2.5.11.2 Algorithm for Life Test Plans

Demonstration Test Plan

Minimum Value to be Demonstrated ---- Goal

1. Scale (weibull or Exponential distribution) or Location (other distribution)
2. Percentile (\(t_p\)) at Percent (\(p\))
3. Reliability (\(R(t_0)\)) at Time (\(t_0\))
4. MTTF (Mean time to failure)

The distribution parameter is first calculated （table below) if the chosen goal of the test plan is either 2, 3 or 4.

Distribution	Parameters	Percentile (\(t_p\)) is provided	Reliability (\(R(t_0)\)) is provided	MTTF is provided
Normal	location: \(\mu\) scale: \(\sigma\)	\[\mu = t_p - \sigma \Phi_{nor}^{-1}(p/100)\]	\[\mu = t_0 - \sigma \Phi_{nor}^{-1}(1 - R(t_0))\]	\[\mu = MTTF\]
Lognormal	location: \(\mu\) scale: \(\sigma\)	\[\mu = \text{ln}(t_p) - \sigma \Phi_{nor}^{-1}(p/100)\]	\[\mu = \text{ln}(t_0) - \sigma \Phi_{nor}^{-1}(1 - R(t_0))\]	\[\mu = \text{ln}(MTTF)-\sigma^2/2\]
Exponential	scale: \(\theta\)	\[\theta = \frac{t_p}{-\text{ln}(1-p/100)}\]	\[\theta = \frac{t_0}{-\text{ln}(R(t_0))}\]	\[\theta = MTTF\]
Smallest Extreme Value	location: \(\alpha\) scale: \(\beta\)	\[\alpha = t_p - \beta \text{ln}( - \text{ln}(1-p/100) )\]	\[\alpha = t_0 - \beta \text{ln}( - \text{ln}(R(t_0)) )\]	\[\alpha = MTTF + c\sigma \]
Weibull	scale: \(a\) shape: \(b\)	\[a= \frac{t_p}{(-\text{ln}(1-p/100))^{1/b}}\]	\[a= \frac{t_0}{(-\text{ln}(R(t_0)))^{1/b}}\]	\[a = \frac{MTTF}{\Gamma(1+b)} \]
Logistic	location: \(\mu\) scale: \(\sigma\)	\[\mu = t_p - \sigma\text{ln}(\frac{p/100}{1-p/100}) \]	\[\mu = t_0 - \sigma\text{ln}(\frac{1-R(t_0)}{R(t_0)}) \]	\[\mu = MTTF \]
Loglogistic	location: \(\mu\) scale: \(\sigma\)	\[\mu = \text{ln}(t_p) - \sigma\text{ln}(\frac{p/100}{1-p/100}) \]	\[\mu = \text{ln}(t_0) - \sigma\text{ln}(\frac{1-R(t_0)}{R(t_0)}) \]	\[\mu = \text{ln}(\frac{MTTF}{\Gamma(1+\sigma)\Gamma(1-\sigma)}) \]

where

\(\Phi_{nor}\): the CDF function for the standard normal distribution.

c: Euler's constant

\(\Gamma\): gamma function

Calculate testing time from sample size

The probability of failure \(p(t)\) for a \(100(1-\alpha)\%\) confidence level is first calculated

\[p(t) = \text{Beta}^{-1}(1 - \alpha, m + 1, N - m)\]

Then \(t\) can be calculated by inverting \(p(t)\)

\(t = \exp(\sigma \Phi^{-1}(p) + \mu)\) for log-location-scale family

\(t = \sigma \Phi^{-1}(p) + \mu\) for location-scale family

where

\(\text{Beta}\): the CDF of beta distribution

\(m\): maximum number of failures allowed

\(N\): sample size

\(t\): testing time

\(\alpha\): significance level

\(\Phi^{-1}\): inverse CDF of the standard distriubtion of the selected distribution

\(\mu\): location parameter of the distribution

\(\sigma\): scale parameter of the distribution

Calculate sample size from testing time

The probability of failure \(p(t)\) for a \(100(1-\alpha)\%\) confidence level is first calculated

\( p(t) = \Phi(\frac{\text{ln}(t)-\mu}{\sigma})\) for log-location-scale distribution

\( p(t) = \Phi(\frac{t-\mu}{\sigma})\) for location-scale distribution

Then \(N\) can be calculated numerically by solving the following equation

\[ binocdf(m, N, p(t)) = \alpha\]

where

\(binocdf\): Binomial distribution

\(\Phi\): CDF of the standard distriubtion of the selected distribution

Probability of passing a test plan

The probability of passing a test plan \(\text{prob}(r)\) as a function of improvement (\(r\)) can be calculated:

\[\text{prob}(r) = binocdf(m, N, p(t,r))\]

where

\(p(t,r)\): the probability of failure in terms of \(t\) and \(r\). It depends on the distribution family:

• \(p(t,r) = \Phi(\frac{\text{ln}(t)-\mu_G-C(r)}{\sigma})\) for log-location-scale distribution
• \(p(t,r) = \Phi(\frac{t-\mu_G-C(r)}{\sigma})\) for log-location-scale distribution

\(r\): ratio of improvement for log-location-scale family; amount of improvement for a location-scale family.

