Time to Event Analysis: An Introduction
Shishir Rao
Introduction
I am currently reading the book Survival Analysis: Techniques for Censored and Truncated Data, Second Edition (John P. Klein and Melvin L. Moescheberger). Although the techniques presented in this book focus on applications in biology and medicine, the same statistical tools can also be applied to disciplines ranging from engineering to economics and demography. I have a background in mechanical engineering and am interested in applying survival modeling concepts to data from reliability engineering, manufacturing and quality assurance. This article is the first of, hopefully, many articles that I intend to write as I finish reading different chapters from the book.
The data set(s) that will be analysed are the ones that have been used as examples in another book: Statistical Methods for Reliability Data, Second Edition (William Q. Meeker, Luis A. Escobar, Francis G. Pascual). Both the books I mentioned are excellent resources for anyone who is interested in learning more about this topic.
In this article, we will analyze vehicle shock absorber failure time data Failure time data is also known as survival data, life data, event-time data or reliability data, depending on the field of study. and estimate a few basic survival quantities. The data contains failure times (in kilometers driven) and the mode of failure, first reported by O’Connor (1985) O’Connor, P. D. T. (1985). Practical Reliability Engineering. Wiley. [54, 610]. We will ignore the mode of failure for now and will only consider whether a failure occurred or not, i.e., censored. In a future article, I plan to use the different failure modes to discuss competing risks for time-to-failure data.
Kaplan Meier Curve
Before we dive into the data and the packages used for the analysis in R, here is a brief introduction to a few basic survival concepts and their mathematical expressions. You don’t strictly need to know these formulas to conduct an analysis using packages in R. This would be of interest to those who prefer to peek a little under the hood to see some of the package’s internal workings.
Survival function $-S(t)-$Cumulative distribution function (CDF) $-F(t) = 1 – S(t)-$ is defined as the probability that a unit’s time to failure ($-T-$) is greater than some time ($-t-$)Time is measured in kilometers in this example.
$$ \displaystyle S\left(t\right)=Pr\left(T>t\right) $$
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