STK 4290 PARAMETRIC LIFETIME MODELING Slides 1 Bo Lindqvist - PowerPoint PPT Presentation

STK 4290 PARAMETRIC LIFETIME MODELING Slides 1 Bo Lindqvist Department of Mathematical Sciences Norwegian University of Science and Technology Trondheim http://www.math.ntnu.no/ ∼ bo/ bo@math.ntnu.no University of Oslo, Spring 2014 Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 1 / 64

LIFETIMES (WIDELY DEFINED) Reliability engineering: Time to failure of a component or a system Number of cycles to failure (fatigue testing) Times between successive failures of a machine Medical research: Time to death of a patient after start of certain treatment Time from entrance to discharge from a hospital Times between successive epileptic seizures for patient Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 2 / 64

RELIABILITY AND SURVIVAL Common technical definition of reliability: The probability that a system or a component will perform its intended task, under given operational conditions, for a specified time period. Lifetime (survival time) in medical research: Time to occurrence of some event of interest for individuals in some population. The event may or may not be “death”, and is often referred to as “failure”. Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 3 / 64

WHY COLLECT AND ANALYZE LIFETIME/SURVIVAL/RELIABILITY DATA? Reliability engineering: Assess reliability of a system/component/product Compare two or more products with respect to reliability Predict product reliability in the design phase Predict warranty claims for a product in the market Medical research: Compare different treatments with respect to survival or recurrence Predict the outcome of an intervention or the life expectancy after the invention Identify risk factors for diseases and assess their magnitude Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 4 / 64

SPECIAL ASPECTS OF LIFETIME ANALYSIS IN STATISTICS Definition of starting time and failure time are difficult Definition of time scale (operation time, calendar time, number of cycles) Censored data (how can we use data from individuals or units for which the event of interest has not occurred within the observation period?) Effect of covariates (demographic, medical, environmental) What if an individual or unit dies or fails of another cause than the one we would like to study? (”competing risks”) Recurrent events – what if the system can fail several times; how to analyze recurring stages of a disease? Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 5 / 64

BALL BEARING FAILURE DATA Data: Millions of revolutions to fatigue failure for 23 units Question: How can we fit a parametric lifetime distribution to these data? 17,88 28,92 33,00 41,52 42,12 45,60 48,40 51,84 51,96 54,12 55,56 67,80 68,64 68,64 68,88 84,12 93,12 98,64 105,12 105,84 127,92 128,04 173,40 Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 6 / 64

BALL BEARING FAILURE DATA (EVENT PLOT) Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 7 / 64

IC DATA (MEEKER, 1987) Questions of interest: How to estimate the distribution of the failure time when there are censored observations? Probability of failure before 100 hours? Failure rate by 100 hours? Proportion failed after 10 5 hours? Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 8 / 64

IC DATA (EVENT PLOT) Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 9 / 64

SURVIVAL OF MULTIPLE MYELOMA PATIENTS Multiple myeloma is a malignant disease characterised by the accumulation of abnormal plasma cells, a type of white blood cell, in the bone marrow. Data (next slide) from Medical Center of the University of West Virginia, USA. Aim: To examine the association between certain explanatory variables or covariates and the survival time of patients in months from diagnosis until death from multiple myeloma). Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 10 / 64

MULTIPLE MYELOMA DATA Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 11 / 64

TYPICAL EXAM EXERCISE CASE Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 12 / 64

RECURRENT EVENTS/REPAIRABLE SYSTEMS Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 13 / 64

VALVE SEAT REPLACEMENT DATA Data on previous slide are collected from valve seats from a fleet of 41 diesel engines. Each engine has 16 valves. (Time unit is days of operation). Questions of interest: Does the replacement rate increase with age? How many replacement valves will be needed in the future? Can valve life in these systems be modeled as a renewal process? Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 14 / 64

ESTIMATED NUMBER OF VALVE SEAT REPLACEMENTS Middle curve is cumulative estimated number of replacements for one engine, as a function of age. Lower and upper curves are 95% confidence limits. Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 15 / 64

LIFETIME The lifetime T of an individual or unit is a positive and continuously distributed random variable. The probability density function (pdf) is usually called f ( t ), the cumulative distribution function (cdf) F ( t ) is then given by � t F ( t ) = P ( T ≤ t ) = 0 f ( u ) du , the reliability (or: survival) function is defined as � ∞ R ( t ) = P ( T > t ) = 1 − F ( t ) = f ( u ) du . t Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 16 / 64

EXAMPLE: EXPONENTIAL DISTRIBUTION λ e − λ t f ( t ) = 1 − e − λ t F ( t ) = e − λ t R ( t ) = Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 17 / 64