\(C(r)\): calculation of \(C(r)\) is listed in the table below:

Distribution Family	Parameters	Percentile (\(t_p\)) is provided	Reliability (\(R(t_0)\)) is provided	MTTF is provided
log-location-scale	\(C(r) = \text{ln}(r) \) \(r = \text{exp}(\mu_1)/\text{exp}(\mu) \)	\(C(r) = \text{ln}(r) \) \( r = t_{p1}/t_p \)	\(C(r) = A(r) \) \(r = R_1(t_0)/R(t_0) \)	\(C(r) = \text{ln}(r) \) \(r = MTTF_1/MTTF \)
location-scale	\(C(r) = r \) \(r = \mu_1-\mu \)	\(C(r) = r\) \( r = t_{p1}-t_{p} \)	\(C(r) = B(r) \) \(r = R_1(t_0)-R(t_0) \)	\(C(r) = r \) \(r = MTTF_1-MTTF \)

where

\[A(r) = \sigma(\Phi^{-1}(1-R_G)-\Phi^{-1}(1-rR_G))\]

\[B(r) = \sigma(\Phi^{-1}(1-R_G)-\Phi^{-1}(1-r-R_G))\]

\(N\): provided sample size or the sample size calculated from testing time

\(t\): provided testing time or the testing time calculated from sample size

Estimation Test Plan

Specify planning values

1. Two distribution parameters (\(\mu\),\(\sigma\))
2. Percentile and scale (\(t_p\),\(p\),\(\sigma\))
3. Percentile and location (\(t_p\),\(p\),\(\mu\))
4. Two percentiles (\(t_{p_1}\),\(p_1\),\(t_{p_2}\),\(p_2\))

The distribution parameters (\(\mu\),\(\sigma\)) are first calculated （table below) if the specified planning value type of the test plan is either 2, 3 or 4.

Distribution Family	Percentile and scale	Percentile and location	Two percentiles
log-location-scale	\[\mu = \text{ln}(t_p) - \sigma\Phi^{-1}(p) \]	\[\sigma = \frac{\text{ln}(t_p) - \mu}{\Phi^{-1}(p)} \]	\(\mu = \frac{\text{ln}(t_{p_1})\Phi^{-1}(p_2)-\text{ln}(t_{p_2})\Phi^{-1}(p_1)}{\Phi^{-1}(p_2)-\Phi^{-1}(p_1)} \) \(\sigma = \frac{\text{ln}(t_{p_2})-\text{ln}(t_{p_1})}{\Phi^{-1}(p_2)-\Phi^{-1}(p_1)} \)
location-scale	\[\mu = t_p - \sigma\Phi^{-1}(p) \]	\[\sigma = \frac{t_p - \mu}{\Phi^{-1}(p)} \]	\(\mu = \frac{t_{p_1}\Phi^{-1}(p_2)-t_{p_2}\Phi^{-1}(p_1)}{\Phi^{-1}(p_2)-\Phi^{-1}(p_1)} \) \( \sigma = \frac{t_{p_2}-t_{p_1}}{\Phi^{-1}(p_2)-\Phi^{-1}(p_1)} \)

Estimate percentile for percent

Estimated value of percentile

For log-location-scale family

• \[t_p = \text{exp}(\mu + \sigma\Phi^{-1}(p)) \]

For location-scale family

• \[t_p = \mu + \sigma\Phi^{-1}(p) \]

Asymptotic variance of percentile

Asymptotic covariance of the MLEs of \(\mu, \sigma\) is first calculated based on the chosen censoring type (uncensored, right censored, interval censored)[1]. Then the asymptotic variance of percentile can be calculated

\[\text{Avar}(t_{p}) = \text{Avar}(\mu) + (\Phi^{-1}(p))^2\text{Avar}(\sigma)+2\Phi^{-1}(p)\text{Acov}(\mu, \sigma)\]

Sample size to estimate percentile \(t_p\)

For log-location-scale family

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(t_{p})}{(\text{ln}(R_T))^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(t_{p})}{(\text{ln}(R_T))^{2}}\), for one-sided confidence interval

For location-scale family

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(t_{p})}{D_T^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(t_{p})}{D_T^{2}}\), for one-sided confidence interval

where

\(D_T\): precision as distance from bound to estimate

\(R_T\): \(1 + D_T/t_p\) for upper bound, \(1/(1-D_T/t_p)\) for lower bound

\(N\): sample size

\(t_p\): estimated percentile

\(\alpha\): significance level

\(\Phi^{-1}\): inverse CDF of the standard distriubtion of the selected distribution

