Thoughts on Simplifying the
Estimation of HIV Incidence
John Hargrove, Alex Welte, Paul Mostert [and others]
Estimates of incident (new) cases are
important in the assessment of changes in an
epidemic, identifying “hot spots” and in
gauging the effects of interventions
HIV incidence most accurately estimated via
longitudinal studies – but these are lengthy,
expensive, logistically challenging.
Do provide a “gold standard” against which to
judge other estimates of HIV incidence
An alternative way of estimating incidence,
involving none of the disadvantages of a
longitudinal study, would be to use a single
chemical test that can be used to estimate
the proportions of recent vs longestablished HIV infections in crosssectional surveys
Idea: identify HIV test where measured
outcome not simply +/- but rather a graded
response increasing steadily over a long period
One such assay is the
BED-CEIA developed
by CDC
2.5
Graph shows result for
a seroconverting client
taken from the
ZVITAMBO study
carried out in
Zimbabwe
BED-CEIA Assay
Case 23903G
Normalised OD
2.0
1.5
1.0
0.5
0.0
0
100
200
300
400
500
Days since last negative
600
700
[14,110 post partum
women followed up at
6-wk, 3-mo, then every
3-mo to two years]
1.6
Theoretical graph of sqrt(OD-n) vs ln(ti, j)
1.2
Selected OD cut-off (B)
Square root of OD-n
0.8
0.4
Negative baseline (A)
.
.
.
.
.
Window (Wi )
0.0
-0.4
Slope = b1,i
-0.8
-1.2
Intercept = b0,i
-1.6
0
1
2
3
4
5
Log time (ti, j days) since last negative
6
7
The idea is to calibrate the BED assay to
estimate the “average” time [or “window”]
taken for a person’s BED optical density [OD]
to increase to a given OD cutoff
In cross-sectional surveys proportion of HIV
positive people with BED < cut-off allows us to
calculate the proportion of new infections –
and thus the incidence.
Estimation of the window period is thus central
to the successful application of the BED
Data from commercial seroconversion panels with
accurately known times of seroconversion indicate
Problem 1.
Delay (~25 days)
between seroconversion and
the onset of
then increase in
BED optical
density
0.9
Window period (W)
0.8
W'
0.7
Date of
seroconversion
Observed OD
Fitted line
0.6
BED ODn
Extrapolated portion
0.5
Seronegative
Seropositive
0.4
0.3
Date of
infection
0.2
Extrapolated time when OD = baseline
D1
0.1
D2
Baseline OD = 0.0476
0.0
-80
-60
-40
-20
0
20
40
60
80
100
Days since BED OD started to increase
120
140
160
180
Min < 0.8; max > 0.8; S > 2; t < 90
BED Optical Density
3.0
2.5
2.0
1.5
1.0
0.5
0.0
0
200
400
600
Days since last negative
800
1000
Problem 2:
Considerable
variability
between clients
in a real
population. No
prospect of
using BED to
identify
individual
recent
infections.
