Censored Data

Samples are often collected in such a way that the exact value of one or more cases is unknown. Such missing information is referred to as censored data. In one source of such censored data, values are known to exceed or fall below some limit. Often, for example, in a study based on the survival times of laboratory animals, a protocol requires that all data be collected within a specified period of time. For an appreciably sized subset of subjects, an exact survival time may not be known, simply because the study ends before the event of interest (such as the animal's death) could be observed. Because the exact survival times for those who lived longer than the length of the study are not known, this illustrates the generation of right-censored data. Another common example occurs when the age of study participants is recorded in exact years for most subjects, but the ages of those younger than 18 are all recorded as less than 18 years. Because the exact ages of persons younger than 18 years are not known, this is an example of leftcensored data. In general, when some, but not all, values are recorded on a continuous scale of measurement, special techniques are required that differ from the many of the common procedures based on what is called maximum likelihood (ML) estimation.

Work with censored data is facilitated by a notational convention for observed values called order statistics. The sample median is an example of an order statistic. Unlike the mean of a size n sample, which is represented by placing a bar above a letter, as in X, the subscripted symbol X(1/2n + 1 =2), when n is an odd number, or [Xi(n=2) + X(1/2n=2 + 1) =2, when n is an even number, designates the median. More generally, by calling on parenthesized subscripts to denote ascending variate value magnitudes, the order of variate values can be designated. For example, the smallest and largest values are denoted respectively as X(1) and X(n). (In case of a tied value, the counterpart of a coin toss distinguishes between variate designations, say, between X(i) and X(i + 1).)