Abstract

Postnatal transmission of human immunodeficiency virus infection through breastfeeding complicates evaluating the efficacy of interventions aimed to reduce mother-to-child transmission risk. Results from trials in Africa evaluating either peripartum antiretroviral therapy or refraining from breastfeeding show an estimated long-term efficacy at 15–24 months of age between 25 and 50 percent. Differences in statistical methods, duration of follow-up, and age at weaning hinder direct comparison between trials. The authors recently outlined theoretically preferred statistical methods for evaluating interventions aimed to reduce risk of mother-to-child transmission of human immunodeficiency virus. When multiple test results and/or supplementary information is available, the more sophisticated methods account for the fact that exact age at infection is unknown, that risk for infection ends at weaning, or that censoring due to death may be informative. The authors apply these methods to four scenarios, using data from four randomized trials carried out in Africa between 1995 and 2000. The authors’ findings suggest that, to estimate the cumulative proportion infected at age 6 weeks, a standard Kaplan-Meier approach is likely to give valid results. For estimation of this proportion at age 18 months, more sophisticated methods, such as the extension of the Kaplan-Meier procedure to interval-censored data and competing risks, would be preferred.

Received for publication May 20, 2002; accepted for publication February 13, 2003.

Mother-to-child transmission of human immunodeficiency virus (HIV) infection can occur before, during, and after delivery. Where prolonged, breastfeeding more than doubles the risk of transmission (1). The overall risk in breastfeeding populations can be reduced from 25–40 percent without (2) to 8–20 percent with interventions (36). Most interventions have concentrated on the administration of antiretroviral therapy in the peripartum period to reduce maternal viral load and/or to serve as postexposure prophylaxis to the neonate. Evaluation of the long-term efficacy of peripartum interventions has been hampered by the additional risk of postnatal acquisition of infection through breastfeeding (7).

Estimated long-term efficacy (at 15–24 months) varied from 25 percent to 50 percent in African trials conducted among breastfeeding populations (1, 810). In these trials, efficacy has been estimated at age 4–6 weeks and then at 12, 15, 18, or 24 months. The statistical methods used in different trials for estimating cumulative transmission rates vary, with either HIV infection or HIV infection and death as the outcome and a range of duration of clinical and laboratory follow-up. The populations in which these trials have been carried out have had various durations of breastfeeding. It is therefore difficult to compare published results from these trials.

To facilitate direct comparison in the efficacy of specific interventions between trials, we developed a consensus approach for the statistical methods that should be considered for analyzing such trials (11). In addition to the standard Kaplan-Meier estimator, we proposed three alternative estimators to account for 1) the unknown exact date of diagnosable HIV infection, 2) the risk of acquiring HIV infection that is ending after weaning, and 3) informative censoring (HIV-infected children are at increased risk of death). We present the results from these four consensus analytical approaches, systematically applied to four trials carried out in breastfeeding populations (1, 3, 4, 9), to assess short- and long-term cumulative infection risk. We compare the point estimates and confidence intervals obtained from these four methods to evaluate the need for using complex statistical methods to estimate the cumulative rate of mother-to-child transmission.

MATERIALS AND METHODS

Four randomized clinical trials in African breastfeeding populations are reanalyzed; in three placebo-controlled trials, the efficacy of a peripartum regimen of antiretroviral therapy, either zidovudine monotherapy (3, 4) or a combination of zidovudine and lamivudine in a different regimen (9), is evaluated. The placebo arm of the Petra Study (9) is compared with the arm receiving the longest regimen of zidovudine plus a combination of zidovudine and lamivudine (arm A). The fourth trial evaluated the use of artificial feeding versus breastfeeding in the absence of antiretroviral therapy (1). For the antiretroviral trials, the primary endpoint was HIV infection while, for the breastfeeding trial, it was HIV-free survival. In all four trials, a blood sample was taken for HIV testing during visits within 48 hours of birth and then at ages 4 (4) or 6 weeks (1, 3, 9), 14 weeks (1), and every 3 months thereafter until the endpoint of the trial, which was 24 months but 18 months in the Petra trial (9). The analyses are restricted to firstborn livebirths.

Pediatric HIV infection was defined as a positive DNA or RNA polymerase chain reaction test result at any age and by a positive serology result at 18 months or older. For both polymerase chain reaction and serology, a negative result can be considered definitive only if cessation of breastfeeding has occurred at least 60 days before the blood sample was taken. In other instances, the result should be considered provisional, as seroconversion can occur up to 2 months after infection (11).

We consider four scenarios defined by the length of follow-up (short or long) and the number of ages at which samples are available for HIV testing (one or multiple) (11).

Scenario 1: short-term efficacy based on a single laboratory assessment and vital status

Polymerase chain reaction results of the single laboratory assessment are used to ascertain infection status at age 6 weeks. The estimated transmission rate is the proportion of children tested at that age who tested positive. We simulate this scenario by restricting data to a maximum age of 59 days. Because relatively few infants would have died from HIV-related causes at 6 weeks, the inherent assumption of uninformative censoring in this analysis is reasonable.

Scenario 2: short-term efficacy based on two laboratory assessments and vital status

Based on blood samples collected around birth and 6 weeks, the cumulative infection rate and probability of HIV-free survival can be estimated by various methods. Weaning is not a competing risk for estimating the probability of HIV diagnosis at such an early age. We simulate this scenario by restricting data to a maximum age of 59 days. We evaluate the resulting estimates by comparison with the corresponding estimates obtained using all available data following scenario 3.

Scenario 3: long-term efficacy based on sequential laboratory assessments and vital status

In this scenario, the cohort has been followed for at least 12 months with blood sample collection for laboratory assessments at several ages. Vital status and duration of breastfeeding are recorded. The cumulative probability of death and probability of breastfeeding are estimated using the Kaplan-Meier method (12). We estimate cumulative infection rates and the probability of HIV-free survival using various methods. We simulate this scenario by restricting data for each study to a maximum age of 590 days.

