# Half-normal distributed DV in generalized linear model

My dependent variable is, by origin, absolute residual* left after some regression; it is distributed half-normally. Now I plan to use Generalized linear model in SPSS (GENLIN) to regress it on some predictors (totally different from those which produced the residuals). What distribution type should I use for the DV? For continuous data, GENLIN offers Gamma, Inverse Gaussian and Tweedie distributions. Which to choose to model the half-normal? Or should I apply special transforms before? And what link function would be most appropriate? What can you advice on that? Thanks.

*More precisely, I'm analysing positive residuals separately and negative residuals separately.

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Why are you using an absolute residual as a dependent variable? If you're interested in how the variance depends on some predictors, there may be better ways of doing it. –  onestop Nov 12 '11 at 13:59
Actually, I'm interested in half of the residuals at a time. I study positive residuals and negative residuals separately, as DVs on their own, with potentially different predictors and models. No, I'm not interested in variance. –  ttnphns Nov 12 '11 at 14:49

I can't tell exactly what you're doing, or why, and it seems a little strange, to be honest. I'm guessing that you have a regression model, and believe that those who are above the line are from a different underlying (latent) group from those who are below the line, and hope to find out something about these 'groups'. Perhaps not, but if so, I'm not sure this is a legitimate strategy. I have difficulty imagining how the distance above a regression line can be determined by one set of explanatory variables, and the distance below the line by a different set. There is a great deal of real noise in data and trying to explain it is more likely to lead to phantoms than knowledge. But I don't mean to scold.

Perhaps a tobit model would be appropriate. The tobit is based on a probit model with an underlying normal distribution, but where some proportion of the data have been censored. Recognize that this means the model is assuming the other half of the data are a part of the picture, but just have been hidden from the model, which is true in your case. The non-zero level of intrinsically unexplainable variance that must exist in your data necessitates that some proportion of the negative residuals really belong to the set of your positive residuals and vice versa, so the assumption the model is making is justifiable. Nonetheless, running two tobits would allow you to try to model your residuals with different explanatory variables.

I have no idea how to conduct a tobit regression in SPSS; I'm not sure the software supports that analysis. If you want to know a little bit more about it, and how to conduct it with other software, UCLA's website has a nice, concise, clear description (of course) of it for R. If you want more theoretical background on it, J Scott Long's book is very accessible. It requires calculus, but he steps through it very gently.

Good luck.

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how the distance above a regression line can be determined by one set of explanatory variables, and the distance below the line by a different set Imagine the residuals are deviations of body weight from normal (median) body weight. Binary variable "endocrine diagnosis" (Present/Absent) will strongly positively correlate with positive residuals, extent of overweight (because of obesity cases), but will not correlate or will weakly negatively correlate with negative residuals, extent of underweight. –  ttnphns Nov 13 '11 at 4:44
That helps to clarify the situation you are asking about. But, I wonder if a simple logistic regression with "endocrine diagnosis" regressed onto BMI (the full range, not just above the median) will work. Patients who are underweight will have a very low probability of ED, but the model can handle that just fine. If you are particularly concerned about the symmetry of the logistic transformation, cloglog or loglog could be used instead. –  gung Nov 13 '11 at 6:36
Deviations (either only positive or only negative) from median body weight have to be DV, and different diagnoses will constitute set of IVs of the model. The only thing that stops me is that I don't know how to be, given that the distribution of DV is thus half-normal. I never worked with such distributed DVs before. –  ttnphns Nov 13 '11 at 9:09
You could try a tobit model, then. If you're just looking at ED vs. no ED (i.e., just 1 factor), you could try a Mann-Whitney U test--it doesn't require assumptions about the distribution. Other than that, I don't know. –  gung Nov 13 '11 at 15:18