Hybrid Censoring Scheme: An Introduction

Transcription

1 Department of Mathematics & Statistics Indian Institute of Technology Kanpur August 19, 2014

2 Outline

3 Outline

4 What is? Lifetime data analysis is used to analyze data in which the time until the event is of interest.

5 Example 1 Time until tumor recurrence. 2 Time until cardiovascular death after some treatment intervention. 3 Time until AIDS for HIV patients. 4 Time until a machine part fails.

6 Type of Data 1 Time to event data is usually restricted to be positive and has a skewed distribution. 2 The data are censored.

7 What is censoring? Censoring is present when we have some information about a subject s event time, but we do not know the exact event time.

8 Cause There are generally three reasons why censoring might occur in a study. 1 A subject/ an item does not experience the event before the study ends. 2 A person is lost to follow up during the study period. 3 An item has been withdrawn from the study. These are examples of right censoring.

9 Outline

10 Type-I Censoring Scheme Type-I censoring occurs when a study is designed to end after a fixed time period T, determined before starting the experiment.

11 Type-I Censoring A total of n units is placed on a life testing experiment. Let the ordered lifetimes of these items be denoted by X 1:n,,X n:n respectively. The test is terminated when a pre-determined time, T, on test has been reached. It is also usually assumed that the failed items are not replaced. Suppose k items fail before time point T, the failure time of the k items are observed.

12 Type-II Censoring A total of n units is placed on a life testing experiment. The test is terminated when exactly r failures occur. The failure time of the r items are observed.

13 Type-I Censoring: Advantage and Disadvantage Advantage: Experimental time is fixed. Disadvantage: Number of failures can be very small.

14 Type-II Censoring: Advantage and Disadvantage Advantage: Number of failures is fixed. Disadvantage: Experimental time can be very large.

15 Outline

16 Scheme The mixture of Type-I and Type-II censoring schemes is known as the hybrid censoring scheme.

17 Type-I HCS A total of n units is placed on a life testing experiment. The lifetimes of the sample units are independent and identically (i.i.d.) random variables. Let the ordered lifetimes of these items be denoted by X 1:n,,X n:n respectively. The test is terminated when a pre-chosen number, r < n, out of n items are failed, or when a pre-determined time, T, on test has been reached, i.e. the test is terminated at a random time T = min{x r:n,t}. It is also usually assumed that the failed items are not replaced.

18 Available Data When the data are Type-I hybrid censored, we have one of the following two types of observations; Case I: {x 1:n < < x r:n } if x r:n T Case II: {x 1:n < < x d:n } if x r:n > T, here d denotes the number of observed failures that occur before time point T.

19 Associated Problems Under parametric assumption of the underlying distribution, estimation of the parameter(s) Finding the distribution of the parameter(s) Finding the confidence interval(s) Finding the optimum censoring scheme (Choosing r and T).

20 Exponential Distribution It is assumed that the lifetime distribution is a one parameter exponential distribution with mean θ, i.e., the PDF is f(x;θ) = 1 θ e x θ ; x > 0, θ > 0

21 Exponential Distribution The likelihood function based on the Type-I Hybrid Censored sample becomes: [ d lnθ 1 d θ i=1 x i:n +(n d)t] if x r:n T l(θ) = r lnθ 1 θ [ r i=1 x i:n +(n r)x r:n ] if x r:n > T.

22 Exponential Distribution Based on the observed sample, it can be easily seen that the MLE of θ does not exist if d = 0, and if d > 0, it is given by θ = 1 d d i=1 x i:n +(n d)t if x r:n T 1 r d i=1 x i:n +(n r)x r:n if x r:n > T.

23 Conditional Distribution of the MLE How to obtain the conditional distribution of θ? Compute the conditonal moment generating function of θ, i.e. M θ D>0 (t) = E(e t θ D > 0)

24 Conditional Distribution of the MLE Using the uniqueness property of the MGF, it can be shown that conditional distribution of θ can be written as a generalized mixture of gamma and shifted gamma distributions.

25 Conditional Distribution of the MLE Now using the moment generating functions of gamma and shifted gamma distribution, one can obtain the PDF of θ for 0 < x < nt as [ r 1 f θ(x) = (1 q n ) 1 +r ( n r ) r d=1k=0 k=1 d C k,d g ( 1) k q n r+k n r +k ( )( ) n d where q = e T/θ, C k,d = ( 1) k q n d+k, d k T k,d = (n d +k)t/d, and g(.) is a gamma PDF. (x T k,d ; dθ ),d +g (x; r ) θ,r ( ) r 1 ( g x T k,r ; r ) ] k 1 θ,r,

26 Conditional Distribution of the MLE It is clear that the PDF of θ, is a generalized mixture of gamma and shifted gamma PDFs, when the mixing coefficients may be negative also. The cumulative distribution function (CDF) and the survival function (SF) of θ can be obtained in terms of incomplete gamma function.

