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Sample size calculation for active-arm trial with counterfactual incidence based on recency assay

Fei Gao, David V. Glidden, James P. Hughes and Deborah J. Donnell

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.


Corresponding author: Fei Gao, Vaccine and Infectious Disease Division, Fred Hutchinson Cancer Research Center, Seattle, WA, USA, E-mail:

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 β ̂ T and Ω ̂ T are the estimated false recency rate and MDRI for the recency assay.

Write W = ( N R N + β ̂ T , N + , Ω ̂ T β ̂ T T , N event , N , enorll ) T . Then, the estimators λ ̂ 0 in (1) and λ ̂ 1 in (2) can be written as

λ ̂ 0 = W 1 ( N W 2 ) W 3

and

λ ̂ 1 = W 4 τ W 5 ,

where W k is the kth element of W for k = 1, …, 5. Therefore, by the delta method, the asymptotic variance of log R ̂ = log λ ̂ 1 log λ ̂ 0 can be written as d T var(W)d, where

d = 1 E W 1 , 1 E N W 2 , 1 E W 3 , 1 E W 4 , 1 E W 5 T = 1 N p ( P R β T ) , 1 N ( 1 p ) , 1 Ω T β T T , 1 N ( 1 p ) r τ λ 1 , 1 N ( 1 p ) r T .

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

P R = β T + λ 0 ( 1 p ) p ( Ω T β T T ) .

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

v a r ( W 1 ) = N p P R ( 1 P R ) + ( 1 p ) ( P R β T ) 2 + σ β ̂ T 2 ( 1 p + N p ) v a r ( W 2 ) = N p ( 1 p ) v a r ( W 3 ) = σ Ω ̂ T 2 + σ β ̂ T 2 T 2 v a r ( W 4 ) = N ( 1 p ) r λ 1 τ { 1 + λ 1 τ p r + λ 1 τ ( 1 r ) } v a r ( W 5 ) = N ( 1 p ) r ( 1 r + p r ) c o v ( W 1 , W 2 ) = N p ( 1 p ) ( P R β T ) c o v ( W 1 , W 3 ) = N p σ β ̂ T 2 T c o v ( W 1 , W 4 ) = N p ( 1 p ) ( P R β T ) r λ 1 τ c o v ( W 1 , W 5 ) = N p ( 1 p ) ( P R β T ) r c o v ( W 2 , W 3 ) = 0 c o v ( W 2 , W 4 ) = N p ( 1 p ) r λ 1 τ c o v ( W 2 , W 5 ) = N p ( 1 p ) r c o v ( W 3 , W 4 ) = 0 c o v ( W 3 , W 5 ) = 0 c o v ( W 4 , W 5 ) = N ( 1 p ) r ( 1 r + p r ) λ 1 τ .

Then, the asymptotic variance of log R ̂ is given by V 0 + V 1 + CV, where

V 0 = d 1 2 var ( W 1 ) + d 3 2 var ( W 2 ) + d 4 2 var ( W 3 ) + 2 d 1 d 3 c o v ( W 1 , W 2 ) + 2 d 1 d 4 c o v ( W 1 , W 3 )

is the asymptotic variance of log λ ̂ 0 ,

V 1 = d 5 2 var ( W 4 ) + d 6 2 var ( W 5 ) + 2 d 5 d 6 c o v ( W 4 , W 5 )

is the asymptotic variance of log λ ̂ 1 , and

C V = 2 d 1 d 5 c o v ( W 1 , W 4 ) + 2 d 1 d 6 c o v ( W 1 , W 5 ) + 2 d 3 d 5 c o v ( W 2 , W 4 ) + 2 d 3 d 6 c o v ( W 2 , W 5 )

is the asymptotic covariance of log λ ̂ 0 and log λ ̂ 1 . Note that

V 0 = P R ( 1 P R ) + ( 1 p ) ( P R β T ) 2 + σ β ̂ T 2 ( 1 p + N p ) N p ( P R β T ) 2 + p N ( 1 p ) + σ Ω ̂ T 2 + σ β ̂ T 2 T 2 Ω T β T T 2 + 2 N 2 σ β ̂ T 2 T ( P R β T ) ( Ω T β T T ) = 1 N p P R 1 P R P R β T 2 + 1 ( 1 p ) + ( 1 p ) σ β ̂ T 2 P R β T 2 + σ Ω ̂ T 2 Ω T β T T 2 + σ β ̂ T 2 Ω T P R T P R β T Ω T β T T 2 , V 1 = 1 + λ 1 τ p r + λ 1 τ ( 1 r ) N ( 1 p ) r λ 1 τ + 1 r + p r N ( 1 p ) r 2 ( 1 r + p r ) N ( 1 p ) r = 1 N ( 1 p ) r λ 1 τ ,

and

CV = 2 N 2 N + 2 p N ( 1 p ) 2 p N ( 1 p ) = 0 .

That is, log λ 0 and log λ 1 have asymptotic covariance zero and the asymptotic variance of R ̂ is given in V 0 + V 1. Particularly, the variance of log λ ̂ 0 can be estimated by

V ̂ 0 = N R N + N R N + N R N + β ̂ T 2 + N N + N + σ ̂ β ̂ T 2 N + ( N N + ) N N R N + β ̂ T 2 + σ ̂ Ω ̂ T 2 Ω ̂ T β ̂ T T 2 + σ ̂ β ̂ T 2 N + Ω ̂ T N R T N R N + β ̂ T Ω ̂ T β ̂ T T 2 ,

and the variance of log λ ̂ 1 can be estimated by

V ̂ 1 = 1 N event .