INTERPRETATION OF DENSITY FUNCTION f ( t ) = F ′ ( t ) � b P ( a < T ≤ b ) = f ( u ) du = F ( b ) − F ( a ) a � t + h P ( t < T ≤ t + h ) = f ( u ) du ≈ f ( t ) · h t Hence, f ( t ) ≈ P ( t < T ≤ t + h ) h Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 18 / 64

HAZARD FUNCTION OF T Suppose we know that unit is alive (functioning) at time t , i.e. T > t . Then it is of interest to consider P ( t < T ≤ t + h | T > t ) = P ( t < T ≤ t + h ) ≈ f ( t ) h P ( T > t ) R ( t ) (Recall conditional probability : P ( A | B ) = P ( A ∩ B ) / P ( B ) . From this we define the hazard function (also called hazard rate or failure rate ) of T at time t by: P ( t < T ≤ t + h | T > t ) = f ( t ) z ( t ) = lim h R ( t ) h → 0 Example: For the exponential distribution we have f ( t ) = λ e − λ t and R ( t ) = e − λ t , so z ( t ) = f ( t ) R ( t ) = λ (not depending on time!) . Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 19 / 64

USEFUL RELATIONS BETWEEN FUNCTIONS DESCRIBING T Since F ( t ) = 1 − R ( t ) we get, f ( t ) = F ′ ( t ) = − R ′ ( t ), and hence ′ ( t ) z ( t ) = f ( t ) R ( t ) = − R R ( t ) Thus we can write, d � � ln R ( t ) = − z ( t ) dt � t ⇒ ln R ( t ) = − z ( u ) du + c 0 � t ⇒ R ( t ) = e − 0 z ( u ) du + c Since R (0) = 1, we have c = 0, so � t 0 z ( u ) du ≡ e − Z ( t ) R ( t ) = e − � t where Z ( t ) = 0 z ( u ) du is called the cumulative hazard function . Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 20 / 64

USEFUL RELATIONS (CONT.) Recall from last slide: � t Z ( t ) = 0 z ( u ) du z ( t ) = Z ′ ( t ) R ( t ) = e − Z ( t ) Since f ( t ) = F ′ ( t ) = − R ′ ( t ), it follows that � t 0 z ( u ) du = z ( t ) e − Z ( t ) f ( t ) = z ( t ) e − (1) For exponential distribution: � t Z ( t ) = λ du = λ t 0 so (1) gives (the well known formula) f ( t ) = λ e − λ t Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 21 / 64

OVERVIEW OF FUNCTIONS DESCRIBING DISTRIBUTION OF LIFETIME T Function Formula Exponential distr = λ e − λ t Density (pdf) f(t) = 1 − e − λ t Cum. distr. (cdf) F(t) = e − λ t Rel/surv function R ( t ) = 1 − F ( t ) Hazard function z ( t ) = f ( t ) / R ( t ) = λ � t Cum hazard function Z ( t ) = 0 z ( u ) du = λ t R ( t ) = e − Z ( t ) = e − λ t f ( t ) = z ( t ) e − Z ( t ) = λ e − λ t Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 22 / 64

EXERCISES 1 Suppose the reliability function of T is R ( t ) = e − t 1 . 7 . Find the functions F ( t ) , f ( t ) , z ( t ) , Z ( t ). 2 Show that if you get to know only one of the functions R ( t ) , F ( t ) , f ( t ) , z ( t ) , Z ( t ), then you can still compute all the other! Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 23 / 64

BATHTUB CURVE Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 24 / 64

MORE ON THE HAZARD FUNCTION P ( t < T ≤ t + h | T > t ) Recall that z ( t ) = lim h → 0 . h Thus z ( t ) h ≈ P ( t < T ≤ t + h | T > t ) = P (fail in ( t , t + h ) | alive at t ) Suppose a typical T is large compared to time unit. Then for h = 1: z ( t ) ≈ P ( t < T ≤ t + 1 | T > t ) = P (fail in next time unit | alive at t ) Thus: Suppose we have n units of age t . How many can we expect to fail in next time unit? e = n · z ( t ) In practice: Ask an expert: “If you have 100 components (of specific type) of age 1000 hours. How many do you expect to fail in the next hour”? Answer is, say, “2”. Looking at e = n · z ( t ) we estimate; 2 ˆ z (1000) = 100 = 0 . 02 Bo Lindqvist Slides 1 STK 4290 PARAMETRIC LIFETIME MODELING () 25 / 64