# Binomial data analysis with all 0 responses for some treatment groups

We have a series of experiments where we measure virus transmission to plants when exposed to virus-infected insects for different time periods, so all of the experiments have similar types of independent and dependent variables. In one experiment, there are 6 time periods (1 to 24 hours) and 25 plants were tested (individually) for each time period. The response for each plant is yes or no (Plants are scored as virus infected). For 2 of the time intervals, all of the plants were negative for virus infection (0/25 for each time interval).

I am using PROC GLIMMIX in SAS for the analyses. For all of the other experiments, using a binary distribution in the model statement gives reasonable results. For the experiment where two of the time intervals had 0 positive plants, if I use a binary distribution in the model statement, the standard errors for the two groups with 0 transmissions are huge, thus distorting the results.

If I use a negative binomial distribution (based on counts of virus positive plants) the results seem reasonable. Since the same number of plants were tested for each time interval, using this approach works, but it differs from the other experiments.

Is there a method to adjust/account for the zeros in treatment groups that would allow the binary distribution return reasonable results?

The problem with zeroes, is that the data do not rule out arbitrarily small proportions. So your prior information must be assessed more carefully, because in this case it still matters. Details which are irrelevant when the prior information is "swamped" by the data can be important. In this type of problem, the population size $N$ becomes important, but a binomial assumes $N\to\infty$, which gives absurd results, if this limit is applied too soon in the calculations (as your standard errors indicates).
In this case, there is relatively straight-forward approximation, you just replace $\frac{0}{25}$ with $\frac{1}{27}$, which is a Bayesian estimate based on a uniform prior for the true fraction of "positive infections". Given that you are using GLIMMIX - doing anything more sophisticated will likely wreck your SAS program.
To be consistent, it may be worthwhile to replace all proportions $\frac{r}{n}$ with $\frac{r+1}{n+2}$ - however it shouldn't influence your results too much.