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

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.

Introduction

Large randomized phase 3 trials of novel biomedical HIV prevention products are typically designed to estimate risk reduction, the population level reduction in HIV incidence due to the intervention. This is typically estimated as the hazard ratio estimated from the Cox Proportional Hazards model (Cox and Oakes 1984). Under ideal settings, such as a highly adherent population with homogeneous risk, the estimate of population level risk reduction is close to the estimate of efficacy, the level of protection an individual can expect to receive from the product. Ideally, efficacy estimates are independent of the clinical trial population and therefore transportable between populations. Instead, risk reduction estimates are conditional on the characteristics of the trial population (Heise et al. 2011). The characteristics governing the generalizability of risk reduction to efficacy cannot fully be controlled, even in the most rigorous clinical trials (Van der Straten et al. 2012). Although adherence to prevention regimens is likely the largest contributor to the divergence of these estimates (Hanscom et al. 2016; Weiss et al. 2008), heterogeneity in risk and exposure may also play a large role (Hernán 2010; O’Hagan et al. 2012). In this paper, we present a method to adjust for time-varying heterogeneity in exposure to HIV with an ultimate goal to estimate per-exposure efficacy.

HIV and other sexually transmitted infection (STI) acquisition outcomes are typically modeled assuming a risk process in continuous time (e.g. with time-to-event models), implicitly assuming that the individuals in the cohort are always at and share the same risk for HIV acquisition. In these models, heterogeneity in risk may be accounted for by adding a frailty term to the survival model (Aalen 1988; Hougaard 1995; Vaupel et al. 1979). A Gamma distribution is commonly used to model the frailty for its mathematical convenience. Specific to HIV infection, Coley and Brown (2016) proposed a compound Poisson frailty model to accommodate the risks from various exposure processes to HIV and allow for a probability that some participants may not be exposed and therefore are not at risk. Although this model allows for heterogeneity in risk between participants and between partnerships for a given participant, it does not account for possible time-varying variability in the exposure process.

Although continuous time models with appropriately defined frailties may account for some heterogeneity in risk, they do not reflect the actual properties of the exposure process. STIs can only be acquired at discrete points in time (during sex acts). The dependence on the occurrence of a sex act for an outcome to occur is well recognized in the fertility literature (Barrett and Marshall 1969; Buck Louis and Sundaram 2012; Kim et al. 2010; Sundaram et al. 2012; Zhou et al. 1996) with many mechanistic models having been developed conditional on the exposure process. However, this approach is more challenging in STI studies since the exposure status (partner’s HIV status) and exposure times are usually immeasurable in practice.

Modeling STI risk on the exposure scale with only partial knowledge of the exposure process has been studied previously. Satten et al. (1994), Vitinghoff et al. (1999) and Shiboski and Jewell (1992) developed statistical models to estimate the per-contact risk of infection and applied these models to studies that targeted high-risk populations and/or collected information on the infection status of sex partners. Other authors have developed models with the objective of evaluating efficacy/effectiveness of an intervention. Yang et al. (2008) developed a Bayesian framework for estimating vaccine efficacy per infectious contact, while adjusting for measurement error in contact-related factors. Wilson (2010) built upon Barrett and Marshall’s model to calculate the overall effectiveness (population-level risk reduction) of an intervention for prevention of STIs using the per-act infectivity and the efficacy estimates from previous studies. This approach provides an alternative estimate of the overall effectiveness of an intervention accounting for a shared exposure process, but it did not account for potential heterogeneity in the magnitude of the individual exposure processes. Zhang and Brown (2014) proposed a model for the risk of HIV infection on the exposure time scale by treating exposure events as a Poisson process with a zero-inflated gamma distribution for the rate of the exposure process.

In this article, we extend the model of Zhang and Brown (2014) for estimating the per-exposure efficacy of an HIV prevention intervention, by relaxing their time-invariant assumption on the exposure process. Similar to previously developed approaches to jointly modeling longitudinal and time to event outcomes (see Rizopoulos 2012 and references contained therein for an overview), we model the time to event dependent upon the mean of an evolving longitudinal process, constructing the likelihood assuming independence of the event times and longitudinal measures conditional upon the mean function ruling the process. Assuming a participant-specific non-homogeneous Poisson process, we build a model for the HIV exposure process. Upon this model, we derive the distribution to time to HIV infection. We propose a Bayesian approach for estimation.

The rest of the article is organized as follows: in the following sub-section, we describe our motivating example, the MTN-020/ASPIRE study, a recent HIV prevention trial. In section “Methods”, we describe the joint model for the time to infection and the exposure process, incorporating a probability of no risk, followed by an outline of the Bayesian estimation procedure. The proposed model is assessed by simulation studies and illustrated with the data from the ASPIRE study in subsequent sections Simulation studyJoint modeling of exposure and time to HIV infection. We finalize with a “Discussion” section.

The MTN-020/ASPIRE study

The MTN-020/ASPIRE study was a multi-center, randomized, double-blind, placebo-controlled Phase 3 trial to evaluate the safety and efficacy of a vaginal matrix ring (VR) containing 25 mg of dapivirine for the prevention of HIV-1 infection in HIV-negative women. The study enrolled 2,629 healthy, sexually active, non-pregnant, HIV-1–seronegative women between the ages of 18 and 45 years from 15 sites in Malawi, South Africa, Uganda and Zimbabwe. Participants were randomized to receive either the 25 mg dapivirine vaginal ring (VR) or a placebo VR. The trial results were summarized in Baeten et al. (2016). The HIV incidence rate in the dapivirine group was estimated to be 27% lower (95% confidence interval 1–46% lower; p=0.046) when compared to the placebo group, or 37% lower (95% confidence interval 12–56% lower; p=0.007) if two sites, that had very low retention and adherence, are excluded.

