Nonparametric Estimation of the Incubation Period of AIDS with Left Truncation and Right Censoring

In the natural history of Human Immunodeficiency Virus Type-1 (HIV-1) infection, many studies included the participants who were seropositive at time of enrollment. Estimation of the unknown times since exposure to HIV-1 in the prevalent cohorts is of primary importance for estimation of the incubation period of Acquired Immunodeficiency Syndrome (AIDS). To estimate incubation period of AIDS we used prior distribution of incubation times, based on a external data as suggested by Bacchetti and Jewell (1991, Biometrics, 47,947-960). In the present study, our estimate was nonparametric based on a method proposed by Wang, Jewell and Tsai (1986, Annals of Statistics, 14, 1597-1605).


Introduction
The incubation period of the disease Acquired Immunodeficiency Syndrome (AIDS) is considered to be the time between infection with Human Immunodeficiency Virus type-1 (HIV-1) and diagnosis of AIDS.Understanding the nature of incubation period in the natural history of AIDS will facilitate improvements in timing and evaluation of therapies as well as in projecting the course of the epidemic.In the context of AIDS cohort studies, a great deal of effort has been made to provide accurate estimate of the incubation distribution, using both prospective and retroprospective data (Brookmeyer andGail, 1988, Brookmeyer andGoedert, 1989;Munoz et al., 1989;DeGruttola and Lagakos, 1989;Bacchetti, 1990;Kuo, Taylor and Detels, 1991;Munoz et al., 1992;and Medley et al., 1988;Kalbfleisch and Lawless, 1989).The limitation of these follow-up studies was that these cohort included subjects who were already HIV seropositive at the time of infection, commonly known as prevalent cohort.The prevalent cohorts have limited use in elucidating the natural history of AIDS because crucial information on the duration of infection is absent (Brookmeyer and Gail, 1987).However, this difficulty can be overcome if a prior infection distribution can be estimated from external data.HIV antibody tested on stored sera can provide information on the distribution of seroconversion times, which are thought to be close to infection times (Horshburg et al., 1989).Therefore, incubation time from seroconversion to AIDS diagnosis can be nonparametrcally estimated by using external information on seroconversion (Bacchetti and Jewell, 1991).An appropriate Kaplan-Meier estimator for the left-truncated data have been described by Turnbull (1976), Woodroofe (1985) and Wang, Jewell and Tasai (1986).
The aim of the present study was to estimate the incubation time of AIDS for a prevalent cohort by method proposed by Wang, Jewell and Tasai (1986).

Study Population
The study enrollment was initiated in April 1987 (Sricant et al., 1994) and followed through December 1994 at National AIDS reference center, Vellore, Tamil Nadu (India).In this study a total of 32 patients were followed up from the date of earliest positive Western blot test to the development of AIDS as shown in figure 1.All the patients were seropositive at enrollment.

Assumption, Notation and Methods
Left-truncation results when only response-free subjects are recruited into a study.If W i is the time from exposure to (potential) study entry for an individual i, it will be impossible to observe a response in individual i, if the incubation time is smaller than W i .We refer W i as truncation time.All follow-up studies recruit AIDS-free HIV-1 infected individuals.If E is study entry date and Sc i is the HIV-1 infection date of subject i, as measured by seroconversion, then If subject i had developed AIDS fewer than W i time units, then that subjects would not be AIDS-free and thus would be excluded (truncated) from the cohort.
Let X be the random variable representing the time between HIV-1 infection and AIDS diagnosis with associated survival function S * which is assumed to be left continuous.Let (W, C) be the random variables describing the left-truncation and right censoring time respectively.Further, to estimate AIDS-free survival time, Wang, Jewell and Tasai (1986) have described appropriate Kaplan-Meier estimator for the left-truncated data.Therefore, we estimate the conditional probability of AIDS-free survival beyond time x (i.e., time between HIV infection and AIDS), given AIDS-free survival at time W , that is Here, d j is the number of AIDS diagnosed cases at time x j , and n j is the number in the risk set at time x j (i.e.number of HIV-1 positive at the time of enrollment in the study) Similarly, for Greenwood's formula (1926) or Nelson-Aalen (1978) estimator, only AIDS cases that occur beyond W are considered.
The asymptotic normality of Ŝ * (x) has been established by Wang, Jwell and Tasai (1986), who also derived the asymptotic variance of Ŝ * (x).Further, the confidence interval for Ŝ * (x) are obtained by Hall and Wellner (H-W band) (1980) or Nair (1984) methods.These results justify the use of the Kaplan-Meier estimator and its variance estimator with left-truncated data after the appropriate adjustment of the risk set.

