The authors have declared that no competing interests exist.
Conceived and designed the experiments: JH. Analyzed the data: JH HE CVS. Contributed reagents/materials/analysis tools: JH HE GM CVS. Wrote the paper: JH CVS.
BED estimates of HIV incidence from crosssectional surveys are obtained by restricting, to fixed time
The BED Capture Enzyme ImmunoAssay (BEDCEIA or simply BED) measures the increasing proportion of antiHIV1 IgG in total IgG following HIV seroconversion
In practice, however, application of the BED method has resulted in overestimates of HIV incidence
The situation has clarified recently, however, with the demonstration that it is neither necessary nor desirable to estimate the mean recency duration over the whole life of a patient
In so doing we investigate whether there is an optimum way of estimating the mean recency duration or whether several estimating procedures provide similar answers and whether, then, simple approaches will provide adequate answers. We also ask how estimates of the mean recency and incidence are affected by our choice of cutoff and whether these effects differ with our choice of estimation method.
Since all of the methods investigated below have been used previously in the literature, we do not in general attempt to provide formal statistical justification for their use, except where we have suggested modifications to the methods. Instead we contrast the resulting estimates in terms of their means and variances under different sets of input conditions, and then discuss under what conditions there could be reasons for preferring some estimators over others.
Mean recency duration was estimated from data produced during the Zimbabwe Vitamin A for Mothers and Babies (ZVITAMBO) Trial, in Harare, Zimbabwe. All details regarding the study design, data collection and ethical clearance have been described previously
Inclusion criterion  
BED samples per case  All  
1  167  – 
2  89  28 
3  35  25 
4  21  17 
5  24  17 
6  8  5 
7  8  7 
8  1  1 
Total  353  100 
Briefly, between October 1997 and January 2000, 14,110 women and their babies were recruited within 96 hours of giving birth, tested for HIV at recruitment and at followup visits at 6weeks, and 3, 6, 9, 12 …. 24months. All available HIV positive samples from seroconverting mothers and from mothers who tested HIV positive at baseline, or at the 12month visit, were tested by BED: subsets of these data were used to estimate mean recency duration. The time distribution of seroconversions during the first 12months postpartum was also used to estimate HIV incidence
Predicted values obtained from: A. LMM. Linear regression of the square root of OD values against log time (
For BED data obtained from the analysis of crosssectional survey data, two independent derivations
Fitting the nonlinear function given by
where
For predefined time
Mean recency durations (with 95% confidence intervals) estimated using: A. Nonlinear mixed modeling (NLMM); linear mixed modeling (LMM); the proportion of recent infections among seroconverters tested at one year postpartum (
The graph shows a scatter plot of all BED OD values obtained from seroconverting women from the ZVITAMBO study, where the time between the last negative and first positive HIV tests did not exceed 120 days and where the woman provided at least four HIV positive samples. Horizontal line marks a preset OD cutoff of 0.8; vertical lines mark a preset cutoff of
For cases that are HIV negative at time 0 and tested again at time
Method  Mean recency duration(95% CI) (days)  Coefficient of variation (%) 
i. NLMM  196 (188–204)  2.0 
ii. LMM  191 (174–208)  4.7 
iii. Survival analysis  192 (168–216)  6.4 
iv. Ratio 
192 (168–216)  6.4 
v. Graphical  193  – 
The optical density cutoff was fixed at 0.8 for all methods, minimum of two samples per case were required and the maximum allowable time between the last negative and first positive HIV tests was 120 days.
Notice that
NS 

Mean recency (95% CrI) (days)  CoV(%)  
80  2  32  176 (165–187)  3.2 
80  3  27  179 (166–191)  3.5 
80  4  23  193 (179–207)  3.7 
120  2  100  196 (188–204)  2.0 
120  3  71  199 (191–208)  2.2 
120  4  47  194 (183–205)  2.9 
160  2  109  193 (185–200)  2.0 
160  3  78  196 (188–204)  2.1 
160  4  51  192 (182–202)  2.6 
The optical density cutoff was fixed at 0.8 for all estimates. The coefficient of variation (CoV) is defined as the standard error of the estimate divided by the mean, expressed as a percentage.
It has been argued that a good estimate for the mean recency duration will ensure equality between BED and followup estimates of incidence (
with variance given by:
The value of ε was estimated as the proportion of cases with a BED OD<
HIV incidence (with 95% confidence intervals) among women during their first year postpartum in the ZVITAMBO Trial, calculated using estimates of the mean recency duration from nonlinear mixed modeling (NLMM), linear mixed modeling (LMM), survival analysis (SA) and graphical analysis (Graph).