Follow-up visits, with evaluation of HIV-1 sero-conversion, were scheduled monthly. Collection of self-reported recent sexual activity was done quarterly, as well as in some interim unscheduled visits. Storage of blood plasma was done at quarterly visits and at the enrollment and last product-use visits. Archived plasma samples from enrollment, from visits before seroconversion, and from the last product-use visit, were tested for HIV-1 RNA on polymerase-chain-reaction (PCR) assays. Using these results, the time to the first visits with detectable HIV-RNA was obtained for all sero-converted participants.

Methods

In the population targeted for enrollment in MTN-020/ASPIRE, HIV is primarily acquired during vaginal intercourse with an HIV-infected partner. These exposures occur at discrete points in time and can be modeled based on self-reported data using counting processes. Detection of HIV infection occurs after the infection event itself through a predetermined testing sequence (independent of the exposure process), so that only a window of time in which the infection occurred is known. In this section, we present a joint model of the exposure process and the time to HIV detection. The aim of this model is to estimate the per-exposure efficacy of an HIV prevention intervention. We start by presenting a non-homogeneous Poisson process for the exposure process with a novel approach to modeling the mean that is (i) flexible enough to allow that the rates vary over time and between participants, while (ii) allowing for pooling (borrowing) of information across participants. Time to HIV infection is then derived based on a model for per-act risk of HIV infection that allows for modeling of per-exposure efficacy. Finally, we present a likelihood based on interval censoring for the infection outcome and propose a Bayesian approach to estimation.

Joint modeling of the exposure process and time to infection

Modeling the rate of sexual contacts

For each participant, indexed by $i=1,\dots ,m$, let $N_i=\left\{N_{it},t>0\right\}$ be a counting process, with N_it denoting the participant’s number of vaginal sex acts in the interval (0, t]. To allow a participant’s rate of vaginal sex acts to change over time, we define N_it to be a non-homogeneous Poisson process with rate function λ_i (t). Thus, for participant i at time t, sex acts occur at an average rate of λ_i (t) per unit time. Also, the number of vaginal sex acts within an arbitrary interval $A=\left(t_1,t_2\right)$, denoted by N_i (A), follows a Poisson distribution $\left[N_i\left(A\right)\sim \text{Poi}\left({\mu }_i\left(t_2\right)-{\mu }_i\left(t_1\right)\right)\text{,}\text{where}{\mu }_i\left(t\right)={\int }_0^t{\lambda }_i\left(u\right)\text{d}u\right]$. We note that, by definition, μ_i (t) is a non-negative, non-decreasing continuous function, usually called the mean function of the Poisson process.

We propose a flexible approach to modeling μ_i (t), using I-splines (Ramsay 1988). These consist of non-negative, monotonic non-decreasing spline bases, so that linear combinations of these, when using non-negative coefficients, result in non-negative, monotonic non-decreasing functions. Let $\left\{I_k(\cdot )\right\},k=1,\dots ,q$ be the q-dimensional I-spline basis functions defined on $\left[0,T_{\max }\right]$, where T_max is the maximum follow-up time among all participants. Then, to account for the heterogeneity in between participants’ processes, we define participant-specific mean functions

$${\mu }_i\left(t\right)=\sum _{k=1}^q{e}^{{\nu }_{ik}}{I}_k\left(t\right)\text{,}$$ (1)

where ${\left(e^{{\nu }_{i1}},\dots ,e^{{\nu }_{iq}}\right)}^T$ is a q-dimensional vector of coefficients for the I-spline basis defining each participant’s unique counting process. To reflect an assumption of ‘a priori’ exchangeability among participants (for example, among those with certain common baseline characteristics X_i), we assume $v_i^T={\left({\nu }_{i1},\dots ,{\nu }_{iq}\right)}^T\sim {\text{MVN}}_q\left(M\left(X_i\right),V\left(X_i\right)\right)$, where ${\text{MVN}}_q$ denotes a q-variate normal distribution with mean M (X_i) and covariance matrix V (X_i).

To link the latent counting process to potentially observed quantities, we assume that each participant self-reports their recent sexual activity at follow-up visits. As in our motivating example, we assume that participants are asked to report the number of vaginal sex acts in the past seven days. Let t_ij with j=1, 2, …, J denote all the visit times for participant i and let Z_ij denote the number of sex acts reported by the participant i at time t_ij. We assume that the visits are spaced far enough so that the periods of seven days for which participants report their sexual activity $\left(t_{ij}-\frac{1}{52},t_{ij}\right)$ do not overlap so that, conditional on the Poisson process defined above, the observations Z_ij are independent and follow a Poisson distribution with mean rate given by $\left({\mu }_i\left(t_{ij}\right)-{\mu }_i\left(t_{ij}-\frac{1}{52}\right)\right)$. Incorporating the modeling of μ_i (t) as in (1), we have:

$$Z_{ij}|v_i\sim \text{Poisson}\left(\sum _{k=1}^q{e}^{{\nu }_{ik}}\left[I_k\left(t_{ij}\right)-I_k\left(t_{ij}-\frac{1}{52}\right)\right]\right),\text{for}j=1,\dots ,J_i\text{.}$$ (2)

Modeling time to HIV infection

In this section we introduce a model for the time to infection based on repeated potential exposures as given by each participant’s counting process of sex acts. For simplicity, we assume that the vast majority of sex acts occurring are penile-vaginal, and that other types of sex acts are rare in the population or have a negligible probability of transmission.

For each participant, we assume a time-invariant risk of acquiring HIV at each sexual contact. Similar to a hazard function modeling the instantaneous risk of an event at time t, conditional on being event-free just before time t, we define a per-act risk of infection at each sex act, conditional on not having been infected in previous sex acts. For a potential exposure (vaginal sex act) event occurring at time t, we define the risk of HIV infection as

$${\rho }_i=E_i{\rho }_0\exp \left(\gamma G_i\right)\text{,}$$ (3)

where ρ₀ denotes a baseline risk of infection from a vaginal sex act with an infected partner, i.e. risk of infection in the absence of the experimental intervention or other modifiers, E_i is a latent time-invariant indicator of exposure to HIV, G_i denotes the random intervention assignment for participant i (0 for control and 1 for the experimental intervention) and γ denotes the per-exposure relative risk of HIV infection (experimental intervention versus control). Thus, the per-exposure efficacy of the experimental intervention, our main parameter of interest, is given by 1 − e^γ.

