A reliability analysis technique for Quantal-Response data

Reliability Engineering and System Safety 41 (1993) 239-243

A reliability analysis technique for QuantalResponse data Ching-Piao Hwang & Huei-Yaw Ke Chung Shan Institute of Science and Technology, PO Box 90008-9-7, Lung Tan, Taoyuan, Taiwan, ROC (Received 15 August 1991; accepted 8 March 1993)

In order to perform one-shot item storage life and reliability evaluation, a new technique is introduced to analyze QuantaI-Response data. First, a modified model is developed to overcome the inherent restrictions when the Maximum Likelihood Method is used to analyze Quantal-Response data. Due to the modification of the model, a simple iterative procedure is then adopted to improve the precision of estimates, with the test data. A numerical example is given to demonstrate the applicability of this technique. It is shown that the proposed technique will provide a practical tool to evaluate the storage life and reliability of one-shot items.

Because of its versatility in fitting time-to-failure distributions of a rather extensive variety of complex systems, the Weibull distribution has frequently been used for reliability and life test data analysis. However, a major deterrent for wider use of the Weibull distribution is the difficulty in estimating its parameters. Also, the calculations involved are not always simple. ~,2 The Maximum Likelihood (ML) Method is the most widely used method for estimating Weibull parameters. The applicability of this method has been discussed in detail by Lagakos. 3 Many scholars, such as Cohen, 4'5 Gibbson and Vance, 6 and Ringer and Sprinkle,7 have used this method to analyze progressively censored test data. Other authors, such as Nelson, ~ Dubey, s and Elperin and Gertsbakh, 9 have explored the use of the Weibull distribution for ML analysis on Quantal-Response data. However, as is mentioned in Nelson's book,~ the iterative procedures for finding the ML estimates of the Weibull parameters may fail to converge. This is therefore still an unsolved problem in this field. This paper presents the application of the ML method to a three-parameter Weibull distribution on life data fitting and reliability estimates by QuantalResponse data analysis. A modified model is used to overcome the inherent restrictions on finding the estimates and a simple iterative procedure is further adopted to improve the precision of estimates. An analysis method is also introduced to estimate the product reliability. Finally, a numerical example is given to illustrate the applicability of the proposed technique.

1 INTRODUCTION Life tests are generally conducted in two areas, i.e., operational life tests and storage life tests. For operational life tests, most life data are complete since the life of each sample unit is continuously monitored and recorded. Such life data consist of the failure time of each test sample. For large samples, the experiment may be terminated after a single stage or through several stages of censoring. Some of the surviving specimens are eliminated from further observation, and the remaining specimens are continuously observed until failure or until censoring is done at subsequent stage. However, most military products being stored or deployed are usually not under continuous surveillance and failure can only be found by inspection. Therefore, one knows only whether the failure time of a product is before or after the inspection time. Two types of such inspection data are (1) QuantalResponse data (test data for one-short item)--each product can be inspected or tested only once; and (2) Interval data--if a product fails, one knows only that the failure occurred between the present inspection time and the previous one. Also, if a product does not fail on its present inspection, one knows only that its failure time will be beyond the inspection time. Such inspection data are often wrongly analyzed as multiple censored data.

Reliability Engineering and System Safety 0951-8320/93/$06.00 © 1993 Elsevier Science Publishers Ltd, England. 239

240

Ching-Piao Hwang, Huei- Y a w Ke

2 PARAMETER DISTRIBUTION

The procedures of ML estimation using the modified model will be briefly delineated as follows. First, the logarithmic likelihood function ,7J is given

ESTIMATION FOR LIFE

Suppose inspections are performed progressively in k stages at times ~, where T,. > T,_~, i = 1, 2 . . . . . k. At the ith stage, f, and r~ denote the number of failed specimens found during inspection and the number of surviving specimens which are removed from further observation (i.e. censored), both of which are assumed to be selected randomly. It follows that k

N=~]

(f+5)

as k

~' = 2 ldIpdf{ti}] + Z Jr,,lnlcdf{ T,}] /.:1

(5)

i=l

Then, considering the Taylor series expansion for in(l + x ) ,

(1)

ln(l + x ) = ~ j l ] ( - 1)/+1xt

(6)

and substituting eqns (3) and (6) into eqn (5) yields ¢T = n ln w - n ln v + ( w - 1 ) 2

In(tj-u)

j=l

k

L = l-I [cdf{ T~}]~[1 - cdf( Till r' i

I

+ ~ r, In[1 - cdf{~}]

i=,1

where N is the total number of specimens originally placed under test. The likelihood function corresponding to QuantalResponse data is

i k

(2)

1

where cdf is the cumulative distribution function associated with the probability density function (pdf) of the three-parameter Weibul] distribution, i.e., pdf{t}=(w)(t-u)W-'exp[

(tv_U)"]

