Archive for September, 2012

Considerations for the Mathematical Modelling Of Survival

September 21, 2012

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.


Basic concepts in Survival Analysis

September 20, 2012

Survival Analysis is a statistical methodology for the occurrence of discrete events unfolding in continuous (or discrete) time. Such events are experienced one or more times during the lifetime of individual subjects whose life histories are available for examination. Examples  include: death of an individual, acute illness (e.g. myocardial infarction), time to resolution of an infection after antibiotics, marriage and divorce, acceptance of a scientific paper. Still, events are not limited to medical, sociological contexts, humans or even biological systems. In fact,  statistical methods for lifetime data form the core of the field of reliability analysis, which focuses on the failure of industrial systems ranging from light bulbs to nuclear safety systems and from software to airplanes.
In all survival analytic activities the term it survival time or  time-to-event is used to denote the time elapsed from an initiating event until the event of interest. In many cases, the initiating event will not coincide with the “birth” of the individual i.e. the time the latter came into existence, but will be some other type of discrete event that is of interest to the investigator. In medical contexts, this could be the time of diagnosis of a particular disease, while in industrial applications an initiating event may be defined by the beginning of a reliability test. Furthermore, the same body of survival time data may be analyzed from alternative viewpoints which adopt a different definition of the initiating event and an origin of the time scale. Even though the relative nature of the survival time is not usually explicitly acknowledged, its importance for proper statistical modelling of real world problems cannot be overemphasized. In particular the issue of left truncation will arise when multiple initiating points are contemplated. Under left truncation individuals whose survival time is less than a threshold cannot studied because they do not survive long enough to be observed. For example when studying stroke, only those individuals that survive the acute (pre-hospital) phase and reach the hospital can be included in a study.
Yet another feature that distinguishes survival analysis from other statistical methods, is censoring. The latter arises because of the nature of collecting of survival times. Typically one has to wait for individuals to experience the event of interest and when a study is terminated only a subgroup of individuals will have experienced that event. Hence the survival, times will be known exactly only for them; the only thing that is known for the lifetimes of the remaining individuals is that they exceed certain values (right censoring).