A Procedure for Estimating the Unconditional Cumulative Incidence Curve and its Variability for the Human Immunodeficiency Virus

A backcalculation procedure is presented for estimating both the cumulative incidence of the human immunodeficiency virus (HIV)—the total number of seropositive individuals as a function of calendar time—and the unconditional sampling distribution of this estimate. This estimation framework explicitly accounts for corrections for reporting lags in the most recent AIDS diagnosis data and imposes no functional form restrictions, besides a roughness penalty, on the resulting estimate of the cumulative incidence function. The construction of the sampling distribution for this estimate ‘integrates out’ the variation due to the use of estimated reporting lag and incubation distributions to obtain the unconditional distribution. The estimation procedure amounts to solving an inequality‐restricted generalized least squares estimation problem subject to smoothness priors on the regression coefficients. We find that both the point estimate of the cumulative incidence curve for HIV and its associated upper and lower confidence bound paths remain stable over a wide range of estimation scenarios. In addition, not accounting for the use of estimated incubation and reporting lag distributions in the procedure for estimating the cumulative incidence curve can lead to a substantial underestimation of its sampling variability.