Why GLM Poisson model predict negative value for count data?

I have a count dataset with mean=3.2, and a little bit Zero-inflated. X1      X2      X3  Y
Food3   Low     13  2
Food3   High    27  1
Food2   Low     13  1
Food1   Medium  27  1
Food1   High    20  8
Food3   Low     20  1
Food1   High    13  5
Food2   Medium  13  4
Food1   Low     13  0
Food2   High    20  6
Food1   Medium  13  2
Food1   Low     13  1
Food1   Low     13  1
Food3   Low     13  1
Food2   Medium  13  5
Food1   Medium  27  0
Food3   Low     13  2
Food1   Medium  20  3
Food3   Medium  13  7
Food1   Low     20  1
Food3   Medium  13  5

I fitted the GLM model with Poisson family:

model1 <- glm(formula=Y~X1+X2+X3+X1:X2+X1:X3+X2:X3,

The summary(model1) output showed a little bit overdispersion, I also tried to fit glm.nb() negative binomial GLM. But the problem for this model is, there are some negative predictions, both for Poisson GLM and negative binomial. How these could be from, and how should I fix this problem? • I guess you need to push those values through exp(). Mar 19 '18 at 10:15
• @NickCox, thank you very much! Could you please give more detail information? I'm not sure how to do it. Mar 19 '18 at 14:59
• I am not a routine R user but I would be amazed if there's not a way to get predictions on your original counted scale. If not I would bet wildly that exp() is an R function. Please note that how to do anything in R or any other software is off-topic here unless the question is at root statistical. Mar 19 '18 at 17:25
• Thanks @NickCox, you are of course correct on all accounts! I have answered the question here because I felt that the root is indeed more a statistical question about GLMs than about the R implementation in particular. I have tried to reflect that in my answer. Mar 20 '18 at 11:21

The Poisson GLM fits a model $y_i \sim \text{Pois}(\mu_i)$ with $\log(\mu_i) = x_i^\top \beta$, i.e., a log links the expectation $\mu_i$ to the so-called "linear predictor" $x_i^\top \beta$, often denoted $\eta_i$ in the GLM literature. Hence, at least two types of predictions may be of interest based on the coefficient estimates $\hat \beta$: the predicted link $\hat \eta_i = x_i^\top \hat \beta$ and the predicted expectation $\hat \mu_i = \exp(\hat \eta_i) = \exp(x_i^\top \hat \beta)$. The latter are typically of more interest in applications while the former are often employed in (diagnostic) graphics because they are on a linear scale.