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Appendix
Word count: 502
Calculating HIV incidence and multivariate HIV incidence rate ratio using
the BED assay results
Pascale LISSOUBAa*, Dirk TALJAARDb, Dino RECHb, Veerle DERMAUX-MSIMANGc, Camille
LEGEAIa, David LEWIS c,d, Beverley SINGHc, Adrian PUREN c,d and Bertran AUVERTa,e,f*
BED window period and long-term specificity
Let CBED be the BED cut-off, W the BED window period in months and ρ2 the long-term specificity.
We define the relationship between CBED and W as follows: When following up a group of HIVnegative individuals participating in a study for a period of 2W before testing, CBED is selected such
that the number of tested recent seroconverters is equal to the number of participants who became
HIV-positive in the period W before testing. Using the McDougal et al. terminology [1], this definition
implies that the sensitivity is equal to the short-term specificity. Using another relationship between
CBED and W, McWalter et al. have obtained sensitivity equals to short-term specificity plus one minus
long-term specificity [2].
From the Orange Farm male circumcision data [3], we found a window period of 185 days (95%CI
146 to 227) when using a cut-off of 0.80 for BED. We also calculated CBED for W varying from 3
months to 18 months in 3-month increments. The results indicated that W and ρ2 vary linearly with
CBED according to the following equations: W =8.0914 CBED and ρ2=1-0.05748 CBED.
Estimation of HIV incidence rate
For HIV negative participants at baseline followed up during the window period, which is W/12 in
years, the HIV incidence rate (IR) in person-years can be estimated by dividing the number of
2
participants becoming HIV infected during W by the total duration of participants’ follow-up. The
follow-up is W for participants remaining HIV-negative. Assuming a uniform distribution of HIV
infections during W, it can be estimated by W/2 for those becoming HIV-positive. It follows that the
IR in person-years can be estimated by:
IR 
NR
in which N- is the number of participants who are HIV negative at the end of
W NR W
N
12
2 12
the follow-up period and NR is the real number of those infected by HIV during W. Taking into
account the misclassifications, NR is calculated by
N TR  N  (1  ρ 2 )
, in which N+ is the number of
(2ρ 2  1)
HIV positive participants and NTR the number of those tested recent seroconverters by BED [1, 4]. It
follows that:
N TR  N  (1  ρ 2 )
(2ρ 2  1)
IR 
(Formula 1)
W N TR  N  (1  ρ 2 ) 1 W
N
12
(2ρ 2  1)
2 12
Estimation of HIV incidence rate ratio
Using Formula 1, we obtain for the incidence rate ratio (IRR) between a group A and a group B, the
following formula:




 N TR  N  (1  ρ 2 ) 

N TR  N  (1  ρ 2 ) 
 N- 

2(2ρ 2  1)

 Group A
IRR 
(Formula 2)




 N TR  N  (1  ρ 2 ) 

N TR  N  (1  ρ 2 ) 
 N- 

2(2ρ 2  1)

 Group B
Formula 2 does not depend on W but assumes that W and ρ2 are identical for groups A and B.
3
The participants’ HIV statuses were observed at the end of a period equal to W. These data were
modeled using an exponential proportional hazards model in which the baseline hazard is considered
constant during W. This theoretical model was implemented by running a Poisson log-linear model to
obtain IR and IRR with or without adjustment for cofactors [5-7]. As indicated by Formula 1, an offset
of log(
W
W
) has to be used for those tested recent seroconverters and of log( ) for HIV-negative
12
24
participants. Because
k
1
2ρ 2  1

N TR  N  (1  ρ 2 )
can be written as kN TR with
(2ρ 2  1)
N  (1  ρ 2 )
, prior weights of k have to be used for those tested recent seroconverters
N TR (2ρ 2 - 1)
and of 1 for those HIV-negative.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
McDougal JS, Parekh BS, Peterson ML, Branson BM, Dobbs T, Ackers M, et al.
Comparison of HIV type 1 incidence observed during longitudinal follow-up with
incidence estimated by cross-sectional analysis using the BED capture enzyme
immunoassay. AIDS Res Hum Retroviruses 2006,22:945-952.
McWalter TA, Welte A. A comparison of biomarker based incidence estimators. PLoS
One 2009,4:e7368.
Fiamma A, Lissouba P, Oliver A, Singh B, Laeyendecker O, Quinn T, et al. Can HIV
incidence testing be used for evaluating HIV intervention programs? A reanalysis of
the Orange Farm male circumcision trial. BMC Infectious Diseases 2010,10:137.
McDougal JS. BED estimates of HIV incidence must be adjusted. Aids 2009,23:20642065; author reply 2066-2068.
Frome EL. The analysis of rates using Poisson regression models. Biometrics
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Berry G. The analysis of mortality by the subject-years method. Biometrics
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Holford TR. The analysis of rates and of survivorship using log-linear models.
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