Use of backcalculation for estimation of the probability of progression from hiv infection to aids

Backcalculation has been used to estimate the rate of past HIV infection and to predict future AIDS incidence. In this study we examine another use: estimating the probability of progression from HIV infection to AIDS as a function of time from infection. Given observed AIDS incidence data, the technique of backcalculation estimates the most likely number of persons infected with HIV in the past. Assumptions about probability of progression from HIV infection to AIDS are necessary. By varying these assumptions and examining the resulting goodness of fit to the AIDS incidence data, we can theoretically estimate parameters of progression. We report on implementation of this method and examine its practical utility in deciding among four competing progression models specified on a priori grounds. The four specific models comprise three Weibull distributions with medians of 8, 10, and 12 years, respectively, and one model that beings as a Weibull with 8 year median but where the hazard is level after 3.5 years. To employ asymptotic maximum likelihood methods, we define a two parameter family of progression models that includes all four a priori models. One parameter sets the scale for an initial Weibull progression (the shape parameter being fixed for all models), and the other specifies a levelling point after which the hazard remains constant. AIDS incidence data from Canada's national surveillance system provided the empiric data for this evaluation. First we corrected these data for reporting delay by Poisson modelling of the delay distribution. We used three parametric families of infection curves: step‐function, log‐logistic, and logistic. The results support the hypothesis of an early levelling of the hazard function. When we fixed the scale parameter to that of the Weibull curve with 8 year median, the maximum likelihood estimate of the levelling point was 2.7 years, and a clearly superior fit was produced compared to a pure Weibull progression with the same scale parameter (likelihood ratio chi‐square of 10.6 on 1 degree of freedom, p = 0.001). The maximum was indistinguishable in fit from the levelling point of 3.5 years hypothesized in advance (chi‐square = 0.30, d.f. = 1, p = 0.58). Backcalculation, however, could not determine the Weibull scale parameter itself because the likelihood was quite flat as a function of this parameter. We conclude that one must determine the parameters governing the initial shape of the hazard function from other kinds of data. We interpret the model with levelling of the hazard as an empirical description of progression in a real population rather than as natural history in a homogeneous population. We make comparisons with other recently proposed models of progression.