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Short version: is there any statistical method that can be used to develop estimates of abundance, spatial distribution, and/or rate of spread of a species when my dataset only includes presence data and not absence? I am not dealing with fossils in the strict sense, but I speculate that if such a method exists it might be found in the field of paleontology.

Long version: I recently undertook a small pilot project trying to gather data on the distribution of a newly introduced species of moth in the US, and possibly to project backwards and estimate the date and site of introduction. We only became aware of this species in 2009, but I searched insect photography databases (mostly contributed by amateurs: bugguide.net, mothphotographersgroup.org, etc.) and found that there were many photos of this beast, dating back to 2004, and covering 8 or 9 states in the southeastern U.S.

It got me thinking about how such a dataset could be used. These photos are functionally exactly like a fossil record, in that positive data points are sparse but intrinsically very informative because they represent a definitive date and locality for the target organism. On the other hand, true negative data is even more sparse (meaning that data exists, showing that the organism wasn't present at a certain place and time). I have a lot of a third kind of "data," and I'm not sure if it can be used at all: the absence of positive data.

On one hand, if no positive data exists for a certain site, it says as much about the conditions for "fossil" (or photo) creation and subsequent collection as it does about the distribution of the species. For fossils I imagine the conditions are geological/topographical, whereas for photos they relate to the density of potential photographers, and the likelihood that they'd post a photo where I can find it.

On the other hand, no data is not the same as no information, right? If I standardize for human population density, citizen scientists across the US should be equivalent in their probability of encountering a moth and photographing it. If moth density per photographer is high, there will be a greater chance of a photo being taken. If moth density is low, there is less chance of a photo. If the moth is truly absent, there is zero chance of a photo. Put differently, I don't think it is random* that all the photos are from southeastern states, and that more photos were found from some states than others among that group. I just don't know if there is a way to statistically support this based on the type of data that I have.

*Full disclosure: I know it isn't random. I didn't want to add another level of complication to my question, but the distribution of its host tree is well documented and represents an absolute boundary for this moth.

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  • $\begingroup$ It seems like a lot of what you have will depend on how representative the photos are of the actual moth population in each photographer's area. Do the photographers actively seek out new species, or might they follow their whims to photograph "pretty" or "interesting" moths? Is this hobby widespread or is it big in some states and not others? Are the areas where the host trees grow more or less friendly to casual hikers (weather, elevation-wise)? That is: do the photos rise above a convenience sample or can you model "convenience"? $\endgroup$
    – Wayne
    Apr 23, 2012 at 18:45
  • $\begingroup$ You've hit on what I think is a key point. Amateurs post photos for three main reasons: for help with ID, or to share things that are new, or that are pretty. I benefited in working with a moth, and a novel one at that. Moths are a attractive, with lots of dedicated hobbyists. Plus, amateur insect photographers are biased toward snapping photos of things that are unfamiliar. A similar effort to track track common species of medical importance was much less successful. The german cockroach just doesn't inspire a lot of unsolicited photos, and so it is extremely underrepresented. $\endgroup$
    – Mark Fox
    Apr 24, 2012 at 17:13

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There's an extensive literature on this area in spatial ecology; but it comes down to what kind of assumptions you're willing to make. A very common method for estimating spatial distributions based on abundance data is a maximum entropy approach, take a look at the software and publications

There's a variety of other algorithms designed for this purpose as well (using methods such as neural nets), mostly falling into the class of machine learning methods where you are simply testing how well a black box can reproduce the data (i.e. in cross-validation between a training set and test set) rather than estimating underlying models. Consequently it is difficult to say anything about the statistical appropriateness of the approaches outside of the datasets on which they have been tested.

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  • $\begingroup$ +1. +2 if I could. I just had a chance to glance at the first linked paper and it's very similar to Mark Fox's problem. I'm a novice at statistics, but the paper looks well-reasoned and the environmental dataset it uses may well be helpful to Mark Fox as well. (Naive question: people talk about "only positive data", but I wonder how often wildlife studies actually have negative data. How sure can you be that there are NO deer in those woods?) $\endgroup$
    – Wayne
    Apr 23, 2012 at 18:56
  • $\begingroup$ Thank you for the references and the summary. I figured I wasn't the first ecologist to experience this dilemma, but without an idea of the appropriate search terms I struck out in my literature searching, so this is a big help. I probably won't have the time or resources to undertake the approaches you mention, but it does somewhat resolve my concern - that beyond simple description, there must be more information, or more quantifiable information, in my dataset that would contribute to a more rigorous estimation of the species' distribution. $\endgroup$
    – Mark Fox
    Apr 24, 2012 at 16:56

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