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I'm creating some multiple regression models on some national statistic data checking whether there is a divide in infant deaths between the north and sound of the UK.

I have created models for males, females and all the data (males+females). This data contains deaths, population, divide(north/south) and gender per year from 1965-2016.

As an example using the female data, I at first created a Poisson model, this was the output:

glm(formula = deaths ~ Divide + year + offset(log(population)), 
    family = poisson(link = "log"), data = nsfemaleMerge)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-6.5420  -2.1335   0.4779   2.3226   5.7547  

Coefficients:
              Estimate Std. Error z value Pr(>|z|)    
(Intercept) 64.5738825  0.3792783  170.25   <2e-16 ***
DivideSouth -0.1726417  0.0054365  -31.76   <2e-16 ***
year        -0.0348758  0.0001914 -182.23   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 38817.15  on 103  degrees of freedom
Residual deviance:   890.77  on 101  degrees of freedom
AIC: 1817.8

Number of Fisher Scoring iterations: 4

As you can see there is overdispersion present. I then did a quasipoisson and negative Binomial mode, quasipossion resulted in:

glm(formula = deaths ~ Divide + year + offset(log(population)), 
    family = quasipoisson(link = "log"), data = nsfemaleMerge)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-6.5420  -2.1335   0.4779   2.3226   5.7547  

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) 64.5738825  1.1243161   57.43   <2e-16 ***
DivideSouth -0.1726417  0.0161158  -10.71   <2e-16 ***
year        -0.0348758  0.0005673  -61.47   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasipoisson family taken to be 8.787408)

    Null deviance: 38817.15  on 103  degrees of freedom
Residual deviance:   890.77  on 101  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4

and negative binomial resulted in:

Call:
glm.nb(formula = deaths ~ Divide + year + offset(log(population)), 
    data = nsfemaleMerge, init.theta = 142.5507992, link = log)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-2.3861  -0.6426   0.1220   0.7028   2.2366  

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) 62.6865274  1.1806850   53.09   <2e-16 ***
DivideSouth -0.1810315  0.0177523  -10.20   <2e-16 ***
year        -0.0339236  0.0005934  -57.17   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for Negative Binomial(142.5508) family taken to be 1.023361)

    Null deviance: 3655.61  on 103  degrees of freedom
Residual deviance:  104.97  on 101  degrees of freedom
AIC: 1262.2

Number of Fisher Scoring iterations: 1

I then compared the parameter estimates and standard errors:

enter image description here

The standard errors for quasipoisson and NB seem extremely close whereas for poisson seem to be underestimated.

I also did an ANOVA:

Analysis of Deviance Table

Model 1: deaths ~ Divide + year + offset(log(population))
Model 2: deaths ~ Divide + year + offset(log(population))
Model 3: deaths ~ Divide + year + offset(log(population))
  Resid. Df Resid. Dev Df Deviance Pr(>Chi)
1       101     890.77                     
2       101     890.77  0     0.00         
3       101     104.97  0   785.81   

My question is how do I choose which model is best to use? Also how would I represent my model or visualise it say for a research paper?

I'm still getting into stats so any help would be appreciated, thanks!

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1 Answer 1

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Good question.

I like to look at the deviance goodness of fit to assess which model to use. You can see in your Poisson model that your residual deviance is quite large as compared to the degrees of freedom. This results in a rejection of the null for a deviance goodness of fit test.

On the other hand, the negative binomial model seems to fit the data quite well (104.97 on 101 degrees of freedom results in a p-value around 0.3). To me, that says the negative binomial model is the model to choose.

For visualization, you could maybe do something like an esplot but to be honest, the best way to visualize the model will depend on what narrative you're trying to make.

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  • $\begingroup$ I noticed that the degrees of freedom for negative binomial was a lot smaller compared to the other two models, where did you get the P value of arounf 0.3 from? (I must be blind haha). Thanks I'll look into ESplot, also, are the coefficient values signifying anything? $\endgroup$
    – yahyalogde
    Jun 16, 2019 at 20:29
  • $\begingroup$ Re: the p value, just look up how to do a deviance goodness of fit test. Coefficients will mater after you pick a model. $\endgroup$ Jun 16, 2019 at 20:35
  • $\begingroup$ I see thanks, I'm now getting around 0.3 also, thanks for the help! Looks like I'm choosing Negative Binomial for all my models. Just to make sure, I'm assuming the null hypothesis is that the model is correctly specified, a p-value of 0.3 means that hypothesis can't be rejected hence signifying the the model is actually correctly specified? $\endgroup$
    – yahyalogde
    Jun 16, 2019 at 21:25
  • $\begingroup$ More or less, yes. The way to think about the DGOF test is as follows: If you fail to reject the null, then that means your model behaves comparably to a "saturated model" (one that fits the data perfectly). Since your model is simpler than the saturated model, then you should prefer your model due to parsimony. If you reject the null in the DGOF, then your model does not model the data as well as a saturated model. You can read more about DGOF in any glm book. $\endgroup$ Jun 16, 2019 at 21:57

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