# Interpreting interaction among 2 categorical IV in quasi-poisson regression

In my dataset, I'm looking at the impacts of developmental and immune phenotypes on morbidity- specifically, I want to determine if developmental phenotype has an effect on the difference in morbidity between individuals with different immune phenotype classes. I have a DV consisting of over-dispersed count data (morbidity) and two binary categorical IVs: developmental phenotype class (1= normal, 2= disrupted) and immune phenotype class (1= normal, 2= pro-inflammatory).

I ran a quasi-poisson regression in R with the following script and results

> summary(p.dat.c)
Call: glm(formula = Morbid ~ Immune * Dev, family = quasipoisson(link = "log"),
data = dat)
Deviance Residuals:
Min       1Q   Median       3Q      Max
-3.2660  -0.7216   0.0795   0.6883   1.5805
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)    1.6740     0.1016  16.482   <2e-16 ***
Immune2        0.2417     0.1156   2.092   0.0398 *
Dev2          -0.1699     0.2556  -0.665   0.5083
Immune2:Dev2   0.1345     0.2918   0.461   0.6462
Null deviance: 92.529  on 78  degrees of freedom
Residual deviance: 85.540  on 75  degrees of freedom

And the exponents of the coefficiences are:

coef1  se.coef1 exponent
(Intercept)   1.6739764 0.1015649 5.333333
Immune2       0.2417199 0.1155558 1.273437
Dev2         -0.1698990 0.2556012 0.843750
Immune2:Dev2  0.1345155 0.2918120 1.143982

I don't understand how to interpret the interaction term between immune and developmental phenotypes- I get that the pro-inflammatory phenotype increases morbidity, whereas the disrupted penotype decreases morbidity (although not significantly). Do pro-inflammatory individuals with disrupted developmental phenotypes have increased morbidity (again, not significantly)? It's my first time doing this sort of analysis, so I just want to make sure I'm getting it right.

In a Poisson model, the expected count of morbidities is

$$E[y \vert x,b]=\exp(\alpha + \beta \cdot x + \eta \cdot b +\gamma \cdot x \cdot b).$$

You can re-write this as

$$E[y \vert x,b]=\exp(\alpha) \cdot \exp( \beta \cdot x) \cdot \exp(\eta \cdot b) \cdot \exp(\gamma \cdot x \cdot b).$$

In your model, $$E[y \vert x=0,b=0]=\exp(1.6739764)=5.33$$ morbidities for someone with normal phenotype classes since $$\exp(0)=1$$ and both $$x$$ and $$b$$ are zero.

If you add pro-inflammatory immune calss on top of that, the 5.33 gets multiplied by $$\exp(0.2417199 \cdot 1)=1.27$$, so the expected morbidities becomes $$5.33\cdot 1.27 = 6.77$$.

If you add disrupted developmental phenotype class instead, that multiplicative factor is $$\exp(-0.1698990 \cdot 1)=0.84$$, so 5.33 becomes 4.47. The negative sign on the Poisson coefficient indicates a decrease in morbidity, and the fact that the exponentiated coefficient is less than one is consistent with that: multiplication by a number less than 1 leads to shrinkage.

Finally, if you have both pro-inflamatory and distrupted phenotype classes, the 5.33 becomes $$5.33\cdot 1.27 \cdot0.84\cdot \exp(0.1345155)=5.33\cdot1.27\cdot0.84\cdot1.14=6.48.$$ In other words, the negative effect of disrupted dev is somewhat offset by having the pro-inflammatory immune phenotype.

• @StephBerge Did this clear things up? Jun 7, 2020 at 21:55
• Everything is really clear and makes sense until your last sentence- did you mean that the negative effect of the pro-inflammatory immune phenotype is somewhat offset by having the disrupted development phenotype? Jun 8, 2020 at 20:42
• @StephBerge The disrupted dev phenotype shrinks the expected morbidities (since we have multiplication by $0.84<1$), but if the person also has the pro-inflammatory, the interaction kicks in, and you get $84 \cdot 1.14 = .96<1$). That's still shrinkage associated with disrupted dev, but it is smaller when the person also has the pro-inflammatory phenotype. The second phenotype moderates negative effect of the first. You can also think of interaction as amplifying the effect of pro-inflammatory when the person has the disrupted dev phenotype. Does that make more sense? Jun 8, 2020 at 21:43
• It took me awhile to get back to this answer, but this was a brilliant explanation- thank you so much! Aug 14, 2020 at 19:00
• No need to thank me, you can just select it as the answer. Aug 14, 2020 at 19:02