sorry, I am not an expert in statistic but would like to compare at least two bimodal histograms. Do you suggest any particular test to do it? Thanks in advance.

sorry, I am not an expert in statistic but would like to compare at least two bimodal histograms. Do you suggest any particular test to do it? Thanks in advance.

I think you’re right. In one dimension (which I should have noted above is the only case in which the Dip test can be used) the existence of a unimodal distribution does imply a bimodal negation. In two dimensions (if I’m visualizing things correctly) it probably doesn’t, because the negation is likely to be a unimodal “donut” around a bivariate normal distribution, for example.

I’m not sure I’m correctly understanding your use of the word “modeled” in the title of the thread. In my mind, a bimodal distribution would likely be modeled by a mixture of distributions chosen to reflect the observed data (or nonparametrically estimated by a kernal estimate, perhaps). I suspect that’s too static an interpretation for your purposes.

Based on your reference to the distribution of species, it seems that you might be interested in a test that works for two dimensions (or more, if the species is aquatic, I guess). The MAP and RUNT tests (or the Excess Mass test, if progress has been made in the years since I last looked) may be appropriate there, though I don’t know of any instantiation of either of them. Silverman’s test was described for one dimension but can be extended to any number of dimensions, in Silverman’s words, “mutatis mutandis” (“presto change-o”?). I believe that all three of these can also be used to test for more that two modes (one versus more than one; that settled, two versus more than two etc.).

I think you’re right. In one dimension (which I should have noted above is the only case in which the Dip test can be used) the existence of a unimodal distribution does imply a bimodal negation. In two dimensions (if I’m visualizing things correctly) it probably doesn’t, because the negation is likely to be a unimodal “donut” around a bivariate normal distribution, for example.

I’m not sure I’m correctly understanding your use of the word “modeled” in the title of the thread. In my mind, a bimodal distribution would likely be modeled by a mixture of distributions chosen to reflect the observed data (or nonparametrically estimated by a kernal estimate, perhaps). I suspect that’s too static an interpretation for your purposes.

Based on your reference to the distribution of species, it seems that you might be interested in a test that works for two dimensions (or more, if the species is aquatic, I guess). The MAP and RUNT tests (or the Excess Mass test, if progress has been made in the years since I last looked) may be appropriate there, though I don’t know of any instantiation of either of them. Silverman’s test was described for one dimension but can be extended to any number of dimensions, in Silverman’s words, “mutatis mutandis” (“presto change-o”?). I believe that all three of these can also be used to test for more that two modes (one versus more than one; that settled, two versus more than two etc.).

I was thinking, there is a bimodal distribution for every unimodal distribution — the negation. For example, if the distribution of a species is unimodal, most are modeled that way with temperature or rainfall, then the distribution of points where the species does not occur should be bimodal, shouldn’t it? That said, if you were to reverse the values of the dependent variables in a logistic regression, 0 for 1 and 1 for 0, how would that change the result?

I was thinking, there is a bimodal distribution for every unimodal distribution — the negation. For example, if the distribution of a species is unimodal, most are modeled that way with temperature or rainfall, then the distribution of points where the species does not occur should be bimodal, shouldn’t it? That said, if you were to reverse the values of the dependent variables in a logistic regression, 0 for 1 and 1 for 0, how would that change the result?

I don’t know of any packages that include the option, but code for Hartigan’s Dip test can be found at http://lib.stat.cmu.edu/apstat/217. It is written in Fortran, but if you have programming experience (or know someone who does), it won’t be too difficult to modify it into a another language, if need be.

When I used the code several years ago there was an error somewhere – something very simple like a parenthesis that wasn’t closed, or was closed in the wrong place. Unfortunately, I don’t remember where it was. It may have been fixed since then.

You should also know that the p-tables provided by Hartigan & Hartigan in their (1985?) paper are very conservative. You can improve the power of the test by constructing a “best unimodal” estimate of the density underlying your observations, creating simulated samples (of the same size as the sample you wish to test) from that estimate, and running the Dip test on each sample to create a distribution of Dip values that occur under the best unimodal assumption. If the Dip for your observed values exceeds the x-th percentile (where x is your alpha) of that constructed distribution you can say that you have rejected unimodality. Computationally heavy, but possible.

Well. I never thought I’d use that bit of knowledge. Hope it helps.

I don’t know of any packages that include the option, but code for Hartigan’s Dip test can be found at http://lib.stat.cmu.edu/apstat/217. It is written in Fortran, but if you have programming experience (or know someone who does), it won’t be too difficult to modify it into a another language, if need be.

When I used the code several years ago there was an error somewhere – something very simple like a parenthesis that wasn’t closed, or was closed in the wrong place. Unfortunately, I don’t remember where it was. It may have been fixed since then.

You should also know that the p-tables provided by Hartigan & Hartigan in their (1985?) paper are very conservative. You can improve the power of the test by constructing a “best unimodal” estimate of the density underlying your observations, creating simulated samples (of the same size as the sample you wish to test) from that estimate, and running the Dip test on each sample to create a distribution of Dip values that occur under the best unimodal assumption. If the Dip for your observed values exceeds the x-th percentile (where x is your alpha) of that constructed distribution you can say that you have rejected unimodality. Computationally heavy, but possible.

Well. I never thought I’d use that bit of knowledge. Hope it helps.