The exposure indicator (E_i) allows that a participant, although sexually active, may never be exposed to HIV infection from sexual partners and therefore carry 0 risk of acquiring HIV infection. We assume that $E_i\sim \text{Bernoulli}\left({\varphi }_i\right)$, where the mean, ${\varphi }_i$, depends on baseline characteristics of the participant (X_i), through a vector of coefficients α:

$$\log \left(\frac{{\varphi }_i}{1-{\varphi }_i}\right)={\alpha }^{\text{T}}{X}_i\text{.}$$ (4)

For participants with $E_i=0$, their risk of HIV infection remains zero throughout follow-up $\left({\rho }_i=0\right)$. Thus, conditional on $E_i=0$, the time to HIV infection is given by:

$$S_i\left(t|G_i,{\rho }_0,\gamma ,{\nu }_i,E_i=0\right)=1.$$

Note this model assumes that participants without HIV-infected partners are not at risk of HIV infection, this is, that the probability of transmission from other routes is negligible.

For participants who have HIV infected partners, we can assume that there is a finite number of exposures (sex acts) that would occur before the participant is infected. We denote as H_i the sex act at which HIV infection would occur for participant i, given $E_i=1$. Assuming that the number of sex contacts follows a non-homogeneous Poisson process with mean function ${\mu }_i\left(t\right)$, then distribution of the time to infection T_i, which is not observed directly, can be derived as the arrival time of the $H_i^{th}$ event in the exposure process. This is, T_i is such that $N_i\left(T_i\right)=H_i$. If we assume that the process of sex contacts for the ith participant is independent from their partner’s HIV status, i.e., the mean value functions ${\mu }_i\left(t|E_i=1\right)={\mu }_i\left(t|E_i=0\right)={\mu }_i\left(t\right)$, then the distribution of T_i given $H_i=h_i$ can then be derived as follows. First, the survival distribution can be expressed as

$$S_i\left(t|h_i,{\mu }_i\left(t\right)\right)=\text{Pr}\left(T_i>t\right)=\text{Pr}\left(N_i\left(t\right)\le h_i-1\right)=\sum _{l=0}^{h_i-1}\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^l}{l!}\text{,}$$ (5)

leading to the following probability density function (see Appendix A.1 for a detailed derivation):

$$f_i\left(t|h_i,{\mu }_i\left(t\right)\right)=-\frac{\text{d}{S}_i\left(t\right)}{\text{d}t}=\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^{h_i-1}{\lambda }_i\left(t\right)}{\left(h_i-1\right)!}$$ (6)

Recalling the assumption that each participant’s per-act risk of infection is constant over time, we have that H_i follows a Geometric distribution:

$$\mathrm{Pr}\left(H_i=h_i|{\rho }_i\right)={\left(1-{\rho }_i\right)}^{h_i-1}{\rho }_i,\text{for}{h}_i=1,2,\dots \infty \text{.}$$ (7)

For simplicity, we integrate out the latent variable H_i from (6) (see details in Appendix A.1). Conditional on having an HIV infected partner, the distribution and density functions of T_i are given by

$$\begin{array}{c}{f}_i\left(t|G_i,{\rho }_0,\gamma ,{\mu }_i\left(t\right),E_i=1\right)=\sum _{n_i=1}^{\infty }{f}_i\left(t|n_i\right)f_i\left(n_i|G_i,{\rho }_0,\epsilon \right)\\ ={\rho }_0{e}^{\gamma G_i}{\lambda }_i\left(t\right)\exp \left(-{\rho }_0{e}^{\gamma G_i}{\mu }_i\left(t\right)\right)\text{,}\end{array}$$ (8)

$$\begin{array}{c}{S}_i\left(t|G_i,{\rho }_0,\gamma ,{\mu }_i\left(t\right),E_i=1\right)=\underset{t}{\overset{\infty }{\int }}{f}_i\left(u|G_i,{\rho }_0,\epsilon \right)\text{d}u\\ =\exp \left(-{\rho }_0{e}^{\gamma G_i}{\mu }_i\left(t\right)\right)\text{.}\end{array}$$ (9)

Incorporating the non-parametric modeling of ${\mu }_i\left(t\right)$ using I-splines, as in (1), we have that the survival function can be expressed as:

$$S_i\left(t|G_i,{\rho }_0,\gamma ,{\nu }_i,E_i=1\right)=\exp \left[-{\rho }_0{e}^{\gamma G_i}\sum _{k=1}^q{e}^{v_{ik}}{I}_k\left(t\right)\right]\text{.}$$ (10)

Integrating out E_i, the survival function of the time to HIV infection T_i can be expressed as

$$S_i\left(t|G_i,{\rho }_0,\gamma ,v_i,\alpha ,X_i\right)=\left[1-{\varphi }_i\left(\alpha ,X_i\right)\right]+{\varphi }_i\left(\alpha ,X_i\right)\exp \left[-{\rho }_0{e}^{\gamma G_i}\sum _{k=1}^q{e}^{v_{ik}}{I}_k\left(t\right)\right]\text{.}$$ (11)

