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 long-

established HIV infections in cross-sectional surveys

Idea: identify HIV test where measured outcome not simply +/- but rather a graded

response increasing steadily over a long period

BED-CEIA AssayCase 23903G

Days since last negative

0 100 200 300 400 500 600 700

Nor

mal

ised

O

D

0.0

0.5

1.0

1.5

2.0

2.5

One such assay is the BED-CEIA developed

by CDC

Graph shows result for a seroconverting client taken from the

ZVITAMBO study carried out in

Zimbabwe

[14,110 post partum women followed up at

6-wk, 3-mo, then every 3-mo to two

years]

Theoretical graph of sqrt(OD-n) vs ln(ti, j)

Log time (ti, j days) since last negative0 1 2 3 4 5 6 7

Sq

ua

re r

oo

t o

f O

D-n

-1.6

-1.2

-0.8

-0.4

0.0

0.4

0.8

1.2

1.6

Selected OD cut-off (B)

Negative baseline (A)

Slope = b1,i

Intercept = b0,i

..

..

.

Window (Wi )

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 withaccurately known times of seroconversion indicate

Problem 1.

Delay (~25 days) between sero-conversion and the onset

of then increase in BED optical

density

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-80 -60 -40 -20 0 20 40 60 80 100 120 140 160 180

Days since BED OD started to increase

BE

D O

Dn

Observed OD

Fitted line

Extrapolated portion

Sero-negative

Baseline OD = 0.0476

Extrapolated time when OD = baseline

Date ofseroconversion

Date ofinfection

2

'

1

Window period ()

Sero-positive

Min < 0.8; max > 0.8; S > 2; t < 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000Days since last negative

BED

Opt

ical

Den

sity

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)

1 2 3 4 5 6 7 8

Frequency 167 89 35 21

24 8 8 1

Problem 4: The available data for a given client quite often do not span the OD cut-

off. 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

Min < 0.8; max > 0.8; S > 2; t < 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ityMax < 0.8; S > 2; t < 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

Min > 0.8; S > 2; t < 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

S = 2; t < 90

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

Max < 0.8; S > 2; 90 <= t < 120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

ptic

al D

ensi

ty

Min > 0.8; S > 2; 90 <= t < 120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

ptic

al D

ensi

ty

S = 2; 90 <= t < 120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

ptic

al D

ensi

ty

Min < 0.8; Max > 0.8; S > 2; 90 <= t<120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

ptic

al D

ensi

ty

Max < 0.8; S > 2; t >= 120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

Min > 0.8; S > 2; t >= 120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

S = 2; 120 < t < 182

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

Min < 0.8; Max > 0.8; S > 2; t >=120

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

S = 2; t >= 183

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 200 400 600 800 1000

Days since last negative

BE

D O

pti

ca

l De

ns

ity

We need to consider how variation in

samples per client, t0 , and failure to span the

cut-off affect our estimate of the window period.

How to approach problem?

Scatter-plot of the data?

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.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 100 200 300 400 500 600 700 800 900 1000

Time since seroconversion

BE

D O

ptic

al d

ensi

ty

A.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

loge days since last negative test

Sq

ua

re r

oo

t o

f O

D v

alu

es

Alternative which uses trend in BED OD

is suggested by an approximately linear relationship between square root of OD and

time-since-last-negative HIV test (t).

Allows a regression approach taking out

variance due to t and to difference between

clients

110

130

150

170

190

210

230

250

270

290

0.65 0.75 0.85 0.95 1.05 1.15

OD Cut-off

Win

do

w (

da

ys)

Minimum 3

Minimum 4

Minimum 5

130

140

150

160

170

180

190

200

210

40 60 80 100 120 140 160 180 200Maximum days last negative to first positive

Win

do

w (

da

ys)

13

2149 53

55 60

68

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

0.0

0.4

0.8

1.2

1.6

2.0

2.4

0 100 200 300 400 500 600 700 800

Days since last negative

Opt

ical

den

sity

Are we even using the right transformation?And should we be using the time of last

negative HIV test as the origin

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.

A.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

loge days since last negative test

Squa

re ro

ot o

f OD

val

ues

C.

-7

-6

-5

-4

-3

-2

-1

0

1

0 1 2 3 4 5 6 7

loge estimated days since seroconversion

log

e O

D v

alue

s

11445X

14557A

15513K

15801X

16715D

16853F

17926A

18101N

20606K

20674F

21556F

23903G

23983A

B.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 1 2 3 4 5 6 7

loge estimated days since seroconversion

Squa

re ro

ot o

f OD

val

ues

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

Our problem is that we do not know when seroconversion occurred.We only know the time of the last HIV negative test.

And the greater the delay between last negative and first positive tests the greater the

uncertainty

True window173 days

0.0

0.2

0.4

0.6

0.8

1.0

-160 -80 0 80 160 240Days since function intersects baseline level

Opt

ical

den

sity

Examples of times when HIV -ve tests might have been taken

Offset = 0 days

y = 0.334x - 0.768

R2 = 0.976Window = 126 d

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

3 4 5 6 7log e (days since last negative)

squa

re r

oot (

OD

)Offset = 100 days

y = 0.53x - 2.08

R2 = 1.00Window = 196 d

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

3 4 5 6 7log e (days since last negative)

squa

re r

oot (

OD

)

For zero offset the window is UNDER-estimated; for 100-day offset it is OVER-

estimated

True window period

120

140

160

180

200

220

0 40 80 120 160Offset (days)

Est

ima

ted

win

do

w

This approach to window estimation is clearly not optimal since the window estimate changes with the timing of the last HIV-

negative testBut can we do any better?

If OD increases as a power function fit:

or equivalently

where a and b are constants, t is the time since the last negative and t0 is

the estimated time of seroconversion.

bttaOD )( 0

)ln()ln()ln( 0ttbaOD

We use the data to estimate a, b and t0 by non-linear 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

Estimation of HIV window period since SC using Turnbull's algorithm

window period (days)

estim

ated

exc

eedi

ng p

roba

bility

0 100 200 300 400

0.0

0.2

0.4

0.6

0.8

1.0 Turnbull window estimates RunsAll 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)

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.