Non-significant p-values for factor levels with only 0s in negative binomial glm using glm.nb() in R

Question

I am trying to fit a negative binomial GLM to fish catch data with month of the year (factor) as my explanatory variable. I have selected the month with the greatest number of catches as my reference level and fitted the model using the glm.nb() function from the MASS package.

model1<-glm.nb(catch~month+offset(log(Duration_hours*NumberHooksDeployed)),
 catchdata)

Using anova(model1) shows that the factor month is significant and my pseudo $R^2$ is about 55%. The problem I have is that months with no catches at all have p-values greater than 0.05 (close to 1), even though I would expect them to be significantly lower than the reference level. Months with only very few catches are significant on the other hand. The output from the predict() function makes sense, so I think the model should be fine. Is there a problem with the likelihood test R uses, or am I fitting the wrong model to my data? From other posts I figured that I might have to use a likelihood ratio rather than Wald's test; any suggestions on how to do that in R? Any help would be greatly appreciated. See model output below, the months with 0 catches are May and June:

Call:
glm.nb(formula = NumCaught ~ factor(month) + offset(log(Effort_mins * 
Nhooks)), data = f[f$Area == "PW", ], init.theta = 1.181267049, 
link = log)

Deviance Residuals: 
Min       1Q   Median       3Q      Max  
-1.8445  -0.7766  -0.2148   0.0000   1.5386  

Coefficients:
                     Estimate Std. Error z value Pr(>|z|)    
(Intercept)            -2.647e+00  3.192e-01  -8.292  < 2e-16 ***
factor(month)May       -2.754e+01  1.024e+05   0.000 0.999785    
factor(month)June      -2.790e+01  3.122e+05   0.000 0.999929    
factor(month)September -2.516e+00  6.516e-01  -3.861 0.000113 ***
factor(month)December  -1.112e-01  4.403e-01  -0.253 0.800543    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

(Dispersion parameter for Negative Binomial(1.1813) family taken to be 1)

Null deviance: 80.420  on 45  degrees of freedom
Residual deviance: 36.178  on 41  degrees of freedom
AIC: 140.81

Number of Fisher Scoring iterations: 1


          Theta:  1.181 
      Std. Err.:  0.525 

2 x log-likelihood:  -128.815 

Answer 1

This is happening because your May and June dummy variables perfectly predict outcomes of zero. The "true" MLEs for May and June would be negative infinity, but the log-likelihood increases at a slower and slower rate and eventually the optimizer gives up. Since the log-likelihood is almost flat at the supposed MLE, the standard errors are estimated to be huge when they are really meaningless. It is the equivalent of a separation problem in logistic regression.