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AIDS:
doi: 10.1097/QAD.0b013e3282f2a960
Epidemiology and Social

Improved HIV-1 incidence estimates using the BED capture enzyme immunoassay

Hargrove, John Wa,f; Humphrey, Jean Ha,c; Mutasa, Kudaa; Parekh, Bharat Sd; McDougal, J Steved; Ntozini, Roberta; Chidawanyika, Henrya; Moulton, Lawrence Hc; Ward, Briane; Nathoo, Kusumb; Iliff, Peter Ja; Kopp, Ekkehardf

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Author Information

From the aZVITAMBO Project

bUniversity of Zimbabwe Faculties of Medicine and Science, Harare, Zimbabwe

cJohns Hopkins Bloomberg School of Public Health, Baltimore, Maryland, USA

dCenters for Disease Control and Prevention, Atlanta, Georgia, USA

eResearch Institute of the Montreal General Hospital, Montreal, Quebec, Canada

fDST/NRF Centre of Excellence in Epidemiological Modelling and Analysis, SACEMA, Stellenbosch, South Africa.

Received 2 March, 2007

Revised 12 July, 2007

Accepted 18 July, 2007

Correspondence to Dr J. Humphrey, ZVITAMBO, 1 Borrowdale Road, Borrowdale, Harare, Zimbabwe. E-mail: jhumphrey@zvitambo.co.zw

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Abstract

Objective: To validate the BED capture enzyme immunoassay for HIV-1 subtype C and to derive adjustments facilitating estimation of HIV-1 incidence from cross-sectional surveys.

Design: Laboratory analysis of archived plasma samples collected in Zimbabwe.

Methods: Serial plasma samples from 85 women who seroconverted to HIV-1 during the postpartum year were assayed by BED and used to estimate the window period between seroconversion and the attainment of a specified BED absorbance. HIV-1 incidences for the year prior to recruitment and for the postpartum year were calculated by applying the BED technique to HIV-1-positive samples collected at baseline and at 12 months.

Results: The mean window for an absorbance cut-off of 0.8 was 187 days. Among women who were HIV-1 positive at baseline and retested at 12 months, a proportion (&epsiv;) 5.2% (142/2749) had a BED absorbance < 0.8 at 12 months and were falsely identified as recent seroconverters. Consequently, the estimated BED annual incidence at 12 months postpartum (7.6%) was 2.2 times the contemporary prospective estimate. BED incidence adjusted for &epsiv; was 3.5% [95% confidence interval (CI), 2.6–4.5], close to the 3.4% estimated prospectively. Adjusted BED incidence at baseline was 6.0% (95% CI, 5.2–6.9) and, like the prospective estimates, declined with maternal age. Unadjusted BED incidence estimates were largely independent of age; the pooled estimate was 58% higher than adjusted incidence.

Conclusion: The BED method can be used in an African setting, but further estimates of &epsiv; and of the window period are required, using large samples in a variety of circumstances, before its general utility can be gauged.

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Introduction

While HIV-1 prevalence estimates are the most commonly available surveillance data, HIV-1 incidence provides a more useful estimate of epidemiological trends, a more sensitive indicator for evaluating the impact of interventions, and a more accurate prediction of the number of infected people in a population who will progress to AIDS and require treatment. Incidence data are also required prior to trials aimed at prevention to provide a baseline measure and to estimate sample size requirements. However, incidence data are rarely available because their collection requires difficult, protracted, and expensive follow-up of large cohorts.

As an alternative, laboratory assays can distinguish recent from long-term HIV-1 infections from changes in antibody characteristics after seroconversion [1]. In a cross-sectional prevalence survey, the numbers recently infected, and HIV-1 negative, can be used to estimate HIV-1 incidence. One such method, the BED capture enzyme immunoassay (BED) technique, developed by the Centers for Disease Control and Prevention (CDC), is based on the increasing proportion of anti-HIV-1 IgG in total IgG following seroconversion [2]. People are classified as ‘recent’ seroconverters if their blood samples test positive by a standard HIV-1 enzyme-linked immunosorbent assay and have a normalized absorbance below a given cut-off on the BED assay.

