New Challenges in HIV Research: Combining Phylogenetic Cluster Size and Epidemiological Data

2.1 Data sources

The SPOT study. SPOT is an HIV-testing program with a research component targeting MSM in Montreal, Quebec. Beginning in July 2009, SPOT offered free, anonymous, HIV point of care rapid tests to MSM at a community-based testing site close to Montreal’s gay village. Individuals were recruited to the research component of SPOT provided they met the inclusion criteria: self-identification as male; at least 18 years of age; resident of Quebec, speaking and understanding French or English; anal sex with another man in the past 12 months; and unknown HIV status at the time of testing. SPOT promotion was also undertaken through outreach activities organized by the RÉZO community organization. Previously, it has been reported that a large number of participants learned of SPOT from friends (Otis et al. 2016).

In addition to free rapid tests, the SPOT project administered an anonymous questionnaire eliciting information on socio-demographic characteristics, HIV testing behavior, and behavorial/lifestyle information; phylogenetic analyses were undertaken on anyone found to be HIV+. We analyzed the data from 1803 men recruited up until 2013, 36 of whom were found to have HIV.

Two inferential challenges are encountered in SPOT. The first is one of sampling: A sampling frame is of course not available for the target population. SPOT is an HIV testing resource, primarily, and hence attracts individuals in an approach that could be likened to convenience-sampling. However, as detailed in the section that follows, we have access to data that were sampled probabilistically.

The second challenge faced is one of measurement error. The Quebec Genotyping Cohort does not contain all phylogenetic information for all individuals with HIV in the province of Quebec. Individuals may not be included for a variety of reasons including not having been tested (either at all, or only outside of the province), having viral load less than 400 copies per ml (Brenner and Moodie 2012), or having been diagnosed well before the inception of the genotyping program. Consequently, measurement error occurs in defining the cluster size and this measurement error is characterized by a systematic under-counting of the true cluster size due to the absence of the phylogenetic information on the HIV status or phylogenetic cluster of the individuals in the province (Parveen, Moodie, and Brenner 2015). However the under-counting exists is only for those men who are HIV-positive; anyone free of HIV has a cluster size of 0.

Thus, to obtain the valid inferences in investigating correlates of large phylogenetic clusters, we proceed with an analysis that accounts for the non-probabilistic recruitment scheme employed by SPOT, and measurement error in cluster size. We propose doing so through venue-based sampling weighting using an external source of data in combination with a new method of measurement error correction designed specifically for settings in which validation data are unavailable and measurement error may depend on another covariate, namely HIV status: simulation-extrapolation conditional on covariates (Parveen, Moodie, and Brenner 2017).

The ARGUS study.^[1] The data used in this study were collected from the second wave of ARGUS (Lambert et al. 2009), a second generation surveillance study designed to monitor trends in HIV, sexually transmitted infections, and risk behaviors among MSM living in the province of Quebec that occurred in 2008-2009. Participants were recruited using time-location sampling, or venue-based sampling, from venues including saunas, bars, coffee shops, sports and recreational groups where gay men interact, as well as a fixed study site (i.e. a non-mobile research site that is not a social venue). At each venue (except for the fixed site), individuals were randomly sampled from among those present. Information was collected on the frequency with which such a venue was attended so that individuals could be reweighted according to the inverse of the likelihood of having been sampled. From the 42 sampling locations in the province of Quebec (37 of which were located in Montreal), 1873 individuals were recruited in the period around when the individuals in SPOT were recruited. A self-administered questionnaire was given, and a blood sample was also collected to perform screening tests for HIV, syphilis and Hepatitis C virus. The ARGUS questionnaire focused on a participant’s socio-demographic characteristics, structure of his social network, sexual and other lifestyle activities. However, phylogenetic cluster size was not available for ARGUS participants at the time of the analysis. As ARGUS employed venue-based sampling, the data from ARGUS respondents, when appropriately reweighted, may better capture the target population of MSM in an urban Quebec setting.

