Quality Progress - December 2016 - 63


was used to estimate these parameters

percentage defective after burn-in to be

from the data.

0.09%, that is, (0.045) x (1 - 0.98) x 100

The probability plot in Figure 2 is identical to that in Figure 1, but now shows

with an upper 97.5% confidence bound of
0.112%, that is, (0.056) x (1 - 0.98) x 100.

the ML fit to the data using the preceding
mixture model. This model appears to fit

Further discussion

the data reasonably well.

Software. ML estimation of mixtures of

The model parameter estimates are

distributions with censored data requires

shown in Table 2. The proportion p of the

special software. Fortunately, modern

defective subpopulation is 0.045 (with a

software packages increasingly allow

95% confidence interval of 0.037 to 0.056).

you to fit such models. We used the JMP

This meant that there must be a recall of

software in our study. Reliasoft Weibull++

potentially vulnerable devices from the field

also provides this capability.

because the lower 97.5% confidence bound
on p of 3.7% exceeded the specified 3%.

DATA AND
TECHNICAL
DETAILS

The data set used in this study, along
with technical details of fitting the
Weibull mixture model in JMP, are included in Online Figure 1 and Online Table
1, which can be found on this column's
webpage at www.qualityprogress.com.

modes, acting independently.
The actual (observed) failure time on a

Mixture models, segmentation

particular device is the minimum of these

analyses and competing failure mode

(hypothetical) failure times. When, in addi-

analyses. When data come from two or

tion, the specific failure mode of the failed

nerable devices also is necessary because

more populations and when the subpopu-

devices can be identified, we recommend

the upper 97.5% confidence bound on p

lation to which individual devices belong

separate analyses for each failure mode

of 5.6% falls well below 20%. Moreover,

can be identified, we recommend using

(and combining the results), as discussed

the 98 percentile of the time-to-failure

segmentation analysis, as described in

in an earlier column.5

distribution, fD(t), of the subpopulation

"Segmentation Analysis of Bleed Systems,"

of defective units was estimated to be 33.2

and an earlier column.4

Moreover, a burn-in of potentially vul-

th

minutes, with an upper 97.5% confidence

In this column's example, neither the
subpopulation nor the specific failure

Another model arises when units come

mode was known, leading to the mixture

bound of 84.5 minutes, providing the

from the same population, but there are

model that we have described. We should

needed information for setting the dura-

several causes of failure (several or com-

emphasize, however, that physically mean-

tion of the burn-in test.

peting failure modes). The most common

ingful segmentation analysis and individual

Finally, we note that because we esti-

assumption in this case (in part, because it

failure mode analysis are generally more

mate 4.5% of the total units to be defective,

makes the analysis simple) is that all units

informative than mixture model analysis.

we would estimate the total population

are susceptible to each of these failure

0.04
0.024
0.014
0.007
0.0035
0.0016
0.0008
0.0004
0.00016
0.00008
0.00004
0.0001 0.001 0.01
0.1
1
10
100
Time to dielectric breakdown (minutes)

Mixture model fit on
a Weibull probability
plot / FIGURE 2

Probability

Probability

Weibull probability plot of
time-to-failure data from
dielectric breakdown test
and ML fit / FIGURE 1

Thus, we urge practitioners to strive

0.04
0.024
0.014
0.007
0.0035
0.0016
0.0008
0.0004
0.00016
0.00008
0.00004
0.0001 0.001 0.01
0.1
1
10
100
Time to dielectric breakdown (minutes)

December 2016 * QP 63


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