Joint modeling of time-varying HIV exposure and infection for estimation of per-act efficacy in HIV prevention trials

Elizabeth R. Brown; Clara P. Dominguez Islas; Jingyang Zhang

doi:10.1515/scid-2019-0016

Published by De Gruyter September 24, 2020

Joint modeling of time-varying HIV exposure and infection for estimation of per-act efficacy in HIV prevention trials

Elizabeth R. Brown , Clara P. Dominguez Islas and Jingyang Zhang

From the journal Statistical Communications in Infectious Diseases

https://doi.org/10.1515/scid-2019-0016

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Abstract

Objectives: Using the MTN-020/ASPIRE HIV prevention trial as a motivating example, our objective is to construct a joint model for the HIV exposure process through vaginal intercourse and the time to HIV infection in a population of sexually active women. By modeling participants’ HIV infection in terms of exposures, rather than time exposed, our aim is to obtain a valid estimate of the per-act efficacy of a preventive intervention.Methods: Within the context of HIV prevention trials, in which the frequency of sex acts is self-reported periodically by the participants, we model the exposure process of the trial participants with a non-homogeneous Poisson process. This approach allows for variability in the rate of sexual contacts between participants as well as variability in the rate of sexual contacts over time. The time to HIV infection for each participant is modeled as the time to the exposure that results in HIV infection, based on the modeled sexual contact rate. We propose an empirical Bayes approach for estimation. Results: We report the results of a simulation study where we evaluate the performance of our proposed approach and compare it to the traditional approach of estimating the overall reduction in HIV incidence using a Proportional Hazards Cox model. The proposed approach is also illustrated with data from the MTN-020/ASPIRE trial. Conclusions: The proposed joint modeling, along with the proposed empirical Bayes estimation approach, can provide valid estimation of the per-exposure efficacy of a preventive intervention.

Keywords: empirical bayes estimation; HIV prevention trials; I-splines; non-homogeneous Poisson process; per-exposure efficacy

Corresponding author: Elizabeth R. Brown, Fred Hutchinson Cancer Reseasrch Center, 1100 Fairview Avenue North, M2-C200Seattle, WA, 98109-1024, USA. Phone: +1 206 667 1731, E-mail: erbrown@fredhutch.org

Funding source: National Institute of Allergy and Infectious Diseases

Award Identifier / Grant number: UM1AI068633

Award Identifier / Grant number: UM1AI068615

Award Identifier / Grant number: UM1AI106707

Funding source: Eunice Kennedy Shriver National Institute of Child Health and Human Development

Funding source: National Institute of Mental Health

Funding source: U.S. National Institutes of Health (NIH)

Acknowledgments

The ASPIRE study was designed and implemented by the Microbicide Trials Network (MTN). The MTN is funded by the National Institute of Allergy and Infectious Diseases through individual grants (UM1AI068633, UM1AI068615 and UM1AI106707), with co-funding from the Eunice Kennedy Shriver National Institute of Child Health and Human Development and the National Institute of Mental Health, all components of the U.S. National Institutes of Health (NIH). The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH. The vaginal rings used the ASPIRE study were supplied by the International Partnership for Microbicides.

Appendix

A.1 Derivation of expressions in section “Methods”

The density function fi(t|ni) is derived as follows:

(A.1)fi(t|ni)=−dSi(t)dt=−(−λi(t)e−μi(t)+∑j=1ni−11j!(−e−μi(t)μi(t)jλi(t)+e−μi(t)μi(t)j−1jλi(t)))=λi(t)(e−μi(t)+∑j=1ni−1e−μi(t)μi(t)jj!−∑j=1ni−1e−μi(t)μi(t)j−1(j−1)!)=λi(t)(∑j=0ni−1e−μi(t)μi(t)jj!−∑j=0ni−2e−μi(t)μi(t)jj!)=e−μi(t)μi(t)ni−1λi(t)(ni−1)!.

Expression in (8) and (9) are derived by integrating out the latent variable N_i from (6), as follows:

(A.2)fi(t|Gi, ρi, ε)=∑ni=1∞fi(t|ni)fi(ni|Gi, ρ0, ε)=∑ni=1∞e−μi(t)μi(t)ni−1λi(t)(ni−1)!(1−ρ0eλGi)ni−1ρ0eλGi=ρ0eλGiλi(t)exp(−ρ0eλGiμi(t))∑ni=1∞exp(−(1−ρ0eλGi)μi(t))[(1−ρ0eλGi)μi(t)]ni−1(ni−1)!=ρ0eλGiλi(t)exp(−ρ0eλGiμi(t))

and

(A.3)Si(t|Gi, ρ0, ε)=∫t∞fi(u|Gi, ρ0, ε)du=∫t∞ρ0eλGiλi(u)exp(−ρ0eλGiμi(u))du=exp(−ρ0eλGiμi(t)).

