptemplgEi gt 1 10192004 Unit 18 r Stat 567 r Ramon Leon 15 Acceleration Factors for the SAFT Arrhenius Model Table 182 gives the Temperature Differential Factors TDF TDFlt 11605 11605 DemPLowK EemPHign K Figure 183 gives AFtempH ghtempLowiE exp Ea gtlt TDF We use AFtemp Aftemptemp 5 when temp1 and E are understood to be respectively product use temperature and reuLLioIiespeLiiiL aLLivaLion energy 10192004 Unit 18 7 8151567 7 Ramon Leo39ri TimeAcceleration Factor as a Function of Figure 183 Temperature Factor T1meyAcce1erauun Facwr Temperamre Dxfferemlm Famor Umt 18 r Stat 567 r Ramon Leo39n 10192004 Nonlinear Degradation ReactionRate Acceleration Consider the simple chemical degradation path model Dttemp 1395C x 1 7 exp 7723 x AFtemp gtlt t where R is the rate reaction at Lise temperature templr and for temp gt tempyr Aftemp gt 1 n For DX gt 0 failure occurs when PU temp gt Dr Equating DTtemp to Pf and solving for failure time Ttemp gives 1 n 1 tempm W39Og 1 Till AFtemp AFtemp where Mnempp is failure time at use conditions Ttemp I This is an SAFT model 10192004 Unit 18 r Stat 567 r Ramon Leon l8 SAFT Model from Nonlinear Degradation Path Powerdvop m dB 10192004 DU temp PX x 1 7 exp 77 x Af emp x t Dwlt0 an DagreesC 195 Degras c 237 Dagrees c 2000 4000 6000 8000 Hours Unil18 Stat 567 Ram n Latin 19 The ArrheniusLognormal Regression Model The Arrheniuselognormal regression model is PriT S t temp hm W where it 30 311 1 11605temp K 11605temp C 27315 and 31 E is the activation energy I U is constant I This implies that 17tempg 17temp gtlt AFtemp iDiBZum Unit 18 r Stat 5B7 r Ramc m Leon 2D 10192004 Hours Example ArrheniusLognormal Life Model Iognemp1 30 311 334200 logT o 1xUZ Z N01 106 1057 1047 1037 2 no 10 7 m m 1017 40 so so me 120 140 DegreesC Umt 18 r Stat 567 r Ramon Leo39n SAFT Model from Linear Degradation Paths am Degreesc m Dagrees c 391 390 Power drop m dB 195 Degrees c 237 Digrees c I I 0 2000 4000 6000 8000 Hours 10192004 Umt 18 e 8151567 e Ramon Leo39n 22 Linear Degradation ReactionRate Acceleration Note 2 3 exprxlrxrm Q xliexp7x ifx is small If RI x A Ftemp x t is small so that PO is small relative to PX then 7K x 1 7 exp 77 gtlt Aftemp x tl Pttemp x 12px A7Eemp gtlt I R x Aftemp gtltt is approximately linear in t 10192004 Unll l8 7 815567 r Ramon Leo39n Linear Degradation ReactionRate Acceleration Aiso some degradation processes are iiiiear in time T7ttemp Rf gtlt AFlttemp X r u Faiiure occurs when 00 temp gt Df Equating D391temp to M and soiving for faiiure time temp 1 tempy A7temp wnere Therapy DfR s faiitire time at Lise conditions Themp a This is an SAFT modei and for exaiiipie 1Cempyr m WEIBLU implies 1cempN WEIB it 7 iogAFtempa iEIiBZEIEIA Unit 18 7 5131567 7 Ramc m LEE iri NonSAFT Degradation ReactionRate Acceleration Non SAFT Degradation ReactionRate Acceleration Consider the more complicated chemical degradation path 39Dttemp Dix x 1 7 exp i39Rlp x AF1tempgtlt t D2X x 1 7 exp 771W x Af2temp gtlt t Rll Rap are the rates of the reactions contributing to fail ure This is not an SAFT model Temperature affects the two degradation processes differently inducing a nonlinearity into the acceleration function relating times at two different temperatures 10192004 Unit 18 7 8151567 7 Ramon Leon Voltage Acceleration and Voltage Stress Inverse Power Relationship Depending on the failure mode voltage can be raised to gt Increase the strength of electric fields This can accele erate some failureecausing reactions gt Increase the stress level