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I'm trying to make predictions about a quantity on a per-subject basis.

If I aggregate my complete sample I can get very good fit for distributions like gamma or Weibull, so I can make some assumptions about how the dependent is distributed.

Can I use that information somehow to improve my predictions (like including a prior to my model)?

Ideally I would perform hierarchical modeling and include the subject as a random effect, but my dataset is too large to do something like this.

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  • $\begingroup$ Do you mean that when you consider response w/o predictors, the distribution of response is similar to gamma or Weibull? $\endgroup$ – Nik Tuzov Jul 18 '16 at 17:51
  • $\begingroup$ Yes. This is a regression problem, so if I just plot out the response (histogram or density), I see an exponentially (?) decaying function. I can then try to fit a couple of distributions and gamma or Weibull get better fits than exponential or Poisson. $\endgroup$ – Bar Jul 18 '16 at 18:16
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In general, looking at just the response, $Y$, and trying to identify the corresponding distribution is meaningless. That's because when you model the response as a function of predictors, you assume that for each value of predictor, $x_i$, the corresponding value of $Y_i$ comes from a certain distribution whose parameters are defined by $x_i$. Therefore the set of response values you have are generated not by one specific distribution, but by a mixture of distributions.

For instance, consider a simple model $Y = x + \epsilon$, $\epsilon \sim N(0, 1)$. Suppose my dataset has 100 observations for x = 10 and 200 observations for x = 150. If I create a histogram of $Y$ alone, I will see a bimodal distribution that is very different from Normal. However, saying that I should use some bimodal distribution to model $Y$ doesn't make sense. Instead, I should use Normal distribution whose mean depends on the covariate.

Doing what you did can make sense only if you have replicates. In my example you could create a histogram separately for x = 10 and x = 150, then you would see the correct picture: for a given value of the covariate, the response distribution is Normal.

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  • $\begingroup$ What do you mean by replicates in this case? $\endgroup$ – Bar Jul 19 '16 at 16:45
  • $\begingroup$ Values of response that correspond to the same value of predictor. In my example there are 100 replicates for x = 10 and 200 replicates for x = 150. $\endgroup$ – Nik Tuzov Jul 19 '16 at 16:48

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