Data Analysis
To estimate the time of HIV-1 infection in 32 patients, we made several assumptions, based on the unique dynamics of the HIV-1 infection in Tamil Nadu region (Kilmarx et al., 2000).
In 1985, recipients of multiple blood transfusion and donors were investigated and were not found have any evidence of HIV-1 infection among them (Simoes et al., 1985).In February 1986, the presence of HIV-1 antibody were detected among female prostitutes in the city of Chennai in Tamil Nadu (Simoes et al., 1993).These evidences indicate that HIV-1 seroconversion among the heterosexual exposures prior to January 1983 were extremely rare.Consequently, the longest unobserved infection time for seroprevalent participants is unlikely to be greater than 52 months.Since most of the enrollment occurred during a one month period, we took W i as 52 months for all i (W i < 52).Further AIDS-free individuals were right censored as of April 31, 1990.
Figure 2 gives Ŝ * (x) for W i < 52 together with 95% two sided EP and H-W confidence bands for Ŝ * (x).The EP band shown corresponds to a L = 0.2 (L stands for lower bound) and a U = 0.8 (U stands for higher bound) with critical value e α = 2.9674.The H-W band shown was computed with value h α = 1.2928.From Figure 2, it can be seen that EP band is narrower in the tails, where as H-W bands performs narrower in the middle.

Discussion
Many prospective studies on natural history of HIV-1 infection are based on prevalent cohort which provide crucial information on the course of the disease progression.The members of the prevalent cohort are by definition at the risk of disease at the time of recruitment; hence the prevalent cohort will supply more atrisk person-time and disease events for the analysis than incident cohort (Muñoz et al., 1992).Effective survival analysis of the incubation period, however, requires the establishment of onset of HIV-1 infection.When the distribution of measurement of origin of a survival time or the time of disease onset is unknown, it can be first estimated based on external data or the data concerning infection time (Bacchetti and Jewell, 1991).To complete the missing date of seroconversion in the seroprevalent cohort, we used the data concerning the infection time based on the unique dynamics of the HIV-1 infection in Tamil Nadu region.
The median time from seroconversion to the development AIDS was 57 months in the present study.Progression time in HIV-1 infected Indians were more rapid with only 34% remaining AIDS-free after five years of seroconversion which is similar to the reports from other developing countries like Uganda, Kenya and Haiti (Morgan et al., 1997;Nagelkerke et al., 1990;and Deschamps et al., 2000).The developed countries have much longer progression times with 78%-85% of HIV-1 infected individuals remained AIDS-free after five years from seroconversion (Flegg, 1994;Lee et al., 1991).
The EP and H-W bands are competitive, with the former being narrower in the tail and latter being narrower in the middle.The relative performance of EP bands to the H-W bands gets better with censoring and asymptotic critical values provide reasonable approximation in finite samples for both bands (Nair, 1984).
In conclusion, the progression of disease in patients infected with HIV-1 in India seems to be rapid.This is most likely to be due to late recognition of AIDS-related conditions in the early phase of an epidemic and delayed medical care for patients.To improve the progression times, greater attention to the early diagnosis and treatment of opportunistic infection is required.

Figure 1 :
Figure 1: Time interval for the earliest diagnosis of HIV-1 infection to the development of AIDS in 32 patients.

Figure 2 :
Figure 2: Cumulative AIDS-free survival of HIV-1 infected individuals after seroconversion with 95% EP and H-W confidence bands.