Techniques for MNAR without Imputation Oftentimes I see that people address missingness from the perspective of missing at random (MAR). However, there are cases where missing not at random (MNAR) in which one can sometimes be handled with stuff like imputation. However, I'm thinking of another case which is more complex but likely not uncommon.
Let's say participants are recruited from a study which had two years of recruiting participants. In the first year, a survey was administered with 20 questions. However, the second year required 30 questions because ceiling effects became very evident. Now if one was to run an analysis using these two years, there would be obvious issues because one year's participants did not get the full range of questions.
What would be the appropriate way of handling this data? Should one only use the 20 questions used for both years? Should the person running the analysis impute the data? What are ethical ways of handling this data? Given it would be substantial (at least 50% of the population wouldn't have answered 10 questions), I would think this would be important to address.

The particular situation you describe could potentially be handled using methods for censored data. I.e. data where you know don't know the exact value, but it's greater than some value ("right censoring"), smaller than some value ("left censoring") or in some interval ("interval censoring").
However, that only makes sense, if the same question gets asked with a truncated response scale (let's say 0 to 50) and without a truncated scale (let's say 0 to 100), and if (that's a big IF) the participants interpret the question the same way each time (i.e. in the same circumstances they would answer X on the truncated scale and X on the untruncated, just when X is >50, then they respond 50 on the truncated scale). In that circumstance, an answer of 50 on the truncated scale simply means $\geq to 50$. That's exactly the situation, for which methods for censored data (here right censored) were developed.
I would be a bit nervous about doing that, because I'd worry that people would treat/interpret the scale differently, in which case it becomes very difficult to say something across the two different questionnaires.
Generic methods for censored data of course only make sense when censoring occurs completely at random or in a way that can be explained by the observed data ("at random"; of course the things influencing censoring should be included in the analysis model to deal with it, otherwise you end up in the next scenario below). If you are in a situation where it occurs dependent on the value. What do I mean by that? Let's imagine you ask people their salary and people with a salary <100k USD/year always tell you their salary, people between 100 to 200k tell you their salary 50% of the time and 50% of the time tell you ">100k", but people with salaries >200k always tell you >100k, you have a "missing not at random" time of censoring that is really problematic (unless you have other information that can make you distinguish these groups). My initial thought was that a split in time for the new scale would not be that kind of situation.