Scenario 4: long-term efficacy based on a single laboratory assessment and vital status

Follow-up data are available only at age 18 months (11), and results based on the tests of survivors to this age are used to estimate cumulative rates of transmission. In this evaluation, we restrict HIV diagnosis information to samples obtained at age 505–559 days. Similar to scenario 1, we first estimate the proportion of children who were tested at that age who were HIV positive and second also include all deaths before age 18 months as possibly infected, giving an upper bound of the rate of infection (A), while in B the estimated number of infected children among those untested is added to the numerator of this proportion (11).

Efficacy E(t) of an intervention can be defined as the preventive fraction in the exposed group (group 1) compared with the reference group (group 0): E = 1 – (Ri(t)/R0(t)), where Ri(t) is the estimated cumulative rate of transmission in group i defined as P(T1 < t), where T1 denotes the random variable for time of HIV diagnosis. In the case of competing risks (see below), we define R(t) = Pr(T1 < t and T1 < T2), where T2 is the random variable for weaning time. When HIV-free survival is the outcome of interest, Ri is 1 minus the probability of HIV-free survival. We estimate short-term efficacy of these peripartum interventions at 6 weeks of age; long-term efficacy is estimated at 18 months. An endpoint is defined either by diagnosed HIV infection or by HIV infection or death.

When HIV test results are available at several ages, the conventional estimate of the cumulative proportion with an endpoint at each age of interest is obtained from the Kaplan-Meier procedure (12). This procedure assumes that the exact age at which an endpoint occurs is known and that censoring is uninformative (risk for an endpoint is independent of risk for censoring). Unless HIV is diagnosed at birth, the age at which HIV could be diagnosed is interval censored, because it is known to be only between the age at the last negative test (or age 0, if there was no negative test) and the age at the first positive test. Because HIV-infected children are at increased risk of early death, death may represent informative censoring when the endpoint is diagnosed HIV infection. In addition, the Kaplan-Meier estimator does not take into account the fact that the risk of HIV infection ends after a child is weaned. If some weaned children are censored before a specified age of interest, the Kaplan-Meier estimate of the cumulative proportion infected can be biased (11).

We evaluate three extensions of the Kaplan-Meier procedure; all allow the event date to be interval censored. One extension also takes into account the fact that infection risk ends after weaning, and another, that death may represent informative censoring. Each extension procedure yields a nonparametric maximum likelihood estimator (in some sense). Like the derivation of the Kaplan-Meier procedure, the derivation for each procedure starts by determining the support set, the set of intervals (and possibly points) on which the estimator can change value (see the Appendix). Each estimator is then derived from an iterative numerical procedure. The estimator is not unique on the support set (unless the change in value is estimated to be zero) but is known to be between only two values (corresponding to the jump in the Kaplan-Meier estimator).

To remove the assumption of an exact endpoint time, we use Turnbull’s extension of the Kaplan-Meier procedure to interval-censored data (13). The survival (or cumulative failure) estimate from Turnbull’s procedure is similar to the corresponding estimate from the Kaplan-Meier procedure, except that the estimate is undefined on intervals on which the change in the estimate is positive. The principal advantage of Turnbull’s method is that it is likely to provide a correct estimate of variance, since we acknowledge that the event time is not directly observed. Turnbull’s method assumes that censoring is uninformative and does not account for weaning as a competing risk, in the sense described next, although weaning information could be used to reduce the interval during which HIV infection could have been diagnosed. The algorithm used to compute this estimator is described in the Appendix. Lindsey and Ryan (14) analyze two data sets to compare the Turnbull estimator with the two extreme Kaplan-Meier estimators (assuming that the event dates were at the left or right endpoints of the event interval). With heavily censored data, the Kaplan-Meier estimates need not bracket the Turnbull estimate.

In using the Kaplan-Meier procedure, we made the conventional midpoint assumption that the age at which infection was diagnosed occurred at the midpoint between the ages at the last negative and first positive tests; for HIV-free survival, we assumed that the event date was the midpoint between the ages at the last negative test and death for children who died without being diagnosed with HIV. In using the Turnbull procedure to estimate HIV-free survival, we assumed that the event occurred between the age at last negative test (birth if no negative test) and the age at death for these children.

Hudgens et al. (15) extended Turnbull’s procedure to include one or more competing risks; alternatively, this can be considered an extension of the estimator with exact event times and competing risks (16). Weaning can be considered as a competing risk for estimating the cumulative proportion of children infected. In the terminology used with the Kaplan-Meier procedure, an uninfected child is removed from the risk set after weaning. The procedure of Hudgens et al. estimates the cause-specific cumulative incidence function, here P(T1 < t and T1 < T2), assuming that censoring is uninformative. The cumulative incidence estimate from this procedure is conditional on the pattern of weaning; it does not estimate cumulative incidence in the absence of weaning. For each failure cause, the cumulative incidence estimator is similar to the cumulative failure estimate from Turnbull’s algorithm. The algorithm used to compute this estimator is an extension of Turnbull’s algorithm (see the Appendix). It can be shown that the competing risks estimate can be greater or less than the Kaplan-Meier estimate based on only the HIV test results, depending on how long children continue to be tested after weaning, when other children are censored compared with when children are weaned, and the numbers of weaned and censored children (see the Appendix).

Hughes and Richardson extended Turnbull’s procedure to allow informative censoring (17). Their procedure does not account for competing risks. They use a semiparametric procedure to estimate the joint distribution of HIV infection time and death time for interval-censored data. The joint distribution of death and HIV diagnosis is modeled by assuming that 1) the conditional probability of HIV diagnosis given death has a Weibull distribution, and 2) death can be modeled by a nonparametric distribution. Then the cumulative probability of diagnosed HIV infection is estimated by the corresponding marginal distribution of the joint distribution (see the Appendix).

