Odd results from Poisson regression in R I have obtained some strange results for the following data when conducting Poisson regression in R.
> RHT1b
   f x1 y1 y2
1 f1  0 35  1
2 f2  2 70  4
3 f3  0  5  1
4 f4  9 37  4
5 f5  0  3  0

> summary(amod2b)

Call:
glm(formula = x1 ~ y2, family = poisson, data = RHT1b)

Deviance Residuals: 
       1         2         3         4         5  
-0.00008  -1.71860  -0.00008   1.36550   0.00000  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)   -26.61    9977.85  -0.003    0.998
y2              7.08    2494.46   0.003    0.998

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 24.9766  on 4  degrees of freedom
Residual deviance:  4.8182  on 3  degrees of freedom
AIC: 15.485

Number of Fisher Scoring iterations: 18

I had an issue with the data earlier where x1 row 4 read "8", and row 5 read "1". When I ran the regression the results seemed reliable and were nearly significant. After correcting the data to how it is shown above and updating the model, I have results which literally do not make sense to me. 
My question is, why has this happened? Is is possible the value of 1 changing to 0 exceeded some sort of threshold of zero counts that the Poisson model can't handle? Would a negative binomial model be more appropriate?
 A: Plot the profile likelihood functions for the intercept & coefficient to see what's going on.  Basically you can rule out high values for the intercept, & low values for the coefficient; but for any low value of the intercept you can find a high value of the coefficient with which your data will be consistent.  If you want better estimates, you don't need to fit a different model to these data, but to collect more data.
In reply to your comment: Say for the y2 coefficient, take fixed values (something like 1, 2, ... , 8) & fit a GLM for each (you'll just be fitting the intercept).  Take the log-likelihood for each GLM and plot against the value of the coefficient.
What you'll see is that there's no maximum to the log-likelihood - it plateaus.  The tell-tale signs in your results are ...
(1) 18 iterations to get to the "maximum" likelihood fit - R is actually just calculating that the likelihood hardly changes between the last iterations.
(2) Ridiculously large standard errors for the coefficient estimates.  This is because they're estimated from the rate of change of the slope at the "maximum", & are only any use when there is a maximum & the log-likelihood function is more or less a parabola. (Look up Wald tests.)
You still have a difference of 20 between the residual & null deviances, which is a lot on one degree of freedom - though it would probably be better to simulate than to assume a chi-square distribution.  At least small or negative values of the y2 coefficient don't look plausible.
Note that zero-inflated or negative binomial models will add parameters, when your data aren't enough to pin down a Poisson !