eg voltage stress relative to declining voltage strength An empirical model for life at Volt relative to use conditions Voltr iS 1 voltp Volt 1 139 l 7 139 l I V0 12 Afvolt ltVOlt39gt V0 tl where Afvolt AFltVoltvoltz39 51 7 1 Voltl39 volt 1 Af 391t 4 1t load no V0 V0 1 1 Tvolt voltr and 31 is a material characteristic TVOlETVolt39 are the failure times at increased voltage and use conditions 10192004 Unit 18 Stat 567 Ramon Leon Inverse Power RelationshipWeibull Model The inverse power relationshipiWeibull model is f 7 PrT g Ivolt dgtsev 7 where M 30 aim and u 1 IogVoItage Stress T assumed to b constant 10192004 Umt 18 7 8151567 7 Ramon Leo39n 27 Example Weibull Inverse Power Relationship Between Life and Voltage Stress ogtvolt 30 31 chgelvqm logT 0 1xUZ ZGumbel01 1067 1057 w 1047 1037 w 102 50 101 um 50 100 150 200 250 350 kVmm 10192004 UmH8yStat567yRamo n Leon Other Commonly Used LifeStress Relationships 10192004 Other commonly used SAFT models have the simple form TOW T l ARI where 471 A7ltl l l39l31 expdl17r 81 is a material characteristic Examples include Cycling rate 1 logfrequency Current density 1 Iogcurrent Size 139 Iogthiclltness Humidity 1 1 IogRH RH is relative humidity100 Humidity 2 r IogRH1 7 RH Some of these models are empirical For a locationescale timeetoefailure distribution u 30 112 Unil18 Stat 567 Ram n Le n Eyring Temperature Relationship Arrhenius relationship obtained from empirical observation Eyring developed physical theory describing the effect that temperature has on a reaction rate r 7E Rtemp 0 gtlt Atemp gtlt exp 7 LB gtlt temp K Atemp is a function of temperature depending on the specifics of the reaction dynamics 0 and E are constants Applications in the literature have used Atemp temp KW with a fixed value of m ranging between m O Boce caletti et al 1989 m 5 Klinger 1991a to m 1 Nelson 1990a and Mann Scnafer and Singpurwalla 1974 Difficult to identify m from limited data Eyring showed how to include other accelerating variables 10192004 Unil18 Slal 567 Ram n Le n The Eyring Regression Model eg for Weibull or Lognormal Distribution The Eyring temperatureeacceleration regression model is PrT g temp ltlgt M where a 7m Ogtemp C 27315 30 311 w H 11605Cemp quotC 27315 331 E is the activation energy m is usually given a is constant but usually unknown With m gt 0 Arrhenius provides a useful first order approxie mation to the Eyring model with conservative extrapolation to lower temperatures 10192004 Unil18 Stat 567 Ram n Lean 31 1 01 92004 Humidity Acceleration Models Useful for accelerating failure mechanisms involving corro sion and certain other kinds of chemical degradation Often used in conjunction with elevated temperature Most humidity models have been developed empirically Empirical and limited theoretical results for corrosion on thin films Gillen and Mead 1980 Peck 1986 and Klinger 1991b suggest the use of RH instead of Ply vapor pressure as the independent or experimental variable in humidity relationships when temperature is also to be varied RH is the preferred variable because the change in life as a function of RH does not depend on temperature That is UQLife 7 URII hemp 7 or no statistical interaction Unit 18 Stat 567 Ramon Leon Humidity Regressnon Relationships Consider the WeibuIlIognormal lifetime regression model lo r 7 PrU39 S humidlty Cbsev g gt H T where p 30 3111 and r is constant Letting O lt RH lt 1 denote relative humidity possible humidity relae tionships are gt rl RH Intel empirical p I