We used SAS (SAS Institute, Inc., Cary, North Carolina) and S-Plus 2000 (Insightful Corporation, Seattle, Washington) code to compute the Kaplan-Meier and Turnbull estimators. SAS code for computing the competing risks estimator of Hudgens et al. is available at the following website: http://www.fhcrc.org/labs/self/hudgens/mgh.html. This code uses the bootstrap to estimate standard errors. The code for the Hughes-Richardson method is written in S-Plus software for Unix (SCO Group, Lindon, Utah) and Fortran (Free Software Foundation, Inc., Cambridge, Massachusetts) and is available from this website: http://lib.stat.cmu.edu/jasasoftware/hughes-r.

RESULTS

The number of livebirths available for analysis and the number available for the different scenarios for estimating short-term (at 6 weeks for scenarios 1 and 2) or long-term (at 18 months for scenarios 3 and 4) efficacy are given in table 1. In the Ditrame trial, children who were known to be uninfected at least 2 months after weaning were no longer tested. Hence, some uninfected children may not have had a test result at 18 months. In the Petra trial, 18 months was the endpoint, and all children not lost to follow-up and alive (about 67 percent of the total) would have been tested at this age. In other trials with an endpoint at 24 months, some children may not have been tested at 18 months but were tested later. With the use of all available data (scenario 3), the proportion of children who died without an HIV test was low except for the placebo arm in the Retro-Ci trial.

For each trial (arms combined), table 2 shows for selected ages the percentage of children who had been weaned, the number and percentage of children who had been censored, and the number and percentage of infected children diagnosed with HIV infection. If nearly all infections are diagnosed by the age at which an appreciable proportion of children are weaned and if a relatively small proportion of children are censored before that age, then the competing risks procedure is unlikely to be necessary. In each arm of each trial, more than half (51–80 percent) of infected children were identified by 3 months of age. At most, 4.1 percent of children were weaned by age 3 months, except for 22.5 percent in Petra. The proportion censored at age 3 months through loss to follow-up or death was 6.4–13.0 percent. For most children with diagnosed HIV infection, the length of the interval between the last negative and first positive tests was relatively short. For each arm of each trial, this interval was longer than 3 months for only 5.5–12.5 percent of these children (data not shown).

Table 3 presents point estimates of cumulative infection risk and of short-term, or early, efficacy, with confidence interval lengths for the point estimates in the placebo group, using only data through age 6 weeks. Results from scenarios 1 and 2 are compared with those of scenario 3, in which all available information is used and which is the reference for this analysis. Scenario 1 point estimates are generally similar to those from scenario 2. For each trial, the three methods give similar estimates of efficacy except for that based on the Kaplan-Meier procedure in the Agence Nationale de Recherches sur le SIDA (ANRS) trial. The cumulative infection risk point estimates in scenario 2 were greater than the corresponding estimates using all data (scenario 3) except for Petra, although some differences are small. We made similar comparisons with HIV-free survival as the endpoint and found similar results (data not shown).

Table 4 presents the point estimates for cumulative infection risk and long-term efficacy, as well as confidence interval length for the point estimate in the placebo group, using data through age 18 months. Differences among the first three procedures depend on the ages at diagnosis of infection, weaning, and censoring (table 2). Estimates from the Hughes-Richardson procedure will differ from those of the other procedures if death represents informative censoring. The first three procedures yield very similar estimates of the cumulative infection risk and confidence intervals for the cumulative infection risk except for the Ditrame study, for which the competing risks procedure yields larger point estimates and a longer confidence interval. The estimates from the Hughes-Richardson procedure are very similar to those from the other procedures except the Ditrame trial.

For the Ditrame placebo group, the cumulative infection risk estimated from the competing risks procedure has a jump from 0.33 at 560 days to 0.49 at 591 days. The estimates shown for the Agence Nationale de Recherches sur le SIDA trial are at 568 days; we used linear interpolation to obtain the point estimate and confidence interval shown in table 4. This illustrates that an estimate obtained from interval-censored survival analysis need not be unique at a time of interest. The Ditrame estimates also demonstrate that the competing risks cumulative infection risk estimate can be greater than the Kaplan-Meier or Turnbull estimates.

Table 5 presents similar comparisons for scenario 3 with HIV-free survival as the endpoint. All three methods give very similar point estimates and confidence interval lengths.

For scenario 4, use of the observed proportion diagnosed with infection generally provided a substantially smaller estimate of the cumulative infection risk than the data from scenario 3, because of the relatively large number of infected children who were not tested at 18 months (table 1). Most estimates of HIV-free survival from scenario 4 (table 5) were similar to the estimates from scenario 3, with similar confidence interval lengths.

DISCUSSION

We compared four statistical methods to estimate cumulative mother-to-child HIV infection risk, or HIV-free survival probability, using data from four clinical trials conducted in African breastfeeding populations (11). In these populations, estimates of the long-term efficacy of peripartum interventions could differ substantially from those of short-term efficacy because of breast milk transmission (1). These populations also have high infant mortality rates even in uninfected children, and loss to follow-up was potentially a major concern in each of the trials. The statistical methods used varied in whether they account for age at a positive HIV test as an interval-censored event, weaning as a competing risk for postnatal HIV infection, and death as potential informative censoring. Scenario 3 was considered the referent, because it allowed for all available information collected in a standardized fashion. We compared the methods with respect to point estimates, confidence interval length, and ease of use. We also assessed whether the cumulative infection rate could be estimated if HIV status were available only at the age at which the study endpoint is defined.