In a special case when β T =0 and σ β ̂ T 2 = 0 , i.e., the false recent probability for the recency test is zero, the variance estimator of log R ̂ is given by

1 N R + 1 N + 1 N event + σ ̂ Ω ̂ T 2 Ω ̂ T 2 .

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 σ Ω ̂ T 2 = σ β ̂ T 2 = 0 . Without considering variability associated with the recency assay properties, we will show that the asymptotic variance of Z is a constant (with respect to N) that departs from 1 under alternative hypothesis.

Note that W = ( N R N + β ̂ T , N + , Ω ̂ T β ̂ T T , N event , N , enorll ) T . In the special case with σ Ω ̂ T 2 = σ β ̂ T 2 = 0 , there is no variability associated with β ̂ T and Ω ̂ T , such that β ̂ T = β T and Ω ̂ T = Ω T . Write W 6 = N R and W * = ( W 1 , W 2 , W 4 , W 5 , W 6 ) T . Then, the test statistic is given by

Z = A B ,

where

A = log λ ̂ 1 log λ ̂ 0 log R 0 = log ( W 1 ) + log ( N W 2 ) + log ( Ω T β T T ) + log W 4 log W 5 log τ log R 0 , B = N R N + N R N + N R N + β T 2 + N N + N + 1 N event = W 6 W 2 W 6 W 2 W 1 2 + 1 W 2 + 1 N W 2 + 1 W 4 .

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

A ̃ = log E W 1 + log { N E W 2 } + log ( Ω T β T T ) + log E W 4 log E W 5 log τ log R 0 = log λ 1 log λ 0 log R 0 B ̃ = E W 6 E W 2 E W 6 E W 2 E W 1 2 + 1 E W 2 + 1 N E W 2 + 1 E W 4 = 1 N P R ( 1 P R ) p ( P R β T ) 2 + 1 p ( 1 p ) + 1 ( 1 p ) r λ 1 τ .

We apply the delta method to find the asymptotic mean of Z is given by A ̃ / B ̃ , and the asymptotic variance of Z is given by d Z T var ( W * ) d Z , where

d Z = 1 B ̃ 1 E W 1 , 1 E N W 2 , 1 E W 4 , 1 E W 5 , 0 T A ̃ 2 B ̃ 3 / 2 2 E W 6 E W 2 E W 6 E W 2 E W 1 3 , E W 6 2 E W 2 2 E W 1 2 1 E W 2 2 1 E N W 2 2 , 1 E W 4 2 , 0 , E W 2 2 E W 6 E W 2 E W 1 2 T = 1 N B ̃ d Z 1 A ̃ 2 N 2 B ̃ 3 / 2 d Z 2 .

where

d Z 1 = 1 p ( P R β T ) , 1 1 p , 1 ( 1 p ) r τ λ 1 , 1 ( 1 p ) r , 0 T

and

d Z 2 = 2 P R ( 1 P R ) p 2 ( P R β T ) 3 , P R 2 p 2 ( P R β T ) 2 p 2 + ( 1 p ) 2 p 2 ( 1 p ) 2 , 1 ( 1 p ) 2 r 2 λ 1 2 τ 2 , 0 , 1 2 P R p 2 ( P R β T ) 2 T .

Since B ̃ is the variance of A, we have d Z 1 T var ( W * ) d Z 1 / N 2 = B ̃ and

var ( Z ) = d Z 1 T var ( W * ) d Z 1 N 2 B ̃ A ̃ d Z 1 T var ( W * ) d Z 2 N 3 B ̃ 2 + A ̃ 2 d Z 2 T var ( W * ) d Z 2 4 N 4 B ̃ 3 = 1 + A ̃ 4 ( N B ̃ ) 3 A ̃ d Z 2 4 ( N B ̃ ) d Z 1 T var ( W * ) N d Z 2 .

Note that A ̃ is a constant related to the relationship of 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 A ̃ = 0 , 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, A ̃ = log R 1 log R 0 , and

E ( Z ) = log R 1 log R 0 V 0 ( λ 0 ) + V 1 ( R 1 λ 0 ) .

Note that var(W*) is proportional to N and B ̃ is proportional to 1/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

V R 1 = d R 1 T V W d R 1 ,

where

d R 1 = 1 B ̃ R 1 d Z 1 log R 1 log R 0 2 B ̃ R 1 3 / 2 d Z 2 , B ̃ R 1 = P R ( 1 P R ) p ( P R β T ) 2 + 1 p ( 1 p ) + 1 ( 1 p ) r λ 0 R 1 τ ,

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

var ( W 6 ) = N p P R ( 1 p P R ) , c o v ( W 1 , W 6 ) = N p P R ( 1 P R ) + N p ( 1 p ) ( P R β T ) P R , c o v ( W 2 , W 6 ) = N p ( 1 p ) P R , c o v ( W 4 , W 6 ) = N p ( 1 p ) P R r λ 1 τ , c o v ( W 5 , W 6 ) = N p ( 1 p ) P R r .

V R 1 is the calculated asymptotic variance of Z under alternative hypothesis R=R 1, in the special case with σ Ω ̂ T 2 = σ β ̂ T 2 = 0 . When the variabilities of β ̂ T and Ω ̂ T cannot be ignored, V R 1 serves as an approximation of the asymptotic variance of Z.

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Received: 2021-04-26
Revised: 2021-09-27
Accepted: 2021-09-30
Published Online: 2021-11-10

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