# Choose the appropriate model for accelerated life testing

Use probability plots to determine whether model assumptions are appropriate for the accelerated levels. Minitab provides these probability plots in its Accelerated Life Testing analysis:
• Probability plot for each accelerating level based on the fitted model
• Probability plot for standardized residuals
• Exponential probability plot for Cox-Snell residuals

You can use these probability plots to check the following assumptions:

1. Verify that the distribution is appropriate.

If the plot points are close to the fitted line, then the chosen distribution fits the data adequately. Use the Anderson-Darling (adjusted) goodness of fit measure to compare the fit of different distributions. Lower AD values indicate a better fitting distribution.

If the plot points are close to the fitted line in the probability plot based on individual fitted values, but a lack-of-fit is found in the other diagnostic probability plots, then either the transformation or the assumption of equal shape (Weibull or exponential) or scale (other distributions) parameter is inappropriate.

2. Verify the assumptions about equal shape or equal scale parameters.

One model assumption is that the shape (Weibull or exponential) or scale (other distributions) parameters are the same for all levels of the accelerating variable. To confirm this assumption, examine the probability plot at each level of the accelerating variable based on individual fitted values.

If the fitted distribution lines on the plot are approximately parallel, then the assumption of an equal shape (Weibull or exponential) or scale (other distributions) parameter is valid for the accelerating levels. There is no way to empirically verify this assumption at design conditions; therefore, you should use engineering knowledge to evaluate the assumption.

3. Choose the appropriate transformation of the accelerating variable.

Usually, the relationship between the accelerating variable and time to failure involves transforming the accelerating variable. Choosing the appropriate transformation is very important because the assumption is very hard to validate for the accelerated levels and impossible to validate for design levels of the accelerating variable. Along with the collected data, you will need to use engineering knowledge about the relationship between failure time and the accelerating variable.

Minitab provides four main relationships between failure time and the accelerating level:
Arrhenius: X = [11604.83 / (Degrees Celsius + 273.16)]
Based on the Arrhenius Law, it states that the rate of a simple chemical reaction depends on temperature. The Arrhenius relationship is often used to describe items that fail due to degradation caused by a chemical reaction.
Common applications of the Arrhenius transformation are electrical insulations and dielectrics, semiconductor devices, solid state devices, and plastics.
Inverse Temperature: X = [1 / (Degrees Celsius + 273.16)]
Simple relationship that assumes failure time or log failure time is inversely proportional to the Kelvin temperature.
The inverse temperature relationship is not as common as the Arrhenius relationship. The results will be the same as those for the Arrhenius model. However, the coefficients have different interpretations.
Ln (Power) Relationship: X = ln(accelerating variable)
Used for modeling the life of products operating under constant stress. The log relationship is most often used in combination with a log-based failure-time distribution that results in what is known as an inverse power relationship.
Common applications of the log transformation are electrical insulations, dielectrics in voltage endurance tests, metal fatigue, and ball bearings.
Linear Relationship: X = accelerating variable
No transformation is needed.
A change in failure time or log failure time is directly proportional to a change in the accelerating variable.

In all cases, if the plot points are close to the fitted line then the model fits the data adequately. Examine the Anderson-Darling (adjusted) goodness of fit measure to compare the fit of different models. Lower AD values indicate a better fitting model.

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