# Quasi-poisson or negative binomial regression with continuous dependent variable?

My dependent variable is originally count data. Because of several corrections it became continuous variable (originaly my data are pellet-group counts (for estimating deer density), corrected for research plot slope, avereged across three seasons, converted to deer density (number per sq. km)). Can I still apply quasi-Poisson or negative binomial GLM? Especially nb-GLM appear to be appropriate for my data structure (histogram and mean/variance relationship are adequate) and R processes the models without any warning and gives reasonable results. Could there be any catch? Rounding the data to integer does not seem to be solution since I would lose part of information and all corrections would be meaningless.

• Do you still have access to the original data? I would recommend including all of the data (you can either control for across-season differences by adding season as a parameter, or ignore it -- which will have the effect as averaging, but be statistically correct). Correcting for slope should be done as part of the model, and the correction for study area should be done with an offset as in @user13317's answer below. – Ben Bolker Jan 28 '14 at 18:43

Rounding your response variable to an integer is NOT OK. For simplicity, lets assume you're conducting a Poisson regression. What you're modeling is the following:

\begin{align*} E(Y|x) &= \beta^{T}x + \beta_{0} \\[0.5em] \log \left( \frac{\mbox{No. of Deer}}{\mbox{Area}} \right) &= \beta^{T}x + \beta_{0} \\[0.5em] \log(\mbox{No. of Deer}) - \log(\mbox{Area}) &= \beta^{T}x + \beta_{0} \\[0.5em] \log(\mbox{No. of Deer}) &= \beta^{T}x + \beta_{0} + \log(\mbox{Area}) \end{align*}

In R, this is done use the following command:

glm(No. of Deer ~ x + offset(log(Area)), family=poisson(link=log), data=data.frame)


This allows you to use Poisson (or Quasi-Poisson or Negative Binomial) regression for a continuous response, even though the No. of Deer is still a count. Your parameter estimates (i.e., $\beta_{0}$ and $\beta \mbox{s}$) will be on the log scale, so just exponentiate them to obtain estimates on the raw scale. Also, be careful of the parametrization used for the negative binomial distribution, if you decide to go with negative binomial regression.

• Actually there were many more corrections/conversions, which makes situation much more complicated. We counted pellet groups on 240 plots in three seasons and then: 1. on each plot we multiplied counts with the factor that accounts for plot inclination, 2. for two of the seasons we had to multiply counts/values with factor that acoounts for fast decay rate, 3. for each plot in each season we calculated deer density (here three factors were took into account: for plot size, for accumulation time (specific for each plot in each season), for defecation rate)... – Matija S. Jan 28 '14 at 19:43
• ...4. we calculated average (for three seasons) deer density on the plot, 5. because we had two plots in each unit, we calculated average for those two plots and got deer density per unit (120 units)... – Matija S. Jan 28 '14 at 19:44
• ...As you see there were many corrections and multiplying (some season specific, some plot specific, some season/plot specific and some universal). Finaly we got continuous values , but they still have typical characteristics of counts (some zeros (12/120), and "Poisson-like" distribution with squared variance approximately proportional to mean, which implies use of nb-GLM. – Matija S. Jan 28 '14 at 19:44