27 Observations Because of the explicit expression of the PDF of the MLE, different moments can be easily obtained. It is observed that the MLE of θ is a biased estimate of θ. It can also be observed that as T, f θ(x) = g (x; r ) θ,r. It is the well known result that 2r θ has a chi-square θ distribution with 2r degrees of freedom. Substitution r = n, we get the PDF of the MLE of θ for the usual Type-I censored case.

28 Exact Confidence Intervals A set of two sided 100(1-α)% confidence intervals for θ: [ ] 2S χ 2,α/2, if d = 0 2 [ 2S 2S, χ 2 χ 2d+2,α/2 2 2d,1 α/2 [ 2S 2S, χ 2 χ 2r,α/2 2 2r,1 α/2 ] ] if 1 d r 1, if d = r, here S is the total time on test, i.e. d i=1 x i:n +(n d)t if x r:n T S = d i=1 x i:n +(n r)x r:n if x r:n > T,

29 Comments The authors provided a formal proof in the with replacement case, and mentioned that the proof can be extended for the without replacement case also. It is not very clear that how the proof will be in the without replacement case, as one of the major assumption in their proof ( with replacement case) is that for 0 j r 1, P{j items fail at the decision time} = e nt/θ (nt/θ) j. j! Clearly, the above assumption is no longer valid for the without replacement case and their proof very much depend on that assumption.

30 Exact Confidence Interval Based on the exact distribution of θ, and based on the fact that P θ ( θ > b) is an increasing function of θ for fixed b, the exact two sided 100(1-α)% symmetric confidence interval of θ, then θ L and θ U can be obtained by solving the following two non-linear equations; α 2 = F θ L ( θ), 1 α 2 = F θ U ( θ). Here F θ (x) = P( θ x). θ L and θ U need to be computed numerically by solving the above two non-linear equations.

31 Bayes Estimates Bayesian solution seems to be a very natural choice in this case, and fortunately it turns out to be quite simple. Prior: Draper and Guttman assumed that θ has a inverted gamma prior with the following PDF; π(θ) = λβ Γ(β) θ (β+1) e λ/θ ; θ > 0, here β > 0, and λ > 0 are the hyper parameters.

32 Based on the above prior, the Bayes estimate with respect to squared error loss function is if (d +β) > 1, where θ Bayes = (S d +λ) (d +β 1), nt if d = 0 S d = d i=1 x i:n +(n d)t if 1 d r 1 r i=1 x i:n +(n r)x r:n if d = r. Under the noninformative prior β = λ = 0, the Bayes estimate matches with the MLE.

33 Credible Interval Interestingly, a 100(1-α)% credible interval of θ can be easily obtained as ( 2(S d +λ), χ 2 2(d+β),α/2 2(S d +λ) χ 2 2(d+β),1 α/2 ), if d +β > 0, and here χ 2 m,α denotes the point of the central χ 2 distribution with m degrees of freedom that leaves an area α in the upper tail.

34 Two-Parameter Exponential Distribution Recently some work has been done on two-parameter exponential distribution, i.e. when both the location and scale parameters are present, i.e. the individual item has the PDF f(x;θ,µ) = 1 θ e 1 θ (x µ) ; θ > 0, x > µ. It is observed that for n 2, the MLEs of both θ and µ exist, and they can be seen easily seen as µ = x 1:n, and the MLE of θ is same as before and it can be obtained by replacing x i:n with x i:n x 1:n.

35 Difficult Part Although, the MLEs of θ and µ can be obtained in explicit form, it is not easy to obtain the joint PDF of µ and θ when they exist. The joint moment generating function of θ to µ is possible to obtain, but from their the inversion has not yet been possible. The PDFs of θ and µ can be obtained along the same line. From the marginal PDFs, the corresponding confidence intervals can be obtained.

36 Bayesian Inference Prior: In this case it is assumed that λ = 1/θ has a Gamma distribution and µ has a non-informative prior. Based on these priors, the posterior distribution of λ given µ is also a gamma distribution. The posterior distribution of µ is l(µ Data) 1 (A 0 µ) n+d; µ < x 1:n. Therefore, it is possible to generate samples from the joint posterior distribution function of λ and µ, and that can be used to compute Bayes estimate and also to construct the credible intervals.