(3) cdf{t} = 1 - exp I

(, ~ u ) " ]

where u, w and v are the location, shape and scale parameters of the Weibull distribution respectively. As mentioned above, the iterative procedures for finding the ML estimates of the Weibull parameters may fail to converge for this model. For example, if there are just two intervals, there is a chance that the observed fraction of the failed specimen over the censored specimen in the second (latter) interval is less than that failed in the first (former) interval; one may then face the problem of non-convergence while applying the ML estimation method. This is more likely with small samples than with large ones. In order to overcome the inherent restrictions on finding the estimates for QuantaI-Response data, a modified model is thus developed. The likelihood function associated with this model is

1 (6 - u)"' - ~ f, Uj=I i=1

] exp /=l

1)

(7~ - u ) w

- Z r , - i=1

(7)

V

Differentiating the logarithmic likelihood function with respect to u, w, v and letting the derivatives equate to zero, the associated likelihood equations are obtained as follows: (l-w)

(tj-u)-'+-Q,..--g,.,,o.o=O j= 1

12

n & 1 - - + ~.~ l n ( t i - u ) - - Q . . , W

j= I

(8)

U

1 + - H,,,,.,,. = 0

U

(9)

U

n 1 1 - -v + ~ 0,,.,,- ~ H, .,,. ,,.,, =

0

(10)

where the notations Qh,g and Hs,.,¢.. are defined as Qh,g =- ~ (ti - u)~-h[ln(ti - u)] ~ j=l k

+ ~ ri(Ti - U)'-h[ln(~ -- U)]u

(11)

i=1 k

/4~,, ~,,--- ~ f ( r ~ -

.)'~. . . . '

i=1

k

L = I~I pdf{tj} ~I [cdf{~} j; /=:1

x [ln(T, - u)] c

i-'l

x 11 - cdf{~}] r'

l ~ exp l=:1

(12) ~-!

(4)

Simultaneous solution of eqns (8)-(10) gives the

In this modified model, the first n inspection times at which failure of product is observed (i.e., inspection times with failure) are assumed to be the pseudo failure times tj, j = 1 . . . . . n. This modification is made to ensure the convergence of finding the estimates. In order to fit the three-parameter Weibull distribution, the first three inspection times with failure are chosen, i.e. n = 3 ~ in this paper.

maximum likelihood estimates ~, ~ and O. Since the closed-form solutions of eqns (8)-(10) are not feasible, an iterative procedure must be used to solve these equations. Due to the high convergence rate, the constrained modified quasilinearization method 1° is adopted to solve these simultaneous equations, then a simple iterative procedure is introduced to improve the precision of estimates.

A reliability analysis technique for Quantal-Response data First, using the estimates obtained from eqns (8)-(10), we find the times t[, i = 1, 2, 3, which are the points where the values of the cumulative distribution function are only one half of the values of the cumulative distribution function counted at the pseudo failure times respectively (i.e., cdf{t~}= lcdf{ti} where ti, i = 1, 2, 3, are the pseudo failure times). Then, replacing each original pseudo failure time ti by the average of ti and t~ and keeping the other experimental results unchanged (i.e., replacing t~ by ½(tz+ t[) for i = 1, 2, 3 and keeping ti unchanged for i > 3), we obtain a set of new estimates. Similarly we find the times tT, i = 1, 2, 3, which possess one half of the cumulative distribution value of those counted at the original pseudo failure times respectively (i.e., cdf{t~'} = ½cdf{ti}). Upon replacing every new pseudo failure time by the average of ti and t7 (i.e., replacing ½(ti+t[) by ½(ti+t~) for i = 1, 2, 3 and keeping ti unchanged for i > 3), we have another set of estimates. If the iteration process continues, we finally obtain one set of pseudo failure times which are close to one half of the cumulative distribution value of those counted by the first three inspection times with failure. The latest set of estimates is considered to give the averge estimated failure times between time 0 and the first three inspection times with failure respectively. These estimates will provide more conservative values than those obtained from the original modified model.

3 RELIABILITY

ANALYSIS

Based on the ML estimates of u, w, v, the point estimator of product reliability at required storage time t is /~(t)=exp[

( t - ~ ~)~]

(13)

It is obvious that in order to calculate the one-sided confidence limit of the reliability estimate, the variance-covariance matrix of the modified model for Weibull parameters is needed. In this paper, the approximate asymptotic variance-covariance matrix C is used and is obtained by inverting the local Fisher information matrix M, i.e. C = M -l

(0/~]2Var{ti}+

Var{/~(fi, if, 0)} = \ 8 a l 13/~\ 2

\8~1

Var{*}

I aft,, 1 aft\ Off

mii = mii

--=-( 025f ) \ c3yi 3yj/f,P293

+ 2(8R](3R]cov{*,

O}

sample size is large. ~t''2 Since /~ one may treat In(/~) as normally the one-sided lower approximate limit for/~ is l

Re = l~/exp{Zr[Var{l~}]~/2/t~}

(15)

where Z r is the 100yth standard normal percentile.

