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I have a univariate time series which is an index of susceptibility to failure, and a binary variable which indicates whether a failure actually occurred in a given time window or not. I want to carry out a statistical test that would quantify the performance of the susceptibility index (e.g., compared to a null model where there is no correlation between the time series and the binary variable, or to another competing index). I am looking for methods to achieve this goal.

I understand that if the failure is a very rare event obtaining a statistical significance may be hard, but even before that, my main problem is that the data points I sample from the time series are not independent (e.g., if the index is very high at a given time period, it's very likely high also in the next period). Therefore the length of the time window that I employ should be important. Any ideas?

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  • $\begingroup$ This sounds like a survival/failure analysis kind of problem. However, I'm not quite clear on why this issue of windows arises. Is there a choose of temporal scales? $\endgroup$ – Iterator Aug 5 '11 at 22:53
  • $\begingroup$ Could you elaborate a bit how it relates to survial/failure analysis? The issue of window is related to how I represent my data. If I sample pairs of susceptibility index for the period [ t,t + dt) and a binary variable whether a failure occurred during the period, I can arbitrarily increase my significance values by using smaller and smaller dt. $\endgroup$ – Ruggiero Spearman Aug 7 '11 at 0:05
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    $\begingroup$ Can you have just 1 failure overall? If so, then it is akin to relating death to age or relating failure of an object to one or more covariates, including age. The "textbook" method for this is to use a technique known as the Cox proportional hazards model. $\endgroup$ – Iterator Aug 7 '11 at 2:31
  • $\begingroup$ Btw, I am not saying that the correct answer is the Cox prop. hazard model, but you might take a look to see if the assumptions are relevant. In general, take a look at survival models. $\endgroup$ – Iterator Aug 7 '11 at 2:33
  • $\begingroup$ By the way, I am proposing an approach that leans toward modeling, though a statistical test appropriate to the model is doable. On the other hand, your original question relates to just using a statistical test. On re-reading, I wonder if I misunderstood - are you solely interested in a statistical test? $\endgroup$ – Iterator Aug 7 '11 at 2:37
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If by "failure" you mean something that can only occur once for a subject without occurring again, use the Cox proportional hazards model. If your "failure" can occur more than once for a given subject, use a shared frailty model, which is related to a multilevel logistic regression.

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