8.
Assessing Product Reliability
8.4. Reliability Data Analysis 8.4.2. How do you fit an acceleration model?


This section will discuss the following:  
Estimate acceleration model parameters by estimating cell \(T_{50}\) values (or \(\alpha\) values) and then using regression to fit the model across the cells 
How to fit an Arrhenius Model with Graphical Estimation
Graphical methods work best (and are easiest to describe) for a simple onestress model like the widely used Arrhenius model $$ t_f = A \cdot \mbox{ exp } \left( \frac{\Delta H}{kT} \right) \, , $$ with \(T\) denoting temperature measured in degrees Kelvin (273.16 + degrees Celsius) and \(k\) is Boltzmann's constant (8.617 × 10^{5} in eV/K). When applying an acceleration model to a distribution of failure times, we interpret the deterministic model equation to apply at any distribution percentile we want. This is equivalent to setting the life distribution scale parameter equal to the model equation (\(T_{50}\) for the lognormal, \(\alpha\) for the Weibull and the MTBF, or \(1/\lambda\) for the exponential). For the lognormal, for example, we have $$ T_{50} = A \cdot \mbox{ exp } \left( \frac{\Delta H}{kT} \right) \, , $$ $$ \mbox{ln } T_{50} = y = \mbox{ln } A + \Delta H \left( \frac{1}{kT} \right) \, . $$ This can be written as $$ y = a + bx $$ with $$ b = \Delta H \,\,\,\,\, \mbox{ and } \,\,\,\,\, x = \frac{1}{kT} \, . $$ So, if we run several stress cells and compute \(T_{50}\) values for each cell, a plot of the natural log of these \(T_{50}\) values versus the corresponding \(1/kT\) values should be roughly linear with a slope of \(\Delta H\) and an intercept of \(\mbox{ln } A\). In practice, a computer fit of a line through these points is typically used to obtain the Arrhenius model estimates. Remember that \(T\) is in Kelvin in the above equations. For temperature in Celsius, use the following for \(1/kT\): 11605/(\(t\)°C + 273.16). An example will illustrate the procedure. 

Arrhenius model example 
Component life tests were run at three temperatures:
85 °C, 105 °C and 125 °C. The lowest temperature cell was populated with 100 components; the 105 °C cell had 50 components and the highest stress cell had 25 components. All tests were run until either all the units in the cell had failed or 1000 hours was reached. Acceleration was assumed to follow an Arrhenius model and the life distribution model for the failure mode was believed to be lognormal. The normal operating temperature for the components is 25 °C and it is desired to project the use CDF at 100,000 hours. Test results: Cell 1 (85 °C): 5 failures at 401, 428, 695, 725 and 738 hours. Ninetyfive units were censored at 1000 hours running time. Cell 2 (105 °C): 35 failures at 171, 187, 189, 266, 275, 285, 301, 302, 305, 316, 317, 324, 349, 350, 386, 405, 480, 493, 530, 534, 536, 567, 589, 598, 599, 614, 620, 650, 668, 685, 718, 795, 854, 917, and 926 hours. Fifteen units were censored at 1000 hours running time. Cell 3 (125 °C): 24 failures at 24, 42, 92, 93, 141, 142, 143, 159, 181, 188, 194, 199, 207, 213, 243, 256, 259, 290, 294, 305, 392, 454, 502 and 696. One unit was censored at 1000 hours running time. Failure analysis confirmed that all failures were due to the same failure mechanism (if any failures due to another mechanism had occurred, they would have been considered censored run times in the Arrhenius analysis). Steps to Fitting the Distribution Model and the Arrhenius Model:


Solution for Arrhenius model example  Analysis of Multicell Arrhenius Model Data:
The following lognormal probability plot was generated for our data so that all three stress cells are plotted on the same graph. Note that the lines are somewhat straight (a check on the lognormal model) and the slopes are approximately parallel (a check on the acceleration assumption). The cell \(\mbox{ln } T_{50}\) and sigma estimates are obtained from linear regression fits for each cell using the data from the probability plot. Each fit will yield a cell \(A_0\), the \(\mbox{ln } T_{50}\) estimate, and \(A_1\), the cell sigma estimate. These are summarized in the table below.
The three cells have 11605/(\(t\) °C + 273.16) values of 32.40, 30.69 and 29.15 respectively, in cell number order. The Arrhenius plot is With only three cells, it is unlikely a straight line through the points will present obvious visual lack of fit. However, in this case, the points appear to line up very well. Finally, the model coefficients are computed from a weighted linear fit of \(\mbox{ln } T_{50}\) versus 11605/(\(t\) °C + 273.16), using weights of 5, 35, and 24 for each cell. This will yield a \(\mbox{ln } A\) estimate of 18.312 (\(A = e^{18.312} = 0.1115 \times 10^{7}\)) and a \(\Delta H\) estimate of 0.808. With this value of \(\Delta H \), the acceleration between the lowest stress cell of 85 °C and the highest of 125 °C is $$ \mbox{exp } \left[ 0.808 \times 11605 \times \left( \frac{1}{358.16}  \frac{1}{398.16} \right) \right] = 13.9 \, , $$ which is almost 14 × acceleration. Acceleration from 125 °C to the use condition of 25 °C is 3708×. The use \(T_{50}\) is $$ e^{18.312} \times e^{0.808 \times 11605 \times 1/298.16} = e^{13.137} = 507383 \, . $$ A single sigma estimate for all stress conditions can be calculated as a weighted average of the three sigma estimates obtained from the experimental cells. The weighted average is (5/64) × 0.908 + (35/64) × 0.663 + (24/64) × 0.805 = 0.74. The analyses in this section can can be implemented using both Dataplot code and R code. 

Models involving several stresses can be fit using multiple regression  Two stress models, such as the temperature/voltage
model given by
$$ t_f = A \cdot V^{\beta} \cdot \mbox{ exp } \left( \frac{\Delta H}{kT} \right) \, , $$
need at least four or five carefully chosen stress cells to estimate all
the parameters. The
Backwards L design
previously described is an example of a design for this model.
The bottom row of the "backward L" could be used for a plot testing the
Arrhenius temperature dependence, similar to the above Arrhenius example.
The right hand column could be plotted using
\(y = \mbox{ln } T_{50}\) and \(x = \mbox{ln } V\),
to check the voltage term in the model. The
overall model estimates should be obtained from fitting the multiple regression model
$$ Y = b_0 + b_1 X_1 + b_2 X_2 \, , $$
with
$$ \begin{array}{rl}
Y = \mbox{ln } T_{50}, & b_0 = \mbox{ln } A \\
& \\
b_1 = \Delta H, & X_1 = 1/kT \\
& \\
b_2 = \beta, & X_2 = \mbox{ln } V \, .
\end{array} $$
Fitting this model, after setting up the
data vectors, provides estimates for \(b_0, \, b_1\), and \(b_2\).
Three stress models, and even Eyring models with interaction terms, can be fit by a direct extension of these methods. Graphical plots to test the model, however, are less likely to be meaningful as the model becomes more complex. 