Theoretically, a valid estimate of the cumulative infection risk and its variance at age 18–24 months would be obtained from the extension of the Kaplan-Meier estimator to interval-censored data with competing risks (14) or from the Hughes-Richardson procedure (17). The first procedure gives a valid estimate when there are few deaths among children not known to be infected and, hence, death as informative censoring can be ignored. The second procedure gives a valid estimate if nearly all infections are detected before many children have been weaned and if testing continues after weaning, so that weaning as a competing risk for HIV infection can be ignored. Use of the competing risks approach assumes that reliable weaning information is available, while the Hughes-Richardson model requires reliable data on the occurrence and time (but not cause) of death. If point estimates or variances from these methods disagree, the choice of the appropriate method depends on evaluating the relations among the distributions of ages at HIV diagnosis, weaning, and censoring, as well as the proportion of children who died without an HIV diagnosis. Both this assessment and an evaluation of the distribution of the length of time between last negative and first positive tests are necessary if consideration is given to use of a simpler statistical procedure.

Overall, the application of these methods (1113, 15, 17) across the four trials shows remarkable consistency in the estimated short-term efficacy of the various interventions. Within the trials, for the assessment at 6 weeks of age, a simple approach based on the results of only one laboratory test and information on whether or not the infant was still alive provided estimates not too dissimilar to those obtained by methods using all available data. If more than one laboratory test result is available, we recommend the standard Kaplan-Meier procedure. Although the Turnbull procedure remains valid under more general conditions than the Kaplan-Meier procedure, the latter is easier to use and yields nearly identical estimates of early efficacy at 6 weeks of age, because intervals between tests are short. Note that in these methods, the exact age at a positive test does not estimate the “time to infection” but “time to detection” of HIV, which is dependent on the testing schedule (11).

For assessment of long-term efficacy in terms of HIV infection risk and HIV-free survival, we found good agreement between the point estimates and confidence intervals from the competing risks and Hughes-Richardson analyses. We would thus recommend the competing risks approach if there are relatively few deaths among children without an HIV test. This approach allows children who are weaned and no longer at risk of becoming infected to be excluded from the denominator. An HIV-free survival approach is preferred if there are many untested children or if deaths in the intervention arm are of particular concern, such as in the infant feeding trial. In this case, based on our findings and on theoretical grounds, we recommend using the Turnbull procedure; the Hughes-Richardson method could also be considered but is more complex to use.

This work compares statistical methods for estimating the efficacy of interventions in four clinical trials. Results of this comparison provide useful guidance for choosing a method of analysis in any intervention trial aimed to prevent HIV mother-to-child transmission in a breastfeeding population. Reports for any trial should clearly state the HIV testing schedule and other information about the numbers of children included in or excluded from the various assessment ages. In particular, the number of children who were lost to follow-up or who died without HIV testing by the ages of assessment used is needed to interpret the results. Second, the length of intervals between HIV tests and particularly between the last negative and the first positive test should be described in summary format: If many of these intervals are long, a Turnbull analysis, possibly including competing risks, is needed. If most of these intervals are short (less than 3 months), a standard Kaplan-Meier approach to the analysis should suffice. Third, if weaning is typically at an advanced age (after 12 months), a relatively long interval between the last negative and the first positive tests will be more common, making it more essential to run a Turnbull analysis. If HIV testing stops after weaning, a competing risks analysis is also recommended. If a large number of deaths occurs among infants with uncertain HIV infection status, then the Hughes-Richardson approach would be appropriate, although this procedure is technically complex. Because these methods can yield different point estimates and confidence interval lengths, it is essential to specify the method of analysis in the study protocol, rather than choosing the method after analyzing the data.

Fourth, we recommend that HIV infection be used to assess short-term efficacy, while HIV-free survival should also be used to assess long-term efficacy. For the assessment of long-term efficacy, the results from the comparative analysis presented here strongly support the previously expressed concern that depending on the results of only one test between 15 and 24 months of age is not appropriate and will give substantially biased results in the estimated efficacy, as loss to follow-up and deaths are unlikely to be evenly distributed between trial arms (11). Where testing at only 15–24 months is feasible, attempts should be made to record the vital status of all children and the age at cessation of breastfeeding, as HIV-free survival is likely to provide a less biased estimate of the efficacy of an intervention in the long-term. This is particularly pertinent for the evaluation of public health programs, where the investment for follow-up and laboratory testing will be limited.

We originally recommended analytical procedures based on theoretical considerations and available statistical methods (11). We have now refined these recommendations on the basis of application of these methods to data from four intervention trials, thus providing a powerful approach to the evaluation of interventions to reduce mother-to-child transmission of HIV in breastfeeding populations. These findings could be used to better assess the merit, at least in terms of efficacy, of different interventions in public health programs in Africa.

ACKNOWLEDGMENTS

The authors thank Ruthie Odum, Centers for Disease Control and Prevention, for data management of the Centers’ Retro-Ci trial.

APPENDIX

Definition of the support set for survival analysis using interval-censored data

If the ith person has an event, denote the interval during which the event is known to occur as [Li, Ri]. For our data, L and R are the ages at the last negative and the first positive HIV tests, respectively, or the age at weaning (in which case, L = R). If the ith person is censored, this interval is [Li, ∞].

With this notation, the support set for an estimator is defined as follows. Order all the interval endpoints (both the Ls and the Rs) from smallest to largest. The support set consists of those intervals [a, b] such that, for some i and j (we allow i = j), a = Li, b = Rj, and for all k and l, Lkand Rl are not in (a, b). Thus, the support intervals are the smallest intervals [Li, Rj] that contain no data interval endpoints. If an event time is known exactly, then that time defines an “interval” in the support set. Note that, for each interval-censored observation [Li, Ri], at least one support interval [a, b] must lie within [Li, Ri], and neither Li nor Ri can be in the interior of any support interval.