To use estimated weights in SPOT that are derived from models for the ARGUS venue-based sampling weights, several assumptions are required. First, we must assume that both SPOT and ARGUS are targeting the same population, in this case of MSM. We also require that (i) the covariates that are common to SPOT and ARGUS that were used to predict the sampling weights are the only variables that predict the frequency with which an individual might visit a gay social venue, and (ii) that we can correctly specify the dependence of the sampling weight on those covariates. Under these assumptions, SPOT participants are conditionally exchangeable with ARGUS participants, and may be consistently weighted to estimate population-level parameters from the reweighted sample.

2.2 Statistical methods

Venue-based sampling. Venue-based sampling a sampling technique targetting hard-to-reach populations (Gustafson et al. 2013), including MSM. Venue-based sampling typically begins with an on-field team interviewing key service providers and members of the target population to identify a range of venue-day-time (VDT) units in which to locate the members of the target population as the sampling frame. The research team then visits the venues, counting the number of individuals present, prepares a list of potentially eligible VDT units, and estimating the population size for each VDT unit. Upon building the sampling frame, the sample is selected in two stages. In the first stage, venues are selected as primary sampling units using simple or stratified sampling with probabilities proportional to the estimated size of the population captured in each venue. In the second stage, a sample of participants from the selected venues is drawn using systematic (random) sampling (Magnani et al. 2005). There are many advantages to venue-based sampling, such as (i) it allows the calculation of a selection probability for each individual in the sample; (ii) unlike convenience sampling, it greatly diminishes the arbitrary selection of venues or individuals, and provides a replicable sampling selection method; and (iii) it does not require a comprehensive enumeration of individual members of the target population so long as all members of the population can be assumed to be reached at the sampled venues at different times. Venue-based sampling is not without costs: it requires intensive fieldwork to visit and map VDTs. Moreover, bias or low generalizability can occur if key venues are missed, or members of the target population do not (as assumed) frequent the venues included in the sampling frame.

We calculated sampling weights in ARGUS based on the reported frequency of attending the venue from which a man was sampled. E.g. men who were sampled from a café received a weight 60, 15 or 3.75 if they reported visiting cafés where MSM socialize less than 1 time per month, 1–3 times per month, or at least 1–3 times per week, respectively. These weights were obtained as the inversely proportional to the frequency with which the venues were attended, as follows: attendance of a particular venue-type such as a café

1. less than once/month gave a weight of 60 (which we may think of as being derived from [1/(60days)]−1, if less than monthly can be taken to every two months);

2. 1–3 times/month gave a weight of 15;

3. 1–3 times/week or more gave a weight of 3.75.

Weights were calculated similarly for men recruited from each of the social sites included in the VDT sample (bars, saunas, etc.). For the fixed study site, all participants were recruited, and hence the sampling weight for these individuals was 1. Therefore, each participants in the ARGUS study received a weights of 1, 3.75, 15, or 60.

As ARGUS and SPOT are assumed to recruit from the sample population, we leveraged the information in the venue-based sampling weights from ARGUS to create venue-based sampling weights for SPOT. Specifically, we built a model to estimate venue-based weights for SPOT using the weight from ARGUS study as dependent variable in a regression model which took all common covariates in the SPOT and ARGUS studies as predictors (see Table 1). Then, using the covariates in SPOT, we predicted the most probable weight for each SPOT participant. In our primary analysis, we used a multinomial logistic regression; a sensitivity analysis using linear regression to model the venue-based weights. Note that the number of sex partners in ARGUS was counted for the last 6 months whereas it was counted in SPOT for the last 3 months. We, therefore, multiplied the number of sex partners in SPOT by 2.

Simulation-extrapolation conditional on covariates. Over the last three decades, several measurement error correction methods including as regression calibration (Gleser 1990; Carroll, Ruppert, and Stefanski 1995), multiple imputation (Rubin 1987; Cole, Chu, and Greenland 2006), and simulation-extrapolation (Cook and Stefanski 1995; Carroll, Ruppert, and Stefanski 1995) have been proposed. Each of these approaches, with the exception of simulation-extrapolation, requires either a validation sample or replicate data for some fraction of the observed sample. In the SPOT study, phylogenetic cluster size is under-counted due to unobserved (untested) individuals. Therefore, obtaining a validation sample is both ethically and practically infeasible as it would require the testing of all members of the population of interest (in our case, all MSM resident in the province of Quebec). Under this circumstance, that is in the absence of validation or replicated data, simulation-extrapolation is the most promising avenue for correcting the undercounting of the cluster size data.

