# How to normalize event times in Survival Analysis?

For simplicity, lets assume that we have 30 cars, same model and same production year, with different mileages, i.e.: car-1 has 102k miles in the odometer, car-2 has 153k, car-3 has 45k, etc.

1. We start measuring the time now and we fully observe these cars until they all break down permanently - so there is no censuring.

2. We calculate the survival curve with Kepler-Meier formula

3. We use this curve to predict when the next 30 cars of the same model is going to break down. These cars too have different initial mileage condition compared to the elements used to calculate the survival curve.

My question is how do we normalize the curve/times to be fair to the initial condition?

*I would like to stay away from Cox Proportional Hazard Model.

• Are you interested in failure rate per unit time or failure rate per distance driven? Also, how does your sampling scheme deal with cars (of the same model and production year) that might have already failed before your study began?
– EdM
Commented Apr 24, 2020 at 14:37
• @EdM I think I am looking for failure rate per unit time. At time $t_0$ all cars are operational but have different mileage and we observe them i.e.: for the next 5 years $t_f$ where they all have failed. Commented Apr 24, 2020 at 14:45
• @EdM Failure rate per unit distance would be interesting to see as well - I would really appreciate to see how you do it Commented Apr 24, 2020 at 14:47

In principle, failure rate as a function of miles driven is easy. You just use miles driven instead of time along the x-axis. At each event/failure mileage, you calculate the fraction of cars that have driven at least that far that failed at that mileage. Then put the information together from the individual failure mileages to get the cumulative survival probability as a function of miles driven. That's standard Kaplan-Meier analysis.

If you want failure rate as a function of time I don't see a good way to correct for miles driven without some type of regression approach like Cox proportional hazards, an accelerated failure time model, or some parametric survival model. If you normalize the time axis by the number of miles driven I think that you are effectively just going back to using miles driven as the x-axis. If you use miles driven as a covariate with time along the x-axis you would probably want to include it as a time-varying covariate rather than just use its value at study start, to account for any subsequent changes in driving patterns.

If you have a large enough study I suppose you could use a stratified approach, for example doing separate Kaplan-Meier type analysis with time along the x-axis for groups of cars driven 0-40K, 40-80K, 80-120K miles, etc. Then get the predictions for each new case from the corresponding group. That wouldn't account for subsequent changes in driving behavior, however.

You really don't need to avoid censoring unless for some reason it's informative. Kaplan-Meier and regression-based analyses use all of the information available for censored cases, up to their times (distances) of last observation. Furthermore, if you wait until the last failure you might never finish your study. Cars can last a very long time. I just turned in a 1999 model-year car that was still operating. Depending on your age, there's a good chance that the last car will outlast you.

One more warning: as you describe the study, you don't seem to be taking into account any cars of that model and year that failed before $$t_0$$. If so, that could lead to substantial bias.

• What would the strategy be if we use parametric survival model i.e: Weibull distribution - how do you account for the different initial condition - assume same driving pattern. Commented Apr 24, 2020 at 17:21
• @EdvBeq In that case I would just include mileage at $t_0$ as a continuous covariate. I suspect that there will be some sort of inverse relationship between mileage at $t_0$ and time to failure. You thus might have to play with transformations of it, but that's the case for any type of regression. Or model it as a restricted cubic spline and let the data tell you the relationship.
– EdM
Commented Apr 24, 2020 at 19:43