(17a)

cii= cii =- Cov{f, .9j}; Cov{pi, Pi} = Var{pi}

(17b)

where (y,, Yz, Y3) =- (u, w, v). The mq are obtained by differentiating eqns (8)-(10) while ignoring the equality to zero on the right: ff m,, = (ff - 1) ~ (ti - t~)-2 + -;- (rb - 1)Qz.o j=l

~2 ~(ff0 -

1 ) nl.2,(,.o

--

+ 02

(18a)

1-t2 2 0

, , ,1

m,2= ~ ( t j - f i ) - ' - = Q , , , - ~ v Q , . o i=l

U

1

+-H,b

^

@-

~

. ,,o,o+ ~ H, .i.,.o

^

02H2.,.,.,

(18b)

m,3=~v2Q,.i,--~vHl.,.,l.,l+~3Hz.,.,,,l

(18c)

n 1 ^ 1 1 m = = ~5 + TvQo.2 - -0/~' .,,02,, + ~-7 /q2 ...02,

(lSd)

1

m23-

1

^

020,.l+~snL,.I,o 2 . n Q0,0- 02

1

^

i5H2.0.1.,

2

1 /)2oo~

03 /~1,0,0,0 + 0-"4

, , ,

(18e) (18f)

If it is preferable to take Y3-= O=-(v) ~/~ than y3=--v; then the point estimator of product reliability at required storage time t is transformed into /~(t) = e x p [ - ( ~ - - ~ ) ~ ]

(14) provided that the must be positive, distributed. Then, 1007% confidence

(16)

The elements of M and C are given by

m33 = ~-~

and the variance of the reliability estimation function can be approximated as

241

(19)

The aforementioned second-order derivative should be modified by the variable transformation 8~ 30

8~ --. 3v

(wv (w-')/w)

(20)

3z ~ 02~ 802 = Or2 • (wv(W-lVw)2 + - - . [w(w - 1 > (w-2)'w] (21) 8v and the variance of the reliability estimation function

Ching-Piao Hwang, Huei-Yaw Ke

242 in eqn (14) is approximated as

Table 2. Summary of estimates from iterative procedures

Var{/~(fi, ~, 0)} ~ ( ~ - ) 2 w • e x p [ - 2 ( ~ - ~ - ) w]

•

Var{fi} + [ln(~--~)

+ff~Var{O}--t-fi

In

No. of iteration

~

~g,

0 ~ (fQ'"

1 2 3 4 5

19.28 20.13 24.30 25.61 25.79

1.94 2.21 2.21 2.75 2.75

85.43 80.35 79.16 76.56 75-77

Var{ff} Cov{fi, if}

2wa) Coy{a, O} - 2 (22)

4 NUMERICAL

Table 3. Confidence limit of reliability

EXAMPLE

In order to illustrate the applicability of the proposed technique, an example will be given in this section. This example wil show how to analyze the storage life test results of a certain product which are QuantalResponse type. The experimental results are listed in Table 1. From column 5 of Table 1, the fraction of failed specimen (FFS) over the censored specimen at each censored stage does not increase with time; therefore, by the traditional QuantaI-Response data model, the ML fitting for parameters of the Weibull distribution will not converge. In order to overcome this restriction, the proposed modified model is used and the first three inspection times with failure (33, 33, 36 in this example) are taken as the pseudo failure times. Utilizing the iterative procedures, one set of more conservative estimates can be obtained as shown in Table 2. If the required storage time is assumed as 60, the point estimate of reliability (/?) at the required storage time is/?(60) = 0.894 and the approximate asymptotic variance-covariance matrix of the parameter estim-

100y%

Rt

80.0 90.0 95.0 99.9

0-889 0-878 0-869 0.853

ates for this example are calculated and shown as

I

Var{a}

Cov{•, ~} Var{~}

Coy{a, 0 } 1 Cov{ff, O} Sym. Var{O} 3.995 x 10 6 -5.377 x 10 -4 -5.929 )< lO - 2 ] = 6"484 X 10 -2 --4"603 X 10-3 / Sym. 2.891 × 1 0 - 3 d

Substituting these estimated variance and covariance values into eqn (14), the estimated variance of the reliability function is obtained. Finally, utilizing eqn (15), the one-sided lower 100y% confidence limit for R can be estimated. Table 3 gives four different values of the one-sided lower 100y% confidence limit for /~ at different y levels.

5 CONCLUSIONS Table 1. Experimental results Stage

1 2 3 4 5 6 7 8 9 10 11

Inspection time (months)

N u m b e r of censored specimens

N u m b e r of failed specimens

FFS

30 33 36 39 42 45 48 54 60 66 72

2 2 3 4 5 4 5 4 4 4 2

0 2 2 0 2 1 2 3 0 1 0

0.000 1.000 0.667 0.000 0-400 0.250 0.400 0.750 0-000 0-250 0.000

In this paper, a technique for the application of the Maximum Likelihood Method for the three-parameter Weibull distribution on life distribution fitting and for the reliability estimation on Quantal-Response data analysis is introduced. This technique, including a modified model and a simple iterative procedure, can always provide a convergent solution for the Weibull parameters estimation. In addition, expressions for the approximate asymptotic variances and covariances of the modified model for Weibull parameters are derived. With these results, the reliability estimate at the required storage time and the associated approximate lower confidence limit are obtained. It is shown that this technique is a very practical tool for military and industrial applications.

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