# Interpretation of negative binomial GLM

In this model, the interpretation of the continuous variable tmax for an example would be: a increase 1 unit of tmax (exp(coef)=1.06) increases in 6% the incidence of disease in a month. Considering that casos = monthly number of cases of the disease, and populacao variable being used as offset representing the population in each city (municipio).

Is this interpretation correct?

summary(m1<- glm.nb(casos ~ 0 + municipio + precip_ant + tmax + tmax_ant + umid + umid_ant + enxu_2 + offset(log(populacao)), data = dataset))

Call:
glm.nb(formula = casos ~ 0 + municipio + precip_ant + tmax +
tmax_ant + umid + umid_ant + enxu_2 + offset(log(populacao)),
data = dataset, init.theta = 2.105944887, link = log)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-2.1763  -1.0107  -0.6286   0.3874   4.2286

Coefficients:
Estimate Std. Error z value Pr(>|z|)
municipio6 -2.566e+01  1.698e+00 -15.110  < 2e-16 ***
municipio1 -2.406e+01  1.706e+00 -14.108  < 2e-16 ***
municipio2 -2.424e+01  1.707e+00 -14.205  < 2e-16 ***
municipio3 -2.530e+01  1.696e+00 -14.914  < 2e-16 ***
municipio4 -2.525e+01  1.701e+00 -14.846  < 2e-16 ***
municipio5 -2.524e+01  1.702e+00 -14.829  < 2e-16 ***
precip_ant  1.750e-03  6.414e-04   2.728 0.006373 **
tmax        6.291e-02  1.922e-02   3.273 0.001066 **
tmax_ant    1.600e-01  1.995e-02   8.020 1.06e-15 ***
umid        2.665e-02  1.230e-02   2.166 0.030297 *
umid_ant    5.555e-02  1.454e-02   3.820 0.000134 ***
enxu_2      3.154e-01  2.074e-01   1.521 0.128384
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Negative Binomial(2.1059) family taken to be 1)

Null deviance: 41285.6  on 1008  degrees of freedom
Residual deviance:  1002.5  on  996  degrees of freedom
AIC: 2702.5

Number of Fisher Scoring iterations: 1

Theta:  2.106
Std. Err.:  0.308

2 x log-likelihood:  -2676.499

> exp(coef(m1))
municipio6   municipio1   municipio2   municipio3   municipio4   municipio5   precip_ant         tmax     tmax_ant
7.171204e-12 3.553106e-11 2.963580e-11 1.033429e-11 1.082516e-11 1.089931e-11 1.001751e+00 1.064932e+00 1.173543e+00
umid     umid_ant       enxu_2
1.027009e+00 1.057120e+00 1.370790e+00


In a Negative Binomial (NB) regression model with no offset, the 6.291e-02 coefficient would represent the change in the log of the expected count of the monthly number of cases of the disease for a one-unit increase in the corresponding predictor variable tmax, while holding all other predictor variables constant. i.e. the expected count would be multiplied by ($$\exp(6.291\text{E-02})$$=) 1.06 for a one-unit increase in tmax. Because though our NB regression model has an offset this increase is against the expected rate. The post uses the term: "incidence of disease" which I think is somewhat open to interpretation, it is probably to explicitly say: "incidence rate of disease" but aside from that the interpretation is good to go.