Zero-inflated negative binomial distribution function NaN warning

I am trying to fit my data to a zero-inflated negative binomial model but one of my 3 independent variables (Exposure) seems to be causing NaNs to be produced (see very end of zeroinfl call) when the SE is being calculated in the summary function. I have also tried running a negative binomial hurdle model and am running into a similar issue.

str(eggTreat)
'data.frame':   455 obs. of  4 variables:
 $ Exposure : Factor w/ 2 levels "C","E": 2 2 2 2 2 2 2 2 2 2 ...
 $ hi_lo    : Factor w/ 2 levels "hi","lo": 2 2 2 2 2 2 2 2 2 2 ...
 $ Egg_count: int  0 0 0 0 0 0 0 0 0 0 ...
 $ Food     : Factor w/ 2 levels "1.5A5YS","5ASMQ": 2 2 2 2 2 2 2 2 2 2 ...

mod.zeroinfl <- zeroinfl(Egg_count ~ Food+Exposure+hi_lo | Food+Exposure+hi_lo, data=eggTreat,
+                          dist="negbin")
> summary(mod.zeroinfl)

Call:
zeroinfl(formula = Egg_count ~ Food + Exposure + hi_lo | Food + Exposure + hi_lo, data = eggTreat, dist = "negbin")

Pearson residuals:
     Min       1Q   Median       3Q      Max 
-0.65632 -0.47163 -0.28588  0.02976  9.00804 

Count model coefficients (negbin with log link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -0.04435    0.14393  -0.308   0.7580    
Food        -1.12486    0.22267  -5.052 4.38e-07 ***
Exposure    -2.34990    0.38684  -6.075 1.24e-09 ***
hi_lo       -0.44893    0.19524  -2.299   0.0215 *  
Log(theta)  -0.24387    0.22639  -1.077   0.2814    

Zero-inflation model coefficients (binomial with logit link):
              Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.830e+01         NA      NA       NA
Food        -5.768e+00  5.628e+04       0        1
Exposure     4.612e-01         NA      NA       NA
hi_lo       -7.477e+00  9.963e+05       0        1
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Theta = 0.7836 
Number of iterations in BFGS optimization: 21 
Log-likelihood: -350.2 on 9 Df
Warning message:
In sqrt(diag(object$vcov)) : NaNs produced

function (object, ...) 
{
  object$residuals <- residuals(object, type = "pearson")
  kc <- length(object$coefficients$count)
  kz <- length(object$coefficients$zero)
  se <- sqrt(diag(object$vcov))

Solution

This problem is typically caused by complete separation; using this search term, or searching for the Hauck-Donner effect, will show you that the problem is that there is some linear combination of your predictor values that perfectly separates the zeros and non-zeros (since the predictor variables in your zero-inflation are all categorical, this translates to a combination of categories where all the values are zero or non-zero).

I would take a look at with(eggTreat, table(eggcount>0, Food, Exposure, hi_lo)) (arrange the arguments in whatever order makes the table easiest to read).

Typical symptoms include:

large values of the parameters (e.g. |beta|>10); in this case your intercept is -18.3, which gives a predicted zero-inflation probability of 1e-8 in the baseline category (two of the other values are also large, although not nearly as extreme as the intercept)
extremely large standard errors (Food, hi_lo), leading to z-values that are effectively zero and p-values of effectively 1
or ... the NA values you're seeing

There are various solutions to this problem:

different forms of regularization or Bayesian priors
compute the p-values using model comparison/likelihood ratio tests

Zero-inflation model coefficients (binomial with logit link):
              Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.830e+01         NA      NA       NA
Food        -5.768e+00  5.628e+04       0        1
Exposure     4.612e-01         NA      NA       NA
hi_lo       -7.477e+00  9.963e+05       0        1