Logistic discrimination of mixtures of M. tuberculosis and non-specific tuberculin reactions

Interpretation of the Mantoux test for tuberculous infection can be complicated by cross-reactions caused by infection with non-specific mycobacteria. Thus, the distribution of positive indurations is a mixture of two distributions. To estimate tuberculous infection prevalence, the marginal distribution of indurations needs to be separated into its component distributions. Observations from several populations with different mixes of the two types of infection are required. Homogeneity across populations of distributions of indurations for each type of infection is assumed. A logistic model is specified for the probability of having tuberculous infection conditional on the observed induration size. No other assumptions about the two distributions are made. Maximum likelihood is used to estimate the logistic function. Goodness-of-fit criteria are discussed. The method is applied to a series of tuberculin surveys carried out in (South) Korea. Estimated infection prevalence agrees reasonably well with several ad hoc criteria. The goodness-of-fit test rejects underlying assumptions of homogeneity. One reason appears to be a decline over time in induration sizes caused by tuberculous infection. However, not all reasons for this rejection are obvious. The proposed method of mixture analysis provides an additional tool for the interpretation of prevalence survey data where the diagnostic test lacks specificity as a result of cross-reactions.