8.3.1.2. Lognormal or Weibull tests

8. Assessing Product Reliability
8.3. Reliability Data Collection
8.3.1. How do you plan a reliability assessment test?

8.3.1.2. Lognormal or Weibull tests

Planning reliability tests for distributions other than the exponential is difficult and involves a lot of guesswork

Planning a reliability test is not simple and straightforward when the assumed model is lognormal or Weibull. Since these models have two parameters, no estimates are possible without at least two test failures, and good estimates require considerably more than that. Because of censoring, without a good guess ahead of time at what the unknown parameters are, any test plan may fail.

However, it is often possible to make a good guess ahead of time about at least one of the unknown parameters - typically the "shape" parameter ( $\sigma$ for the lognormal or $\gamma$ for the Weibull). With one parameter assumed known, test plans can be derived that assure the reliability or failure rate of the product tested will be acceptable.

Lognormal Case (shape parameter known): The lognormal model is used for many microelectronic wear-out failure mechanisms, such as electromigration. As a production monitor, samples of microelectronic chips taken randomly from production lots might be tested at levels of voltage and temperature that are high enough to significantly accelerate the occurrence of electromigration failures. Acceleration factors are known from previous testing and range from several hundred to several thousand.

Lognormal test plans, assuming sigma and the acceleration factor are known

The goal is to construct a test plan (put $n$ units on stress test for $T$ hours and accept the lot if no more than $r$ failures occur). The following assumptions are made:

The life distribution model is lognormal
Sigma = $\sigma_0$ is known from past testing and does not vary appreciably from lot to lot
Lot reliability varies because $T_{50}$ values (the lognormal median or 50th percentile) differ from lot to lot
The acceleration factor from high stress to use stress is a known quantity "$A$"
A stress time of $T$ hours is practical as a line monitor
A nominal use $T_{50}$ of $T_u$ (combined with $\sigma_0$) produces an acceptable use CDF (or use reliability function). This is equivalent to specifying an acceptable use CDF at, say, 100,000 hours to be a given value $p_0$ and calculating $T_u$ via:
An unacceptable use CDF of $p_1$ leads to a "bad" use $T_{50}$ of $T_b$, using the same equation as above with $p_0$ replaced by $p_1$

The acceleration factor $A$ is used to calculate a "good" or acceptable proportion of failures $p_a$ at stress and a "bad" or unacceptable proportion of fails $p_b$: $$ p_a = \Phi \left( \frac{\mbox{ln } (AT / T_u)}{\sigma_0} \right) , \,\,\,\,\, p_b = \Phi \left( \frac{\mbox{ln } (AT / T_b)}{\sigma_0} \right) \, , $$ where $\Phi$ is the standard normal CDF. This reduces the reliability problem to a well-known Lot Acceptance Sampling Plan (LASP) problem, which was covered in Chapter 6.

If the sample size required to distinguish between $p_a$ and $p_b$ turns out to be too large, it may be necessary to increase $T$ or test at a higher stress. The important point is that the above assumptions and equations give a methodology for planning ongoing reliability tests under a lognormal model assumption.

Weibull test plans, assuming gamma and the acceleration. factor are known

Weibull Case (shape parameter known): The assumptions and calculations are similar to those made for the lognormal:

The life distribution model is Weibull

Gamma = $\gamma_0$ is known from past testing and does not vary appreciably from lot to lot
Lot reliability varies because $\alpha$ values (the Weibull characteristic life or 62.3 percentile) differ from lot to lot
The acceleration factor from high stress to use stress is a known quantity "$A$"
A stress time of $T$ hours is practical as a line monitor
A nominal use $\alpha$ of $\alpha_u$ (combined with $\gamma_0$) produces an acceptable use CDF (or use reliability function). This is equivalent to specifying an acceptable use CDF at, say, 100,000 hours to be a given value $p_0$ and calculating $\alpha_u$
An unacceptable use CDF of $p_1$ leads to a "bad" use $\alpha$ of $\alpha_b$, using the same equation as above with $p_0$ replaced by $p_1$

The acceleration factor $A$ is used next to calculate a "good" or acceptable proportion of failures $p_a$ at stress and a "bad" or unacceptable proportion of failures $p_b$: $$ p_a = 1 - \mbox{exp } \left[ -\left( \frac{AT}{\alpha_u} \right)^{\gamma_0} \right] , \,\,\,\,\,\,\, p_b = 1 - \mbox{exp } \left[ -\left( \frac{AT}{\alpha_b} \right)^{\gamma_0} \right] \,\, . $$ This reduces the reliability problem to a Lot Acceptance Sampling Plan (LASP) problem, which was covered in Chapter 6.

Planning Tests to Estimate Both Weibull or Both Lognormal Parameters

Rules-of-thumb for general lognormal or Weibull life test planning

All that can be said here are some general rules-of-thumb:

If you can observe at least 10 exact times of failure, estimates are usually reasonable - below 10 failures the critical shape parameter may be hard to estimate accurately. Below 5 failures, estimates are often very inaccurate.
With readout data, even with more than 10 total failures, you need failures in three or more readout intervals for accurate estimates.
When guessing how many units to put on test and for how long, try various reasonable combinations of distribution parameters to see if the corresponding calculated proportion of failures expected during the test, multiplied by the sample size, gives a reasonable number of failures.
As an alternative to the last rule, simulate test data from reasonable combinations of distribution parameters and see if your estimates from the simulated data are close to the parameters used in the simulation. If a test plan doesn't work well with simulated data, it is not likely to work well with real data.