[Please note that the following article — while it has been updated from our newsletter archives — may not reflect the latest software interface and plot graphics, but the original methodology and analysis steps remain applicable.]
Although often viewed with fear and loathing by reliability engineers, a high value of the Weibull distribution’s shape parameter beta is not necessarily a cause for concern.
The Weibull Distribution and Beta
The Weibull distribution is widely used in the analysis and description of reliability data. This statistical model, first introduced by Waloddi Weibull in the middle of the 20th century, is very popular due to its flexibility. The distribution’s shape parameter, often denoted as the Greek letter beta contributes to this flexibility. The value that this parameter takes on can be quite descriptive with regard to the data being analyzed. For example, the value of beta describes how the failure rate changes over time. Values of beta less than 1 indicate that the failure rate is decreasing with time and are commonly associated with "infantile failures" or failures that occur at a relatively early time due to egregious flaws. Values of beta approximately equal to 1 indicate a failure rate that does not vary over time and are associated with failures of a random nature. Values of beta greater than 1 indicate that the failure rate is increasing with time and are associated with failures related to mechanical wearout modes.
Since many organizations perform reliability analyses of mechanical items, the phenomenon of data sets with beta values greater than 1 is not uncommon. However, when the calculated value of beta turns out to be substantially larger than 1 (for example, 6 or larger), a sense of unease may set in for the reliability engineer. Such high beta values are sometimes regarded as an indication that the product or the test is flawed in some way. As the following example serves to illustrate, however, this is not necessarily the case.
Examples of a "Harmless" High Beta
Suppose that an engineer is considering designs for a small vacuum pump that is to be incorporated into a larger system. The reliability allocation for the system dictates that the vacuum pump must have a 95% reliability at 1000 hours with a 90% confidence level for the system to reach its target goal. The engineer secures a sample of ten units from a likely supplier and puts them under a life test to determine whether that particular pump design will meet the specification. The results of this life test are shown in the table below.
Using ReliaSoft Weibull++ and the maximum likelihood estimation (MLE) analysis method, the engineer estimates the Weibull parameters for the data as eta = 1665.5 hours and beta = 11.6. This may alarm the engineer as the beta value of 11.6 is relatively high. However, when he doggedly perseveres with his analysis, the engineer discovers that the 90% confidence reliability value at 1000 hours is 96.3%. This result is more than enough to meet the specification and, despite the high value of beta, there was nothing wrong with the test or the product.
Explanation of High Betas that are not a Problem
The Weibull shape parameter is a measure of the variability of the data; a high beta value implies a low variability. Consequently, if test units have a high value of beta, it indicates that they will fail within a relatively small time span. This should not be a problem as long as the onset of that time span begins relatively far out on the time scale. In other words, a high value of the shape parameter (beta) is not a problem in itself, as long as the corresponding value of the scale parameter (often denoted as eta) is high enough to allow the product to achieve an acceptable overall reliability.
In fact, for repairable systems, components with high beta values may actually be preferred because the lack of variability can increase the efficiency of a preventive maintenance program. Less variability means that failures occur in a more "controlled" manner and therefore a better optimum replacement interval for preventive maintenance can be quantified. For example, it would be ideal for a preventive maintenance program to have a component that always fails at exactly 1,000 hours of operation. The optimum replacement time would therefore be just before the expected failure, at 999.9 hours.
Another concern that has been associated with reliability data sets with high values of beta is that the relatively steep slope makes it difficult to discern patterns in the data, such as outliers or breaks in the data, on the probability plot. While this is true, it is not a sufficient reason to reject a set of data. Most of the problems that could potentially be discerned by viewing the pattern in the probability plot would hopefully be detected elsewhere in the engineering and testing process. For example, breaks in the pattern of data may indicate that multiple failure modes are active. While a steep slope on a probability plot may tend to obscure such a pattern, the presence of multiple failure modes would likely have been observed during the test or by a concerted failure analysis program. Similarly, outliers can often be identified merely by looking at the raw data. In general, it is not a good idea to depend on the pattern of a probability plot to supplant more dedicated engineering and analysis efforts.
Although there may be genuine concerns about data sets with high values of beta, the fact that a data set has a high value of beta is not necessarily cause for alarm as long as the associated value of h is high enough to offset the lack of variability inherent in data sets with high beta values. In the end, it is more important to analyze the overall behavior of the data and whether or not the product’s test results meet the requirements than to focus solely on the value of a single parameter.