The simulation-extrapolation method developed by Cook and Stefanski (1995) is a simulation based technique for estimating and reducing bias due to additive measurement error. Simulation-extrapolation is an estimation procedure consisting of a simulation step and an extrapolation step. Estimates are obtained by adding additional measurement error (in known increments) to the mis-measured data, computing estimates from the contaminated data, establishing a trend between these estimates and the variance of the added measurement errors, and extrapolating this trend back to the case of no measurement error. This method requires the knowledge of the distribution of the measurement error (which may be known in cases such as a laboratory assay, or estimated using external or validation data). Let Ui, i=1,…,n is the unobserved true explanatory variable, and Xi an error-prone version. Consider Xi=Ui+δi, where δi∼N(0,σδ2) and δi is independent of Ui,Yi. Simulation-extrapolation proceeds in two steps. In the first simulation step, artificial measurement error is added to Xi and B new covariates Xi,b(λk) are generated through Xi,b(λk)=Xi+λkδib, where b=1,…,B; k=1,…,K and i=1,…,n for values of λk chosen by the analyst and {δi,b}b=1B are independent computer simulated random numbers from N(0,σδ2). It can be shown that the variance of Xi,b(λk) is σU2+(1+λk)σδ2, which increases with λk. For each λk, let βb^(λk) denote the vector of naïve estimates obtained by regressing Y on Xi,b(λk). Using B estimates for each λk, an average estimate can be obtained as B−1∑b=1Bβ^b(λk). By regressing βb^(λk) on λk, and extrapolating back to λk=−1, we find the estimate β^(−1) corresponding to the error σU2+(1+λk)σδ2=σU2 (i.e. the error free setting). Typically, βb^(λk) is regressed on λk assuming either a quadratic or a non-linear relationship e.g. a lowess smoother (Cleveland 1979). Parveen, Moodie, and Brenner (2017) extended simulation-extrapolation to accommodate measurement error distributions that (i) depend on other covariate and (ii) need not to have mean zero, called simulation-extrapolation conditional on covariates (SIMEX-CC). They expressed the imperfect measurement of Ui as Xi=Ui−δi, where the conditional distribution of the error δi given a correctly measured variable Zi was P(δi|Zi=zi)=fz with a finite mean (μδ) and variance (σδ2). In SIMEX-CC, the simulated covariate Xi,b(λk) was taken to be

where μδ=E(δi|Zi); b=1,…,B; k=1,…,K and i=1,…,n. Note that this measurement error specification can be made to accommodate settings where Xi is always less than or equal to Ui if, for example, we take Xi=Ui−δi with δi a non-negative random variable.

In the SPOT data setting, we consider Ui to be the (unobserved) true cluster size and Zi to be the HIV status of the participants. All HIV negative participants belong to a cluster size zero, and there is no measurement error in this cluster size. Measurement error was only present in the cluster size of HIV positive participants. Thus measurement error in cluster size depends on another covariate: the HIV status of the participants and, as such, we applied the SIMEX-CC method.

To undertake our analysis of the epidemiological correlates of phylogenetic cluster size in the SPOT data, we considered a weighted regression of Y on Xi,b(λk) in the simulation step of SIMEX-CC method, where weights were the venue-based sampling weights estimated with the external information provided from ARGUS, as described above.

3.1 Methods

In the present analysis, our main focus was on the association between the phylogenetic cluster size and the number of sex partners in the SPOT data. We adopted log-linear models to assess whether there was any evidence of a relationship between cluster size on number of sex partners, when adjusting for age, ethnicity, education, and income as potential confounders. Selected characteristics (unweighted) of the study samples from SPOT and ARGUS subjects are presented in Table 1; for a detailed breakdown of key covariates in the SPOT study by HIV status, please see Supplementary Materials Table 1, where similar distributions of characteristics are observed between the HIV positive and negative participants. As noted in the previous section, to adjust for the SPOT sampling scheme in the analysis stage, we used external information from the ARGUS study. The distribution of covariates varies between the SPOT and ARGUS studies, with ARGUS participants being more commonly of French-Canadian origin, less likely to hold a university degree, and more likely to have snorted or smoked cocaine.

