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 loglog 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 loglog link function. Also specified is the CLPARM=PL option, which requests the profilelikelihood 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 53.13.1.
Constant Risk of Infection 
Model Information  

Data Set  WORK.SERO  
Response Variable (Events)  R  
Response Variable (Trials)  N  
Offset Variable  X  Log of Midpoint of Age Range 
Model  binary cloglog  
Optimization Technique  Fisher's scoring 
Number of Observations Read  9 

Number of Observations Used  9 
Sum of Frequencies Read  1175 
Sum of Frequencies Used  1175 
Response Profile  

Ordered Value 
Binary Outcome  Total Frequency 
1  Event  193 
2  Nonevent  982 
InterceptOnly Model Convergence Status 

Convergence criterion (GCONV=1E8) satisfied. 
Deviance and Pearson GoodnessofFit Statistics  

Criterion  Value  DF  Value/DF  Pr > ChiSq 
Deviance  41.5032  8  5.1879  <.0001 
Pearson  50.6883  8  6.3360  <.0001 
Analysis of Maximum Likelihood Estimates  

Parameter  DF  Estimate  Standard Error 
Wald ChiSquare 
Pr > ChiSq 
Intercept  1  4.6605  0.0725  4133.5626  <.0001 
X  1  1.0000  0  .  . 
Parameter Estimates and ProfileLikelihood Confidence Intervals 


Parameter  Estimate  95% Confidence Limits  
Intercept  4.6605  4.8057  4.5219 
Output 53.13.1 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 backtransforming 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 goodnessoffit 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 loglog 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% profilelikelihood 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 53.13.2.
Infectious Rates and 95% Confidence Intervals 
Analysis of Maximum Likelihood Estimates  

Parameter  DF  Estimate  Standard Error 
Wald ChiSquare 
Pr > ChiSq  
Group  1  1  3.1048  0.3536  77.0877  <.0001 
Group  2  1  4.4542  0.4083  119.0164  <.0001 
Group  3  1  4.2769  0.2358  328.9593  <.0001 
Group  4  1  4.7761  0.2674  319.0600  <.0001 
Group  5  1  4.7165  0.2238  443.9920  <.0001 
Group  6  1  4.5012  0.1606  785.1350  <.0001 
Group  7  1  5.4252  0.2296  558.1114  <.0001 
Group  8  1  4.9987  0.2008  619.4666  <.0001 
Group  9  1  4.1965  0.1559  724.3157  <.0001 
X  1  1.0000  0  .  . 
Parameter Estimates and ProfileLikelihood Confidence Intervals 


Parameter  Estimate  95% Confidence Limits  
Group  1  3.1048  3.8880  2.4833 
Group  2  4.4542  5.3769  3.7478 
Group  3  4.2769  4.7775  3.8477 
Group  4  4.7761  5.3501  4.2940 
Group  5  4.7165  5.1896  4.3075 
Group  6  4.5012  4.8333  4.2019 
Group  7  5.4252  5.9116  5.0063 
Group  8  4.9987  5.4195  4.6289 
Group  9  4.1965  4.5164  3.9037 
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 53.16. 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*( 1exp(exp(Estimate)) ) ); LowerCL =round( 1000*( 1exp(exp(LowerCL )) ) ); UpperCL =round( 1000*( 1exp(exp(UpperCL )) ) ); run;
Number Infected per 1,000 People 


Age 
Point 
95% Confidence Limits 

Group 
Estimate 
Lower 
Upper 
1 
44 
20 
80 
2 
12 
5 
23 
3 
14 
8 
21 
4 
8 
5 
14 
5 
9 
6 
13 
6 
11 
8 
15 
7 
4 
3 
7 
8 
7 
4 
10 
9 
15 
11 
20 