I am fitting survival models on time-to-event data representing the number of days of delay in paying invoices from the expiration date (negative values represents advance payments). The data consists of some exploratory variables such as the customer, invoice amount, revenue type, expiry month, ecc.
The main goal is to do predictions, but instead of a point estimate I need to compute for new invoices quantities such as the probability of payment in a given month, or the probability of delays greater than x days etc. all of which can be calculated from a survival curve.
Since I also have negative values, referring to payments prior to the due date, in order to be able to use these models I had to make all the values of the outcome greater or equal to 0 removing the minimum observed value (and after estimating the survival curve add it up).
This operation does not completely convince me, because an invoice can be paid before the due date, but not before the issue date, and therefore for each invoice I have a maximum possible number of days in advance of payment (or minimum possible delay value) corresponding to the difference between due date and issue date, information which is not taken into account by the model. Certainly I can correct the survival curve by dividing the survival probabilities beyond this minimum possible time t by the survival probability evaluated at it (which more generally is the method I use when I want to calculate the survival probabilities conditional on a minimum number of days of delay), but I was wondering if there was a better way to deal with this problem.
I tried to work with the number of days from the date of issue instead of the delay from the due date so as not to have the problem of negative values, and using the number of days between due date and issuing date as a regressor, however the predictive performance are much worse.
Edit: I am using Cox regression (coxph funcion on R) and evaluating performance through concordance in training and test set. I suspect that the problems in using the date of issue rather than due date (to avoid negative values) is that the time interval between these two dates varies according to the invoice, and most of the invoices are paid close to due date. I added the length of this time interval (due date - date of issue) among the predictors, and of course the estimated coefficient is highly significant (the greater this interval, the further the expected payment is from the date of issue), but maybe the relationship with time from payment to issue is not the one assumed by the Cox model. I think that I need to treat it as a sort of varying intercept rather than a predictor.