What type of Generalized Linear Model can handle high-to-low-variance heteroscedasticity?

I am trying to model the relationship between a continuous response variable (sample-corrected species-diversity estimates) and a continuous predictor variable (geographic spread). I have log-transformed both variables to make the relationship linear.

I am investigating the use of Generalized Linear Models because my response variable is strongly heteroscedastic: ,

However, discussions about how to address heteroscedasticity in GLMs all seem to concern cases where variance in the response increases as the predictor variable increases. My problem is the reverse — high variance in species diversity at low values of geographic spread, decreasing as geographic spread increases.

Is there a GLM link function that can cope with this type of heteroscedasticity?

• Can you explain how this "sample-correction" was done? – Glen_b Mar 15 '16 at 0:11
• It's quite complicated (bio.mq.edu.au/%7Ejalroy/SQS.html), but the upshot is that counts are turned into continuous values. – Roger Mar 15 '16 at 7:47
• You can't transform discrete counts into continuous values. Maybe continuous values are being generated in some way based on discrete variables. Unfortunately the link is less not at all clear about what is being done mathematically (let alone why). Is the procedure actually explained somewhere? (documenting inputs and outputs to R functions is of little value) ... – Glen_b Mar 15 '16 at 8:36
• what concerns me is that counts generally have variance that increases with mean. If I can understand what was done to these counts I (or someone else) might be able to suggest a suitable way to analyze them. – Glen_b Mar 15 '16 at 8:45
• It's not a transformation — it's a method that ensures even coverage of underlying species-abundance distributions. The values are continuous because it's the geometric mean of a large number of subsampling trials. The variance decreases with larger geographic areas because I'm subsampling area from a continental region, and there's more opportunity for wide variation in species composition with smaller geographic subsamples. – Roger Mar 15 '16 at 8:50