It can be shown (see Appendix A.1) that the hazard ratio of HIV infection at time t associated to the experimental intervention is given by:

$$\begin{array}{c}\text{HR}\left(t|{\rho }_0,\gamma ,\nu ,\hspace{0.17em}\alpha ,X_i\right)=\frac{f_i\left(t|G_i=1,{\rho }_0,\gamma ,\nu ,\alpha ,X_i\}\right)}{S_i\left(t|G_i=1,{\rho }_0,\gamma ,\nu ,\alpha ,X_i\}\right)}/\frac{f_i\left(t|G_i=0,{\rho }_0,\gamma ,\nu ,\alpha ,X_i\}\right)}{S_i\left(t|G_i=0,{\rho }_0,\gamma ,\nu ,\alpha ,X_i\right)}\\ =e^{\gamma }\exp \left[{\rho }_0\left(1-e^{\gamma }\right)\left(\sum _{k=1}^q{e}^{{\nu }_{ik}}{I}_k\left(t\right)\right)\right]\\ \times \frac{\left[1-{\varphi }_i\left(\alpha ,X_i\right)\right]+{\varphi }_i\left(\alpha ,X_i\right)\exp \left[-{\rho }_0\left(\sum _{k=1}^q{e}^{{\nu }_{ik}}{I}_k\left(t\right)\right)\right]}{\left[1-{\varphi }_i\left(\alpha ,X_i\right)\right]+{\varphi }_i\left(\alpha ,X_i\right)\exp \left[-{\rho }_0{e}^{\gamma }\left(\sum _{k=1}^q{e}^{{\nu }_{ik}}{I}_k\left(t\right)\right)\right]}\end{array}$$ (12)

From (12), we can see that the proportional hazards assumption holds only under extreme scenarios. For example, when the experimental intervention is not different from the control ($\gamma =1$) or when the baseline per-act risk of HIV infection is null for all participants (${\rho }_0=0$). The assumption also holds under the ideal scenario that ${\varphi }_i=1$, this is, when participants in a study are all exposed to HIV. Although this assumption holds in some cases (like clinical trials in HIV serodiscordant couples) it is not usually the case for other target populations.

Building the likelihood function

As in any HIV prevention trial, we assume that the exact time to HIV infection, T_i, is not observable. Instead, we assume that a time interval $\left(t_i^{\left(L\right)},t_i^{\left(R\right)}\right)$ and a binary indicator ${\delta }_i$ are observed, where ${\delta }_i=1$ indicates that HIV infection occurred between $t_i^{\left(L\right)}$ and $t_i^{\left(R\right)}$ (and 0 otherwise), $t_i^{\left(L\right)}$ is the time of the last negative HIV test ($t_i^{\left(L\right)}=0$ if there is no negative HIV test) and $t_i^{\left(R\right)}$ is the time of the first positive RNA HIV test ($t_i^{\left(R\right)}=\infty$ if ${\delta }_i=0$).

We will use $\Psi$ to denote the vector of parameters of interest: $\Psi ={\left({\rho }_0,\gamma ,{\nu }_i^{\text{T}},\hspace{0.17em}\alpha \right)}^{\text{T}}$. The contribution to the likelihood function from participant i is constructed from the observed interval censored time-to-event $\left\{t_i^{\left(L\right)},t_i^{\left(R\right)},{\delta }_i,G_i,\hspace{0.17em}{X}_i\right\}$ and the longitudinal self-reported data on recent sexual activity $\left\{z_{ij},j=1,\dots ,J_i\right\}$:

$$\begin{array}{c}{L}_i\left(\Psi ;t_i^{\left(L\right)},t_i^{\left(R\right)},{\delta }_i,G_i,X_i,Z_{ij}\right)={\left[\text{Pr}\left(t_i^{\left(L\right)}<T_i<t_i^{\left(R\right)}|G_i,{\rho }_0,\gamma ,\hspace{0.17em}{\nu }_i,\alpha ,\hspace{0.17em}{X}_i\right)\right]}^{{\delta }_i}\\ \times {\left[\text{Pr}\left(t_i^{\left(L\right)}<T_i|G_i,{\rho }_0,\gamma ,\hspace{0.17em}{\nu }_i,\hspace{0.17em}\alpha ,\hspace{0.17em}{X}_i\right)\right]}^{1-{\delta }_i}\\ \times \prod _{j=1}^{J_i}\text{Pr}\left(Z_{ij}=z_{ij}|{\nu }_i\right)\end{array}$$ (13)

Here, we highlight the assumption that the counting process of sex acts is independent of the participants’ probability of having an HIV infected partner, their group assignment, or baseline characteristics. Thus, conditioning on ${\nu }_i$, the time to HIV infection is independent of the participant’s counting process of sex acts.

Assuming that each participants’ counting process and time to infection is independent from others’, the full likelihood is obtained by multiplying the individual contributions from (13).

Bayesian estimation

We propose a Bayesian approach to estimate the parameters of the model presented in the previous section Joint modeling of the exposure process and time to infection. The prior distribution proposed for the main parameter of interest (ε) is fairly vague, while the prior for the per-exposure risk of HIV infection (${\rho }_0$) is based on relevant literature. For the ancillary parameter vectors $\alpha$ and $\nu$ we propose an empirical Bayes approach, by using priors that incorporate estimates from the target population. In this section we describe, in general terms, our proposed priors for the analysis of the ASPIRE trial. Similar priors were used for the simulation study.

Prior distributions

Per-exposure efficacy$\left(1-e^{\gamma }\right)$: For the parameter γ we propose a normal prior centered at zero and with a large variance. This allows for both favorable and unfavorable effects of the experimental intervention. For the analysis of our motivating example, and in our simulation study, we have used $\gamma \sim N\left(0,100\right)$, where $N\left(a,b\right)$ refers to the normal distribution with mean a and variance b.

Per-exposure risk of HIV infection$\left({\rho }_0\right)$: Estimates of the risk of HIV transmission per sexual contact with an HIV infected partner, as found in the literature, are very heterogeneous. To inform this prior, we extracted estimates of the male-to-female probability of transmission from various different sources. We looked into comprehensive systematic reviews (Boily et al. 2009; Patel et al. 2014) and selected estimates reported from studies on sero-discordant couples or retrospective partner studies conducted in low-income countries. We excluded other types of studies that not enroll couples because the estimated probability of transmission per act is bound to underestimate the true per-exposure probability. We also added a more recent study on sero-discordant couples not included in the systematic reviews (Hughes et al. 2012), as well as estimates from a modeling study (Powers et al. 2011) in which the risk of transmission is estimated for various stages of the HIV infection. Estimates from the selected studies are shown in Figure 1A. Considering that an unknown proportion of the infections in the study could occur from partners at early stages, the prior distribution for ${\rho }_0$ was selected to cover a large range of values. For convenience, we defined the prior on the log-odds scale:

$$\{\begin{array}{c}{\theta }_0\sim N\left(-4.95,\frac{1}{4}\right)\text{,}\\ {\rho }_0=\frac{\exp \left({\theta }_0\right)}{1+\exp \left({\theta }_0\right)}\text{.}\end{array}$$ (14)

Figure 1:

(A) Estimates from retrospective sero-discordant couple studies and partner studies used to inform the prior on the baseline per-exposure probability of infection $\left({\rho }_0\right)$, along with estimates from a modeling study. (B) Selected prior distribution for the per-exposure risk of HIV infection, ${\rho }_0$.

