Quantitative Accelerated Life Testing Data Analysis
An Overview of Basic Concepts and Directory of Other Resources

In typical life data analysis, the practitioner analyzes life data of a product's sample operating under normal conditions in order to quantify the life characteristics of the product and make predictions about all of the products in the population. For a variety of reasons, manufacturers may wish to obtain reliability results for their products more quickly than they can with data obtained under normal operating conditions. Instead, they may use quantitative accelerated life tests to capture life data for the product under accelerated stress conditions, which cause the products to fail more quickly.

This document presents an overview of basic concepts in quantitative accelerated life testing data analysis and some suggestions for additional research on the subject. We assume that you already have a basic understanding of life data analysis concepts. 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 (like HALT, HAST, torture tests or shake and 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 quantify the life of the product and 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 products under test. With usage rate acceleration, which is appropriate for products that do not operate continuously under normal conditions, the analyst operates the products 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 (like 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, which include a mathematical model to "translate" the overstress probability density functions to normal use conditions. The analysis techniques for data from quantitative overstress accelerated life tests are discussed in this guide.

Stress Types and Stress Levels
In an effective quantitative accelerated life test, the analyst 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 then applies the stress(es) at various increased levels and measures 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 life distribution (e.g. Weibull, exponential or lognormal) that best fits the data at each stress level. This results in an overstress probability density function (pdf) for each accelerated stress level. Another mathematical model, the life-stress relationship, is then required to estimate the probability density function (pdf) at the normal use stress level based on the characteristics of the pdfs 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 (pdf) from life data obtained at increased stress levels. These models describe the path of a life characteristic of the distribution from one stress level to another. The life characteristic can be any life measure, such as the mean or median, expressed as a function of stress. For example, for the Weibull distribution, the scale parameter, (eta), is considered to be stress-dependent and the life-stress model for data that fits the Weibull distribution is assigned to eta.

You 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). The temperature-humidity and temperature-nonthermal relationships are combination models that allow you to analyze data with two stress types (e.g. temperature and voltage or 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. ReliaSoft's ALTA PRO is the only commercially available software package capable of analyzing data with time-varying stresses.  

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. In addition, for the failure rate, reliability/unreliability and pdf plots, the information can be displayed for a given stress level in a two-dimensional plot or for a range of stress levels in a three-dimensional plot (e.g. failure rate vs. time vs. stress). Some frequently used metrics include:

• Reliability Given Time: The probability that a product 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 product 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 (and an 88% chance that it will operate successfully). 
• Mean Life: The average time that the products 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.
• 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 warranty time 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 that 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.
• Pdf Plot: A plot of the probability density function (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, like 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.