Turnbull’s generalization of the Kaplan-Meier estimator to interval-censored data

Let T be the random time at which an event occurs, and let F be the cumulative distribution function for T, so that F(t) = Pr(Tt). By writing down the general form of the likelihood based on F, it is easy to see that F cannot increase outside the support set, and that F is not defined uniquely on the support set. Using Turnbull’s notation, let the support set consist of the M intervals [qj, pj], and let sj be the mass associated with the jth interval (sj ≥ 0, Ssj = 1). Thus, s (the vector of the masses) is a discrete probability distribution. With this notation, if t is not in the support set, then

graphic

Turnbull shows that the maximum likelihood estimator of F is obtained from the probability distribution s* that satisfies the self-consistency property. Let N be the number of persons contributing data. Let µij(s) be the proportion of the mass in [Li, Ri] that is contributed by [qj, pj]. Let

graphic

Then s is self-consistent if

sj = πj(s) for all j.

An iterative algorithm can be used to find s*. Let sk be the value of s at the kth step. Choose an initial value for s, s0, such as sj = 1/M for all j. Given sk, compute the numerator and denominator of equation 1, and set sjk+1 = πj(sk). Continue this iteration until the desired accuracy is achieved. Turnbull shows that this algorithm converges to the maximum likelihood estimator of F. Lindsey and Ryan give simple code for this iterative computation (14).

The competing risks estimator for interval-censored data

In the notation of Hudgens et al., suppose that there are J possible failure types; for our data, J = 2 (HIV diagnosis and weaning). Define a support set {[qjk, pjk]} for each failure type j; for weaning, the support set will be a set of points. The support set is defined as for a single failure type: For failures of type j, use only failures of that type and censored observations. We need to estimate the cumulative incidence functions Ij(t) = Pr(Tt, and failure of type j). For the jth failure type, define a set of nonnegative masses sj = {sjk} such that the sum of all masses for all failure types is 1. Given sj and t not in the support set for the jth failure type, the estimate is

graphic

The maximum likelihood estimator is obtained from a self-consistent estimator s* using an iterative procedure similar to that used to obtain Turnbull’s estimator. Not all cumulative incidence estimates need be defined at a point at which the Turnbull estimate (for all failure types combined) is defined. In general, the sum of the cumulative incidence estimates need not be equal to the cumulative failure estimate from Turnbull’s procedure at a point at which all of these estimates are defined.

The semiparametric estimator of the joint distribution of infection and death (Hughes and Richardson)

Let t and s indicate the time of HIV infection and death, respectively. Using the notation of Hughes and Richardson, we summarize the data on the times (ti, si) for the ith infant by the pair of intervals graphic , where graphicand graphic are the left and right endpoints of the interval containing the HIV infection time sifor infant i (allowing for a special mass point for children never infected), and graphicand graphic are the left and right endpoints of the interval containing the time of death tifor infant i. Suppose infants are followed 18 months after birth. Then t ϵ graphic , where the special mass point “18+” is used for children that were alive at 18 months, and s ϵ {[0,18], NI}, where NI is a special mass point for children uninfected at death or 18 months (whichever is first). The joint distribution of HIV infection time and death time is given as w(s,t) = f(t) g(s|t).

Hughes and Richardson model f(t), the distribution of death time, nonparametrically and g(s|t), the conditional distribution of HIV infection times given death times, as a mixture of point mass at birth and a Weibull distribution with parameters that vary with death time. The observed log-likelihood:

graphic

is maximized by alternately maximizing the parameters of f(t) conditional on the current estimate of g(s|t) and then maximizing the parameters of g(s|t) conditional on the current estimate of f(t). The marginal distribution of HIV infection is then easily derived from this joint distribution.

Dependence of the estimated cumulative incidence of HIV diagnosis on the order of diagnosis and weaning times

Assume that “events” occur at times 1, 2, 3, 4. For sequence 1, the order of events is weaning, censored, HIV diagnosis, and censored; for sequence 2, censored, weaning, HIV diagnosis, and censored. We wish to estimate cumulative HIV diagnosis at time 4. For simplicity, assume that the time of HIV diagnosis is known exactly (we do not consider the interval-censored case). Assume that weaned children continue to test negative. Assume that there are a total of N children, with nt children with an “event” at time t.

We consider three cumulative incidence estimates: 1) a competing risks analysis; 2) the standard Kaplan-Meier procedure, when the last HIV test for weaned children is shortly after weaning (before the next time with an “event”); and 3) the standard Kaplan-Meier procedure, when the last HIV test for weaned children is at time 4 (continued testing of weaned children). Appendix table 1A shows estimated cumulative incidence at time 4 for the two situations and three estimates.

In situation 1, it is clear that cumulative incidence estimate 3 is less than estimate 1, and both are less than estimate 2.

In situation 2, it is easy to show that the estimates from 1 and 3 above are less than that from 2 above (testing stops at weaning), but clearly the estimate from 1 above can be greater than or less than that from 3, depending on the values of n1 and n2.

Correspondence to Dr. François Dabis, INSERM U. 593, Institut d’Epidémiologie, Santé Publique et Développement, Université Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France (e-mail: francois.dabis@isped.u-bordeaux2.fr).

TABLE 1.