BED enables HIV-1 incidence estimation from cross-sectional surveys [2,3] and it has been used in the United States in case-based surveillance systems [4,5]. However, several studies in Africa and Asia, where additional clinical and epidemiological information is not available, produced BED estimates of HIV-1 incidence two to three times higher than those measured prospectively or derived from prevalence surveys [6]. Consequently, UNAIDS recommended in December 2005 that, pending further research on the assay, the BED technique should not be used during routine HIV-1 surveillance for estimating absolute incidence or monitoring trends [6].

This study used archived plasma samples from postpartum women enrolled in the Zimbabwe Vitamin A for Mothers and Babies Trial (ZVITAMBO) conducted in 1997 to 2001, to calibrate the BED assay for the first time among people infected with the clade C HIV-1 virus predominant in southern Africa. The BED assay was then used to estimate HIV-1 incidence over the first 12 months postpartum and to compare this estimate with the prospectively measured incidence over the same period. An analytic adjustment was developed that brings the BED-derived estimate close to the cohort estimate.

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Methods

Data source

This project used archived plasma specimens and data collected during the ZVITAMBO trial [7–10]. Briefly, 14 110 postpartum mothers were recruited within 96 h of delivery between November 1997 and January 2000 and followed at 6 weeks, 3 months and at 3-monthly intervals thereafter, for at least 1 year.

The ZVITAMBO trial protocol was approved by the Medical Research Council of Zimbabwe (MRC-Z), the Medicines Control Authority of Zimbabwe, the Johns Hopkins Bloomberg School of Public Health Committee on Human Research (CHR), and the Montreal General Hospital Ethics Committee (MGHEC). BED analysis of archived specimens was approved by MRC-Z, CHR, MGHEC, and the CDC Program Ethics Review Board.

All available HIV-1-positive samples for the following groups were analysed by BED: (i) samples from 4437 of the 4495 baseline HIV-1-positive women (99%); (ii) of the 353/9562 women who seroconverted during follow-up, all HIV-1-positive samples from 97 women who provided at least three such samples; (iii) samples from 2749/3010 women (91%) who were HIV-1 positive at baseline and who were also seen at the 12 month postpartum visit; (iv) 226 (97%) samples of the 234 women who were baseline HIV-1 negative and who seroconverted by (and provided seropositive samples at) the 12 month visit. Samples from 53 women HIV-1 indeterminate at baseline were excluded from this analysis.

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Window estimation

Linear mixed effects modelling of the absorbance values for panels from seroconverting women were used to estimate the mean period Ω′ for the BED absorbance to increase from the level typical of HIV-1-negative clients (at time tN) to a predefined absorbance cut-off (C; at time tC): Ω′ = tCtN [2]. The period Ω′ has been identified in the past with the BED ‘window period’ (Ω): this is the period between seroconversion (at time tS) and tC. Since, however, there is no measurable increase in BED absorbance for ∼ 25 days after seroconversion (BS Parekh, unpublished data), Ω′ underestimates Ω; therefore, the true window was provisionally taken as Ω = Ω′ + 25 days.

The reliability of the regressions used to estimate Ω′ increases with the number (S) of samples available per client, but the numbers of clients providing a minimum of n samples declines with increasing n, as does the total number of samples in the regression. To investigate the effect of data selection on the estimated window for given C, separate regressions were applied to the data where S ≥ 3, ≥ 4, or ≥ 5. For a given client's data to be included in the analysis for given C, the sample absorbances had to span the range above and below C. Data were analyzed using Stata 8.0 (Stata, College Station, Texas, USA) and Microsoft Excel.

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Incidence estimators

Follow-up estimates (ÎF) of incidence (I) from the ZVITAMBO study were expressed as the cumulative incidence or risk that an HIV-1-negative person seroconverts in a 12 month period [9]; all incidence estimates in this study are expressed in this way.