Transformation of the unbalanced longitudinal data produces a linear mean structure and allows, by solving for
HIV incidence (with 95% confidence intervals) in women during the period prior to their recruitment into the ZVITAMBO Trial. A. Nonlinear mixed modeling (NLMM); linear mixed modeling (LMM); survival analysis (SA). B. The proportion of recent infections among seroconverters tested at one year postpartum (
where
The coefficient of variation for BED HIV incidence estimates obtained using the ZVITAMBO baseline data, as a function of the preset optical density cutoff (
(NLMM)
where
We investigated a variant of this function:
where
Assuming no underlying parametric model for the recency duration, the SA approach is followed when recognizing the data as being double interval censored. The exact times of seroconversion and of reaching a predefined OD cutoff are not known, but intervals for each are obtained from the data. This creates an interval of the shortest and longest possible recency durations for each individual. Sweeting
They also noted that carrying out a univariate survival analysis of the double interval censored data, as if they were single interval censored, assumes an incorrect likelihood function. We consider an alternative approach where we approximate the time of seroconversion to be the midpoint between the times of the last negative and first positive HIV tests. Given that, for our data, the maximum time between these tests was set at 120 days (average 83 days), the average error in the assumed date of seroconversion should be small. The data are then single interval censored and Turnbull’s modification of the ProductLimit Estimator yields a survival function which, when integrated over [0,
For seroconverting cases that have been HIV positive for less than time
Data were analyzed using Microsoft Excel, R version 2.14.1
Of 14,110 women recruited, 9562, 4495 and 53 mothers tested HIV negative, positive and indeterminate, respectively. During followup, 353 of the initially HIV negative mothers were seen to seroconvert: the numbers of times that each of these cases was seen, and tested for HIV and for BED optical density (OD), are shown in
At 12months postpartum, 6829 of the baseline HIV negative cases were retested: 6595 still tested HIV negative and
The
Previously published survival analysis of the followup HIV test data provided an estimate of the probability (
Of the cases testing HIV negative at baseline, and then tested again at 12months,
Accordingly, further reports will contrast only
The likely form of the increase of the BED optical density with time since seroconversion can only be well judged from results for those individuals who are seen a number of times over an extended period.
BED data for all qualifying seroconverting cases were analyzed using LMM and NLMM. For the NLMM method, estimates of
For estimates at each
Whereas it was not possible to provide confidence intervals for this simple method the estimates of
At the commonly used cutoff of
Estimates of the mean recency duration were fairly insensitive to the way in which data were selected. When the minimum allowable number of samples per client was varied between 2 and 4, and
Whereas we have, for completeness, examined the way in which various estimators perform over a large range of
BED data from the ZVITAMBO Trial, for women testing HIV positive both at baseline and at 12months postpartum, were used to estimate ε. Data for these cases cannot therefore be used to obtain BED estimates of incidence over this period, since this incidence estimate, obtained from
BED estimates of the incidence over this period can, however, be obtained legitimately via
For all values of
Despite the differences in CoVs between the estimates of
The way in which our choice of C, and thus of
The contrasts between estimating methods seen in the postpartum incidence estimates are, as expected, largely repeated for the baseline analysis (
For the NLMM method, which produces estimates of
The consistent trend in the baseline HIV estimates with changes in
For
Estimates of
The
Our findings support previous work suggesting that the use of SA will be problematic for estimating
For values of the OD cutoff between
The choice of cutoff is then decided by a tradeoff of the advantages of increasing
Notwithstanding the results of the previous section, the NLMM estimates of baseline HIV incidence in
Thus, with reference to
The results in
On the above interpretation, the results in
We caution that the present study is based on the application of various methods to a single set of data, all derived from postpartum women, from a single city in Zimbabwe and all infected with a single clade of HIV. The results apply, moreover, only to the BED method. Similar studies are needed to establish the extent to which our results can be generalized in other settings and using other biomarker methods.
The estimation of the mean recency duration for cases which have been HIV positive for some defined finite period
The authors are grateful to Dr Jean Humphrey for ongoing access to the ZVITAMBO data, and to Dr Michael Sweeting for generously providing the code required for carrying out the NLMM analysis. We are very grateful to Gavin Hitchcock, Reshma Kassanjee and Alex Welte for critical comments on the manuscript and to two anonymous reviewers who caused us to substantially rethink our analysis and make significant improvements to the manuscript.