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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.

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    $\begingroup$ Are you sure that you need both the extreme sentences and the more minor sentences in the same model? Are you sure that it is reasonable and practical to assume that the extremes and the moderates are distributed identically, but only have means that vary with the X's? Do you think that the mean/variance relationship holds over the whole sample? $\endgroup$
    – generic_user
    Commented Apr 26, 2015 at 18:20
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    $\begingroup$ The issue you're referring to is the variance family for the glm (assuming I do that and not something like a hurdle model) and I would agree that the gaussian family does not correctly represent dispersion in the model. Negative binomial is probably "correct" but this, to me, is all separate from issues regarding the dispersion in the predictors which is what this question is all about. Perhaps I don't understand your point, but how would disaggregating the analysis arbitrarily by the outcome resolve the overdispersion and leverage issues in the predictors? $\endgroup$
    – whauser
    Commented Apr 26, 2015 at 18:31
  • $\begingroup$ I really don't know, hence the comment rather than the answer. But yeah, I'm talking about GLM fundamentals, where you basically assume that your data is identically distributed. Any set of data can be considered to be identically distributed if you pick a tortuous-enough distribution, making the whole thing a bit tautological. In practice, with non-infinite data, hurdle models might make more sense? $\endgroup$
    – generic_user
    Commented Apr 26, 2015 at 18:50
  • $\begingroup$ I agree that a hurdle model is worth considering and I am actually working to fit one now, but it really doesn't address the issue arising from the extreme values in the predictors. It addresses issues with the censoring from below and above.. but I am certain the hurdle model will not fit well and probably won't even converge if these extreme values are left as is. $\endgroup$
    – whauser
    Commented Apr 26, 2015 at 22:21
  • $\begingroup$ I also see what you're saying about the distribution of the data but it seems to me that the outcome is the result of a formulaic scoring process and some human judgement. To the extent that a mathematical formula is involved you would think the outcome would be made easier to model but this is proving not to be the case. $\endgroup$
    – whauser
    Commented Apr 26, 2015 at 22:23

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