When the BED analysis was applied to HIV-1-seropositive ZVITAMBO samples, women whose sample had an absorbance < 0.8 were classified as ‘recent’ infections and HIV-1 incidence estimated from:

Equation (Uncited)
Equation (Uncited)
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with a 95% confidence interval (CI) of Î0 ± 1.96Î0/√R and where ϖ is Ω/365, R is the number of recent infections and N the number HIV-1 negative [11].

This estimator can be adjusted for the sensitivity and specificity of the BED test:

Equation (Uncited)
Equation (Uncited)
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where f = [(R/P) + ρ2 − 1]/[(R/P)(σ − ρ1 + 2ρ2 − 1], P is the total HIV-1 positive, σ is the sensitivity of the BED test, ρ1 is the specificity of the BED test over the period [Ω, 2Ω], and ρ2 is the specificity of the BED over all times > 2Ω [11].

If σ is ≈ ρ1 and ϖ is 0.5, then rearranging Equation (2), setting &epsiv; = 1 − ρ2, gives the simpler equation:

Equation (Uncited)
Equation (Uncited)
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where T = P + N. The estimators Î0, ÎF, ÎI, and ÎII are compared below. The variance for ÎII is derived in the Appendix (below); no equivalent variance estimate is available for ÎI.

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Results

Selection of data used for windows estimation

As in previous studies, cases were excluded, for the purposes of estimating the window only, if the minimum observed absorbance was greater than the chosen cut-off C [2]; this excluded 12 of the 97 plasma panels from seroconverting women (Table 1). Various subsamples of the remaining 85 panels were used to investigate the effect of data selection on window estimation.

Table 1
Table 1
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Distribution of individual window periods

Individual window periods were estimated for 41 seroconverting women each providing ≥ 4 HIV-1-positive samples, with the time between the last negative and the first positive HIV-1 test < 100 days (Table 1). While 35/41 (85%) of estimated window periods were < 365 days, all of the remaining windows were > 405 days and 4/41 (10%) had windows > 525 days (Fig. 1). Note that the window was taken as Ω = Ω′ + 25 days.

Fig. 1
Fig. 1
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Estimating the mean window period

In estimating the mean time (Ω′) taken for the BED absorbance to increase to C, clients such as those in the tail of Fig. 1 are generally excluded because their maximum observed absorbance was < C [2]. This approach is appropriate, and has been followed here, if trying to estimate the mean value of Ω′ for the main peak in Fig. 1. However, cases typical of the tail will be present in cross-sectional surveys, though not individually identifiable. These cases are misclassified as recent seroconverters and are the principal cause of HIV-1 incidence being overestimated by the BED method (see below).

The estimate of Ω′ increased linearly with C (Fig. 2a) but varied little with S. While Ω̂′ was up to 6.4% greater for an S value of 4 than for one of 3, requiring S to be 5 resulted in a reduction in Ω̂′. Window estimates are inherently less reliable for large values of the time (t0) between the last negative and first positive sample, because of the increasing uncertainty of the time of seroconversion, and for small t0, because of the decreasing number of qualifying clients. However, for t0 of 75–175 days (with S = 3 and C = 0.8), Ω̂′ varied only between 159 and 163, despite the number of clients varying by a factor of three (Fig. 2b). For C = 0.8, and S = 3 or 4, Ω̂′ = 162 days (95% CI, 148 − 179) with ϖ′ = Ω′/365 = 0.444. Given the delay between seroconversion and the onset of the increase in BED absorbance, the window (Ω) is estimated as Ω̂′ = 162 + 25 = 187 days (ϖ = 0.512).

