Probability model for the time to event when the origin is not known

Question

I am interested in a situation when the time to event is observed partially. In the right-censoring situation, we know the start time of the disease $s$ but do not observe the death time $d$, because of censoring time $c_r$. So time interval $c_r-s$ is observed.

What situation am I in, when the death is observed at time $d$, but the start time of the disease $s$ is not known due to censoring time $c_l$? So the time interval $d-c_l$ is observed.

For the right-censoring the contribution to the likelihood is $S_\theta(c_r-s)$, where $S_\theta(t)$ is a parametric survival function. What would be the contribution to the likelihood in the second situation: is it $1-S_\theta(d-c_l)$?

Consider an example of consumer buying a product. Suppose we have data from time $c_l$ to $c_r$ about repeated purchases and we model inter-purchase time. Three cases are possible:

1. The time of the current purchase is observed but the time of the next purchase is after $c_r$ (not observed) - right-censoring.
2. The time of the previous purchase is before $c_l$ (not observed) but the time of the current purchase is observed - the situation under question. I could just ignore this observation, but it might be informative (especially in a situation with rare purchases).
3. The interval between the two purchases falls into $(c_l,c_r)$ - full observation.

Answer 1

If you are modeling the inter-purchase time as explained at the end of your question, with time = 0 for the survival origin reset at each purchase, then the inter-purchase time would seem to be right censored even in Case 2 (the first purchase after study start).

A right-censored observation is one for which you know a lower limit for the time between time = 0 and the event. The definition of time = 0 thus matters a lot.

If time = 0 is defined as the time of the previous purchase (and thus is reset at each purchase), then you do have a lower limit to the elapsed inter-purchase time in Case 2. You know that the prior purchase (if any) was prior to study start, so the corresponding prior inter-purchase time was at least the time elapsed since study start. In terms of inter-purchase time that's as right censored as the time between the last purchase and the end of the study (Case 1). The value of what you are modeling, the inter-purchase time, has a lower limit but no upper limit, the definition of right censored.

A few warnings that come quickly to mind: This would not necessarily hold for other analysis approaches that use a different definition of time = 0. You might have problems with modeling actual customers who made no purchases during the study period; the length of the study period could set a truncation limit. Complications would arise if you were incorporating covariate values at time = 0 into your model and didn't know the covariate values for purchases that happened prior to study start.

Answer 2

It does not easily surrender to survival analysis. I strongly doubt there is any analytical form. Additional data may need to be used to infer $s$ as an imputation task.

If for example we only know that the patient died on 2021-05-01 and that they fell ill some time before 2021-02-01, then clearly the survival time from illness to death must be over 3 months. But we don't know at which stage of the illness they died. They may have lived with it for years, months, or just days. For instance, I have understood that breast cancer can kill even after 10 years. If we assume these 3 months were soon after falling ill, then the earlier months of the disease are estimated to be more dangerous. If later months, then those are more dangerous. But we don't have any reason to assume that these were the early months or the late months. This is unlike in right censoring, where we know that the patient survived the early months at the very least.

In a missing data specification, we might have the joint likelihood $p(S, d - S|x) = p(S|x)p(d - S|S)$, where $x$ can contain some data by which we can impute $S$ to be in harmony with the rest of the dataset.

PS.

We could easily handle left censoring where $s < d < c$ and $d$ is unknown via $p(L_d < c - s) = 1 - S(c_r - s)$. We can also handle left truncation where patients are only included if they first survive up to some study entry time $e$, so that $s < e < d$ and all of them are known. Then we get $p(L_d = d - s|L_d > e - s)$.