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All,

I am trying to create a regression model where the (continuous) outcome is multimodal:

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The outcome is the retail price of a certain product, and prices tend to fall around distinct amounts (750, 1000, 1250, 1500, etc). There are, however, a few prices in between so the prices are not distinct.

I have run a linear model with satisfying results, though the extra prices between the modes give me pause. I also tried binning the prices down to a few groups representing the modes and it works somewhat well.

Is there a better or worse way to model this? is there some sort of better or worse methodology for binning the outcome?

Thank you

Macro
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PeteL
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  • Hey Pete, I added the plot for you. It may be a little bit more informative if you could lower the bin size (using, e.g. breaks=50 in your call to hist() in R) so we can really see the shape of the distribution. Also, I agree with what Peter Flom says below, which is similar to what I said to you when we were talking about this earlier, so it may be helpful to see a histogram of the residuals of an ordinary linear model to see whether Peter's answer does solve your problem. – Macro Oct 19 '12 at 21:24
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    In line w/ Peter's point (below) re the residuals, but not the outcome, needing to be normally distributed, this: what-if-residuals-are-normally-distributed-but-y-is-not may be helpful. – gung - Reinstate Monica Oct 19 '12 at 21:32
  • This is one of the top search results for multimodal regression. I suspect what most people mean when they search for this is regression with a multimodal response for y given the predictor values, x. For this there are a few different approaches such as Gaussian mixture regression where you use density estimation to estimate the density p(x,y) and p(x) then find p(y|x) and mixture density networks where you have a mixture model such as a Gaussian mixture where the components and mixing coefficients are the output of a neural network and depend on the level of the predictor x. – tail_recursion Mar 24 '22 at 05:06
  • There are also conditional mixture models such as mixtures of linear regression models see PRML and mixtures of experts, hierarchical mixtures of experts. – tail_recursion Apr 13 '22 at 00:50

1 Answers1

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OLS regression does not assume that the dependent variable is normally distributed, nor even unimodal. It makes assumptions about the error term, as estimated by the residuals.

Many variables exhibit "clumping" at certain round numbers and this is not necessarily problematic for regular regression.

Categorizing, or binning, continuous data is very rarely a good idea. However, if there are very few prices between the round numbers, this may be a case where it does make sense. If you do this, then the OLS model should no longer be used, but ordinal logistic regression (or some other ordinal model) instead.

Peter Flom
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  • Peter, based on the description, it sounds like there is some inherent discreteness in the data - "prices tend to fall around distinct amounts (750, 1000, 1250, 1500, etc)" - which makes it less likely that the errors are approximately normal (unless there are only discrete predictors or something like that). If you did not want to categorize the data and still wanted to model this with a regression approach, what would you suggest? – Macro Oct 19 '12 at 21:27
  • Hi @Macro . I guess some sort of robust regression, although I can't say, without research, which type would be best. – Peter Flom Oct 19 '12 at 21:39
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    I think this is a good answer, but I also think that it neatly avoids the reality: we do not do linear regression for multimodal data (or at least it's not generally suggested). Can this be explained using the distribution of residuals? It would seem to me that the residuals are no longer normally distributed when we have multiple modes – information_interchange May 10 '19 at 15:14
  • The normal error regression model which is the model taught in most regression analysis courses and books (and on which most of the inference possible with linear models is based) DOES imply that the $Y_i$ are independent normal random variables, with mean $E[Y_i] = \beta_0 + \beta_1 X_i$ and variance $\sigma^2$. See Applied Linear Regression Models by Kutner 5th Edition Ch 1.8 Comment 4, one of the most referenced regression textbooks. Most of the book is about the normal error regression model and from my exposure to other books they are mostly about the normal error regression model too. – tail_recursion Apr 13 '22 at 00:47