Trial profile of four clinical trials evaluating interventions to prevent mother-to-child transmission of human immunodeficiency virus in African breastfeeding populations, 1995–2000*

Trial arm ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo Zidovudine  Placebo Zidovudine  Breastfeeding Formula feeding  Placebo Zidovudine and lamivudine‡ 
Stillbirths (no.)    
Livebirths (no.) 201 200  133 128  197 204  349 363 
Ever breastfed (no.) 198 (98.5)§ 193 (96.5)  132 (99.2) 127 (99.2)  189 (95.9) 53 (25.9)  265 (75.9) 265 (73.0) 
Scenario 1            
No test and death (no.)  10   
Children tested (no.) 131 129  110 111  136 138  255 282 
HIV+† (no.) 35 26  24 14  32 15  43 16 
Scenario 2            
No test and death (no.)    
Children tested (no.) 156 145  122 121  164 179  267 288 
Cumulative HIV+¶ (no.) 42 28  26 15  37 16  45 16 
HIV–† and death (no.)    
Scenario 3            
No test and death (no.)    
Children tested (no.) 198 191  124 126  191 193  339 353 
Cumulative HIV+ (no.) 59 40  35 27  54 28  71 49 
HIV– and death (no.) 10   14 18  11 12 
Scenario 4            
No test and death (no.) 43 32  29  40 35  37 22 
Children tested (no.) 89 106  72 76  62 74  227 250 
HIV+ (no.) 17 10  12  17 12  30 24 
Trial arm ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo Zidovudine  Placebo Zidovudine  Breastfeeding Formula feeding  Placebo Zidovudine and lamivudine‡ 
Stillbirths (no.)    
Livebirths (no.) 201 200  133 128  197 204  349 363 
Ever breastfed (no.) 198 (98.5)§ 193 (96.5)  132 (99.2) 127 (99.2)  189 (95.9) 53 (25.9)  265 (75.9) 265 (73.0) 
Scenario 1            
No test and death (no.)  10   
Children tested (no.) 131 129  110 111  136 138  255 282 
HIV+† (no.) 35 26  24 14  32 15  43 16 
Scenario 2            
No test and death (no.)    
Children tested (no.) 156 145  122 121  164 179  267 288 
Cumulative HIV+¶ (no.) 42 28  26 15  37 16  45 16 
HIV–† and death (no.)    
Scenario 3            
No test and death (no.)    
Children tested (no.) 198 191  124 126  191 193  339 353 
Cumulative HIV+ (no.) 59 40  35 27  54 28  71 49 
HIV– and death (no.) 10   14 18  11 12 
Scenario 4            
No test and death (no.) 43 32  29  40 35  37 22 
Children tested (no.) 89 106  72 76  62 74  227 250 
HIV+ (no.) 17 10  12  17 12  30 24 

* For definitions of scenarios, see Materials and Methods.

† ANRS, Agence Nationale de Recherches sur le SIDA (France); Ditrame, Diminution de la Transmission Mere-Enfant; CDC, Centers for Disease Control and Prevention; NIH, National Institutes of Health; UNAIDS, United Nations AIDS (acquired immunodeficiency syndrome) agency; HIV+, human immunodeficiency virus positive; HIV–, human immunodeficiency virus negative (HIV-uninfected children).

‡ Arm A (Petra).

§ Numbers in parentheses, percentage.

¶ Cumulative no. of HIV-infected children.

TABLE 2.

Proportions of African children weaned,* censored,† and diagnosed with human immunodeficiency virus infection at selected ages, by trial, 1995–2000

Age ANRS‡ 049a Ditrame‡ (two arms combined)  CDC‡ Retro-Ci, Abidjan (two arms combined)  NIH,‡ Nairobi (breastfeeding arm only)§  UNAIDS‡ Petra (two arms combined) 
% weaned Censored  Infected  % weaned Censored  Infected  % weaned Censored  Infected  % weaned Censored  Infected 
No.  No.  No.  No.  No.  No.  No.  No. 
3 months 3.0 25 6.4  79 80  2.6 17 6.8  49 79  4.1 24 13.0  37 68.5  22.5 51 7.2  65 51 
6 months 12.6 33 8.5  85 86  5.9 20 8.0  49 79  8.9 37 20.0  42 77.7  37.7 67 9.4  87 70 
12 months 54.1 54 13.9  92 93  23.2 38 15.2  56 90  24.8 63 34.0  50 92.6  64.5 94 18.4  109 91 
18 months 73.6 135 34.7  99 100  82.5 46 18.4  62 100  65.0 113 61.0  54 100.0  85.9 353 49.6  120 100 
Age ANRS‡ 049a Ditrame‡ (two arms combined)  CDC‡ Retro-Ci, Abidjan (two arms combined)  NIH,‡ Nairobi (breastfeeding arm only)§  UNAIDS‡ Petra (two arms combined) 
% weaned Censored  Infected  % weaned Censored  Infected  % weaned Censored  Infected  % weaned Censored  Infected 
No.  No.  No.  No.  No.  No.  No.  No. 
3 months 3.0 25 6.4  79 80  2.6 17 6.8  49 79  4.1 24 13.0  37 68.5  22.5 51 7.2  65 51 
6 months 12.6 33 8.5  85 86  5.9 20 8.0  49 79  8.9 37 20.0  42 77.7  37.7 67 9.4  87 70 
12 months 54.1 54 13.9  92 93  23.2 38 15.2  56 90  24.8 63 34.0  50 92.6  64.5 94 18.4  109 91 
18 months 73.6 135 34.7  99 100  82.5 46 18.4  62 100  65.0 113 61.0  54 100.0  85.9 353 49.6  120 100 

* Estimated using a Kaplan-Meier analysis among ever breastfed children.

† Censored: tested at least once, no positive test, last negative test before 590 days, and not weaned or weaned less than 60 days at last negative test. Includes children who are censored as a result of death.

‡ ANRS, Agence Nationale de Recherches sur le SIDA (France); Ditrame, Diminution de la Transmission Mere-Enfant; CDC, Centers for Disease Control and Prevention; NIH, National Institutes of Health; UNAIDS, United Nations AIDS (acquired immunodeficiency syndrome) agency.