To obtain the expression for the Hazard Ratio, we first derive the pdf of t from the survival function in (11):

(A.4)fi(t|Gi, ρ0, ε, {νik}, α, Xi, {Ik(⋅)})=ddtSi(t|Gi, ρ0, ε, {νik}, α, Xi, {Ik(⋅)})=(1−ϕi(α, Xi))ρ0(1−εGi)(∑k=1qeνikIk′(t))e−ρ0(1−εGi)(∑k=1qeνikIk(t)),

where Ik′(t) is the derivative function of the kth I-spline basis function. Evaluating (A.3) and (A.4) for G_i=1 and G_i=0, we obtain the expression in (12) from section “Methods”.

A.2 Sensitivity analyses

For the parameter ρ0=exp(θ0)/(1+exp(θ0)), we set different prior distributions, with θ0∼N(−4.95, κ×1/4) and κ taking values: 1.11, 1.33, 2 and 4. Figure A.1 shows the resulting prior distribution of ρ0, along with the different posteriors produced for ρ0, γ and ε. Table A.1 shows summaries of these posterior distributions.

$Figure A.1: Prior distribution for the per-act risk of HIV infection (ρ0${\rho }_{0}$) and posterior distributions of main model parameters.$

Figure A.1:

Prior distribution for the per-act risk of HIV infection (ρ0) and posterior distributions of main model parameters.

Table A.1:

Summaries of posterior distributions of ρ0 and ε resulting from different priors of ρ0.

		Posterior distribution summaries HPD
		Mean	SD	2.5%	50%	97.5%	95% CI
ρ0	κ=1.00	0.41	0.11	0.20	0.40	0.65	0.19	0.63
	κ=1.11	0.41	0.12	0.21	0.41	0.66	0.20	0.64
	κ=1.33	0.41	0.12	0.19	0.40	0.66	0.18	0.64
	κ=2.00	0.36	0.12	0.16	0.36	0.61	0.14	0.59
	κ=4.00	0.33	0.13	0.13	0.32	0.60	0.11	0.57
ε	κ=1.00	0.44	0.13	0.15	0.45	0.65	0.18	0.67
	κ=1.33	0.44	0.12	0.15	0.46	0.64	0.19	0.67
	κ=2.00	0.42	0.13	0.13	0.44	0.63	0.16	0.65
	κ=4.00	0.42	0.13	0.13	0.43	0.63	0.16	0.65

References

Aalen, O. O. 1988. “Heterogeneity in Survival Analysis.” Statistics in Medicine 7: 1121–37.10.1002/sim.4780071105Search in Google Scholar

Baeten, J. M., T. Palanee-Phillips, E. R. Brown, K. Schwartz, L. E. Soto-Torres, V. Govender, N. M. Mgodi, F. Matovu Kiweewa, G. Nair, F. Mhlanga, S. Siva, L.-G. Bekker, N. Jeenarain, Z. Gaffoor, F. Martinson, B. Makanani, A. Pather, L. Naidoo, M. Husnik, B. A. Richardson, U. M. Parikh, J. W. Mellors, M. A. Marzinke, C. W. Hendrix, A. van der Straten, G. Ramjee, Z. M. Chirenje, C. Nakabiito, T. E. Taha, J. Jones, A. Mayo, R. Scheckter, J. Berthiaume, E. Livant, C. Jacobson, P. Ndase, R. White, K. Patterson, D. Germuga, B. Galaska, K. Bunge, D. Singh, D. W. Szydlo, E. T. Montgomery, B. S. Mensch, K. Torjesen, C. I. Grossman, N. Chakhtoura, A. Nel, Z. Rosenberg, I. McGowan, and S. Hillier. 2016. “Use of a Vaginal Ring Containing Dapivirine for HIV-1 Prevention in Women.” New England Journal of Medicine 375: 2121–32.10.1056/NEJMoa1506110Search in Google Scholar