The resulting prior distribution for ${\rho }_0$, shown in Figure 1B, has a large support, with 2.5th and 97.5th percentiles at 0.27 and 1.85%, respectively. Figure 1B also shows the variations of this prior (by inflating the variance of the normal distribution for ${\theta }_0$ in 14) that we used to evaluate the sensitivity of the resulting inference to prior for this parameter.

Probability of participant exposed to HIV$\left({\alpha }_i\right)$: As initially stated in the previous section Modeling time to HIV infection, we propose a logistic model linking the probability of exposure to certain baseline characteristics. Here, we use the participant’s baseline age and site. A non-informative prior distribution is elicited for the coefficient of age $\left({\alpha }_{\text{AGE}}\sim N\left(0,4\right)\right)$. For the prior distribution of the site-specific coefficients $\left({\alpha }_{\text{SITE}}\right)$, we propose an empirical Bayes approach, by using priors that are informed by observed data. Given an estimate of the site’s proportion of women exposed to HIV $\left({\hat{\theta }}_{\text{SITE}}\right)$, we set the following prior:

$${\alpha }_{\text{SITE}}\sim N\left(\text{log}\left(\frac{{\hat{\theta }}_{\text{SITE}}}{\left(1-{\hat{\theta }}_{\text{SITE}}\right)}\right),4\right)$$ (15)

For the analysis of the ASPIRE trial, we use the proportion of HIV-1 infected women among the potential participants screened at each site as an estimate of ${\theta }_{\text{SITE}}$. In our simulation study, we use the proportion of women with $E_i=0$ as obtained in each simulation.

Parameters defining the Poisson counting process of sexual acts$\left({\nu }_i\right)$: As specified in the previous section Modeling the rate of sexual contacts, we propose a multivariate prior distribution with mean $M\left(X_i\right)$ and covariance matrix $V\left(X_i\right)$ for the parameter vector ${\nu }_i$. Taking an empirical Bayes approach in the analysis of the ASPIRE trial, we fit site-specific linear mixed-effect models on the number of self-reported sexual acts in the past week Z_ij, with $\Delta I_k\left(t_{ij}\right)$, $k=1,\dots ,q$ as fixed-effect covariates and random intercepts (at the participant level) to account for repeated measures. Using the estimated mean vector and covariance matrix of the coefficients from this mixed-effect model and Monte Carlo simulation we obtained the distribution of ${\nu }_i$.

Estimation

We propose estimation by Markov Chain Monte Carlo (MCMC) methods. The joint model described in section “Methods” was coded using JAGS (Plummer 2017). To simplify the coding of the model, we make use of the latent (non-observable) variables Methods, H_i and E_i (the participant’s number of exposure until infection and exposure to HIV indicator, respectively).

Simulation study

In order to assess the validity of estimate from our proposed model, we conducted a simulation study. Our main interest is to test if the model can successfully use the information from the frequency of sex contacts to inform exposure to HIV and the time to HIV infection, and thus be able to produce a better estimate of the per-act relative risk. For this reason, we use relative simple scenarios, with little to no variability between participants.

Simulation settings

The counting process for each participant’s sex acts is simulated as a non-homogeneous Poisson process with the mean function $\mu \left(t\right)$ built from an I-Spline model with 5 degree of freedom and the vector linear coefficients ${\left(28.9,70.2,55.5,53.9,32.5\right)}^T$. The number of sex acts in the past seven days are simulated at each visit from a Poisson distribution with a rate calculated from (2). The probability of being exposed to HIV for each subject (ϕ) was set as 25%. The per contact risk of HIV infection (${\rho }_0$) and the intervention efficacy log risk ratio (γ) were set constant and the same for all participants. Nine different scenarios were explored, as given by the combination of three different values of the baseline per-act risk of infection ${\rho }_0$ (0.1, 0.7 and 2%) and three values of per-contact effectiveness ε (0.3, 0.5 and 0.8). We recall that $\epsilon =1-e^{\gamma }$.

Each simulation represents a realization of a randomized clinical trial (RCT) with a sample of size 500 participants, with half randomly assigned to the active arm and evenly split into three sites. For simplicity, only one baseline covariate, age (in years), is included in the simulation study, simulated from a truncated normal distribution with mean of 25 and standard deviation of 10 and restricted between 18 and 45. Each subject is followed up every four weeks with up to 36 follow-up visits. The trial is stopped when a total of 100 events (HIV infections) is reached.

For each simulated RCT, a Cox proportional hazard model is fitted. The propose joint model Methods is fitted with three options for the degrees of freedom for the I-splines (q=4, 5 and 6). Prior distributions as described in section “Bayesian estimation” are used. The efficacy estimates ($\hat{\epsilon }$) resulting from the Cox PH model and the proposed Bayesian Model (posterior mean and median) are reported, along with 95% confidence intervals, as given by a Wald-type interval for the former and the highest posterior density (HPD) credible interval for the latter. We report the empirical average and MSE of the point estimates and the empirical probability coverage of the interval estimates.

The simulations were implemented using R (R Core Team 2017, Version 3.6.0), with the rjags package (Plummer 2016, Version 4.2.0). Two chains of MCMC estimates are produced, and results from 10,000 samples (after burning 5,000 samples) are used for estimation of the posterior distribution.