\(\Phi_{nor}^{-1}\): inverse CDF of the standard normal distribution

\(\mu\): location parameter of the distribution

\(\sigma\): scale parameter of the distribution

Estimate reliability for time

Estimated value of reliability

• \[R(t) = 1-\Phi(Z)\]

where

\(Z\): \(\frac{\text{ln}(t)-\mu}{\sigma}\) for log-location-scale family; \(\frac{t-\mu}{\sigma}\), for location-scale family

\(\Phi\): CDF of standard distribution of the selected function

Asymptotic variance of Z

Asymptotic covariance of the MLEs of \(\mu, \sigma\) is first calculated based on the chosen censoring type (uncensored, right censored, interval censored)[1]. Then the asymptotic variance of \(Z\) can be calculated

\[\text{Avar}(Z) = \frac{1}{\sigma^2} (\text{Avar}(\mu) + Z^2\text{Avar}(\sigma)+2Z\text{Acov}(\mu, \sigma))\]

Sample size to estimate reliability \(R(t)\)

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(Z)}{D_T^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(Z)}{D_T^{2}}\), for one-sided confidence interval

where

\(D_T\): \(\Phi^{-1}(D_{lower} + \Phi(Z)) - Z\) for the lower bound; \(Z - \Phi^{-1}(D_{upper} + \Phi(Z)) \) for the upper bound

\(D_{lower}, D_{upper}\): precision as distance from bound to estimate

\(N\): sample size

\(\alpha\): significance level

\(\Phi^{-1}\): inverse CDF of the standard distriubtion of the selected distribution

\(\Phi_{nor}^{-1}\): inverse CDF of the standard normal distribution

\(\mu\): location parameter of the distribution

\(\sigma\): scale parameter of the distribution

Accelerated Life Test Plan

Specify planning values

1. Intercept and slope (\(b_0\),\(b_1\))
2. (Percentile, percent, stress) and slope (\(t\),\(p\),\(x\),\(b_1\))
3. (Percentile, percent, stress) and intercept (\(t\),\(p\),\(x\),\(b_0\))
4. Two percentiles (\(t_1\),\(p_1\),\(x_1\),\(t_2\),\(p_2\),\(x_2\))

The intercept and slope (\(b_0\),\(b_1\)) are first calculated （table below) if the specified planning value type of the test plan is either 2, 3 or 4.

Distribution Family	Percentile and slope	Percentile and intercept	Two percentiles
log-location-scale	\[b_0 = \text{ln}(t) - \sigma\Phi^{-1}(p) - b_1x \]	\[b_1 = \frac{\text{ln}(t)-\sigma\Phi^{-1}(p)-b_0}{x} \]	\(b_0 = \frac{x_2\text{ln}(t_1)-x_1\text{ln}(t_2)+\sigma(x_1\Phi^{-1}(p_2)-x_2\Phi^{-1}(p_1))}{x_2-x_1} \) \(b_1 = \frac{\text{ln}(t_2)-\text{ln}(t_1)+\sigma(\Phi^{-1}(p_1)-\Phi^{-1}(p_2))}{x_2-x_1} \)
location-scale	\[b_0 = t - \sigma\Phi^{-1}(p) - b_1x \]	\[b_1 = \frac{t-\sigma\Phi^{-1}(p)-b_0}{x} \]	\(b_0 = \frac{x_2t_1-x_1t_2+\sigma(x_1\Phi^{-1}(p_2)-x_2\Phi^{-1}(p_1))}{x_2-x_1} \) \(b_1 = \frac{t_2-t_1+\sigma(\Phi^{-1}(p_1)-\Phi^{-1}(p_2))}{x_2-x_1} \)

Location parameter at accelerating variable

• \[ \mu = b_0 + b_1\times x \]

where

• \(x\): the transformed accelerating variable \(AR\).

Linear: \(x = AR\)

Arrhenius: \(x = \frac{11604.53}{AR+273.16}\)

Inverse temp: \(x = \frac{1}{AR+273.16}\)

Ln (Power): \(x = \ln(AR)\)

Estimate percentile for percent

Estimated value of percentile

For log-location-scale family

• \[t_p = \text{exp}(\mu + \sigma\Phi^{-1}(p)) \]

For location-scale family

• \[t_p = \mu + \sigma\Phi^{-1}(p) \]

Best allocation for each stress level

Asymptotic covariance of the MLEs of \(\mu, \sigma\) is first calculated for each accelerating variable (refer to section). Then it can be converted to the asymptotic covariance of the MLEs of \(\sigma, b_0, b_1\)[1]. The overall asymptotic covariance is a weighted sum of the asymptotic covariances associated with each accelerating variable, where the weights are determined by the allocation at each stress level.