37 Weibull Distribution Estimation and the associated inferential procedures are quite different when the lifetime distribution of the individual item is a Weibull random variables. Let us assume that the shape and scale parameters as α and λ respectively, i.e. it has the following PDF; f(x;α,λ) = α λ ( x λ ) α 1e (x/λ) α, x > 0, where α > 0 and λ > 0 are the natural parameter space.

38 Available Data Remember we have one of the following two types of observations; Case I: {x 1:n < < x r:n } if x r:n T Case II: {x 1:n < < x d:n } if x r:n > T, here d denotes the number of observed failures that occur before time point T.

39 Likelihood Based on the available sample as described in the beginning of this section, the likelihood function for Case I and Case II are ( α r r l(α,λ) = λ) and l(α,λ) = ( α λ i=1 ) d d i=1 ( xi:n λ ) α 1e { r i=1 (x i:n/λ) α +(n r)(x r:n/λ) α } ( xi:n ) α 1e { d i=1 (x i:n/λ) α +(n r)(t/λ) α } λ if d > 0 e n(t/λ)α if d = 0 respectively. The MLEs can be obtained by maximizing the log-likelihood function with respect to the unknown parameters α and λ.

40 MLE of the Unknown Parameters As expected the MLEs of α and λ cannot be obtained in explicit form. It has been observed that the MLE of α can be obtained by finding the unique solution of a fixed point type equation i=1 h(α) = α, where for Case I ( r [ r ]) 1 h(α) = r lnx i:n + 1 x α u(α) i:nlnx i:n +(n r)xr:nlnx α r:n i=1 i=1 and for Case II ( d [ d ]) 1 h(α) = d lnx i:n + 1 x α u(α) i:nlnx i:n +(n d)t α lnt i=1

41 MLE: Contd. r ( r i=1 xα i:n +(n r)xα r:n) 1 for Case I u(α) = ( d ) 1 d i=1 xα i:n +(n d)tα for Case II Simple iterative scheme can be used to solve for α, the MLE of α, from the above equation. Start with some initial guess of α, say α (0), obtain α (1) = h(α (0) ), and proceeding in this manner we can obtain α (k+1) = h(α (k) ). Stop the iteration procedure, when α (k+1) α (k) < ǫ, some pre-assigned tolerance level. Once α is obtained then λ, the MLE of λ, can be obtained as λ = u( α) 1/ α.

42 Approximate MLEs To avoid the numerical computation, approximate maximum likelihood estimates have been proposed, which have explicit expressions. It has been used in many cases particularly, when the family is a location scale family. Although, Weibull family is not a location scale family, but log Weibull family is a location scale family.. Suppose a random variable X has Weibull distribution with PDF as given before, then Y = lnx has the extreme value distribution with PDF f Y (y;µ,σ) = 1 σ e(y µ)/σ e(y µ)/σ ; < y <, here µ = lnλ, and σ = 1/α.

43 AMLEs The main idea is to work with Y = lnx, when X has a Weibull distribution. Make a Taylor series approximation to the two normal equations upto first order terms, and obtain the estimates in explicit forms.

44 AMLEs Let us denote y i:n = lnx i:n, and T = T. The likelihood function becomes: where L(µ,σ) 1 r σ r g(z i:n ) { Ḡ(z r:n ) } n r L(µ,σ) 1 σ d i=1 d g(z i:n ) { Ḡ(V) } n d i=1 Case I Case II g(z) = e z ez, Ḡ(z) = e ez, z i:n = y i:n µ, σ V = T µ, µ = lnλ, σ = 1 σ α.

45 AMLEs The log-likelihood functions become Let us denote y i:n = lnx i:n, and T = T. The likelihood function becomes: l(µ,σ) = r lnσ + l(µ,σ) = d lnσ + r lng(z i:n )+(n r)lnḡ(z r:n ) Case I i=1 d lng(z i:n )+(n d)lnḡ(v) Case II i=1

46 Normal Equations: Case I r r i=1 r i=1 g (z i:n )z i:n g(z i:n ) g (z i:n ) g(z i:n ) +(n r)g(z r:n) Ḡ(z r:n ) +(n r) g(z r:n)z r:n Ḡ(z r:n ) = 0 = 0

47 Normal Equations: Case II d d i=1 d i=1 g (z i:n )z i:n g(z i:n ) g (z i:n ) g(z i:n ) +(n d)g(v) Ḡ(V) +(n d) g(v)z r:n Ḡ(V) = 0 = 0

48 Estimates: µ = A B σ and σ = D + D 2 +4rE 2r

49 Bayesian Inference This problem has a nice Bayesian solution also. For the Bayesian inference the following Weibull PDF has been considered: f(x;α,λ) = αλx α 1 e λxα ; x > 0. Prior: It is assumed that λ has a gamma prior, and α has a PDF which is log-concave, and they are independent. Posterior: In this case the posterior distribution of λ given α is gamma, and the posterior distribution α has a log-concave PDF.

