# Survival (Kaplan Meier) of signal transmission that can both fail and recover

I'm investigating the failure to transmit of a number of signals in time. In my data I often see signals transmitting for quite a long time, but after some time they gradually start failing but still recover quickly, and after some more time they fail more often and for a longer time before recovering, ultimately failing definitively.

To describe this process, I want to use Kaplan Meier survival analysis: a signal at risk is a signal of which I receive data; it is at risk to fail the transmission.

As I described: After failure, the connection can be re-established (recovery / revival).

What are possible methods for this kind of survival analysis?

I propose these few methods of which I don't know what would be more useful or what might be general practice:

1. Do a ‘normal’ survival analysis on the failures. Do a second revival analysis on all the signals that recover after failure.
2. Categorise groups and analyse risk of transition to the next group: ‘at risk’ (normal signal), ‘failure’ (transmission lost) and ‘recovered’ (recovered signal after failure).
3. I could graph the empirical cumulative distribution function for all signals over time. The result can go up or down over time where respectively more or less signals transmit their data or not.
4. I could take the total failure time and total success time and analyse the fraction of the signals at different times.

Ad 1. I'm not sure if this method would be useful for my data. For example: a signal might fail to transmit and recover many times. This would result in a very short survival and high risk of failure in the beginning of time. Also, this would split a very bad signal in lots of very short signals, whereas a perfect, long and continuous signal without failure would count as only one.

Ad 2. How do I do that? I would end up with three categories such that a signal can go from ‘at risk’$\to$‘failure’$\to$‘recovered’($\to$‘failure’$\to$‘recovered’$\to$‘failure’)... etc. This doesn't seem much different from approach 1. Quite dodgy, I could also create lots of categories for ‘at risk’$\to$‘fail 1’$\to$‘recover 1’$\to$‘fail 2’$\to$‘recover 2’$\to$ ... etc. This is probably not the way to go.

Ad 3. I have a feeling this is not survival analysis, but merely a plot of available signals in time. If the risk to fail doesn't both decrease and increase over time, then for $n$ the number of signals, $n\to\infty$, the (empirical) cumulative distribution function would become continuously decreasing. However, since I don't have a virtually infinite number of signals, it is possible that at some point in time the survival curve goes up. How can I then describe hazard rates and survival probabilities? Will the hazard rate simply be negative (e.g. -3/5 signals failed, i.e. in total 3 signals recovered while 5 were at risk) and will survival probability simply increase when more signals recover than fail? This can possibly work, but it doesn't provide information when for example one signal fails, but at the same time another recovers; it would result in no change in survival probability.

Ad 4. A signal of 10 seconds could e.g. fail for 6 seconds in total: 1 second during the first failure, 2 during the second, 3 during the third. This is not simply Kaplan Meier analysis, because the signal goes from $1\to5/6\to3/6\to0$ (in this example). Because my signal survival now can take any continuous value between 0 and 1, this cannot be done with Kaplan Meier anymore, can it? Do I use Cox regression instead, with only time as a predictor?

Do you know any other method(s) to analyse both survival/failure/revival simultaneously or separately?

• use multistate survival models eg cran.r-project.org/web/packages/msm/vignettes/msm-manual.pdf – seanv507 Mar 23 '16 at 8:42
• Thanks, I'll look into that. It seems like quite a read though, with more states than I need (so my problem is simpler than the general cases described in that document). – Erik Mar 23 '16 at 8:44
• I find it useful to think through discrete time survival models (and use) and then move to continuous time. in discrete time you are just estimating a markov transition matrix for each time period – seanv507 Mar 23 '16 at 10:06