# How to interpret coefficients in a Poisson regression with interaction terms?

This question is a prolongation of this question: How to interpret coefficients in a Poisson regression?

If we follow the (almost) exact same routine, but we add correlation between the variablese treatment and improved (just for the sake of my question, which is interpreting the output), we get:

treatment     <- factor(rep(c(1, 2), c(43, 41)),
levels = c(1, 2),
labels = c("placebo", "treated"))
improved      <- factor(rep(c(1, 2, 3, 1, 2, 3), c(29, 7, 7, 13, 7, 21)),
levels = c(1, 2, 3),
labels = c("none", "some", "marked"))
numberofdrugs <- rpois(84, 10) + 1
healthvalue   <- rpois(84, 5)
y             <- data.frame(healthvalue, numberofdrugs, treatment, improved)
test          <- glm(healthvalue~numberofdrugs+treatment+improved + treatment:improved, y, family=poisson)
summary(test)


Note the $\textbf{ treatment:improved}$ term I added inside the glm function.

Now, we get the following output:

    Call:
glm(formula = healthvalue ~ numberofdrugs + treatment + improved +
treatment:improved, family = poisson, data = y)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-2.9261  -0.8733  -0.0296   0.5473   2.3358

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)                      1.553051   0.184229   8.430   <2e-16 ***
numberofdrugs                    0.004298   0.014242   0.302   0.7628
treatmenttreated                 0.007399   0.149440   0.050   0.9605
improvedsome                     0.358897   0.164891   2.177   0.0295 *
improvedmarked                  -0.178360   0.203756  -0.875   0.3814
treatmenttreated:improvedsome   -0.330336   0.265310  -1.245   0.2131
treatmenttreated:improvedmarked  0.050617   0.260203   0.195   0.8458
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 97.805  on 83  degrees of freedom
Residual deviance: 89.276  on 77  degrees of freedom
AIC: 383.29

Number of Fisher Scoring iterations: 5


If we ignore what seems to be insignificant coefficients, I can ask my question:

I understand that, as in the original post, treatment=placebo and improved=none is the base level for those variables, and thus are set to zero. My question is, why does it not exist any interaction terms with the base lavels for treatment=placebo and improved=none?

I thought setting the base levels to zero was just a construct, and in my mind there should still exist correlation between them...(?)