The table below summarizes the lung cancer incident counts (cases) per age group for four Danish cities from 1968 to 1971. Since it's reasonable to assume that the expected count of lung cancer incidents is proportional to the population size, we would prefer to model the rate of incidents per capita.

       city   age  pop cases
 Fredericia 40-54 3059    11
    Horsens 40-54 2879    13
    Kolding 40-54 3142     4
      Vejle 40-54 2520     5
 Fredericia 55-59  800    11
    Horsens 55-59 1083     6
    Kolding 55-59 1050     8
      Vejle 55-59  878     7
 Fredericia 60-64  710    11
    Horsens 60-64  923    15
    Kolding 60-64  895     7
      Vejle 60-64  839    10
 Fredericia 65-69  581    10
    Horsens 65-69  834    10
    Kolding 65-69  702    11
      Vejle 65-69  631    14
 Fredericia 70-74  509    11
    Horsens 70-74  634    12
    Kolding 70-74  535     9
      Vejle 70-74  539     8
 Fredericia   75+  605    10
    Horsens   75+  782     2
    Kolding   75+  659    12
      Vejle   75+  619     7

With \(Y_i\) the count of lung cancer incidents and \(t_i\) the population size for the \(i^{th}\) row in the data, the Poisson rate regression model would be

\(\log \dfrac{\mu_i}{t_i}=\log \mu_i-\log t_i=\beta_0+\beta_1x_{1i}+\beta_2x_{2i}+\cdots\)

where \(Y_i\) has a Poisson distribution with mean \(E(Y_i)=\mu_i\), and \(x_1\), \(x_2\), etc. represent the (systematic) predictor set. Notice that by modeling the rate with population as the measurement size, population is not treated as another predictor, even though it is recorded in the data along with the other predictors. The main distinction the model is that no \(\beta\) coefficient is estimated for population size (it is assumed to be 1 by definition). Note also that population size is on the log scale to match the incident count.

data cancer;
input city $ age $ pop cases; 
lpop = log(pop);
cards;    
 Fredericia 40-54 3059    11
    Horsens 40-54 2879    13
    Kolding 40-54 3142     4
      Vejle 40-54 2520     5
 Fredericia 55-59  800    11
    Horsens 55-59 1083     6
    Kolding 55-59 1050     8
      Vejle 55-59  878     7
 Fredericia 60-64  710    11
    Horsens 60-64  923    15
    Kolding 60-64  895     7
      Vejle 60-64  839    10
 Fredericia 65-69  581    10
    Horsens 65-69  834    10
    Kolding 65-69  702    11
      Vejle 65-69  631    14
 Fredericia 70-74  509    11
    Horsens 70-74  634    12
    Kolding 70-74  535     9
      Vejle 70-74  539     8
 Fredericia   75+  605    10
    Horsens   75+  782     2
    Kolding   75+  659    12
      Vejle   75+  619     7
; run;
proc genmod data=cancer; 
class city(ref='Frederic') age(ref='40-54');
model cases = city age / dist=poi link=log offset=lpop type3;
run;

library(ISwR)
library(dplyr)
data(eba1977)
eba1977 = eba1977 %>% mutate(lpop = log(pop))
model = glm(cases ~ city + age, offset=lpop, family=poisson, data=eba1977)
summary(model)
anova(model, test='Chisq')

> summary(model)

Call:
glm(formula = cases ~ city + age, family = poisson, data = eba1977, 
    offset = lpop)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-2.63573  -0.67296  -0.03436   0.37258   1.85267  

Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -5.6321     0.2003 -28.125  < 2e-16 ***
cityHorsens  -0.3301     0.1815  -1.818   0.0690 .  
cityKolding  -0.3715     0.1878  -1.978   0.0479 *  
cityVejle    -0.2723     0.1879  -1.450   0.1472    
age55-59      1.1010     0.2483   4.434 9.23e-06 ***
age60-64      1.5186     0.2316   6.556 5.53e-11 ***
age65-69      1.7677     0.2294   7.704 1.31e-14 ***
age70-74      1.8569     0.2353   7.891 3.00e-15 ***
age75+        1.4197     0.2503   5.672 1.41e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 129.908  on 23  degrees of freedom
Residual deviance:  23.447  on 15  degrees of freedom
AIC: 137.84

The fitted model is

\(\log\dfrac{\hat{\mu}}{t} = -5.6321-0.3301C_1-0.3715C_2-0.2723C_3 + 1.1010A_1+\cdots+1.4197A_5\)

where \(C_1\), \(C_2\), and \(C_3\) are the indicators for cities Horsens, Kolding, and Vejle (Fredericia as baseline), and \(A_1,\ldots,A_5\) are the indicators for the last five age groups (40-54 as baseline). 

Thus, for people in (baseline) age group 40-54 and in the city of Fredericia, the estimated average rate of lung cancer is

\(\dfrac{\hat{\mu}}{t}=e^{-5.6321}=0.003581\)

per person. For a group of 100 people in this category, the estimated average count of incidents would be \(100(0.003581)=0.3581\).

Does the overall model fit? From the deviance statistic 23.447 relative to a chi-square distribution with 15 degrees of freedom (the saturated model with city by age interactions would have 24 parameters), the p-value would be 0.0715, which is borderline. But the model with all interactions would require 24 parameters, which isn't desirable either. Is there perhaps something else we can try?

Since age was originally recorded in six groups, we needed five separate indicator variables to model it as a categorical predictor. We may also consider treating it as quantitative variable if we assign a numeric value, say the midpoint, to each group.

The following code creates a quantitative variable for age from the midpoint of each age group. It also creates an empirical rate variable for use in plotting. Note that this empirical rate is the sample ratio of observed counts to population size \(Y/t\), not to be confused with the population rate \(\mu/t\), which is estimated from the model.

data cancer;
set cancer;
if age='40-54' then age_mid=47;
if age='55-59' then age_mid=57;
if age='60-64' then age_mid=62;
if age='65-69' then age_mid=67;
if age='70-74' then age_mid=72;
if age='75+' then age_mid=75;
e_rate = cases/pop;
run; 
proc sgplot data=cancer;
reg y=e_rate x=age_mid / group=city;
run; 

proc genmod data=cancer; 
class city;
model cases = age_mid / dist=poi link=log offset=lpop type3;
run;

library(dplyr)
eba1977 = eba1977 %>% mutate(
 age.mid = rep(c(47,57,62,67,72,75), each=4),
 e.rate = cases/pop)
ggplot(eba1977, aes(x=age.mid, y=e.rate, color=city)) + 
 geom_point() + geom_smooth(method='lm', se=FALSE)

model = glm(cases ~ city + age.mid, offset=lpop, family=poisson, data=eba1977)
summary(model)
anova(model, test='Chisq')

> summary(model)

Call:
glm(formula = cases ~ city + age.mid, family = poisson, data = eba1977, 
    offset = lpop)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-4.0037  -0.5442   0.3013   0.7957   2.2684  

Coefficients:
             Estimate Std. Error z value Pr(>|z|)    
(Intercept) -7.982221   0.431070 -18.517   <2e-16 ***
cityHorsens -0.305279   0.181423  -1.683   0.0924 .  
cityKolding -0.350278   0.187705  -1.866   0.0620 .  
cityVejle   -0.246016   0.187769  -1.310   0.1901    
age.mid      0.056759   0.006527   8.696   <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: 129.908  on 23  degrees of freedom
Residual deviance:  46.448  on 19  degrees of freedom
AIC: 152.84