# Logistic regression where there are unreported failures

Hello: Let's say you have a large number of reported results from a cooperative game. The data consists of a number of independent variables, such as number of players, choice of opposition, etc., as well as the dependent variable (binary win/loss). You have been estimating the effect of the independent variables using a logistic regression model. However, you know that there is a bias in the data; games that would have been lost by the players are under-reported because they are often ended (due to frustration/boredom/etc.) before the game is complete, and the data entry form requires complete game information. Therefore, nearly all won games are reported, but some fraction of lost games are not reported.

My question takes three forms. Is there a way to modify the logistic regression model to account for this bias:

1) if you have an independent estimate of the number of unreported losses?

2) if you do not know to what extent games are unreported but think it is uncorrelated with any of the independent variables in the model?

3) If you do not know to what extent games are unreported and think it IS correlated with the variables (e.g. games with 3 players are more unreported than those with 4 or 5)?

This is clearly not the most important situation in the world (for reference, the game is the card game Sentinels of the Multiverse, see: https://mindwanderer.net/sotm/) However, the basic problem seems like something that would come up in research, for example, studying whether a creature is alive or dead at a point in time where for some reason you know you are unable to record some dead specimens. Therefore it seems like there should be some literature on how to deal with it, but I cannot locate any articles/websites dealing with this specific issue. Any references would be appreciated.

EDIT: I think the problem might be similar to that solved by truncated logistic regression, however, that seems to involve situations where observations are missing because of some threshold in the independent variables (e.g. only people above a certain height were included) and not the dependent variable. The example of motor vehicle fatalities seems to come up in my searching for an answer and be a similar situation, but I'm not sure if it is directly applicable.

EDIT2: It also seems like it is related to the issue of correcting for publication bias in meta-analysis, but I'm having a hard time drawing a clear line between a meta-analysis (where you are simply combining a number of studies with ave a an effect sizes and standard errors) and logistic regression.