IogRH Peck empirical gt I logRH1 7 RH Klinger corrosion on thin films For temperature and humidity acceleration possible relae tionships inclu e l1 do dNi 521 2 0quot M 50 5111 9121 2 133111 2 or u 330 321 2 3311112 where 12 11605temp C 27315 10192004 Unit 18 Stat 567 Ram n Le n 33 TemperatureHumidity Acceleration Factors with RH and Temp K and No Interaction o Peck39s relationship RtempRH 7tempURHp RH 131 11605 11605 exp E0 7 RH temply K temp K Aftemp RH o Klinger39s relationship RtempRH 73tempU min 13111 1 35131 X 11605 11605 exp El 7 tempt K temp K Aftemp RH 10192004 Unit 18 Stat 567 Ramon Leon 10192004 Thermal Cycling Fatigue is an important failure mechanism for many prod ucts and materials Mechanical expansion and contraction from thermal cycling can lead to fatigue cracking and failure Applications include gt Powereonpowereoff cycling of electronic equipment and effect on component encapsulement and solder joints gt Takeeoff powerethrust in jet engines and its effect on crack initiation and growth in fan disks gt Powereuppoweredown of nuclear power plants and effect on the growth of cracks in heat generator tubes gt Thermal inkjet printhead delamination could be caused by temperature cycling during normal use Unil18 Slat 567 Ram n Le n CoffinManson Relationship The CoffinManson relationship says that the typical nume ber of cycles to failure is r Atempyi where Atemp is the temperature range and 6 and 51 are properties of the material and test setup This powererule relationship explains the effect that temperature range has on thermalefatigue life For some metals Jl z 2 Letting T be the random number of cycles to failure eg T N gtlt a where r is a random variable the acceleration factor when Atemp relative to the number of cycles when Atempr l5 AfAtemp 139Atemprg7 Atemp quot1 1Atemp 7 Atemplv There may be a Atemp threshold below which little or no fatigue damage is done during thermal cycling 10192004 Unit 18 Stat 567 Ramon Leon Generalized Cof nManson Relationship Empirical evidence has shown that effect of temperature cye cling can depend importantly on tempmaXK the maximum temperature in the cycling eg if tempmaXK is more than 2 or 3 times a metal s melting point The effect of temperature cycling can also depend on the cycling rate eg due to heat buildup An empirical extension of the CofriIrManson relationship is V 6 1 E x 11605 1V i x 1 x exp Atemp i freqgt 3 Demprnax K Where freq is the cycling frequency and Eu is an activation energy Caution must be used when usmg such a model outsme the range of available data and past experience 10192004 Unil18 r Sial 567 r Ramon Le n Unit 4 LocationScaleBased Parametric Distributions Ram on V Leon Notes largely based on Statistical Methods for Reliability Data by WQ Meeker and L A Escobar Wiley 1998 and on their class notes 8262003 Stat 567 Ram n V Le n Unit 4 Objectives Explain the importance of parametric models in the analysis of reliability data Define important functions of model parameters that are of interest in reliability studies Introduce the locationscale family of distributions Describe the properties of the exponential distribution Describe the Weibull and lognormal distributions and the related underlying locationscale distributions 8262003 Stat 567 Ram n V Le n 2 Motivation for Parametric Models Complement nonparametric techniques Parametric models can be described concisely with just a few parameters instead of having to report an entire curve It is possible to use a parametric model