Idea only to
estimate
population
incidence
Problem 3: Often have limited follow-up: of 353
seroconverters in ZVITAMBO, 167 only
produced a single HIV positive sample,
Samples per
client (S)
Frequency
1
2
3
4
5
167 89 35 21 24
6
7
8
8
8
1
Problem 4: The available data for a given
client quite often do not span the OD cutoff. The proportion that fail to do so varies
with the chosen cut-off. Failure to span
increases the uncertainty in estimating the
time at which the OD cut-off is crossed
Problem 5: There is a large variation (27 – 656
days) in the time (t0) elapsing between last
negative and first positive HIV tests. The
degree of uncertainty in the timing of
seroconversion increases with increasing t0
2.5
2.0
1.5
1.0
2.0
1.5
1.0
0.5
0.0
0.0
200
3.0
400
600
Days since last negative
800
0
1000
3.0
BED Optical Density
2.5
2.0
1.5
1.0
1000
800
1000
S = 2; t < 90
1.0
0.0
600
800
1.5
0.0
400
600
2.0
0.5
Days since last negative
400
2.5
0.5
200
200
Days since last negative
Min > 0.8; S > 2; t < 90
0
Max < 0.8; S > 2; t < 90
2.5
0.5
0
BED Optical Density
3.0
Min < 0.8; max > 0.8; S > 2; t < 90
BED Optical Density
BED Optical Density
3.0
0
200
400
600
Days since last negative
800
1000
3.0
Min < 0.8; Max > 0.8; S > 2; 90 <= t
<120
2.5
BED Optical Density
BED Optical Density
3.0
2.0
1.5
1.0
2.5
2.0
1.5
1.0
0.5
0.5
0.0
0.0
0
200
400
600
800
Max < 0.8; S > 2; 90 <= t < 120
0
1000
200
Days since last negative
3.0
Min > 0.8; S > 2; 90 <= t < 120
2.5
BED Optical Density
BED Optical Density
3.0
400
600
800
1000
Days since last negative
2.0
1.5
1.0
0.5
S = 2; 90 <= t < 120
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0
200
400
600
Days since last negative
800
1000
0
200
400
600
Days since last negative
800
1000
3.0
Min < 0.8; Max > 0.8; S > 2; t >=120
2.5
BED Optical Density
BED Optical Density
3.0
2.0
1.5
1.0
2.5
2.0
1.5
1.0
0.5
0.5
0.0
0.0
0
200
400
600
800
Max < 0.8; S > 2; t >= 120
0
1000
200
3.0
Min > 0.8; S > 2; t >= 120
2.5
BED Optical Density
BED Optical Density
3.0
400
600
800
1000
Days since last negative
Days since last negative
2.0
1.5
1.0
0.5
S = 2; 120 < t < 182
2.5
2.0
1.5
1.0
0.5
0.0
0.0
0
200
400
600
Days since last negative
800
1000
0
200
400
600
Days since last negative
800
1000
S = 2; t >= 183
We need to consider
how variation in samples
per client, t0 , and
failure to span the cutoff affect our estimate
of the window period.
3.0
BED Optical Density
2.5
2.0
1.5
1.0
0.5
0.0
0
200
400
600
Days since last negative
800
1000
3.0
BED Optical density
2.5
How to approach problem?
Scatter-plot of the data?
2.0
Makes no use of the
information of the trend
for individual clients and
ignores the fact that the
sequential points for that
client are not independent.
1.5
1.0
0.5
0.0
0
100 200
300 400 500 600 700 800 900 1000
Time since seroconversion
Alternative which uses
trend in BED OD is
suggested by an
approximately linear
relationship between
square root of OD and
time-since-lastnegative HIV test (t).
A.
1.6
Square root of OD values
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
3.5
4.0
4.5
5.0
5.5
6.0
6.5
loge days since last negative test
7.0
Allows a regression
approach taking out
variance due to t and to
difference between
clients
Gives consistent results; in that results independent of
whether we insist on minimum of 3, 4 or 5 samples per
client; and on value of t0 between 75 and 180 days
290
210
270
200
250
230
Window (days)
Window (days)
190
210
190
170
180
170
13
21
53
55
60
160
68
150
150
49
Minimum 3
Minimum 4
130
140
Minimum 5
110
0.65
130
0.75
0.85
0.95
OD Cut-off
1.05
1.15
40
60
80
100 120 140 160 180 200
Maximum days last negative to first positive
Are we even using the right transformation?
And should we be using the time of last negative
HIV test as the origin
2.4
2.0
Optical density
Try instead to do a
preliminary estimate
of the time when OD
starts to increase
by fitting a
quadratic polynomial
to the data. Then
use this estimate as
the origin.
1.6
1.2
0.8
0.4
0.0
0
100 200 300 400 500 600 700 800
Days since last negative
1
A.
1.6
C.
