Poisson Regression

When we perform a table analysis, logistic regression and log-linear models are most likely. Logistic regression deal with the ratio of odds but Poisson does the fixed amount as total number observation.

The main regression equation for this model is as follows: $ log \frac{\mu}{N} = \beta X $

Example

This is data about melanoma cases, in other words skin cancers.

mela=read.csv("data/melanoma.csv")
mel=xtabs(cases~age+region, data=mela)
out=glm(cases~age+region, family=poisson,offset=log(total),data=mela)
summary(out)

##
## Call:
## glm(formula = cases ~ age + region, family = poisson, data = mela,
##     offset = log(total))
##
## Deviance Residuals:
##       1        2        3        4        5        6        7        8  
##  0.4780  -0.4273  -0.5302   0.7469  -1.3610   0.8686  -0.4581   0.3667  
##       9       10       11       12  
##  0.4279  -0.5932   0.8904  -0.8283  
##
## Coefficients:
##              Estimate Std. Error z value Pr(>|z|)    
## (Intercept) -10.65831    0.09518 -111.97   <2e-16 ***
## age35-44      1.79737    0.12093   14.86   <2e-16 ***
## age45-54      1.91309    0.11844   16.15   <2e-16 ***
## age55-64      2.24180    0.11834   18.94   <2e-16 ***
## age65-74      2.36572    0.13152   17.99   <2e-16 ***
## age75+        2.94468    0.13205   22.30   <2e-16 ***
## regionsouth   0.81948    0.07103   11.54   <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: 895.8197  on 11  degrees of freedom
## Residual deviance:   6.2149  on  5  degrees of freedom
## AIC: 92.44
##
## Number of Fisher Scoring iterations: 4

As the result above shows us, all the estimated parameter are significant.

mu=fitted(out)
cbind(mela[,c(1,2)], cases=rpois(12,mu))

##      age region cases
## 1  35-44  south    83
## 2  45-54  south    74
## 3  55-64  south    65
## 4  65-74  south    35
## 5    75+  south    37
## 6    <35  south    47
## 7  35-44  north    77
## 8  45-54  north    87
## 9  55-64  north   104
## 10 65-74  north    59
## 11   75+  north    77
## 12   <35  north    60

2 factors as age groups and regions make 12 different sub-groups. From the estimated values, the number of the expected cases in each group is shown above.

Written on February 29, 2020