Considerations for the Mathematical Modelling Of Survival

From the preceding discussion it should be evident that the scale of measurement of survival time is a relative rather than an absolute one.  Furthermore, the scale of measurement of survival is not limited to  physical time, but may be defined relative to a different reference framework and process. In the analysis of software reliability, survival time may be measured as the number of memory access or floating point operations until the program “crashes”, whereas in industrial accelerated testing, time may be expressed as the number of continuous operating cycles until machine failure. Even in situations in which survival is measured in terms of physical time, the experimental setup may dictate the use of a more or less arbitrary and usually discrete temporal scale. For example in clinical studies of sub-acute and chronic diseases, survival time is usually measured in weeks or months, whereas in certain acute illness a finer discretization of time is more appropriate.

These considerations suggest that a distinction is made between the scale of the observation time  in which survival is measured and the physical lifetime of the individuals under study. The former may be either discrete or continuous, whereas the latter corresponds to a continuous quantity (at least according to some physicists!). In certain situations one will be interested in relating the physical time to the observation time scale, but in other contexts such a relation will be irrelevant. Other problems may necessitate the introduction of additional time scales in addition to the physical and observation time scales. Since some of these scales will be discrete and others continuous, it is desirable to faithfully represent their nature in statistical models. Yet for reasons of analytical simplification and computational tractability one may be forced to discretize a continuous scale or embed a bona fide discrete problem into a continuous framework. Another mathematical subtlety concern the cardinality of the mathematical spaces used to represent the various scales of a survival analysis problem. In problems where death is the event under study the observation time scale will be a proper subset of the set of positive integers or real numbers, since no one lives for ever (except maybe Duncan McLeod in Highlander). In this light, the use of probability distributions in which there is no upper bound and time can range from an (arbitrary) value of zero to infinity is a (usually) useful approximation, that allows the full utilization of differential calculus.

In summary, both discrete and continuous probability frameworks are needed when embarking on survival analysis  depending on the application context and the availability of analytical/numerical tools to get the job done.

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