This is just an Excerpt from a larger document, click here to view the entire document.Types of Censoring
To better analyze this complex issue, we begin with a characterization of the censoring mechanisms. Such characterization can be based on several elements, among them, the status of the entity observed, both at the time we start and at the time we finish our observation. Censoring mechanisms can also be characterized based on whether or not the experiment is terminated at the time of the "event of interest" (e.g., failure or death).
With respect to the status of the entity observed, censoring can occur at either extreme (or at both ends) of the entity life. That is, we may not know exactly at what time the life of the entity started or finished. This happens because the entity in question may have already started operating at the time we begin our observation. Or the life may have not yet finished (e.g., failed) by the time we complete our observation period.
Figure 1 illustrates censoring situations. Line "a" shows an entity that has already been "operating" for some unknown period of time, before we start monitoring it. This case is called "left-censoring." The "X" symbols in Figure 1 represent the points in time when we actually start or finish monitoring the censored entities, other than the beginning (of entity life, at time zero) or the end of the experimental observation period (time T0).
Figure 1. Type I (Time-Truncated) Censoring Cases (Click to Zoom)
Similarly, Line "b" shows an entity that has been monitored since the beginning of its life (i.e., at the start of the experiment) but which we have ceased to observe before the experiment ends (time T0) or it fails. That is, we observe the entity for some time, after which we are not able to monitor it any more. This other type of truncation is known as "right censoring."
We can stop monitoring all the entities, putting an end to the experiment, at some pre-specified time T0, which is independent of the event of interest (e.g., death). The entity in Line "c" has been monitored all along the experiment. Finally, a more complex example is presented in Line "d".
Here, both the beginning and end of the entity "life" are now unknown (interval censored). We can only monitor such entity for some intermediate part of its "life" span. Censoring schemes, where the end of the observation period is not determined by an event of interest (e.g., failure), are referred to as time censoring, time truncation, or suspension in time. Such censoring schemes are not event-driven and are known as Type I. In these schemes, the experiment stopping time (T0) is pre-established and the number of failures observed (i) during the period of experimentation is random.
On the other hand, we may elect to observe a sample of "n" entities until the time of occurrence of some pre-specified event of interest, such as the time of the ith failure or death (i ≤ n) denoted by the Xi in Figure 2. That is:
0 < X1 < X2 < X3 < ..... Xi < ∞
Figure 2. Type II (Event-Driven) Censoring Case (Click to Zoom)
At the time of the ith failure (failure times Xi are denoted in the graph by an arrowhead) we discontinue our observation of the (n-i) sample elements remaining in operation. This other censoring scheme is often referred to as "failure" or "event" truncation and is known as Type II censoring. In these cases, the experiment stopping time (Xi) is random and the number of failures (i) occurred during experimentation is pre-established.
In either censoring scheme (Type I or II) the number "i" of "events" of interest (e.g., death) observed during the experiment is less than the total "n" entities on trial. Some times the distribution of the "lives" of the entities is known. Other times, the probability "p" of occurrence of an event during the observation period (time T0), can be calculated. In such cases, we may be able to model the underlying life (X) distribution and estimate the parameters of interest such as Mean Time to Failure (MTTF or μ), failure rate (FR or θ), tenth percentile of device life (L-10) and calculate confidence intervals (CI) for them.
Other times, the problem of modeling "life" is further complicated and, thus, approached differently than we do here. Some examples of such complications include when failures are (or are not) replaced at the time they occur, or when the distribution of the "lives" is not Exponential. In such cases, the hazard function (instantaneous probability of failure) is time-dependent and there are several additional parameters than we now need to estimate from the data. In addition, having more complex censoring mechanisms, in conjunction with a time-dependent hazard rate, creates many more theoretical difficulties.
In the rest of this START sheet, we discuss some of the issues involved in estimating reliability parameters from Exponentially distributed censored data and present several numerical examples. We first present the case for time-censored experiments. Then, we discuss failure censored ones, of which experiments developed until the first failure occurs, constitute a special case. We end by giving a short bibliography for further study of time and failure censored experiments.
