# Quantitative Accelerated Life Testing Data Analysis

## An Overview of Basic Concepts

In typical life data analysis, the practitioner analyzes life data from a sampling of units operated under normal conditions. This analysis allows the practitioner to quantify the life characteristics of the product and make general predictions about all of the products in the population. For a variety of reasons, engineers may wish to obtain reliability results for their products more quickly than they can with data obtained under normal operating conditions. As an alternative, these engineers may use quantitative accelerated life tests to capture life data under accelerated stress conditions that will cause the products to fail more quickly without introducing unrealistic failure mechanisms.

This document presents an overview of basic concepts in quantitative accelerated life testing data analysis and some suggestions for additional research. ReliaSoft's ALTA software provides a complete array of accelerated life data analysis tools.

## Quantitative Accelerated Life Tests

Accelerated testing methods can be either qualitative or quantitative.
*Qualitative* accelerated tests (such as HALT, HAST, torture tests or
"shake & bake" tests) are used primarily to reveal probable failure modes
for the product so that product engineers can improve the product design.
*Quantitative* accelerated life tests (QALT) are designed to produce the data required for accelerated
life data analysis. This analysis method uses life data obtained under
accelerated conditions to extrapolate an estimated probability density
function (* pdf*) for the product under normal use conditions.

QALT tests can employ usage rate acceleration or overstress acceleration
to speed up the time-to-failure for the units under test. With *usage
rate acceleration*, which is appropriate for products that do not operate
continuously under normal conditions, the analyst operates the units
under test at a greater rate than normal to simulate longer periods
of operation under normal conditions. Data from this type of test can
be analyzed with standard life data analysis techniques. With *overstress
acceleration*, one or more environmental factors that cause the product
to fail under normal conditions (such as temperature, voltage, humidity,
etc.) are increased in order to stimulate the product to fail more quickly.
Data from this type of test require special accelerated life data analysis
techniques that include a mathematical model to "translate" the
overstress *pdfs* to normal use conditions.

## Stress Types and Stress Levels

In an effective quantitative accelerated life test using
overstress acceleration, the
practitioner chooses
one or more stress types that cause the product to fail under normal
use conditions. Stress types can include temperature, voltage, humidity,
vibration or any other stress that directly affects the life of the
product. He/she applies the stress(es) at carefully selected increased levels
and then records the times-to-failure for the products under accelerated
test conditions. For example, if a product normally operates at 290K
and high temperatures cause the product to fail more quickly, then the
accelerated life test for the product may involve testing the product
at 310K, 320K and 330K in order to stimulate the units under test to
fail more quickly. In this example, the *stress type* is temperature and
the *accelerated* *stress levels* are 310K, 320K and 330K. The
*use stress
level* is 290K. [View
a brief introduction to common stress loading schemes and stress profiles
in accelerated life testing.]

Using the life data obtained at each accelerated stress level, the
analyst can use standard life data analysis techniques to estimate the
parameters for the *lifetime distribution* that best fits the data at each stress level
(e.g., Weibull, exponential or lognormal). This results in an
overstress *pdf* for each accelerated stress level.
Another mathematical model, the *life-stress relationship*, is
then required to estimate the *pdf* at the normal use
stress level based on the characteristics of the *pdf*s at each accelerated
stress level.

## Life-Stress Relationships

Statisticians, mathematicians and engineers have developed life-stress relationship models that allow the analyst to extrapolate a use level probability density function from life data obtained at increased stress levels. These models describe the path of a particular life characteristic of the distribution from one stress level to another. The life characteristic can be any life measure expressed as a function of stress. For example, for the Weibull distribution, the scale parameter (eta) is considered to be stress-dependent. Therefore, the life-stress model for data that fits the Weibull distribution is assigned to eta.

The practitioner must choose a life-stress relationship that fits the type of
data being analyzed. Available life-stress relationships include the
*Arrhenius*, *Eyring* and *inverse power law* models. These models are designed
to analyze data with one stress type (e.g., temperature, humidity or
voltage).

Alternatively, the *temperature-humidity* and *temperature-nonthermal* relationships
are combination models that are used to analyze data with two stress
types (e.g., temperature and voltage, temperature and humidity).

The
*general log-linear* and *proportional hazards* models can be used to analyze
data where up to eight stress types need to be considered.

Finally,
the *cumulative damage* (or *cumulative exposure*) model
has been developed to analyze data where the application of the
stress (either at the accelerated stress levels or at the use stress
level) varies with time.

## Calculated Results and Plots

Once you have calculated the parameters to fit a life distribution and a life-stress relationship to a particular data set, you can obtain the same plots and calculated results that are available from standard life data analysis. Some additional results, related to the effects of stress on product life, are also available. Some frequently used metrics include:

**Reliability Given Time:**The probability that a unit will operate successfully at a particular point in time under normal use conditions. For example, there is an 88% chance that the product will operate successfully after 3 years of operation at a given stress level.**Probability of Failure Given Time:**The probability that a unit will be failed at a particular point in time under normal use conditions. Probability of failure is also known as "unreliability" and it is the reciprocal of the reliability. For example, there is a 12% chance that the product will be failed after 3 years of operation at a given stress level (probability of failure/unreliability) and an 88% chance that it will operate successfully (reliability).**Mean Life:**The average time that the unit in the population are expected to operate at a given stress level before failure. This metric is often referred to as "mean time to failure" (MTTF) or "mean time before failure" (MTBF).**Failure Rate:**The number of failures per unit time that can be expected to occur for the product at a given stress level.**Reliable Life**(warranty time): The estimated time when the reliability will be equal to a specified goal at a given stress level. For example, the estimated time of operation at a given stress level is 4 years for a reliability of 90%.**B(X) Life:**The estimated time when the probability of failure will reach a specified point (X%) at a given stress level. For example, if 10% of the products are expected to fail by 4 years of operation at a given stress level, then the B(10) life is 4 years. (Note that this is equivalent to a reliable life of 4 years for a 90% reliability.)**Acceleration Factor:**A unitless number that relates a product's life at an accelerated stress level to the life at the use stress level.**Probability Plot:**A plot of the probability of failure over time. This can display either the probability at the use stress level or, for comparison purposes, the probability at each test stress level. (Note that probability plots are based on the linearization of a specific distribution. Consequently, the form of a probability plot for one distribution will be different than the form for another. For example, an exponential distribution probability plot has different axes than those of a normal distribution probability plot.)**Reliability vs. Time Plot:**A plot of the reliability over time at a given stress level. A similar plot, unreliability vs. time, is also available.A plot of the probability density function (*pdf*Plot:*pdf*) at a given stress level.**Failure Rate vs. Time Plot:**A plot of the failure rate over time at a given stress level. This can display the instantaneous failure rate at a given stress level in a two-dimensional plot or the failure rate vs. time vs. stress in a three-dimensional plot.**Life vs. Stress Plot:**A plot of the product life vs. stress. A variety of life characteristics, such as B(10) life or eta, can be displayed on the plot. This plot demonstrates the effect of a particular stress on the life of the product.**Standard Deviation vs. Stress Plot:**A plot of the standard deviation vs. stress, which provides information about the spread of the data at each stress level.**Acceleration Factor vs. Stress Plot:**A plot of the acceleration factor vs. stress.**Residuals Plots:**Plots of the residual values that have been assigned, via regression analysis, to each point in a data set. These plots provide a tool to assess the adequacy of the model (distribution and life-stress relationship) used to analyze the data set.