Simulations results

Results from nine different scenarios are presented Table 1, out of 1,000 simulation replicates for each scenario. First, we focus on the scenarios where ${\rho }_0=0.7\text{\%}$, as the prior distribution that we set for this parameter is closest to this value. The efficacy of the intervention is strongly underestimated by the Cox PH point estimate and the confidence intervals show coverage probabilities below the nominal level (down to 61.6%). In comparison, estimates obtained from our model are closer in average to the truth (lower MSE) and the credible intervals show coverage probabilities close to nominal level. Importantly, we note that similar results from the proposed model are obtained when using different number of I-spline basis (q=4, 5 or 6) for modeling the exposure process, implying that the model is robust to miss-specification of this parameter.

Table 1:

Summary of the simulation study evaluating the performance of the Cox PH model and the proposed Joint Model (with Empirical Bayes estimation) on estimating the per-act efficacy (ε), for different values of the per-act baseline risk of HIV infection (${\rho }_0$).

	Cox PH model				Bayes estimates from joint model
					Post. mean		Post. median
	Mean	MSE	CP, %		Mean	MSE	Mean	MSE	CP, %
${\rho }_o=0.2\text{\%}$
$\epsilon =0.3$	0.22	0.070	95.0	q=4	0.32	0.066	0.37	0.060	93.0
				q=5	0.32	0.066	0.37	0.060	92.7
				q=5	0.32	0.066	0.37	0.060	92.4
$\epsilon =0.5$	0.41	0.052	93.5	q=4	0.51	0.049	0.56	0.043	90.2
				q=5	0.51	0.049	0.56	0.043	90.5
				q=6	0.51	0.049	0.56	0.043	90.6
$\epsilon =0.8$	0.76	0.014	95.2	q=4	0.82	0.011	0.84	0.010	85.7
				q=5	0.82	0.011	0.84	0.010	85.8
				q=5	0.82	0.011	0.84	0.010	85.7
${\rho }_o=0.7\text{\%}$
$\epsilon =0.3$	0.159	0.053	87.3	q=4	0.245	0.053	0.28	0.046	94.9
				q=5	0.245	0.053	0.28	0.046	95.4
				q=6	0.245	0.053	0.28	0.046	95.4
$\epsilon =0.5$	0.332	0.052	75.9	q=4	0.46	0.031	0.48	0.026	96.1
				q=5	0.46	0.031	0.48	0.026	96.4
				q=6	0.46	0.031	0.48	0.026	96.2
$\epsilon =0.8$	0.676	0.023	61.6	q=4	0.78	0.006	0.79	0.005	95.2
				q=5	0.78	0.006	0.79	0.005	94.7
				q=6	0.78	0.006	0.79	0.005	94.9
${\rho }_o=2.0\text{\%}$
$\epsilon =0.3$	0.129	0.065	83.3	q=4	0.14	0.099	0.17	0.083	92.7
				q=5	0.14	0.099	0.17	0.083	93.6
				q=6	0.14	0.099	0.17	0.083	93.3
$\epsilon =0.5$	0.247	0.090	52.2	q=4	0.38	0.050	0.41	0.041	94.9
				q=5	0.38	0.050	0.41	0.041	94.7
				q=6	0.38	0.050	0.41	0.041	94.2
$\epsilon =0.8$	0.460	0.132	1.9	q=4	0.75	0.008	0.77	0.007	95.3
				q=5	0.75	0.008	0.77	0.007	95.4
				q=6	0.75	0.008	0.77	0.006	95.1

For extreme scenarios where the true baseline risk of infection is relatively large (${\rho }_0=2\text{\%})$ and not consistent with the prior, we observed some bias on the estimates of the per-exposure efficacy (ε), as given by the posterior mean or median. This bias seems to be related to an under-estimation of ${\rho }_0$, as the prior for this parameter has some influence on the posterior (results not shown). However, in terms of the estimation of ε, the proposed model still performs better than the Cox PH model: estimates are less biased, coverage probabilities are closer to nominal level (never below 92%, while Cox PH shows coverage probabilities as low as 1.5%) and the MSE is smaller (except for low values of efficacy, $\epsilon =0.3$). This suggest that estimation of ε can be relatively robust to mis-specification of the prior for ${\rho }_0$.

Finally, for scenarios where the baseline risk of infection is relatively low (${\rho }_0=0.2\text{\%}$), we observe a slight over-estimation of the per-act efficacy (ε) by our proposed model, which also seems related to the influence exerted by the prior of ${\rho }_0$ (in the opposite direction from the scenario described above). This bias seems to translate into coverage probabilities that fall below the nominal level, with the most extreme results (85% coverage probability) seen for the scenario where the true efficacy is largest ($\epsilon =0.8$). As in other scenarios, the Cox PH model seems to underestimate ε, but in less degree, resulting in coverage probabilities that are closer to nominal level. Overall, the performance seems comparable between the two models, except in scenarios where efficacy is high ($\epsilon =0.8$).

Joint modeling of exposure and time to HIV infection in the ASPIRE trial

Prior distributions for γ, ${\rho }_0$ and ${\alpha }_{\text{AGE}}$ were set as described above Bayesian estimation. The prior distribution for site-specific coefficients (${\alpha }_{\text{SITE}}$, $SITE=A,B,\dots ,M$) were obtained from the site’s HIV prevalence as observed at screening. As proposed in section “Bayesian estimation”, we fitted a linear mixed-effect model on the number of self reported sex acts. However, self-reported data were collected only up to 24 months in Sites C and D compared to 34 months in other sites, and the model fitting appears to be problematic for these two sites. Hence, a linear mixed model was fit to the data combining the two sites and Site H since their mean cumulative intensity function estimates are close (shown in Figure 2, and the prior distributions for the ${\nu }_{ik}$ are shared across the three sites. Summaries of the posterior distributions, including the posterior mean and the 95% HDP credible intervals, are reported from 20,000 posterior samples (from two parallel chains), after discarding 10,000 initial ‘burned’ samples.