to extrapolate in time to the lower or upper tail of a distribution Parametric models provide smooth estimates of failuretime distributions In practice it is often useful to compare various parametric and nonparametric analysis of a data set 8262003 Stat 567 Ram n V Le n Function of the Parameters Cumulative distribution function cdf of T Fr6 PrT g t t gt 0 o The p quantile of T is the smallest value t such that Fm 9 2 p Hazard function ofT ft0 i re 7 t O 1 147mg gt 8262003 Stat 567 r Ramon V Leon Functions of the Parameters Continued The mean time to failure MTTF of 139 also known as expectation of T ET I we 9m fox 1 7 Ft 9 ch If f0X tft0dt x we say that the mean of I does not exist Remark If TD 7 ldl if the last time is a failure time so that a reaches 1 at that failure time 8262003 Stat 567 r Ramon V Leon JMP Example V i PmduclLimit Survival Fil aummeu a 39J i a Wu CW 391 Fal ure Plot 1 1 o W 2 2 U 1 gt 7 i 2 3 05 Area above curve E w estimated mean 8 4 1 7 5 U s 5 g u Time Timeto event Time Censnred by Censor 391 Summary Group N Failed N Censored Mean S d Error Combined B 3 555555 334851 8262003 Stat 567 Ram n V Le n 6 Functions ofthe ParametersContinued The variance or the second central moment of T and the standard deviation VarT Axuianmwm SDT 7 VVar 39 Coefficient of variation 72 SDT ET 39 72 8262003 8151567 7 Ramon V Leon LocationScale Distributions Y belongs to the locationiscale family of distributions if the cdf of Y can be expressed as 7 A FigMun PFO S y lt1gt 700 lt y lt so 7 where 7x lt p lt oc is a location parameter and a gt O is a scale parameter 4gt is the cdf of Y when u 0 and a 1 and lt1gt does not depend on any unknown parameters Note The distribution of Z Y 7pJ does not depend on any unknown parameters 8262003 8151567 7 Ramon V Leon Importance of LocationScale Distributions Most widely used statistical distributions are either members of this class or closely related to this class of distributions exponential normal Weibull lognormal loglogistic logistic and extreme value distributions Methods of inference statistical theory and computer software generated for the general family can be applied to this large important class of models Theory for locationscale distributions is relative simple 8262003 Stat 567 Ram n V Le n One Parameter Exponential Distribution Parametrized by the Hazard Rate ft 26 Ft 2 261de 1 6 81 e ht A for t 2 O 1 1 ETz andV T A 6W 12 8262003 Stat 567 Ram n V Le n 1O TwoParameter Exponential Distribution For T x EXP63 Ft6 17egtltp7t797 a 1 J Mill0 XD H lit6 fag l zgtm 1 eaten 0 where 9 gt O is a scale parameter and 7 is both a location and a threshold parameter When 7 0 one gets the welleknown oneeparameter exponential distribution Quantiles r r 7 6 Ioglt1 7p Moments For integer in gt O ET 7 lt m 0 quot Then ET w a VarT 92 8262003 Stat 567 Ramon V Latin 11 Examples of Exponential Distributions Cumulative Distribution Function Probability Density Function 20 9 Y 15 ht 10 05 0 7 7 10 0 395 77777777 w 20 0 8262003 Stat 567 Ram n V Le n 12 Motivation for the Exponential Distribution dSimplest distribution used in the analysis of reliability ata Has the important characteristic that its hazard function is constant does not depend on time t Popular distribution for some kinds of electronic components eg capacitors or robust highquality integrated circuits This distribution would not be appropriate for a population of electronic components having failure causing qualitydefects Might be useful to describe failure