Barrett, J. C., and J. Marshall. 1969. “The Risk of Conception on Different Days of the Menstrual Cycle.” Population Studies 23: 455–61.10.1080/00324728.1969.10405297Search in Google Scholar

Boily, M. C., R. F. Baggaley, L. Wang, B. Masse, R. G. White, R. J. Hayes, and M. Alary. 2009. “Heterosexual Risk of HIV-1 Infection Per Sexual Act: Systematic Review and Meta-Analysis of Observational Studies.” The Lancet Infectious Diseases 9: 118–29.10.1016/S1473-3099(09)70021-0Search in Google Scholar

Buck Louis, G. M., and R. Sundaram. 2012. “Exposome: Time for Transformative Research.” Statistics in Medicine 31: 2569–75.10.1002/sim.5496Search in Google Scholar PubMed PubMed Central

Coley, R. Y., and E. R. Brown. 2016. “Estimating Effectiveness in HIV Prevention Trials with a Bayesian Hierarchical Compound Poisson Frailty Model.” Statistics in Medicine 35: 2609–34.10.1002/sim.6884Search in Google Scholar PubMed PubMed Central

Cox, D. R., and D. Oakes. 1984. Analysis of Survival Data. London, UK: Chapman & Hall/CRC.Search in Google Scholar

Hanscom, B., H. E. Janes, P. D. Guarino, Y. Huang, E. R. Brown, Y. Q. Chen, S. M. Hammer, P. B. Gilbert, and D. J. Donnell. 2016. “Preventing HIV-1 Infection in Women Using Oral Pre-exposure Prophylaxis: A Meta-Analysis of Current Evidence.” Journal of Acquired Immune Deficiency Syndromes (1999) 73: 606.10.1097/QAI.0000000000001160Search in Google Scholar PubMed PubMed Central

Heise, L. L., C. Watts, A. Foss, J. Trussell, P. Vickerman, R. Hayes, and S. McCormack. 2011. “Apples and Oranges? Interpreting Success in HIV Prevention Trials.” Contraception 83: 10–15.10.1016/j.contraception.2010.06.009Search in Google Scholar PubMed PubMed Central

Hernán, M. A. 2010. “The Hazards of Hazard Ratios.” Epidemiology 21: 13.10.1097/EDE.0b013e3181c1ea43Search in Google Scholar PubMed PubMed Central

Hougaard, P. 1995. “Frailty Models for Survival Data.” Lifetime Data Analysis 1: 255–273.10.1007/BF00985760Search in Google Scholar PubMed

Hughes, J. P., J. M. Baeten, J. R. Lingappa, A. S. Magaret, A. Wald, G. de Bruyn, J. Kiarie, M. Inambao, W. Kilembe, C. Farquhar, C. Celum, and the Partners in Prevention HSV/HIV Transmission Study Team. 2012. “Determinants of Per-Coital-Act HIV-1 Infectivity Among African HIV-1-“serodiscordant Couples.” The Journal of Infectious Diseases 205: 358–65.10.1093/infdis/jir747Search in Google Scholar

Kim, S., R. Sundaram, and G. M. Buck Louis. 2010. “Joint Modeling of Intercourse Behavior and Human Fecundability Using Structural Equation Models.” Biostatistics 11: 559–571.10.1093/biostatistics/kxq006Search in Google Scholar

O’Hagan, J. J., M. A. Hernán, R. P. Walensky, and M. Lipsitch. 2012. “Apparent Declining Efficacy in Randomized Trials: Examples of the Thai RV144 HIV Vaccine and South African CAPRISA 004 Microbicide Trials.” AIDS 26: 123–6.10.1097/QAD.0b013e32834e1ce7Search in Google Scholar

Patel, P., C. B. Borkowf, J. T. Brooks, A. Lasry, A. Lansky, and J. Mermin. 2014. “Estimating Per-Act HIV Transmission Risk: A Systematic Review.” AIDS 28: 1509–19.10.1097/QAD.0000000000000298Search in Google Scholar

Plummer, M. 2016. Rjags: Bayesian Graphical Models Using MCMC. URL https://CRAN.R-project.org/package=rjags, R package version 4-6.Search in Google Scholar

Plummer, M. 2017. JAGS Version 4.3.0 User Manual. URL https://martynplummer.wordpress.com/2017/07/18/jags-4-3-0-is-released/.Search in Google Scholar

