Example 51.13 Complementary Log-Log Model for Infection Rates

Antibodies produced in response to an infectious disease like malaria remain in the body after the individual has recovered from the disease. A serological test detects the presence or absence of such antibodies. An individual with such antibodies is called seropositive. In geographic areas where the disease is endemic, the inhabitants are at fairly constant risk of infection. The probability of an individual never having been infected in years is , where is the mean number of infections per year (see the appendix of Draper, Voller, and Carpenter 1972). Rather than estimating the unknown , epidemiologists want to estimate the probability of a person living in the area being infected in one year. This infection rate is given by

The following statements create the data set sero, which contains the results of a serological survey of malarial infection. Individuals of nine age groups (Group) were tested. The variable A represents the midpoint of the age range for each age group. The variable N represents the number of individuals tested in each age group, and the variable R represents the number of individuals that are seropositive.

data sero;
   input Group A N R;
   X=log(A);
   label X='Log of Midpoint of Age Range';
   datalines;
1  1.5  123  8
2  4.0  132  6
3  7.5  182 18
4 12.5  140 14
5 17.5  138 20
6 25.0  161 39
7 35.0  133 19
8 47.0   92 25
9 60.0   74 44
;

For the th group with the age midpoint , the probability of being seropositive is . It follows that

By fitting a binomial model with a complementary log-log link function and by using X=log(A) as an offset term, you can estimate as an intercept parameter. The following statements invoke PROC LOGISTIC to compute the maximum likelihood estimate of . The LINK=CLOGLOG option is specified to request the complementary log-log link function. Also specified is the CLPARM=PL option, which requests the profile-likelihood confidence limits for .

proc logistic data=sero;
   model R/N= / offset=X
                link=cloglog
                clparm=pl
                scale=none; 
title 'Constant Risk of Infection';  
run;

Results of fitting this constant risk model are shown in Output 51.13.1.

Output 51.13.1 shows that the maximum likelihood estimate of and its estimated standard error are and , respectively. The infection rate is estimated as

The 95% confidence interval for , obtained by back-transforming the 95% confidence interval for , is (0.0082, 0.0108); that is, there is a 95% chance that, in repeated sampling, the interval of 8 to 11 infections per thousand individuals contains the true infection rate.

The goodness-of-fit statistics for the constant risk model are statistically significant (), indicating that the assumption of constant risk of infection is not correct. You can fit a more extensive model by allowing a separate risk of infection for each age group. Suppose is the mean number of infections per year for the th age group. The probability of seropositive for the th group with the age midpoint is , so that

In the following statements, a complementary log-log model is fit containing Group as an explanatory classification variable with the GLM coding (so that a dummy variable is created for each age group), no intercept term, and X=log(A) as an offset term. The ODS OUTPUT statement saves the estimates and their 95% profile-likelihood confidence limits to the ClparmPL data set. Note that is the regression parameter associated with Group.

proc logistic data=sero;
   ods output ClparmPL=ClparmPL;
   class Group / param=glm;
   model R/N=Group / noint 
                     offset=X 
                     link=cloglog 
                     clparm=pl;
   title 'Infectious Rates and 95% Confidence Intervals';
run;

Results of fitting the model with a separate risk of infection are shown in Output 51.13.2.

For the first age group (Group1), the point estimate of is , which transforms into an infection rate of . A 95% confidence interval for this infection rate is obtained by transforming the 95% confidence interval for . For the first age group, the lower and upper confidence limits are and , respectively; that is, there is a 95% chance that, in repeated sampling, the interval of 20 to 80 infections per thousand individuals contains the true infection rate. The following statements perform this transformation on the estimates and confidence limits saved in the ClparmPL data set; the resulting estimated infection rates in one year’s time for each age group are displayed in Table 51.16. Note that the infection rate for the first age group is high compared to that of the other age groups.

data ClparmPL;
   set ClparmPL;
   Estimate=round( 1000*( 1-exp(-exp(Estimate)) ) );
   LowerCL =round( 1000*( 1-exp(-exp(LowerCL )) ) );
   UpperCL =round( 1000*( 1-exp(-exp(UpperCL )) ) );
run;