Equation (Uncited)
Equation (Uncited)
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Fig. 2
Fig. 2
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BED incidence estimates over the first 12 months postpartum
Equation (Uncited)
Equation (Uncited)
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Of 9589 women who were HIV-1 negative at baseline, 6829 provided an unequivocal HIV-1 test 12 months later; 6595 tested negative (N) and 234 positive (P). Of the latter, 123 produced a BED absorbance level < 0.8 and were classified as recent infections (R). Using these data in Equation (1) gave an unadjusted HIV-1 BED incidence estimate of 3.5% (95% CI, 2.9–4.2) close to the 3.4% (95% CI, 3.0–3.8) measured prospectively [9]. This result, based on subjects who were HIV-1 positive for ≤ 1 year, involved no error relating to long-term specificity (ρ2) and is consistent with an approximate cancellation of false-positive and false-negative BED results over the period [0, 2Ω]: that is, for ϖ ≈ 0.5, sensitivity and short-term specificity were similar (σρ1), as reported previously [11].

Equation (Uncited)
Equation (Uncited)
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When, however, the baseline-positive samples were added to the above analysis, the essential problem with the BED method was revealed. The new input data were P = 3244 and R = 279. As before, N = 6595 and ϖ = 0.512, from which Î0 was 7.6% (95% CI, 6.7–8.5), which is 2.2 times the follow-up value. Differences between the two estimates arise because a proportion (&epsiv;) of women [142/2749; 5.2% (95%CI, 4.4–6.1)] were seropositive at recruitment into ZVITAMBO and had a 12 month BED absorbance < 0.8; they were, therefore, misclassified as recent seroconverters at least a year after they first tested HIV-1 positive. Of the 142 women, BED absorbance was < 0.8 at baseline in 103 and > 0.8 in 39. When incidence was estimated using Equation (3), setting &epsiv; = 0.052, the misclassified women were removed from the analysis and the estimated incidence was naturally the same as for the seroconverting women: ÎII = 3.5% (95% CI, 2.6–4.5).

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BED incidence estimates over the 12 months prior to recruitment

At baseline, with N = 9562, P = 4495, R = 517 and Ω = 187 days, Î0 = 9.5% (95% CI, 8.7–10.3) while the adjusted estimates ÎI = 5.5% (95% CI, 5.0–6.0) and ÎII = 6.0% (95% CI, 5.2–6.9) did not differ significantly from each other but were 42% and 37%, respectively, lower than Î0. Differences between adjusted and unadjusted BED estimates of incidence were independent of the time of recruitment into the ZVITAMBO trial, but both showed a well marked peak in the middle of the trial period (Fig. 3).

Age–incidence functions differed significantly between estimation methods. Î0 was > 9% per annum for all women < 35 years of age, whereas ÎF, ÎI, and ÎII all declined sharply with age (Fig. 4). The error in Î0 is a function of the proportion of recent infections (R/P), which took values 0.22, 0.13, 0.080, 0.073, and 0.074 for age groups < 20, 20–24, 25–29, 30–34 and ≥ 35 years. Simple manipulations of Equations (1) and (3) show that the error in Î0 increases from 0, when all infections are recent, to a value that increases with prevalence when incidence, and thus R/P, is low. The differences are not a result of age-related changes in &epsiv;; from misclassifications in each of the above age groups we find &epsiv; values of 5.5% (95% CI, 3.3–8.6), 5.5% (95% CI, 4.2–7.1), 4.2% (95% CI, 3.0–5.8), 6.2% (95% CI, 3.6–9.1) and 5.0% (95% CI, 2.2–9.6), respectively, showing no trend with age.

The absence of an effect of age on &epsiv; is consistent with the observation that, for mothers who were HIV-1 positive at baseline and who had CD4 cell counts in the grouped ranges < 200, 200–399, 400–599, 600–799, and ≥ 800 cells/μl, the estimated values of &epsiv; were 4.5% (95% CI, 2.5–7.3), 3.0% (95% CI, 1.9–4.4), 6.1% (95% CI, 4.5–8.1), 6.5% (95% CI, 4.1–9.8), and 5.4% (95% CI, 2.7–9.4), respectively, showing no consistent change, and certainly no increase, in &epsiv; with declining CD4 cell count.

