So I'm having some difficulty fitting a linear model to the data (see other post here glm model fit - can't find a family/link combination that produces good fit).
In particular, I'm worried about the pattern in the residuals vs fitted plot that indicates bias.
Part of the problem, I think, is that I have these absurdly high values on a couple of the most important predictors in the model. I believe these values are legitimate - there's a lot of posts out there on identifying leverage points, outliers, influential observations but nothing about how to incorporate them into the model in a substantively meaningful way.
The outcome is the sentence length an offender will spend in prison. It's censored at 600 months (50 years), a life sentence. Among the predictors are various scores that are reflect the seriousness of the offense, prior record, etc. There are also possible "enhancements" that can add considerably to these various scores and even multiply them. Thus, some poor souls end up with scores that are absurd but probably legitimate.
Even if I account for the censoring on the top end, lots of these people with crazy severity scores get fairly mild treatment. I suspect that the sentencing judge sometimes recognizes that the enhancements are ludicrous, ignores them partly or totally, and sentences the offender in a typical fashion. I'm wondering how to account for this in the model.
A spline doesn't seem to work well because at the high end of these predictors, there are just so few datapoints. I think curvilinear terms have the same problem. Do I truncate them to a more sensible value? Do I include an arbitrary dummy to indicate "excessive enhancement points?" I can log them and this does improve fit but I don't think it substantively addresses the problem. In the past, I've truncated them and then logged them but I really don't feel like this is resolving the problem in a statistically appropriate and substantively meaningful way.
To give you an idea of how bad the problem is, for one of these predictors the 99th percentile is 187.2 and the 15 largest values are as follows: 1368 1375 1591 1810 1959 2115 2292 2458.2 2494.4 2494.4 2524.5 3420 5419
Severe leverage is an understatement.