time for components that exhibit physical wear out only after expected technological life of the system in which the component would be installed 8262003 Stat 567 Ram n V Le n 13 Normal Gaussian Distribution For 1 N NORQL a lt1gtnov U 7 u a 1 i L WNW ltvw i 7xltyltx a n 1W in Where nan lbE exp772 2 and 0mm f are pdf and cdf for a standardized mormai i1 05 7x lt 1 lt x is a iocation parameter a gt O is a scaie param7 eter Cum LL LILL39 Quantiles yp izndgt olrp where b ol p is the p quantiie for a sta dardized normai Moments For mteger m gt D EY 7mm 0 if m is odd 3in EY 7mm mam27 2 m2 if m is even Tims EY 7 ii VarO39 72 8262003 8151567 7 Ramon V Leo39n Examples of Normal Distributions Cumuiaiive Distribution Function Probabiiiw Density Function U U 03 5 i 05 5 08 5 2 4 5 o 7 8262003 8151567 7 Ramon V Leo39ri Lognormal Distribution If 1 w LOGNORQLU then iogU39 NORQLM with i i 7 Fe m om ft 1T idinoi39 w xx 0 71 T om and hm are pdf and mi for a standardized normai eprz is a scaie parameter a gt O is a shape parameier Quantiles tp exp yalttgt ol 12 where 03031 is the 17 antiie or a staimardized normai Moments For integer m gt 0 Ear exp mp 7773022 Eu exp it 022 Vara exp 2p a2 exp02 7 1 EZBZEIEIS Stat 5B7 r Ramc m v Lenin Examples of Lognormal Distributions Cumuiatve Distribuiion Funciion Probabmy Density Function 4 Y 11 hm 3 03 0 1 77 05 0 7777777 08 0 EZBZEIEIS Stat 5B7 r Ramc m v Latin 17 Motivation for Lognormal Distribution The Iognormal distribution is a common model for failure times It can be justified for a random variable that arises from a product of a number of identically distributed independent positive random quantities It has been suggested as an appropriate model for failure times caused by a degradation process with combinations of random rates that combine multiplicatively Widely used to describe time to fracture from fatigue crack growth in metals Useful in modeling failure time of a population of electronic components with a decreasing hazard function due to a small proportion of defects in the population Useful for describing the failuretime distribution of certain degradation processes 8262003 Stat 567 Ram n V Le n 18 Smallest Extreme Value Distribution For Y w SEVLJ 17W 17 5ev 1 y 7 it lt3 sev I J 1 17 L egtltp J gt 739Xiltyltgtc a a KIALU My a lt1gtsevlt 1 7 expl7 expml wsevz explz 7 expzgtl are cdf and pdf for standardized SEV p Om 1 730lt1Llt 00 is a location parameter and a gt O is a scale parameter Quantiles y u lt1gt61Vpo 11 log 7 og1 e p a Mean and Variance EY 11 7 m VarY 02w26 where 39y 2577217 2 31416 8262003 8151567 7 Ramon V Leo39ri Examples of Smallest Extreme Value Distributions Cumulaiive DlSlleuUO Funclinn Probability Densiiy Function 30 35 40 45 so 55 50 l 30 35 40 45 50 55 an i Hazard Fu nation 1 5 G Ll m1 0 7 5 50 0 5 6 50 O O 7777777 7 50 3930 35 40 45 50 55 60 l 8262003 Stat 567 Ram n V Le n Weibull Distribution Common Parameterization a Ft PrT g r 1 iexp 7 J 77 71 1 J t 1 f0 77 exp 7 7 J 7i ii 77 71 3 t Mr 7 7 rgt o 7 ii a gt O is shape parameter n gt O is scale parameter Quantiles 71 7 log 7 Impi Moments For integer m gt 0 EG W 77mr1771a Then ET nr 1 VarT 72 r 1 7 r2 1 x where I H D u quot1exp7uvduv is the gamma function Note When 3 1 then T N EXPO 8262003 Stat 567 Ram n V Le n 21 Examples of Weibull Distributions Cumuiaiive Disiribution Funciion Probabimy Densiiy Function Hazard Funciion B n 08 1 10 1 15 1 8262003 Stat 567 Ram n V Le n 22 Alternative Weibull Parametrization Note If T N WEIBQLJ then Y 109039 