# Abstract

## Objectives

The past decade has seen tremendous progress in the development of biomedical agents that are effective as pre-exposure prophylaxis (PrEP) for HIV prevention. To expand the choice of products and delivery methods, new medications and delivery methods are under development. Future trials of non-inferiority, given the high efficacy of ARV-based PrEP products as they become current or future standard of care, would require a large number of participants and long follow-up time that may not be feasible. This motivates the construction of a counterfactual estimate that approximates incidence for a randomized concurrent control group receiving no PrEP.

## Methods

We propose an approach that is to enroll a cohort of prospective PrEP users and aug-ment screening for HIV with laboratory markers of duration of HIV infection to indicate recent infections. We discuss the assumptions under which these data would yield an estimate of the counterfactual HIV incidence and develop sample size and power calculations for comparisons to incidence observed on an investigational PrEP agent.

## Results

We consider two hypothetical trials for men who have sex with men (MSM) and transgender women (TGW) from different regions and young women in sub-Saharan Africa. The calculated sample sizes are reasonable and yield desirable power in simulation studies.

## Conclusions

Future one-arm trials with counterfactual placebo incidence based on a recency assay can be conducted with reasonable total screening sample sizes and adequate power to determine treatment efficacy.

**Funding source: **National Institutes of Health

**Award Identifier / Grant number: **AI029168

**Award Identifier / Grant number: **AI143357

**Award Identifier / Grant number: **AI143418

**Award Identifier / Grant number: **UM1A1068617

## Appendix A: Derivation of asymptotic variances

Note that *N*, *N*
_{+}, and *N*
_{−} are the numbers of total screened, HIV-positive, and HIV-negative subjects, *p* is HIV prevalence, *r* is the proportion of HIV-negative subjects enrolled to the trial, *N*
_{−,enroll} is the number of HIV-negative subjects enrolled to the trial, and *τ* is the follow-up time, and

Write

and

where *W*
_{
k
} is the *k*th element of *W* for *k* = 1, …, 5. Therefore, by the delta method, the asymptotic variance of
*d*
^{T} var(*W*)*d*, where

Note that *N*
_{+}∼Bin(*N*, *p*), *N*
_{−}=*N* − *N*
_{+},and *N*
_{−,enroll}∼Bin(*N*
_{−}, *r*). The number of test-recent subjects *N*
_{R} can be viewed as from Bin(*N*
_{+}, *P*
_{
R
}), where

The number of incidence cases *N*
_{event} is from Poisson(*N*
_{−,enroll}
*τλ*
_{1}). Then, calculation yields

Then, the asymptotic variance of
*V*
_{0} + *V*
_{1} + *CV*, where

is the asymptotic variance of

is the asymptotic variance of

is the asymptotic covariance of

and

That is, log *λ*
_{0} and log *λ*
_{1} have asymptotic covariance zero and the asymptotic variance of
*V*
_{0} + *V*
_{1}. Particularly, the variance of

and the variance of

In a special case when *β*
_{
T
}=0 and

That is, the variance of the estimated incidence ratio is driven by the numbers of observed events and the variability of MDRI of the recency test.

## Appendix B: Derivation of asymptotic distribution of *Z* under alternatives

In this section, we calculate the asymptotic distribution of *Z* under alternative hypothesis *R* = *R*
_{1}. Particularly, we consider the derivation under a simplified case with
*Z* is a constant (with respect to *N*) that departs from 1 under alternative hypothesis.

Note that
*W*
_{6} = *N*
_{R} and

where

We would like to apply the delta method with respect to *W** to calculate the distribution of *Z*.

Replacing *W*
_{
j
} (*j*=1, 2, 4, 5, 6) by their expectations in the definitions of *A* and *B*, we denote

We apply the delta method to find the asymptotic mean of *Z* is given by
*Z* is given by

where

and

Since
*A*, we have

Note that
*λ*
_{0} and *λ*
_{1}. Particularly, if *λ*
_{1}/*λ*
_{0}=*R*
_{0}, i.e., the true relationship of *λ*
_{0} and *λ*
_{1} follows from the null hypothesis, then
*E*(*Z*)=0, and var(*Z*)=1. When *λ*
_{1}/*λ*
_{0}=*R*
_{1}, i.e., the true relationship of *λ*
_{0} and *λ*
_{1} follows from the alternative hypothesis,

Note that var(*W**) is proportional to *N* and
*N*. Then, the second term of the last expression is a constant with respect to *N*. When this constant is non-zero, the asymptotic variance of *Z* departs from 1. Particularly, the asymptotic variance of *Z* under alternative hypothesis *R*=*R*
_{1} is given by the formula

where

and *V*
_{
W
} is the covariance matrix of *W** divided by *N* (which does not depend on *N*). Note that to calculate *V*
_{
W
}, we make use of the covariance matrix of *W* calculated in Appendix A and

*Z* under alternative hypothesis *R*=*R*
_{1}, in the special case with
*Z*.

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**Received:**2021-04-26

**Revised:**2021-09-27

**Accepted:**2021-09-30

**Published Online:**2021-11-10

© 2021 Walter de Gruyter GmbH, Berlin/Boston