The LOGISTIC Procedure

Example 72.14 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 Y years is $\exp (-\mu Y )$ , where $\mu$ is the mean number of infections per year (see the appendix of Draper, Voller, and Carpenter 1972). Rather than estimating the unknown $\mu$ , epidemiologists want to estimate the probability of a person living in the area being infected in one year. This infection rate $\gamma$ is given by

$\gamma = 1-{e}^{-\mu }$

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 ith group with the age midpoint $A_ i$ , the probability of being seropositive is $p_ i=1-\exp (-\mu A_ i)$ . It follows that

$\log (-\log (1-p_ i)) = \log (\mu ) + \log (A_ i)$

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 $\alpha =\log (\mu )$ as an intercept parameter. The following statements invoke PROC LOGISTIC to compute the maximum likelihood estimate of $\alpha$ . 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 $\alpha$ .

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 72.14.1.

Output 72.14.1: Modeling Constant Risk of Infection

Constant Risk of Infection

The LOGISTIC Procedure

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

Intercept-Only Model Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

-2 Log L = 967.1158

Deviance and Pearson Goodness-of-Fit Statistics
Criterion	Value	DF	Value/DF	Pr > ChiSq
Deviance	41.5032	8	5.1879	<.0001
Pearson	50.6883	8	6.3360	<.0001

Number of events/trials observations: 9

Analysis of Maximum Likelihood Estimates
Parameter	DF	Estimate	Standard Error	Wald Chi-Square	Pr > ChiSq
Intercept	1	-4.6605	0.0725	4133.5626	<.0001
X	0	1.0000	0	.	.

Parameter Estimates and Profile-Likelihood Confidence Intervals
Parameter	Estimate	95% Confidence Limits
Intercept	-4.6605	-4.8057	-4.5219

Output 72.14.1 shows that the maximum likelihood estimate of $\alpha =\log (\mu )$ and its estimated standard error are $\widehat{\alpha }=-4.6605$ and $\widehat{\sigma }_{\widehat{\alpha }}=0.0725$ , respectively. The infection rate is estimated as

$\widehat{\gamma }=1-{e}^{-\widehat{\mu }} =1-{e}^{-{e}^{{\widehat{\beta }}_0}} =1-{e}^{-{e}^{-4.6605}} =0.00942$

The 95% confidence interval for $\gamma$ , obtained by back-transforming the 95% confidence interval for $\alpha$ , 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 ( $p < 0.0001$ ), 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 $\mu _ i$ is the mean number of infections per year for the ith age group. The probability of seropositive for the ith group with the age midpoint $A_ i$ is $p_ i=1-\exp (-\mu _ i A_ i)$ , so that

$\log (-\log (1-p_ i))=\log (\mu _ i) + \log (A_ i)$

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 $\log (\mu _ i)$ is the regression parameter associated with Group $=i$ .

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 72.14.2.

Output 72.14.2: Modeling Separate Risk of Infection

Infectious Rates and 95% Confidence Intervals

The LOGISTIC Procedure

Analysis of Maximum Likelihood Estimates
Parameter		DF	Estimate	Standard Error	Wald Chi-Square	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		0	1.0000	0	.	.

Parameter Estimates and Profile-Likelihood 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 (Group=1), the point estimate of $\log (\mu _1)$ is –3.1048, which transforms into an infection rate of $1-\exp (-\exp (-3.1048))=0.0438$ . A 95% confidence interval for this infection rate is obtained by transforming the 95% confidence interval for $\log (\mu _1)$ . For the first age group, the lower and upper confidence limits are $1-\exp (-\exp (-3.8880)=0.0203$ and $1-\exp (-\exp (-2.4833))=0.0801$ , 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 72.18. 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;

Table 72.18: Infection Rate in One Year

	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