### Excerpt

A cross-sectional design is used for studies measuring the proportion of late diagnoses [2,3]. The proportion of late diagnoses is the number of late diagnoses divided by the total number of new diagnoses in a calendar year. To measure the risk of being diagnosed late among HIV infections, a cohort study is more appropriate, retrospective, or prospective, by following persons living with undiagnosed HIV infection until the event, late diagnosis, occurs.

Figure 1 compares a retrospective cohort study with a cross-sectional study. The retrospective cohort study is a hypothetical one because it is difficult, if not impossible, to determine the time when a person acquired HIV infection. To conduct the hypothetical retrospective cohort study, we would identify persons who acquired HIV infection 10 years ago, and follow them until their diagnosis. On average, undiagnosed/untreated HIV infections develop AIDS in 10 years after infection [4]. In this retrospective cohort study, we assumed that all persons developed AIDS in the 9th year of infection if not diagnosed earlier and no persons remained undiagnosed after 10 years of infection. Late diagnoses were defined as persons diagnosed in their 9th year of infection.

From the retrospective cohort study, we can measure the risk of being diagnosed late, which is the number of persons diagnosed in their 9th year of infection divided by the number of persons who acquired HIV infection 10 years ago, n/Ni. From the cross-sectional study, we can measure the proportion of late diagnoses, which is the number of persons diagnosed in their 9th year of infection divided by the total number of new diagnoses in the year, n/Nd. These two measures have the same numerator, but different denominators. Given the changes in both HIV epidemic and HIV testing efforts, it is unlikely that the total number of new diagnoses, Nd, equals or approximates the number of new infections 10 years ago, Ni. Therefore, the proportion of late diagnoses is neither measuring nor approximating the risk of being diagnosed late among persons who acquired HIV infection 10 years ago.

The retrospective cohort study can measure the risk of being diagnosed late, but the results from the study reflect historical conditions that may no longer apply. A prospective cohort study would be a better study design. To conduct a hypothetical prospective cohort study, we would identify persons living with undiagnosed HIV infection at the beginning of the year, determine the duration of their HIV infection, and follow them until the end of the year.

From the prospective cohort study, we can construct a static life table by duration of infection. The risk of being diagnosed late among persons in their first year of infection is P9, and the proportion of late diagnoses is n10/Σn (Table 1). Mathematically, these two proportions are not equivalent nor do they approximate. Therefore, the proportion of late diagnoses does not measure the risk of being diagnosed late among persons who are newly infected.

With the total number of new diagnoses as the denominator, the proportion of late diagnoses is measuring the probability of duration of infection given being diagnosed, that is, Pr(Duration of infection | Being diagnosed). What we really want to measure is the probability of being diagnosed given the duration of infection, that is, Pr(Being diagnosed | Duration of infection).