w SEVQLJ For T w WEIBQna then FUJLU 1 iexp 7 J sev w 7 T 471 4 m M E 3 exp 3 J 77 7 7 m a where rr 1311og71y and rusez 6Xpz7egtltpz 4gtsev239 1 7 exp7 expz Quantiles Tl 7 3909 11 P1 H eXD p U elvp where lt1gt e p is the p quantile for a standardized SEV ie L O a 1 8262003 Stat 567 Ram n V Le n Motivation for the Weibull Distribution The theory of extreme values shows that the Weibull distribution can be used to model the minimum of a large number of independent positive random variables from a certain class of distributions Failure of the weakest link in a chain with many links with failure mechanisms eg creep or fatigue in each link acting approximately independent Failure of a system with a large number of components in series and with approximately independent failure mechanisms in each component The more common justification for its use is empirical the Weibull distribution can be used to model failuretime data with a decreasing or an increasing hazard rate 8262003 Stat 567 Ram n V Le n 24 Largest Extreme Value Distribution When Y m LEVQL 0 y7 A FHALU pmT ying ic ev1 ll I T lL1 lizr 7 lt y lt g exp exp 7 1 where Digz exp7 exp7z and Oievltl egtltp77egtltp7z are the Cdf and pdf for a standardized LEV p 057 1 distribution 730 lt u lt x is a location parameter and 0 gt O is a scale parameter 8262003 Stat 567 r Ramon V Leo39l i 25 Largest Extreme Value Distribution Continued Quantiles y p 7 a log 7 Iogp Mean and Variance E0quot 1 m VarO39 02w26 where 39y w 5772 n a 31416 Notes a The hazard is increasing but is bounded in the sense that Iimy x hyp0 10 If Y x LEVta then 439 sawing 8262003 Stat 567 r Ramon V Leo39ri 26 Examples of the Largest Extreme Value Distribution Cumuiaiive Dismbuiion Functian Probabiiity Densiiy Funciion Hazard Fu notion ICDO39IQ o 8262003 Stat 567 Ram n V Le n Logistic Distribution For Y N LOGISOLU y FUNK iogi5lt U n 1 y 7 L Kyuw Uiogis l U U 1 z 7 j Mym iogis J gt OC lt y lt Do U U 7x lt p lt ac is a scale parameter r gt O is a shape parameter meg 5 and iogis are pdf and Cdf for a standardized logistic distribution defined by quoti i eXmZ quotO S V 7 g 1 exp2 2 expltsgt lt1 L iogislt 1 8Xpz 8262003 Stat 567 Ram n V Le n Logistic Distribution Continued Quantiles y it wg ism 1 alog where g isp ogp1 717 is the p quantile for a standardized logistic distribution Moments For integer m gt 0 EY 7mm 0 if m is odd and EY 7mm 2am n7 1 712gtm1117m if m is even Thus T2772 3 EY it Val Y 8262003 Stat 567 r Ramon V Leon Examples of Logistic Distributions Cumuiaiivs Disiribuiion Funciion Probabiiity Densiiy Funmion Hazard Funciion A 5 Ll ME 7 1 15 02 2 15 00 1 3 15 8262003 Stat 567 r Ramon V Leo39ri Loglogistic Distribution If Y N LOGISQM then T egtltpY LOGLOGISWU with mm 7 i 170 it iogis 1 Iogt 7 I flt iwgt ltDiogis at fgt0 1 IOgltTgt7 hUMMU E iogisT v eprt is a scale parameter 7 gt O is a shape parameter iogis and moms are Cdf and pdf for a LOGIS0 1 8262003 81515677 Ramon V Leo39ri 31 Loglogistic Distribution Continued Quantiles t1 exp p Ud g b exML Ir1 7p Moments For integer m gt O EUm expmp r1 7710r1 7 mm The In moment is not finite when 17147 2 1 Foralt1 ET eXDUL r1 r r1 7 a and for a lt 12 VarT X2t H1 2n r1 7 247 7 r21 r r21 7 0 8262003 Stat 567 r Ramon V Leo39ri 32 Examples of Loglogistic Distributions Cumuiaiive Disiribuiion Funcnon Probabimy Densiiy Funcan 1 2 0 a m 0 4 0 0 0 2 3 4 t7 1 2 3 4 i Hazard Fu nciion G Ll 7 02 o 04 0 7777777 06 0 8262003 Stat 567 Ram n V Le n 33

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