B.
1.6
0
1.4
1.4
1.2
1.2
1.0
0.8
0.6
loge OD values
Square root of OD values
Square root of OD values
-1
1.0
0.8
0.6
-2
11445X
14557A
15513K
-3
15801X
16715D
16853F
-4
17926A
18101N
0.4
0.4
-5
0.2
0.2
-6
0.0
0.0
-7
20606K
20674F
21556F
3.5
4.0 4.5 5.0 5.5 6.0 6.5
loge days since last negative test
7.0
0
1
2
3
4
5
6
loge estimated days since seroconversion
7
23903G
23983A
0
1
2
3
4
5
6
loge estimated days since seroconversion
Seems to suggest that the true relationship may
actually be a power function.
What it really were? What would we see if we
plotted OD vs time since-last negative
7
Our problem is that we do not know when seroconversion occurred.
We only know the time of the last HIV negative test.
1.0
Optical density
0.8
0.6
0.4
0.2
Examples of times
when HIV -ve tests
might have been taken
True window
173 days
0.0
-160
-80
0
80
160
240
Days since function intersects baseline level
And the greater the delay between
last negative and first positive
tests the greater the uncertainty
1.6
1.6
Offset = 100 days
1.4
1.4
1.2
1.2
square root (OD)
square root (OD)
Offset = 0 days
1.0
0.8
0.6
0.4
0.8
0.6
0.4
y = 0.334x - 0.768
R2 = 0.976
Window = 126 d
0.2
1.0
y = 0.53x - 2.08
R2 = 1.00
Window = 196 d
0.2
0.0
0.0
3
4
5
6
log e (days since last negative)
7
3
4
5
6
log e (days since last negative)
For zero offset the window is UNDER-estimated;
for 100-day offset it is OVER-estimated
7
Estimated window
220
200
True window period
180
160
140
120
0
40
80
Offset (days)
120
160
This approach to window estimation is clearly not
optimal since the window estimate changes with
the timing of the last HIV-negative test
But can we do any better?
If OD increases as a power function fit:
OD  a (t  t 0 )
b
or equivalently
ln( OD)  ln( a)  b ln( t  t 0 )
where a and b are constants, t is the time since
the last negative and t0 is
the estimated time of seroconversion.
We use the data to estimate a, b and t0 by nonlinear regression
For the generated data [without noise] this
approach gives the correct window – regardless
of the time of the last negative test
But for real data in 40% of 61 cases the time of
seroconversion was estimated to be before the
time of the last negative test or after the time
of the first positive.
[Work in progress]
Turnbull survival analysis different approach suggested
by Paul Mostert (Stellenbosch Statistics Department).
This is a slightly more sophisticated variant of the
Kaplan Meier survival analysis. Works on the basis that
the (unknown) times of:
i) seroconversion
ii) OD cut-off
each lie between two known times
The times of the two events are quantified using
interval censoring
0.8
Turnbull window
estimates
Runs
0.2
0.4
0.6
All data (red; 183 d)
2: Excluding max OD <
0.8 (purple; 141 d)
3: Excluding min OD >
0.8 (green; 210 d)
4: Excluding 2 and 3
(blue; 163 d)
0.0
estimated exceeding probability
1.0
Estimation of HIV w indow period since SC using Turnbull's algorithm
0
100
200
window period (days)
300
400
The window length is estimated using a non-parametric
survival technique which makes no assumptions about
any parametric models and underlying distributions. .
No interpolation is used to obtain the cut-off time
where the BED OD reaches 0.8 or the seroconversion
time point. Only time points that will define the
interval boundaries were used, which means that time
points more than four for a specific women were not
fully utilised. However, time points as few as two per
women could be used in this estimation of window
length.
Conclusion
There is still no general agreement on how best to
estimate the window for methods like the BED.
Fortunately most of those described seem to give
fairly similar answers – though it’s not clear to
what extent this is happening by chance.