For 56 women who tested HIV positive at baseline and at 12 months and who were known to have died within the next 12 months, three had a BED absorbance < 0.8 at 12 months [&epsiv; = 5.4% (95% CI, 1.1–14.9)]. Among those not known to have died after 12 months the corresponding proportion was similar, 139/263 [&epsiv; = 5.2% (95% CI, 4.4–6.1)].

Equation (Uncited)
Equation (Uncited)
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Fig. 3
Fig. 3
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Sensitivity of adjusted HIV incidence estimates to assumed values of &epsiv;
Equation (Uncited)
Equation (Uncited)
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The value of ÎII declined with increasing &epsiv; at a rate that increased with HIV prevalence (Fig. 5a). For the ZVITAMBO baseline data (prevalence 32.0%), the value of ÎII declined more rapidly with increasing &epsiv; than for data from Masaka, Uganda (prevalence 11.4%) [12]. If the value of &epsiv; increases such that &epsiv;P is > R then ÎII is < 0. For sufficient further increases in &epsiv;, the denominator in Equation (3) tends to 0 and there is a discontinuity in the function (Fig. 5b). For &epsiv; > (R+ωN)/T, both numerator and denominator are negative and ÎII is > 0. All values of &epsiv; where &epsiv;P is > R are invalid since they imply that the number of false recent cases exceeds the total number of false, plus true, recent cases.

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Discussion

Equation (Uncited)
Equation (Uncited)
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Fig. 4
Fig. 4
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Whereas the BED test has been used successfully in the United States [4,5,11], recent publications question whether it can provide reliable estimates of HIV incidence in Africa [12,13]. The present study shows that the method can be so used, if the value of &epsiv; is accurately and appropriately estimated. Moreover, our results, for postpartum heterosexual African women infected with clade C virus in Zimbabwe, showed remarkable similarities with those from the VAX004 cohort [11], where most clients were homosexual American and European white males infected with clade B virus.

Equation (Uncited)
Equation (Uncited)
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Inclusion of increasing numbers of long-term HIV-1-infected cases in the VAX analysis resulted in increasing overestimates in the unadjusted BED incidence levels [11]. As with the ZVITAMBO situation, it was shown that the overestimates resulted from the inclusion of increasing numbers of long-term infections that were still testing as recent by BED. The rate at which these occurred was nearly identical in the two studies: &epsiv; = 5.0% (95% CI, 1.9–10.6) (6/120) in the VAX study and &epsiv; = 5.2% (95% CI, 4.4–6.1) (142/2749) in ZVITAMBO; application of suitable adjustments resulted in accurate incidence estimates in both cases. There is, therefore, no reason to suggest that the situation in Africa and the United States differs in any qualitative way.

Equation (Uncited)
Equation (Uncited)
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Further research is nonetheless required before the BED method can be used with confidence in general settings. The root cause of overestimates of HIV incidence is the BED's low ‘long-term’ specificity, as measured by &epsiv;. If further studies show that &epsiv; is so variable that it has to be measured for every cross-sectional sample, the BED test would be of little general use.

Equation (Uncited)
Equation (Uncited)
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ZVITAMBO is the only study in Africa where this variability has been investigated with large samples; we found little change in &epsiv; with age, CD4 cell count, or mortality, suggesting a weak association between &epsiv; and disease stage. The current investigation involves, however, only postpartum women from one locality where HIV-1 infection is overwhelmingly clade C. Further studies are needed to see if the apparent stability with age and disease stage holds in different situations, for example in males, nonpregnant women, with different viral clades and coinfection with a variety of clades.

