9.3 - Modeling Rate Data

Example: Lung Cancer Incidence Section

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.

By using an OFFSET option in the MODEL statement in GENMOD in SAS we specify an offset variable. The offset variable serves to normalize the fitted cell means per some space, grouping, or time interval to model the rates. In this case, population is the offset variable.

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;
And, here is the output from the GENMOD procedure:

The SAS System

The GENMOD Procedure

 
Model Information
Data Set WORK.CANCER
Distribution Poisson
Link Function Log
Dependent Variable cases
Offset Variable lpop
 
Number of Observations Read 24
Number of Observations Used 24
 
Class Level Information
Class Levels Values
city 4 Horsens Kolding Vejle Frederic
age 6 55-59 60-64 65-69 70-74 75+ 40-54
 
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 15 23.4475 1.5632
Scaled Deviance 15 23.4475 1.5632
Pearson Chi-Square 15 22.5616 1.5041
Scaled Pearson X2 15 22.5616 1.5041
Log Likelihood   278.4530  
Full Log Likelihood   -59.9178  
AIC (smaller is better)   137.8355  
AICC (smaller is better)   150.6927  
BIC (smaller is better)   148.4380  
 
Algorithm converged.

By adding offset in the MODEL statement in GLM in R, we can specify an offset variable. The offset variable serves to normalize the fitted cell means per some space, grouping, or time interval to model the rates. In this case, population is the offset variable.

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')

Here is the output we get:

> 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?

Considering Age as Quantitative Section

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; 
The SGPlot Procedure 50 55 60 65 70 75 age_mid 0.000 0.005 0.010 0.015 0.020 e_rate Vejle Kolding Horsens Frederic city
With age treated as quantitative, we no longer include it in the CLASS statement.
proc genmod data=cancer; 
class city;
model cases = age_mid / dist=poi link=log offset=lpop type3;
run;

The interpretation of the slope for age is now the increase in the rate of lung cancer (per capita) for each 1-year increase in age, provided city is held fixed.

The SAS System

The GENMOD Procedure

 
Model Information
Data Set WORK.CANCER
Distribution Poisson
Link Function Log
Dependent Variable cases
Offset Variable lpop
 
Number of Observations Read 24
Number of Observations Used 24
 
Class Level Information
Class Levels Values
city 4 Horsens Kolding Vejle Frederic
 
Criteria For Assessing Goodness Of Fit
Criterion DF Value Value/DF
Deviance 19 46.4483 2.4446
Scaled Deviance 19 46.4483 2.4446
Pearson Chi-Square 19 41.4714 2.1827
Scaled Pearson X2 19 41.4714 2.1827
Log Likelihood   266.9526  
Full Log Likelihood   -71.4182  
AIC (smaller is better)   152.8363  
AICC (smaller is better)   156.1696  
BIC (smaller is better)   158.7266  
 
Algorithm converged.
 
Analysis Of Maximum Likelihood Parameter Estimates
Parameter   DF Estimate Standard
Error
Wald 95% Confidence Limits Wald Chi-Square Pr > ChiSq
Intercept   1 -7.9822 0.4311 -8.8271 -7.1373 342.89 <.0001
city Horsens 1 -0.3053 0.1814 -0.6609 0.0503 2.83 0.0924
city Kolding 1 -0.3503 0.1877 -0.7182 0.0176 3.48 0.0620
city Vejle 1 -0.2460 0.1878 -0.6140 0.1220 1.72 0.1901
city Frederic 0 0.0000 0.0000 0.0000 0.0000 . .
age_mid   1 0.0568 0.0065 0.0440 0.0696 75.62 <.0001
Scale   0 1.0000 0.0000 1.0000 1.0000    

Note:The scale parameter was held fixed.

 
LR Statistics For Type 3 Analysis
Source DF Chi-Square Pr > ChiSq
city 3 4.25 0.2357
age_mid 1 80.07 <.0001
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)

The plot generated shows increasing trends between age and lung cancer rates for each city. There is also some evidence for a city effect as well as for city by age interaction, but the significance of these is doubtful, given the relatively small data set.

As we have seen before when comparing model fits with a predictor as categorical or quantitative, the benefit of treating age as quantitative is that only a single slope parameter is needed to model a linear relationship between age and the cancer rate. The tradeoff is that if this linear relationship is not accurate, the lack of fit overall may still increase.

model = glm(cases ~ city + age.mid, offset=lpop, family=poisson, data=eba1977)
summary(model)
anova(model, test='Chisq')
The interpretation of the slope for age is now the increase in the log rate of lung cancer (per capita) for each 1-year increase in age, provided city is held fixed.
> 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