Weibull and Reliability/Failure Time Analysis - Censored Observations
There are several procedures in STATISTICA that can be used to fit the Weibull distribution to data: The Survival Analysis module allows you to fit the Weibull distribution to so-called grouped data, specifically, to a life table. The Stats 2D Graphs menu contains options for producing quantile-quantile (Q-Q) plots and probability-probability (P-P) plots for the Weibull (and many other distributions). From these plots, parameter estimates can be obtained (see, for example, Dodson, 1994, or Hahn and Shapiro, 1967). Of course, the Process Capability Analysis Setup: Raw Data dialog also offers the choice of several distributions, among them the Weibull distribution. The options provided in the Weibull analysis and reliability/failure time analysis facilities (accessible from the Process Analysis Startup Panel) are unique in several ways. Most importantly, they allow you to fit the Weibull distribution to data sets containing censored observations.
- Censoring
- In most studies of product reliability, not all items in the study will fail. In other words, by the end of the study the researcher only knows that a certain number of items have not failed for a particular amount of time, but has no knowledge of the exact failure times (i.e., "when the items would have failed"). Those types of data are called censored observations. The issue of censoring, and several methods for analyzing censored data sets, are also described in great detail in the context of the Survival Analysis module. Censoring can occur in many different ways.
- Type I and II censoring
- So-called Type I censoring describes the situation when a test is terminated at a particular point in time, so that the remaining items are only known not to have failed up to that time (e.g., we start with 100 light bulbs, and terminate the experiment after a certain amount of time). In this case, the censoring time is often fixed, and the number of items failing is a random variable. In Type II censoring the experiment would be continued until a fixed proportion of items have failed (e.g., we stop the experiment after exactly 50 light bulbs have failed). In this case, the number of items failing is fixed, and time is the random variable.
Left and right censoring. An additional distinction can be made to reflect the "side" of the time dimension at which censoring occurs. In the examples described above, the censoring always occurred on the right side (right censoring), because the researcher knows when exactly the experiment started, and the censoring always occurs on the right side of the time continuum. Alternatively, it is conceivable that the censoring occurs on the left side (left censoring). For example, in biomedical research one may know that a patient entered the hospital at a particular date, and that s/he survived for a certain amount of time thereafter; however, the researcher does not know when exactly the symptoms of the disease first occurred or were diagnosed.
Single and multiple censoring. Finally, there are situations in which censoring can occur at different times (multiple censoring), or only at a particular point in time (single censoring). To return to the light bulb example, if the experiment is terminated at a particular point in time, then a single point of censoring exists, and the data set is said to be single-censored. However, in biomedical research multiple censoring often exists, for example, when patients are discharged from a hospital after different amounts (times) of treatment, and the researcher knows that the patient survived up to those (differential) points of censoring.
The methods described in this section are applicable primarily to right censoring, and single- as well as multiple-censored data.
Two- and three-parameter Weibull distribution. The Weibull distribution is bounded on the left side. If you look at the probability density function, you can see that the term x- must be greater than 0. In most cases, the location parameter (Theta) is known (usually 0): it identifies the smallest possible failure time. However, sometimes the probability of failure of an item is 0 (zero) for some time after a study begins, and in that case it may be necessary to estimate a location parameter that is greater than 0. The options available via the Weibull analysis and reliability/failure time analysis facilities (accessible from the Process Analysis Startup Panel) provide several methods for estimating the location parameter of the three-parameter Weibull distribution. In practice, Dodson (1994) recommends to look for downward of upward sloping tails on a probability plot, as well as large (>6) values for the shape parameter after fitting the two-parameter Weibull distribution, which may indicate a non-zero location parameter.