Such studies need to be based on large sample sizes. Recent estimates of &epsiv;, of 0.304 for clade C virus in Zambia and 0.571 for clade A1 in Rwanda, were 6 and 11 times higher, respectively, than our estimate [12]. The sample sizes (23 and 14), however, were < 1% of ours and the 95% CI values were consequently large (0.132–0.529 and 0.289–0.823, respectively). Application of either estimate to the BED data from the study generates impossible incidence estimates (Fig. 5) reflecting inaccuracy in the estimates of &epsiv; rather than any inherent problem in the BED method. Accordingly, for data from Masaka, Uganda, the authors used the ZVITAMBO estimate of &epsiv; of 0.052 and found an adjusted BED HIV-1 incidence over three times higher than follow-up estimates [12]. They concluded that the BED test does not perform reliably in all populations; but their use of a value of &epsiv; not derived from their own region and small sample sizes (and consequently huge CI values for their own estimates of &epsiv;) scarcely provided a fair test.

Apparent overestimates in BED-adjusted incidence from Kenya and Côte d'Ivoire may similarly indicate that the true value of &epsiv; for these situations was greater than the value assumed in the absence of local estimates [14]. These results underline the importance of estimating &epsiv; – with large samples – in a variety of circumstances; and of obtaining for comparison, wherever possible, follow-up incidence estimates. The value of &epsiv; can be measured by applying the BED test to samples of clients known to be HIV-1 positive for more than 1 year and calculating the proportion of these who still test as recent using an appropriate absorbance cut-off.

Estimated incidence varies with the value of the window. Increasing its value from Ω′ = 162 to Ω = 187 days decreased the baseline incidence ÎII by 15%. This difference, the absence of an agreed method for estimating Ω [2,12,13], and resulting uncertainty about the appropriate window in different circumstances, all underline the need for further work to standardize the application of the method and, particularly, to investigate and interpret regional differences in BED-adjusted incidence estimates.

Finally, if adjusted incidence is estimated by ÎI rather than ÎII, we need to derive the appropriate variance for ÎI analogous to the one derived here for ÎII and to consider possible local variability in σ and ρ1 as well as in ρ2 = 1 − &epsiv;. While recent BED estimates of HIV-1 incidence for South Africa appear reasonable, adjustments were carried out using values of σ, ρ1, ρ2, and Ω derived from different sources outside South Africa [15]. Problems experienced elsewhere in Africa underline the importance of using good locally derived parameters [12–14].

Fig. 5
Fig. 5
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Since incidence was not directly measured in ZVITAMBO for the year prior to parturition, no ‘gold standard’ exists for comparison with BED incidence estimates at baseline. The value of ÎII for the year prior to parturition (6.0%) was 1.7 times the adjusted BED incidence estimated for the postpartum year (3.5%) among women in ZVITAMBO, consistent with findings that HIV-1 incidence was 1.8 times higher among pregnant women than in those lactating or not pregnant [16]. The difference might also reflect a cohort effect; people recruited into a trial often show thereafter lower HIV incidence than was typical among the population screened prior to recruitment (see references in McDougal et al. [11]).

Strong support for the proposed analytic adjustment comes from age-stratified incidence estimates (Fig. 4). Unadjusted BED estimates of incidence are not only substantially higher than follow-up estimates but are also age independent for women < 35 years; follow-up and adjusted estimates declined with age as expected.

Previous overestimates of incidence arising from the use of the BED assay are consistent with the widespread existence of a subgroup of people with unusually long window periods. Discussion of the reasons for the existence of the aberrant subgroup is beyond the scope of the present study but is clearly of wider interest. In attempting to estimate HIV incidence using the BED method, it will always be necessary to apply a compensatory procedure to account for this subgroup. This may be achieved in general settings by applying a mathematical adjustment, as suggested here, or in clinical settings, by removing from the analysis people testing BED-recent known to be true long-term infections, including those known to be receiving antiretroviral therapy.

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Acknowledgements

We are grateful to Brian Williams and Eleanor Gouws for extensive advice on data analysis and the production of this paper and to Frances Cowan, Peter Ghys, Richard Hayes, Meade Morgan and Basia Zaba for critical comments on the manuscript.

