Discrete lifetime data

Abstract

In addressing quality issues in an industrial, manufacturing or engineering context, inherently non-negative measures of quality are often used. Statistical methodology for reliability analysis is well developed, at least in the situation where the quality measure is continuous. However, the quality measure may be discrete either through the method of observation or the countable nature of the data. There are very clear theoretical differences between continuous and discrete reliability methods. For example the simple relationship between the failure rate function and the reliability function in the continuous case does not hold in the discrete case. Also, some naive statistical methods can be misleading in the discrete case. For example, a plot of the empirical failure rate will always suggest that the failure rate eventually increases even when the true failure rate is non-increasing. The theoretical differences between continuous and discrete reliability methods are highlighted. The basic properties of some discrete distributions that are potentially useful for lifetime data analysis are presented and the general behaviour of the failure rate function is investigated. Two exact distributions for an empirical failure rate estimator of discrete lifetimes are proposed. The use of the two proposed distributions is illustrated by introducing a new method, the failure rate control chart, to detect departures from a constant failure rate. The properties of the proposed method and other related tests are investigated by simulation. hi dealing with failure time data it is common that there are two or more failure modes, the competing risks phenomenon. The observed data in such a situation will comprise the failure times together with the relevant failure mode for each failure time. In trying to understand the failure process it is reasonable to ask whether the failure modes are acting independently. In the case of continuous failure times, it is well known that it is not possible to answer this question with such data. This is the so-called identifiability crisis in competing risks. However, when the failure times are discrete the hypothesis of independence can sometimes be addressed. Crowder, in his Lifetime Data Analysis paper of 1997, proposed a test in such circumstances, derived its large sample null properties and illustrated its use on a medical data set. The results presented by Crowder are summarised and corrected, and some potential practical shortcomings of his test are discussed. Also, a modified version of the Crowder test is proposed. Simplified forms of Crowder's test statistic and its modified version are proposed. Finally, by recasting Crowder's results in terms of classical contingency tables, the relationship of his test to other tests of independence is highlighted and the properties are investigated by simulation.

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