# Analysis of rates and post hoc test with offset variable using glmer.nb

I am investigating variation in pollinator visitation rate (number of visits per inflorescence) with treatment and time category as fixed factors. Block is a random factor. Following Zuur et al. (2009), I used the number of visits as the response variable with the $\log$(number of inflorescences) as the offset variable in the analysis. A Poisson model was overdispersed, and therefore I opted for a negative binomial model in lme4, as follows:
model1 = glmer.nb(visits ~ treat + timecat + offset(log(infl)) + (1|block))

I am specifically interested in differences in visitation rates between treatments. I therefore performed a post hoc test: OPexp1 = glht(model1,mcp(treat = "Tukey")); plot(cld(OPexp1))

When I plot these results, I get number of visits on the $y$ axis. But what I want is visitation rate (visits per inflorescence).

Is the post hoc test taking into account the number of inflorescences? Or how do I specify that the post hoc test should be performed using visitation rate? I thought that running the model above is analogous to analysing the visitation rate, so are the Tukey's differences between treatments for number of visits also the same for visitation rates?

I assume what is happening is that fitted values are currently expressed as $μ × V$ (where $μ =$ visitation rate and $V =$ number of inflorescences), but how do I specify that they should be expressed as $μ$ (visits per inflorescence) only? On p. 240, Zuur et al. (2009) mentions that this is possible, but I have not been able to find an example.

I am quite new to R, and this is my first post here, as I have been unsuccessful in finding an answer elsewhere, so any advice or kick in the right direction would be much appreciated. Kind regards.