The asymptotic covariance of \(t_{p}\) at designed stress \(x_0\) can be calculated by the following equations

• \[ \text{Avar}(\mu) = \text{Avar}(b_0) + 2x_0\text{Acov}(b_0, b_1) + x_0^2\text{Avar}(b_1) \]
• \[ \text{Acov}(\mu, \sigma) = \text{Acov}(b_0, \sigma) + x_0\text{Acov}(b_1, \sigma)\]
• \[\text{Avar}(t_{p}) = \text{Avar}(\mu) + (\Phi^{-1}(p))^2\text{Avar}(\sigma)+2\Phi^{-1}(p)\text{Acov}(\mu, \sigma)\]

The optimal allocation is obtained via a numerical search that minimizes \(\text{Avar}(t_{p})\).

Sample size to estimate percentile \(t_p\)

For log-location-scale family

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(t_{p})}{(\text{ln}(R_T))^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(t_{p})}{(\text{ln}(R_T))^{2}}\), for one-sided confidence interval

For location-scale family

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(t_{p})}{D_T^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(t_{p})}{D_T^{2}}\), for one-sided confidence interval

where

\(D_T\): precision as distance from bound to estimate

\(R_T\): \(1 + D_T/t_p\) for upper bound, \(1/(1-D_T/t_p)\) for lower bound

\(N\): sample size

\(t_p\): estimated percentile

\(\alpha\): significance level

\(\Phi^{-1}\): inverse CDF of the standard distriubtion of the selected distribution

\(\Phi_{nor}^{-1}\): inverse CDF of the standard normal distribution

\(\mu\): location parameter of the distribution

\(\sigma\): scale parameter of the distribution

Standard error of percentile \(t_p\) for sample size

When the specifications include the sample size, then

For log-location-scale family

• \[SE = \sqrt{\text{Avar}(t_p)/N} * t_p\]

For location-scale family

• \[SE = \sqrt{\text{Avar}(t_p)/N} \]

Estimate reliability for time

Estimated value of reliability

• \[R(t) = 1-\Phi(Z)\]

where

\(Z\): \(\frac{\text{ln}(t)-\mu}{\sigma}\) for log-location-scale family; \(\frac{t-\mu}{\sigma}\), for location-scale family

\(\Phi\): CDF of standard distribution of the selected function

Best allocation for each stress level

Asymptotic covariance of the MLEs of \(\mu, \sigma\) is first calculated for each accelerating variable (refer to section). Then it can be converted to the asymptotic covariance of the MLEs of \(\sigma, b_0, b_1\)[1]. The overall asymptotic covariance is a weighted sum of the asymptotic covariances associated with each accelerating variable, where the weights are determined by the allocation at each stress level.

The asymptotic covariance of reliability \(rel\) at designed stress \(x_0\) can be calculated by the following equations

• \[ \text{Avar}(\mu) = \text{Avar}(b_0) + 2x_0\text{Acov}(b_0, b_1) + x_0^2\text{Avar}(b_1) \]
• \[ \text{Acov}(\mu, \sigma) = \text{Acov}(b_0, \sigma) + x_0\text{Acov}(b_1, \sigma)\]
• \[\text{Avar}(rel) = \frac{\phi(Z)^2}{\sigma^2} \text{Avar}(\mu) + Z^2\text{Avar}(\sigma)+2Z(p)\text{Acov}(\mu, \sigma)\]

where

\( \phi \): PDF of the selected distribution

The optimal allocation is obtained via a numerical search that minimizes \(\text{Avar}(rel)\).

Sample size to estimate reliability \(R(t)\)

• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha/2))^2 \frac{\text{Avar}(Z)}{D_T^{2}}\), for two-sided confidence interval
• \(N=(\Phi_{\text{nor}}^{-1}(1-\alpha))^2 \frac{\text{Avar}(Z)}{D_T^{2}}\), for one-sided confidence interval

where

\(D_T\): \(\Phi^{-1}(D_{lower} + \Phi(Z)) - Z\) for the lower bound; \(Z - \Phi^{-1}(D_{upper} + \Phi(Z)) \) for the upper bound

\(D_{lower}, D_{upper}\): precision as distance from bound to estimate

\(N\): sample size

\(\alpha\): significance level

\(\Phi^{-1}\): inverse CDF of the standard distriubtion of the selected distribution

\(\Phi_{nor}^{-1}\): inverse CDF of the standard normal distribution

\(\mu\): location parameter of the distribution

\(\sigma\): scale parameter of the distribution

Standard error of reliability \(rel\) for sample size

When the specifications include the sample size, then

• \[SE = \sqrt{\text{Avar}(rel)/N}\]

Reference

1. W.Q. Meeker and L.A. Escobar (1998). "Statistical Methods for Reliability Data". ohn Wiley & Sons, Inc