Probability as a function of age from observations over several years

Question

Hoping someone can help me correct the flaws in my logic.

I have a number of water tanks, which may leak. I want to model the probability a water tank leaks given that it has never leaked before as a function of water tank age. Due to the small number of water tanks that do leak, I have to calculate probabilities in broad age bins of 10 years.

I calculate the probability as follows

1. Find all the first leaks and stratify them into year of leak.
2. In each year, calculate the water tank age at first leak.
3. In each year, count the number of first leaks in each 10 year age bin.
4. In each year, calculate the age of all water tanks that have not leaked prior to that year (or ever).
5. In each year, calculate probability in 10 year age bins, as first leak counts divided by the sum of first leak counts and never leaked counts.

This gives me probabilities in 10-year age bins for each year of data I have. I then take the mean over age bins across years to get a final probability as function of age. The splitting by year may seem weird but I do this because non-leaking water tanks need to be sampled at the same age as the leaking water tanks at their first leaks. I know of no other way to deal with this.

There is a lot I am unsure of in the calculation for example

1. Is this business of calculating the probability in each year of data and taking the mean reasonable?
2. Does a flaw in this method arise because I don't track the leakage of water tanks all the way through a 10-year age bin? For example, If I have a water tank aged 2 years old that has never leaked, should I be considering the fact that it may leak in the 8 future years that it remains in that bin?

I am hoping someone can help me with my logic, feel free to dismantle this procedure as long as you can help me work towards a correct method.

Answer 1

This seems like a study that is best analyzed by standard methods of survival analysis. If you know the actual age at which tanks fail and have data on the ages of the tanks that haven't failed, survival analysis overcomes many of the problems that you have encountered and provides additional ways to make your analysis more compelling:

1. You don't need to combine the failure data into 10-year bins. You can use individual age-at-failure data at the precision with which you know those ages.

2. You don't need to sample non-leaking tanks at the same ages as tank failures. For non-leaking tanks all you need is their ages at the last time you had information that they weren't leaking.

3. Treating tanks that haven't yet leaked as censored observations, a standard practice in survival analysis, includes information about them up to their ages when they were last observed, in a way that removes your concern that they might nevertheless fail during the rest of a 10-year period.

4. Survival analysis can be combined with regression analysis that incorporates information about other variables that could be associated with failures, like dimensions, materials, temperatures, and usage of the tanks.

Survival analysis is similar in outline to what you are proposing, but can be done at each age for which a leak was observed: at each such age you see how many tanks in total reached at least that age without a leak, to calculate the probability of a leak developing at that age. There are well-established ways to estimate confidence intervals to gauge the precision of the probability estimates.

So instead of inventing your own methods apply the tools of survival analysis, available (at least in rudimentary form) in most statistical software packages.