50 Bayes Estimates and Credible Intervals It is possible to generate samples from the joint posterior density function of α and λ. Using the generated samples, the Bayes estimates and the associated credible intervals can be obtained.

51 Log-Normal Distribution The PDF of a log-normal distribution is for < µ <, σ > 0; f(x;µ,σ) = 1 2πσx e (lnx µ)2 2σ 2 x > 0.

52 Likelihod Function L(µ,σ) where ( 1 σ ) R exp{ (lnx i:n µ) 2 2σ 2 }{ ( c µ 1 Φ σ )} n R R = { r for Case I d for Case II and c = { lnxr:n for Case I lnt for Case II

53 Log-Likelihod Function l(µ,σ) = R lnσ 1 2σ 2 R { ( c µ (y i:n µ) 2 +(n R)ln 1 Φ σ i=1 )}

54 Normal Equations l µ = 1 σ 2 l σ = R σ + 1 σ 3 R φ(z ) (y i:n µ)+(n R) σφ( z ) i=1 R (y i:n µ)+(n R) z φ(z ) σφ( z ) i=1 = 0 = 0, where z = c µ σ

55 Expectation-Maximization Algorithm 1 Treat this as a missing value problem: 2 Complete observation: The whole sample. (not observable) 3 Missing observations: The censored observations. 4 Try to estimate the missing observations based on the observed samples. 5 X = (x 1:n,...,x R:n ) be the observed data 6 U = (u 1,...,u n R ) be the censored data. 7 (W,U) is the complete data.

56 Complete log-likelihood function l c (µ,σ) = nlnσ 1 2σ 2 In this case R i=1 i=1 (lnx i:n µ) 2 1 2σ 2 [ µ = 1 R ] n R lnx i:n + u i n i=1 n R (lnu i µ) 2 i=1 [ σ = 1 R ] n R (lnx i:n µ) 2 + (lnu i µ) 2 n i=1 i=1

57 E-Step l s (µ,σ) = nlnσ 1 2σ 2 R i=1 (lnx i:n µ) 2 1 2σ 2 n R i=1 E ( (lnu i µ) 2 U i > c )

58 Outline

59 Like conventional Type-I censoring, the main disadvantage of Type-I HCS is that most of the inference results are obtained under the condition that the number of observed failures is at least one, and more over there may be very few failures occurring up to the pre-fixed time T. In that case the efficiency of the estimator(s) may be very low. Because of this alternative hybrid censoring scheme that would terminate the experiment at the random time T = max{x r,n,t} has been proposed. It is called.

60 It has the advantage of guaranteeing that at least r failures are observed at the end of the experiment. If the r failure occurs before time T, the experiment continues up to time point T. On the other hand, if the r-th failure does not occur before time T, then the experiment continues until r-th failure takes place.

61 Comparison Type-I HCS: In this case the termination time is pre-fixed, which is clearly an advantage. However, if the mean lifetime of the experimental item is not small compared to the pre-fixed termination time T, then with high probability, far fewer than r failures may be observed before the termination T. This will definitely has an adverse effect on the efficiency of the inferential procedure based on Type-I HCS.

62 Comparison : In this case, the termination time is a random variable, which is clearly a disadvantage. On the other hand for, more than r failures may be observed at the termination time, and this will result in efficient estimation procedure in this case.

63 Outline

64 Generalized Type-I HCS Suppose n identical items are put on a life test at the time point 0. Fix r,k {1,2,,n} and T (0, ), such that k < r < n. If the k -th failure occurs before time T, terminate the experiment at min{x r:n,t}. If the k-th failure occurs after time T, terminate the experiment at X k:n. It is clear that this HCS modifies the Type-I HCS by allowing the experiment to continue beyond time T if very few failures had been observed up to time point T. Under this censoring scheme, the experimenter would like to observe r failures, but is willing to accept a bare minimum of k failures.

65 Generalized Consider a life-testing experiment in which n items are put on a test. Fix r {1,2,,n}, and T 1,T 2 (0, ), where T 1 < T 2. If the r-th failure occurs before the time point T 1, terminate the experiment at T 1. If the r-th failure occurs between T 1 and T 2, terminate the experiment at X r:n. Otherwise, terminate the experiment at T 2. This hybrid censoring scheme modifies the by guaranteeing that the experiment will be completed by time T 2. Therefore, T 2 represents the absolute longest that the experimenter allows the experiment to continue.

66 Thank You