Powers, K. A., A. C. Ghani, W. C. Miller, I. F. Hoffman, A. E. Pettifor, G. Kamanga, F. E. Martinson, and M. S. Cohen. 2011. “The Role of Acute and Early HIV Infection in the Spread of HIV and Implications for Transmission Prevention Strategies in Lilongwe, Malawi: A Modelling Study.” The Lancet 378: 256–268.10.1016/S0140-6736(11)60842-8Search in Google Scholar

R Core Team. 2017. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. URL https://www.R-project.org/.Search in Google Scholar

Ramsay, J. O. 1988. “Monotone Regression Splines in Action.” Statististical Science 3: 425–441.10.1214/ss/1177012761Search in Google Scholar

Rizopoulos, D. 2012. Joint Models for Longitudinal and Time-to-Event Data: With Applications in R. Boca Raton, FL: Chapman &Hall/CRC.10.1201/b12208Search in Google Scholar

Satten, G. A., T. D. Mastro, and I. M. Longini. 1994. “Modelling the Female-to-Male Per-Act HIV Transmission Probability in an Emerging Epidemic in Asia.” Statistics in Medicine 13: 2097–106.10.1002/sim.4780131918Search in Google Scholar PubMed

Shiboski, S. C., and N. P. Jewell. 1992. “Statistical Analysis of the Time Dependence of HIV Infectivity Based on Partner Study Data.” Journal of the American Statistical Association 87: 360–72.10.1080/01621459.1992.10475215Search in Google Scholar

Sundaram, R., A. C. McLain, and G. M. Buck Louis. 2012. “A Survival Analysis Approach to Modelling Human Fecundity.” Biostatistics 13: 4–17.10.1093/biostatistics/kxr015Search in Google Scholar PubMed PubMed Central

Van der Straten, A., L. Van Damme, J. E. Haberer, and D. R. Bangsberg. 2012. “Unraveling the Divergent Results of Pre-exposure Prophylaxis Trials for HIV Prevention.” AIDS 26: F13–9.10.1097/QAD.0b013e3283522272Search in Google Scholar PubMed

Vaupel, J. W., K. G. Manton, and E. Stallard. 1979. “The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality.” Demography 16: 439–54.10.2307/2061224Search in Google Scholar

Vitinghoff, E., J. Douglas, F. Judon, D. McKiman, K. MacQueen, and S. P. Buchinder. 1999. “Per-contact Risk of Human Immunodeficiency Virus Transmission between Male Sexual Partners.” American Journal of Epidemiology 150: 306–11.10.1093/oxfordjournals.aje.a010003Search in Google Scholar PubMed

Weiss, H. A., J. N. Wasserheit, R. V. Barnabas, R. J. Hayes, and L. J. Abu-Raddad. 2008. “Persisting with Prevention: The Importance of Adherence for HIV Prevention.” Emerging Themes in Epidemiology 5: 8.10.1186/1742-7622-5-8Search in Google Scholar PubMed PubMed Central

Wilson, D. P. 2010. “Interpreting Sexually Transmissible Infection Prevention Trials by Adjusting for the Magnitude of Exposure.” Clinical Trials 7: 36–43.10.1177/1740774509355177Search in Google Scholar PubMed

Yang, Y., P. Gilbert, I. M. Longini, and M. E. Halloran. 2008. “A Bayesian Framework for Estimating Vaccine Efficacy Per Infectious Contact.” The Annals of Applied Statistics 2: 1409.10.1214/08-AOAS193Search in Google Scholar PubMed PubMed Central

Zhang, J., and E. R. Brown. 2014. “Estimating the Effectiveness in HIV Prevention Trials by Incorporating the Exposure Process: Application to HPTN 035 Data.” Biometrics 70: 742–50.10.1111/biom.12183Search in Google Scholar PubMed PubMed Central

Zhou, H., C. R. Weinberg, A. J. Wilcox, and D. D. Baird. 1996. “A Random-Effects Model for Cycle Viability in Fertility Studies.” Journal of the American Statistical Association 91: 1413–22.10.1080/01621459.1996.10476709Search in Google Scholar

Received: 2019-10-15

Accepted: 2020-06-04

Published Online: 2020-09-24

Joint modeling of time-varying HIV exposure and infection for estimation of per-act efficacy in HIV prevention trials

Abstract

Acknowledgments

A.1 Derivation of expressions in section “Methods”

A.2 Sensitivity analyses

References

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