Quality Progress - December 2016 - 62


STATISTICS ROUNDTABLE
high voltage stress and high tempera-

A Weibull distribution is commonly used

(defective) and H (healthy) devices, and

ture was developed to weed out the

as the underlying statistical model in the

each are subject to a different failure-time

vulnerable devices without damaging

analysis of data on time to dielectric break-

distribution. Use fD(t) and fH(t) to desig-

the healthy ones. This test, however,

down. Figure 1 is a Weibull distribution

nate the probability density functions-

was expensive and would be economi-

probability plot of the time to dielectric

assumed to be Weibull distributions in

cally worthwhile (versus scrapping all

breakdown data. The straight line shown in

this example-for the lifetimes of these

potentially vulnerable product) only if p

Figure 1 is the maximum likelihood (ML)

subpopulations.

was less than 20%.

fit of a Weibull distribution to the data. The

If, for example, it was known that 15%

plotted points are for the 188 units that

of the population belonged to the defec-

the appropriate (minimum) time dura-

failed during tests. The unfailed units are

tive subpopulation and the remaining 85%

tion of the burn-in test that would weed

taken as censored observations in this plot

belonged to the healthy subpopulation, the

out at least 98% of the defective units.

and in the subsequent analysis.

probability density function f(t) for prod-

*	 In addition, we wanted to determine

To respond to the preceding questions,

3

The curvature of the plotted observa-

uct life of the entire (that is, combined)

a sample of 3,600 randomly selected

tions suggests that a simple Weibull dis-

population is provided by the so called

devices from the population of affected

tribution does not provide a good model.

"mixture model" for two subpopulations:

devices was subjected to the burn-in test

This is not surprising because a simple

for 50 minutes each. The resulting time-to-

Weibull distribution assumes that all units

failure data are summarized in Table 1.

f(t) = 0.15 fD(t) + 0.85 fH(t).
In our application, the mixing pro-

are subject to a single common failure

portion is unknown and also must be

Initial analysis. Table 1 shows that

mode. In the current application, there is

estimated from the data. Therefore, more

188 units failed and the remaining 3,412

strong evidence suggesting that there are

generally, the mixture model with two

units survived the 50-minute test. There-

two distinct product populations: devices

subpopulations can be expressed as:

fore, 188/3600, or 5.2%, is an initial but

with compromised dielectric insulation

f(t) = pfD(t) + (1 - p) fH(t),

crude, estimate of p. This estimate could

due to exposure to the faulty valve and

in which p (0 ≤ p ≤ 1) is the unknown mix-

be too low because it excluded dielectric

normal (healthy) devices that were not

ing proportion.

breakdown failures that occur presumably

exposed. It was not known prior to the

shortly after 50 minutes. Or the estimate

test, however, whether a particular device

may be too large because it may have

came from the defective (that is, exposed

Semiconductor example:
Mixture model analysis

included possible early failures unrelated

to faulty valve) or the healthy population.

In our application, we used a mixture

to the valve malfunction (that is, failures

Therefore, a segmentation analysis could

model with a Weibull distribution for the

on healthy units). Also, it did not respond

not be performed.

two subpopulations. Thus, there were

to the question concerning the duration of

five parameters to be estimated from the

the burn-in test. Therefore, more sophisti-

Mixture model

data: the mixture proportion p, and the

cated statistical analysis was needed.

Suppose that the product population is

Weibull scale and shape parameters for

comprised of two subpopulations-D

subpopulations D and H. The ML method

Single distribution Weibull analysis.

Summary of time-to-failure
data for the semiconductor
example / TABLE 1

ML estimates of the mixture
model parameters / TABLE 2
Parameter

Minutes

Status

Number of devices

0 to 10

Failed

147

10 to 20

Failed

13

p

Maximum
likelihood (ML)
estimate

95% confidence interval
Lower
bound

Upper
bound

0.045

0.037

0.056

2.53

1.33

4.79

20 to 30

Failed

8

Weibull scale D

30 to 40

Failed

10

Weibull shape D

0.53

0.44

0.64

Weibull scale H

664

47

9,354

Weibull shape H

1.89

0.59

6.05

40 to 50

Failed

10

Greater than 50

Unfailed
(right censored)

3,412

62 QP * www.qualityprogress.com

Note: The confidence intervals are based on the likelihood ratio method.


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