§ Information on duration of breastfeeding was not available for the intervention arm.

TABLE 3.

Estimates of early* efficacy in four randomized clinical trials in African breastfeeding populations, 1995–2000†,‡

Method Trial and results 
ANRS§ 049a Ditrame§  CDC§ Retro-Ci, Abidjan  NIH,§ Nairobi  UNAIDS§ Petra 
Placebo point estimate¶ Zidovudine point estimate¶ Placebo CI§ Relative efficacy point estimate¶  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate¶ Formula feeding point estimate¶ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate¶ Placebo CI Relative efficacy point estimate 
Scenario 1                    
Proportion positive 0.27 0.20 0.15 25  0.22 0.13 0.15 42  0.25 0.13 0.15 49  0.17 0.06 0.09 65 
Scenario 2                    
Kaplan-Meier exact  0.29 0.21 0.15 29  0.24 0.13 0.16 46  0.28 0.17 0.15 39  0.18 0.06 0.09 67 
Turnbull exact 0.29 0.26 0.15 10  0.38 0.22 0.31 42  0.31 0.16 0.22 48  0.16 0.05 0.27 69 
Hughes-Richardson 0.29 0.25 0.15 14  0.29 0.14 0.18 52  0.27 0.18 0.14 34  0.48 0.12 0.62 75 
Scenario 3                    
Kaplan-Meier exact  0.23 0.15 0.12 34  0.22 0.12 0.15 46  0.21 0.09 0.12 58  0.15 0.05 0.08 67 
Turnbull exact 0.24 0.16 0.12 36  0.26 0.15 0.16 41  0.21 0.08 0.19 62  0.17 0.06 0.20 65 
Turnbull competing 0.24 0.15 0.12 38  0.26 0.15 0.16 41  0.22 0.10 0.12 55  0.16 0.06 0.09 63 
Hughes-Richardson 0.24 0.14 0.15 42  0.24 0.12 0.24 48  0.21 0.10 0.11 52  0.19 0.07 0.09 63 
Method Trial and results 
ANRS§ 049a Ditrame§  CDC§ Retro-Ci, Abidjan  NIH,§ Nairobi  UNAIDS§ Petra 
Placebo point estimate¶ Zidovudine point estimate¶ Placebo CI§ Relative efficacy point estimate¶  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate¶ Formula feeding point estimate¶ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate¶ Placebo CI Relative efficacy point estimate 
Scenario 1                    
Proportion positive 0.27 0.20 0.15 25  0.22 0.13 0.15 42  0.25 0.13 0.15 49  0.17 0.06 0.09 65 
Scenario 2                    
Kaplan-Meier exact  0.29 0.21 0.15 29  0.24 0.13 0.16 46  0.28 0.17 0.15 39  0.18 0.06 0.09 67 
Turnbull exact 0.29 0.26 0.15 10  0.38 0.22 0.31 42  0.31 0.16 0.22 48  0.16 0.05 0.27 69 
Hughes-Richardson 0.29 0.25 0.15 14  0.29 0.14 0.18 52  0.27 0.18 0.14 34  0.48 0.12 0.62 75 
Scenario 3                    
Kaplan-Meier exact  0.23 0.15 0.12 34  0.22 0.12 0.15 46  0.21 0.09 0.12 58  0.15 0.05 0.08 67 
Turnbull exact 0.24 0.16 0.12 36  0.26 0.15 0.16 41  0.21 0.08 0.19 62  0.17 0.06 0.20 65 
Turnbull competing 0.24 0.15 0.12 38  0.26 0.15 0.16 41  0.22 0.10 0.12 55  0.16 0.06 0.09 63 
Hughes-Richardson 0.24 0.14 0.15 42  0.24 0.12 0.24 48  0.21 0.10 0.11 52  0.19 0.07 0.09 63 

* Estimate at age of last positive test in the 6-week interval from birth.

† For definitions of scenarios, see Materials and Methods.

‡ Comparison of statistical methods using human immunodeficiency virus infection as the outcome.

§ ANRS, Agence Nationale de Recherches sur le SIDA (France); Ditrame, Diminution de la Transmission Mere-Enfant; CDC, Centers for Disease Control and Prevention; NIH, National Institutes of Health; UNAIDS, United Nations AIDS (acquired immunodeficiency syndrome) agency; CI, total width of 95% confidence interval of point estimate.

¶ Point estimate of infection risk.

TABLE 4.

Estimates of long-term efficacy, using human immunodeficiency virus infection as the outcome, in four randomized clinical trials in African breastfeeding populations, 1995–2000*

Method Trial and results 
ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo point estimate‡ Zidovudine point estimate‡ Placebo CI† Relative efficacy point estimate‡  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate‡ Formula feeding point estimate‡ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate‡ Placebo CI Relative efficacy point estimate 
Scenario 3                    
Kaplan-Meier exact  0.32 0.22 0.14 31  0.30 0.23 0.17 22  0.33 0.17 0.15 49  0.23 0.16 0.09 30 
Turnbull exact 0.33 0.24 0.14 27  0.30 0.23 0.17 23  0.33 0.17 0.14 48  0.22 0.15 0.14 32 
Turnbull competing 0.36 0.30 0.20 21  0.30 0.23 0.17 23  0.34 0.18 0.14 45  0.23 0.15 0.09 35 
Hughes-Richardson 0.34 0.26 0.13 24  0.30 0.22 0.27 25  0.34 0.19 0.14 44  0.26 0.19 0.11 27 
Scenario 4                    
Proportion positive 0.19 0.09 0.16 51  0.17 0.09 0.17 45  0.27 0.16 0.23 41  0.13 0.10 0.09 23 
Method Trial and results 
ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo point estimate‡ Zidovudine point estimate‡ Placebo CI† Relative efficacy point estimate‡  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate‡ Formula feeding point estimate‡ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate‡ Placebo CI Relative efficacy point estimate 
Scenario 3                    
Kaplan-Meier exact  0.32 0.22 0.14 31  0.30 0.23 0.17 22  0.33 0.17 0.15 49  0.23 0.16 0.09 30 
Turnbull exact 0.33 0.24 0.14 27  0.30 0.23 0.17 23  0.33 0.17 0.14 48  0.22 0.15 0.14 32 
Turnbull competing 0.36 0.30 0.20 21  0.30 0.23 0.17 23  0.34 0.18 0.14 45  0.23 0.15 0.09 35 
Hughes-Richardson 0.34 0.26 0.13 24  0.30 0.22 0.27 25  0.34 0.19 0.14 44  0.26 0.19 0.11 27 
Scenario 4                    
Proportion positive 0.19 0.09 0.16 51  0.17 0.09 0.17 45  0.27 0.16 0.23 41  0.13 0.10 0.09 23 