Figure 2:

Mean cumulative intensity function estimated from the site-specific linear mixed model for each site and combining three sites (right) when the degrees of freedom is 4 in the spline model.

Table 2 lists the summary of the posterior distributions of the primary parameters from the different models. Three values for the degrees of freedom in the spline model (q=4, 5, and 6) were used to fit the data. The model with q=5 yields the smallest DIC, although there is little different in the estimates for the per-contact effectiveness the models. Figure 3 displays the Kaplan-Meier survival curves for the dapivirine ring and placebo ring groups in the ASPIRE trial, along survival functions predicted from the selected model. The predicted survival functions appear to approximate the Kaplan-Meier curves closely, indicating that the selected model provides a proper fit to the data.

Table 2:

Results from Empirical Bayes estimation of the Joint Model with three different I-Spline basis for the participant’s sex act processes. The posterior median, along with 95% HPD credible intervals are shown.

Degrees of freedom (q)	Per-contact risk (${\rho }_0$), %	Per-contact efficacy (ε)	DIC
q=4	0.48 (0.25, 0.72)	0.46 (0.18, 0.68)	68125.49
q=5	0.40 (0.19, 0.63)	0.45 (0.18, 0.67)	67691.38
q=6	0.48 (0.24, 0.71)	0.46 (0.18, 0.67)	68938.54

Figure 3:

Kaplan–Meier curve of time to HIV-1 infection, as first detected by RNA test, for the experimental intervention (Dapivirine VR) and the control (Placebo VR) arms of the MTN-020/ASPIRE study, along with the median survival functions obtained from the posterior distributions from the joint model described in section “Methods”.

The results from the selected model (q=5) indicate that compared to participants receiving the placebo VR, the risk of HIV infection per sexual act is 45% (95% CI 18–67%) lower for participants receiving the dapivirine VR. The per-contact risk of HIV infection if using the placebo VR was estimated as 0.40% (95% CI 0.19–0.67%). This estimate is higher than the estimate from the Partners in Prevention study (Hughes et al. 2012), but still in the reasonable range from the aforementioned systematic reviews. Figure 4 shows the study-specific posterior distributions for the participants’ probability of being exposed to HIV $\left({\varphi }_i\right)$, along with the prior distribution for two of the sites (the site that produced the most informative prior and the site that produced the least informative prior). We can see that the probability of exposure, as estimated by the joint model, varies widely across sites. Additionally, although the prior was designed to be mildly informative based on screening data, there appears to be adequate information in the data to overpower the prior.

Figure 4:

ASPIRE Study site-specific posterior distributions of participants’ probability of exposure to HIV (colored lines), along with the two most extreme site-specific prior distributions, the most informative (black solid line) and the least informative (black dashed line).

We conducted a sensitivity analysis on the prior distribution used for the per-act risk of HIV infection $\left({\rho }_0\right)$, by repeating the analysis described above using less precise priors. Specifically, given ${\rho }_0=\exp \left({\theta }_0\right)/\left(1+\exp \left({\theta }_0\right)\right)$, we set the prior for ${\theta }_0$ to be $N\left(-4.95,\kappa \times 1/4\right)$, for different values of κ: 1.11, 1.33, 2 and 4 (convergence issues were starting to show for κ=4). The resulting posterior distributions of γ and ε were very similar to those obtained with κ=1 (full results shown in Appendix A.2).

Discussion

In this article, we have proposed an approach to estimate the per-exposure efficacy of an intervention in HIV prevention trials. This new approach is different from the standard Cox model approach that estimates population-level effectiveness and assumes the same magnitude of HIV exposure for all participants. Instead, our model estimates efficacy of the intervention at an individual exposure to HIV. We applied this model to the MTN-020/ASPIRE phase 3 randomized clinical trial of the dapivirine vaginal ring (VR) for HIV prevention and estimated that the dapivirine VR reduces the per-contact risk of HIV infection by 45% compared to the placebo VR, which is higher than the population-level estimate of 37% from the Cox model .Baeten et al. (2016)

This approach extends the model in Zhang and Brown (2014) by specifying a non-homogeneous Poisson process for the observed longitudinal sexual contacts. The mean value function of the Poisson process is approximated by an I-spline model with non-negative participant-specific coefficients. The coefficients of the individual I-splines are estimated using an empirical Bayes approach. The hyper-parameters in the prior distribution of the log-transformed coefficients (ν_ik in 1), mean vector M_i and covariance matrix V_i, are set as fixed for each site from a linear mixed-effect model fitted to the self-reported numbers of sexual contacts in past week.

There are some key assumptions in this model: (1) the process of the sex contacts is independent of (a) the partner’s HIV status, (b) the HIV testing process and (c) the intervention; and (2) the exposure status to HIV for each participant is constant throughout follow-up.

In HIV prevention trials targeted at high risk populations, the partner is rarely involved in the study, limiting our ability to ascertain partner status and thus making it challenging to evaluate the adequacy of assumption (1a). In our motivating example, the MTN-020/ASPIRE trial, the majority of women reported their partners as being HIV-negative or having unknown status, limiting our ability to ascertain whether sexual frequency differs by exposure. However, mixed effects models of the self-reported number of sex acts showed no statistically significant relationship between partner’s HIV status (as reported by the participant) and frequency of sex acts. Also within the context of HIV prevention trials, it is not unreasonable that HIV testing, occurring at pre-specified periodic visits, would not be affected by the exposure process. Conversely, it is not unlikely that the frequent testing (along with counseling) would have an impact in the frequency of sex acts (assumption 1b). However, any changes in the frequency of sex acts (regardless of the reason) are expected to be accounted with the flexible modeling of process over time, as proposed here. Similarly, regarding assumption (1c), there is the possibility that the intervention assignment could have an effect on the exposure process, in the form of adverse effects having an impact on the frequency of sex acts, for example. However, specifically for the DPV vaginal ring, there has not been any evidence on increased adverse events, from the MTN-020/ASPIRE trial or any other posterior studies. Finally, regarding assumption (2), almost 90% of the participants in MTN-020/ASPIRE reported being with the same primary partner since enrollment.