The sampling weight for ARGUS participants were directly calculated using the frequency of venue attendance. We then use multinomial logistic regression to predict the sampling weights in ARGUS using the variables listed in Table 1, which are the variables common to both the SPOT and ARGUS studies. The fitted model was then used to estimate sampling weights for the SPOT study participants. In the SPOT study, there were some missing data. For the variables age, ethnicity, cluster size, number of sex partners and number of one night sex partners the number of missing data points were 6, 3, 2, 137 and 50 respectively. Since fewer than 8% of data were missing, our primary analysis used complete cases only, and then perform sensitivity analyses following imputation.

To apply SIMEX-CC, we must supply the method with the mean and variance of the measurement error distribution of cluster size of the HIV positive individuals. We estimate this distribution using the following external data: The adult population of Quebec in the year from which the SPOT data were taken was 6,802,700 with an HIV incidence rate of 7/100,000^[2] so that the total number of incident cases of HIV in Quebec can be estimated as 6,802,700 × 0.00007 ≈ 476. It has been suggested that in Canada, approximately 25% of people who are living with HIV do not know their infection status.^[3] In fact, the proportion of MSM in Montreal who are unaware of their HIV status is likely lower (13% in ARGUS), however we opt for the higher bound as previous work has suggested that it is better to overestimate rather than underestimate measurement error variance (Parveen, Moodie, and Brenner 2017). Thus, we estimate that there are 476 × 0.25 ≈ 160 who are “missing" from the genotyping program database, and thus contributing to the under-counting of cluster size. Based on previous studies of cluster size distributions (see, e.g. Lewis et al. 2008; Leigh Brown et al. 2011) and the sizes of clusters in SPOT, we found through trial and error that a Poisson(3) distribution was appropriate to yield a distribution of cluster sizes that was similar those found in the literature and would account for the approximately 160 individuals who are estimated to be missing from the Quebec genotyping program. While the Poisson distribution almost surely mis-specifies the true error distribution in cluster size, sensitivity analyses will be undertaken to assess the sensitivity of our findings to this rather stringent assumption.

We compared an unadjusted analysis to one that implements only SIMEX-CC, one that implements only the sampling weighting, and a final model that combined the sampling weighting and SIMEX-CC. When using SIMEX-CC, we used B = 200 replicates in the simulation step, and calculated the variance using a non-parametric bootstrap procedure with 1000 resamples. When combining both weighting and SIMEX-CC, we first applied the weighting then applied the SIMEX-CC to the weighted estimates. The bootstrap procedure ensures any variability due to weighting is also incorporated into the estimated standard errors.

3.2 Results

In Table 2, we compare four models: (i) a naïve unweighted model (neither adjusted for the sampling scheme nor corrected for measurement error); (ii) a naïve weighted model (not corrected for measurement error, but adjusted for the sampling scheme); (iii) an unweighted SIMEX-CC analysis (not adjusted for sampling scheme but corrected for measurement error); and (iv) a weighted SIMEX-CC (adjusted for both the sampling scheme and measurement error). SIMEX-CC extrapolant functions are given in Supplementary Figures 1–2. We have found no evidence of association between cluster size and the total number of sex partners across all models. However, notable differences appear in the analysis of the number of one-night sex partners: cluster size appears to be associated with the number of one night partners when adjusting for sampling weights, however the association is very weak in magnitude across all models and not significant when measurement error is taken into account. In this analysis, adjusting for the sampling confers greater changes in the estimates than correcting for measurement error – perhaps unsurprisingly as there are relatively few individuals living with HIV and hence whose cluster size is subject to measurement error.

To assess the sensitivity of the results to the measurement error distribution, missing data, and the estimation of the sampling weight, we re-analyzed the SPOT data (i) assuming measurement error was distributed with a mean and variance of 5; (ii) considering all missing variables as the modal (most common) values for binary variables, and median for continuous and count variables; and (iii) modelling the distribution of sampling weights as via linear regression. We found no meaningful changes in the resulting estimates (see Supplementary Table 2–4). In settings like SPOT, where the measurement error affects only a small fraction of the total sample, analysts may wish to focus efforts on addressing any potential biases due to sampling design.