* For definitions of scenarios, see Materials and Methods.

† ANRS, Agence Nationale de Recherches sur le SIDA (France); Ditrame, Diminution de la Transmission Mere-Enfant; CDC, Centers for Disease Control and Prevention; NIH, National Institutes of Health; UNAIDS, United Nations AIDS (acquired immunodeficiency syndrome) agency; CI, total width of 95% confidence interval of point estimate.

‡ Point estimate of cumulative infection risk.

TABLE 5.

Estimates of long-term efficacy, using human immunodeficiency virus infection or death as the outcome, in four randomized clinical trials in African breastfeeding populations, 1995–2000*

Method Trial and results 
ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo point estimate‡ Zidovudine point estimate‡ Placebo CI† Relative efficacy point estimate‡  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate‡ Formula feeding point estimate‡ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate‡ Placebo CI Relative efficacy point estimate 
Scenario 3                    
Kaplan-Meier exact  0.38 0.27 0.17 29  0.38 0.25 0.17 35  0.43 0.30 0.16 30  0.27 0.20 0.10 26 
Turnbull exact 0.38 0.28 0.15 27  0.38 0.23 0.17 44  0.42 0.29 0.15 31  0.26 0.19 0.14 27 
Hughes-Richardson 0.37 0.28 0.13 24  0.37 0.23 0.16 39   0.38 0.27 0.14 29  0.28 0.23 0.13 22 
Scenario 4                    
0.46 0.30 0.17 33  0.41 0.19 0.19 54  0.56 0.43 0.20 23  0.28 0.20 0.11 29 
0.30 0.21 0.13 30  0.31 0.12 0.16 59  0.29 0.23 0.12 20  0.22 0.16 0.09 27 
Method Trial and results 
ANRS† 049a Ditrame†  CDC† Retro-Ci, Abidjan  NIH,† Nairobi  UNAIDS† Petra 
Placebo point estimate‡ Zidovudine point estimate‡ Placebo CI† Relative efficacy point estimate‡  Placebo point estimate Zidovudine point estimate Placebo CI Relative efficacy point estimate  Breastfeeding point estimate‡ Formula feeding point estimate‡ Breastfeeding CI Relative efficacy point estimate  Placebo point estimate Zidovudine and lamivudine (long arm) point estimate‡ Placebo CI Relative efficacy point estimate 
Scenario 3                    
Kaplan-Meier exact  0.38 0.27 0.17 29  0.38 0.25 0.17 35  0.43 0.30 0.16 30  0.27 0.20 0.10 26 
Turnbull exact 0.38 0.28 0.15 27  0.38 0.23 0.17 44  0.42 0.29 0.15 31  0.26 0.19 0.14 27 
Hughes-Richardson 0.37 0.28 0.13 24  0.37 0.23 0.16 39   0.38 0.27 0.14 29  0.28 0.23 0.13 22 
Scenario 4                    
0.46 0.30 0.17 33  0.41 0.19 0.19 54  0.56 0.43 0.20 23  0.28 0.20 0.11 29 
0.30 0.21 0.13 30  0.31 0.12 0.16 59  0.29 0.23 0.12 20  0.22 0.16 0.09 27 

* For definitions of scenarios, see Materials and Methods.

† ANRS, Agence Nationale de Recherches sur le SIDA (France); Ditrame, Diminution de la Transmission Mere-Enfant; CDC, Centers for Disease Control and Prevention; NIH, National Institutes of Health; UNAIDS, United Nations AIDS (acquired immunodeficiency syndrome) agency; CI, total width of 95% confidence interval of point estimate.

‡ Point estimate of cumulative infection risk.

APPENDIX TABLE 1.

Estimated cumultive incidence at time 4

Situation Analysis 1* Analysis 2† Analysis 3‡ 
(n3 + n2n3/(n3 + n4))/N (n3 + (n1 + n2)n3/(n3 + n4))/N (n3 + n2n3/(n1 + n3 + n4))/N 
(n3 + n1n3/(n2 + n3 + n4))/N (n3 + (n1 + n2)n3/(n3 + n4))/N (n3 + n1n3/(n1 + n3 + n4))/N 
Situation Analysis 1* Analysis 2† Analysis 3‡ 
(n3 + n2n3/(n3 + n4))/N (n3 + (n1 + n2)n3/(n3 + n4))/N (n3 + n2n3/(n1 + n3 + n4))/N 
(n3 + n1n3/(n2 + n3 + n4))/N (n3 + (n1 + n2)n3/(n3 + n4))/N (n3 + n1n3/(n1 + n3 + n4))/N 

* Competing risks analysis.

† Standard Kaplan-Meier analysis.

‡ Standard Kaplan-Meier analysis when the last human immunodeficiency virus test for weaned children is at time 4 (continued testing of weaned children).

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