Future work includes some potential extensions of the model. For example, the model could be extended to incorporate the time-varying exposure status, if there are adequate data available on male partners’ HIV status and partnership changes over time. Similarly, the model could be extended to incorporate measurement error in the ascertainment of exact time of infection, to address the delay in HIV detection of available testing methods.

The model is developed for HIV acquisition, but can also be applied to other STD studies with several extensions. Different from HIV, some STDs are curable but can also come back with recurring infection. The model can be extended for the recurrent STD infections by construct the likelihood similar to (13) for each occurrence under certain assumptions. The STD co-infection has been observed to be associated with a higher risk of HIV infection due to the same transmission path. Therefore, another extension could be allow the per-contact risk of HIV infection ρ to change over time depending on the STD status. Such extensions are under further investigations.

Appendix.

A.1 Derivation of expressions in section “Methods”

The density function $f_i\left(t|n_i\right)$ is derived as follows:

$$\begin{array}{c}{f}_i\left(t|n_i\right)=-\frac{d{S}_i\left(t\right)}{dt}\\ =-\left(-{\lambda }_i\left(t\right)e^{-{\mu }_i\left(t\right)}+\sum _{j=1}^{n_i-1}\frac{1}{j!}\left(-e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^j{\lambda }_i\left(t\right)+e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^{j-1}j{\lambda }_i\left(t\right)\right)\right)\\ ={\lambda }_i\left(t\right)\left(e^{-{\mu }_i\left(t\right)}+\sum _{j=1}^{n_i-1}\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^j}{j!}-\sum _{j=1}^{n_i-1}\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^{j-1}}{\left(j-1\right)!}\right)\\ ={\lambda }_i\left(t\right)\left(\sum _{j=0}^{n_i-1}\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^j}{j!}-\sum _{j=0}^{n_i-2}\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^j}{j!}\right)\\ =\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^{n_i-1}{\lambda }_i\left(t\right)}{\left(n_i-1\right)!}\text{.}\end{array}$$ (A.1)

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

$$\begin{array}{c}{f}_i\left(t|G_i,{\rho }_i,\epsilon \right)=\sum _{n_i=1}^{\infty }{f}_i\left(t|n_i\right)f_i\left(n_i|G_i,{\rho }_0,\epsilon \right)\\ =\sum _{n_i=1}^{\infty }\frac{e^{-{\mu }_i\left(t\right)}{\mu }_i{\left(t\right)}^{n_i-1}{\lambda }_i\left(t\right)}{\left(n_i-1\right)!}{\left(1-{\rho }_0{e}^{\lambda G_i}\right)}^{n_i-1}{\rho }_0{e}^{\lambda G_i}\\ ={\rho }_0{e}^{\lambda G_i}{\lambda }_i\left(t\right)\exp \left(-{\rho }_0{e}^{\lambda G_i}{\mu }_i\left(t\right)\right)\\ \sum _{n_i=1}^{\infty }\frac{\exp \left(-\left(1-{\rho }_0{e}^{\lambda G_i}\right){\mu }_i\left(t\right)\right){\left[\left(1-{\rho }_0{e}^{\lambda G_i}\right){\mu }_i\left(t\right)\right]}^{n_i-1}}{\left(n_i-1\right)!}\\ ={\rho }_0{e}^{\lambda G_i}{\lambda }_i\left(t\right)\exp \left(-{\rho }_0{e}^{\lambda G_i}{\mu }_i\left(t\right)\right)\end{array}$$ (A.2)

and

$$\begin{array}{c}{S}_i\left(t|G_i,{\rho }_0,\epsilon \right)={{\int }^{\text{}}}_t^{\infty }{f}_i\left(u|G_i,{\rho }_0,\epsilon \right)\text{d}u\\ ={{\int }^{\text{}}}_t^{\infty }{\rho }_0{e}^{\lambda G_i}{\lambda }_i\left(u\right)\exp \left(-{\rho }_0{e}^{\lambda G_i}{\mu }_i\left(u\right)\right)\text{d}u\\ =\exp \left(-{\rho }_0{e}^{\lambda G_i}{\mu }_i\left(t\right)\right)\text{.}\end{array}$$ (A.3)

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

$$\begin{array}{c}{f}_i\left(t|G_i,{\rho }_0,\epsilon ,\left\{{\nu }_{ik}\right\},\alpha ,X_i,\left\{I_k(\cdot )\right\}\right)=\frac{\text{d}}{\text{d}t}{S}_i\left(t|G_i,{\rho }_0,\epsilon ,\left\{{\nu }_{ik}\right\},\alpha ,X_i,\left\{I_k(\cdot )\right\}\right)\\ =\left(1-{\varphi }_i\left(\alpha ,X_i\right)\right){\rho }_0\left(1-\epsilon G_i\right)\left(\stackrel{q}{\sum _{k=1}}{e}^{{\nu }_{ik}}{I}_k^{\prime }\left(t\right)\right)e^{-{\rho }_0\left(1-\epsilon G_i\right)\left({{\sum }^{\text{}}}_{k=1}^q{e}^{{\nu }_{ik}}{I}_k\left(t\right)\right)}\text{,}\end{array}$$ (A.4)

where $I_k^{\prime }\left(t\right)$ 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 ${\rho }_0=\exp \left({\theta }_0\right)/\left(1+\exp \left({\theta }_0\right)\right)$, we set different prior distributions, with ${\theta }_0\sim N\left(-4.95,\kappa \times 1/4\right)$ and κ taking values: 1.11, 1.33, 2 and 4. Figure A.1 shows the resulting prior distribution of ${\rho }_0$, along with the different posteriors produced for ${\rho }_0$, 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 (${\rho }_0$) and posterior distributions of main model parameters.

Table A.1:

Summaries of posterior distributions of ${\rho }_0$ and ε resulting from different priors of ${\rho }_0$.

		Posterior distribution summaries HPD
		Mean	SD	2.5%	50%	97.5%	95% CI
${\rho }_0$	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