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Our response variable is highly skewed and there is evidence of overdispersion as well. We tried with the Poisson, and Quasi-Poisson models. Both Poisson and Quasi-Poisson models failed to satisfy Pearson's $ \chi^2 $ goodness of fit test. Then we used the negative binomial regression model. We used $pseudo R-squared $ to assess the quality of the model fit. However, we don't know how can we validate the use of the negative binomial regression model like we did for Poisson, and Quasi-Poisson models using the goodness of fit test. Is there any way to check that the negative binomial regression model fits the data well?

Our fit results:

glm(formula = Infected ~ log(1 + PopDensity) + log(1 + GDPPPP) + 
    LifeExpectancy + log(1 + HealthExpend) + Humidity + AvgHigh + 
    AvgLow + AQI + PM1 + PM2 + log(1 + TotalTest), family = negative.binomial(1), 
    data = df, maxit = 10000)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.9231  -0.8082  -0.2303   0.3280   1.8219  

Coefficients:
                       Estimate Std. Error t value         Pr(>|t|)    
(Intercept)            8.992467   6.879656   1.307          0.19633    
log(1 + PopDensity)    0.137216   0.057273   2.396          0.01983 *  
log(1 + GDPPPP)       -1.636393   0.683709  -2.393          0.01995 *  
LifeExpectancy         0.024059   0.056888   0.423          0.67392    
log(1 + HealthExpend)  0.614003   0.326203   1.882          0.06482 .  
Humidity               0.003370   0.010989   0.307          0.76021    
AvgHigh               -0.062498   0.046816  -1.335          0.18710    
AvgLow                 0.061456   0.040870   1.504          0.13809    
AQI                    0.002927   0.003672   0.797          0.42867    
PM1                    0.016954   0.006471   2.620          0.01120 *  
PM2                   -0.021772   0.006481  -3.359          0.00139 ** 
log(1 + TotalTest)     0.809103   0.091449   8.848 0.00000000000237 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Negative Binomial(1) family taken to be 0.8757727)

    Null deviance: 200.335  on 69  degrees of freedom
Residual deviance:  63.787  on 58  degrees of freedom
AIC: 1373

# Pseudo R-squared

pR32 = 1 - p3$deviance /p3$null.deviance
pR32

[1] 0.6815972

Q-Q plot:

enter image description here

Distribution of the response variable, Y:

enter image description here

Simulation using the DHARMA package

enter image description here

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  • $\begingroup$ The conditional distribution of Y is highly discrete, and therefore grossly non-normal. Further, this is the point of the NB model: to model a specific, discrete non-normal distribution. So other than for providing insights into the data (which are better accomplished using other graphs), there no use for the normal q-q plot here. A first thing I would like to see is a frequency distribution of Y. $\endgroup$ Commented Jan 14, 2021 at 11:53
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    $\begingroup$ @BigBendRegion I added the distribution of Y. My concern here is: How can we say that our negative binomial regression model fits the data well? $\endgroup$ Commented Jan 14, 2021 at 12:06
  • $\begingroup$ "Fits well" is always relative, because all models are wrong. At least the frequency distribution provides no red flags. You can try alternative models for discrete Y including different discrete distributions, quadratic X terms and interactions, etc and compare likelihood based statistics (such as AIC) to see which models fit better. $\endgroup$ Commented Jan 14, 2021 at 13:23
  • $\begingroup$ @BigBendRegion Since Pseudo R-squared = 0.68, Can I say that my model was not bad? How can I favor my model? I am not sure why my reviewer commented that my model did not fit the data well! By the way, interactions did not workout well. $\endgroup$ Commented Jan 14, 2021 at 16:24
  • $\begingroup$ Why did the reviewer say it was bad? $\endgroup$ Commented Jan 14, 2021 at 16:30

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