Increment-decrement life-tables

The increment-decrement life-table appears to be a birth-death life-table or more generally an open-population life-table.

This could be appropriate for the hospital acquired infection length of stay calculations because the risk set in the infected state changes over time as new patients become infected and infected patients die in-hospital or are discharged alive.

So potentially we can calculate a life-table for the infected state and a separate life-table for the non-infected state, and then make a comparison.

This would require us to account for censoring.

AFT and LoS

I realised the connection between the hospital length of stay estimation that I've been working on and the microlife work of David Speigelhalter.

At the heart of his approach is the life-table. These give the probability of reaching your next birthday. So that conditional on reaching age x what is the the chance that you don't die before you reach age x+1. Using life-table its possible to calculate an expected lifetime which is the residual remaining life.

It easy to see the connection with expected LoS time in hospital conditional on being in hospital on day x.

He uses an accelerated failure time model (AFT) which is an alternative to the very commonly used Cox proportional hazard model. Whereas the Cox model give the ratio of hazards for infected and non-infected i.e. a change in rate for each day, the AFT gives the amount of time difference for the two groups. That is, for a given percentile (e.g. the median) what are the times at which they occur. For example, when both samples have a 50% survival the difference in time may be x days.

The problem with the AFT model is that you need to specify a failure time distribution unlike the semi-parametric Cox model.

Speigelhalter, when looking at mortality uses the Gompertz distribution because after 30 years old the hazard is exponentially shaped, i.e. the log-transformed data is linear.

The hospital data is not so nicely behaved, unfortunately.

Using the Gompertz distribution an approximation of the remaining lifetime is derived in terms of hazards. Therefore, with a proportional hazard coefficient the difference in lifetime can be estimated.