Sponsorship: The ZVITAMBO project was supported by the Canadian International Development Agency (CIDA) (R/C Project 690/M3688), United States Agency for International Development (USAID) (cooperative agreement number HRN-A-00-97-00015-00 between Johns Hopkins University and the Office of Health and Nutrition - USAID) and a grant from the Bill and Melinda Gates Foundation (Seattle Washington State). Additional funding was received from the Rockefeller Foundation (New York) and BASF (Ludwigshafen Germany).

Disclaimer: The findings and conclusions of this paper are those of the authors and do not necessarily represent the views of the funding agencies.

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References

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4. Nesheim S, Parekh B, Sullivan K, Bulterys M, Dobbs T, Lindsay M, et al. Temporal trends in HIV-1 incidence among inner-city childbearing women in Atlanta: use of the IgG-Capture BED-Enzyme Immunoassay. AIDS Res Hum Retroviruses 2005; 21:537–544.

5. CDC. Using the BED HIV-1 Capture EIA Assay to Estimate Incidence Using STARHS in the Context of Surveillance in the US. Centers for Disease Control and Prevention, Division of HIV/AIDS Prevention. Atlanta, Georgia [CDC] updated October 2007. http://www.cdc.gov/hiv/topics/surveillance/resources/factsheets/BED.htm.

6. UNAIDS Reference Group for Estimates, Modelling and Projections. Statement on the use of the BED-assay for the estimation of HIV-1 incidence for surveillance or epidemic monitoring. Report of a meeting of the UNAIDS Reference Group for Estimates, Modelling and Projections. Athens, Greece, December 13th–15th 2005. Geneva: UNAIDS; 2005. http://www.epidem.org/Publications/BED%20statement.pdf.

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8. Humphrey JH, Iliff PJ, Marinda E, Mutasa K, Moulton LH, Nathoo KJ, et al. Effects of a single large dose of vitamin A, given during the postpartum period to HIV-positive women and their infants, on child HIV infection, HIV-free survival, and mortality. J Infect Dis 2006; 193:860–871.

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10. Zvandasara P, Hargrove JW, Ntozini R, Chidawanyika H, Iliff PJ, Moulton LH, et al. Mortality and morbidity among postpartum HIV-positive and HIV-negative women in Zimbabwe: risk factors, causes, and impact of single-dose postpartum vitamin A supplementation. JAIDS 2006; 43:107–116.

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Appendix
Variance of the estimator ÎII

Let R′ be the number of those who test positive but are not labelled as ‘recent’ according to the BED assay algorithm, so that, R′ + R = P. Then given a fixed sample size T, we have T = N + R′ + R and (N, R′, R) follows a multinomial distribution. Let X = R&epsiv;P and Y = R +ϖN&epsiv;T, so that ÎII = X/Y. Using the ‘delta’ method,

Now,

which we estimate by:

where N̂, R̂′, R̂ are the realized values of the trinomial components.

Similarly,

Equation (Uncited)
Equation (Uncited)
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estimated by:

Equation (Uncited)
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Also,

Equation (Uncited)
Equation (Uncited)
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which, after multiplying and combining terms, becomes:

Equation (Uncited)
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estimated by:

Equation (Uncited)
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Estimating μX and μY by

Equation (Uncited)
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and

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respectively, and combining with the variances and covariances into (10), we obtain Var

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Equation (Uncited)
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. An Excel spreadsheet containing these formulas is available upon request.

We have tested this variance estimator in a series of simulations of size 100 000 each. Each trinomial response is simulated by generation of two binomial variates, and a 95% confidence interval is calculated as

Equation (Uncited)
Equation (Uncited)
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The results are given in TABLE A1.

Equation (Uncited)
Equation (Uncited)
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Back to Top | Article Outline
Keywords:

BED; HIV-1; incidence

Equation (Uncited)
Equation (Uncited)
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Equation (Uncited)
Equation (Uncited)
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Table 1
Table 1
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© 2008 Lippincott Williams & Wilkins, Inc.

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