The article by Balzer et al.1 in the September 2020 issue of Epidemiology was published without its accompanying Appendix. This Appendix is provided below. The citations in the Erratum refer to the references in the main text of Balzer et al.1
For the moment, assume complete measurement and let be an indicator of HIV-positive serostatus at time t; be an indicator of having an HIV diagnosis by time t; be an indicator of ART use at time t, and be an indicator of suppressed viral replication at time t. The UNAIDS 90-90-90 targets are a series of proportions or conditional probabilities:5
Percentage of all HIV-positives who are diagnosed (first-90):
Percentage of diagnosed who are on ART (second-90):
Percentage on ART who are currently suppressed (third-90):
Multiplying together the three “90s” yields the proportion of all HIV-positive persons who are currently suppressed (i.e. population-level suppression):
Since each numerator and denominator is a population-level proportion, we can equivalently express the targets as follows: first-90 = (number previously diagnosed)/(number HIV-positive), second-90 = (number on ART)/(number previously diagnosed), third-90 = (number virally suppressed)/(number on ART), and population-level suppression = (number virally suppressed)/(number HIV-positive).
Therefore, one could directly estimate population-level suppression, as we demonstrated here, or instead estimate each 90-90-90 target and multiply. These two approaches should yield identical results, as demonstrated in our previous work.8,9,27 However, deviations between the direct estimate and the multiplied one can occur when making the MCAR assumption.1–4 Specifically, under MCAR, the denominators of the third-90 and population-level suppression become conditional on having a viral load measured, which is almost always a subset of the population on ART and a subset of the population who is HIV-positive.