/****************************************************************/ /* S A S S A M P L E L I B R A R Y */ /* */ /* NAME: LOGIEX13 */ /* TITLE: Example 13 for PROC LOGISTIC */ /* PRODUCT: STAT */ /* SYSTEM: ALL */ /* KEYS: logistic regression analysis, */ /* binomial response data, */ /* CLOGLOG link */ /* PROCS: LOGISTIC */ /* DATA: */ /* */ /* SUPPORT: Bob Derr */ /* REF: SAS/STAT User's Guide, PROC LOGISTIC chapter */ /* MISC: */ /* */ /****************************************************************/ /***************************************************************** Example 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 antibodies. An individual with such antibodies is termed seropositive. In 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(-uY) where u is the mean number of infections per year (see the appendix of Draper, C.C., Voller, A., and Carpenter, R.G. 1972, "The epidemiologic interpretation of serologic data in malaria," American Journal of Tropical Medicine and Hygiene, 21, 696-703). Rather than estimating the unknown u, it is of interest to epidemiologists to estimate the probability of a person living in the area being infected in one year. This infection rate is 1-exp(-u). The SAS statements below create the data set SERO, which contains the results of a serological survey of malaria infection. Individuals of nine age groups were tested. Variable A represents the midpoint of the age range for each age group, N represents the number of individuals tested in each age group and R represents the number of individuals that are sero- positive. */ title 'Example 13. CLOGLOG Model for Infection Rates'; 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 age midpoint A_i, the probability of being seropositive is p_i=1-exp(-uA_i). It follows that log(log(1-p_i)) = log(u) + 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, b0=log(u) is estimated as an intercept parameter. The following SAS statements invoke PROC LOGISTIC to compute the maximum likelihood estimate of b0. The LINK=CLOGLOG option is specified to request the complementary log-log link function. Also specified is the CLPARM=PL option, which produces the profile-likelihood confidence limits for the b0. */ proc logistic data=sero; model R/N= / offset=X link=cloglog clparm=pl scale=none; title 'Constant Risk of Infection'; run; /* The maximum likelihood estimate of b0 is -4.6605. This translates into an infection rate of 1-exp(-exp(-4.6605))=.00942. The 95\% confidence interval for the infection rate, obtained by back-transforming the 95\% confidence interval for b0, is (.0082, .0011); that is, there is a 95\% chance that 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<.0001), indicating that the assumption of constant risk of infection is not correct. One can fit a more extensive model by allowing a separate risk of infection for each age group. Let u_i be the mean number of infections per year for the ith age group. The probability of seropositive for the ith group with age midpoint A_i is p_i=1-exp(-u_iA_i), so that log(-log(1-p_i)=log(u_i) + log(A_i) In the following SAS statements, the GLM parameterization creates dummy variables for the age groups. PROC LOGISTIC is invoked to fit a complementary log-log model that contains the dummy variables as the only explanatory variables with no intercept term and with X=log(A) as an offset term. Note that log(u_i) is the regression parameter associated with GROUP=i. The DATA statement transforms the estimates and confidence limits saved in the CLPARMPL data set to estimate the infection rates in one year's time. */ 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; 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;